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H: Help with proof
Let $X$ be a metric space, $x \in X$ and $S \subset X$. Then,
I have to prove that $x \in Cl(S)$ if and only if, every open ball of $X$ centered at $x$ has non-empty intersection with $S$.
I managed to do the first half of the proof but got stuck with the converse of it.
I checked my book for the co... |
H: Different answers for probability density function and cumulative density function
I have a function $f(x)=2ae^{-ax}(1-e^{-ax})$, for $x>0, a>0$. This is a pdf. I need to find $P(X>1)$. I have done all my work in such a way that I should get the same answer whether I use the pdf or the cdf to find this probability.... |
H: How to prove that two sets are equaivalent
There's this book iI'm reading and its definition of two sets being equivalent is that they have to have a one-to-one correspondence.
So there's this question I'm trying to prove and it goes:
Prove that if $f:A\to B$ is a function defined on the countable set $A$ then the ... |
H: Under what conditions is the DE exact?
Under what conditions for $\{a,b,k,l\}$ is $$(ax+by)dx+(kx+ly)dy=0$$ exact? Solve the exact ODE.
A differential equation of the form $I(x,y)dx+J(x,y)dy=0$ is exact if there exists a function $F$ such that $\frac{\partial F(x,y)}{\partial x}=I$ and $\frac{\partial F(x,y)}{\part... |
H: How to show that the linear operator is diagonalizable
Let $\mathbf{A}=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathcal{M}_{2\times2}(\mathbb{R})$. Define a linear operator $\mathit{T}$ on $\mathcal{M}_{2\times2}(\mathbb{R})$ such that
$\qquad\qquad\qquad\qquad\qquad\qquad\mathit{T}\left(\mathbf{X}\right)=\mathbf{A... |
H: Where is $\operatorname{Log}(z^2-1)$ Analytic?
$\newcommand{\Log}{\operatorname{Log}}$
The question stands as
Where is the function $\Log(z^2-1)$ analytic
, where $\Log$ stands for the principal complex logarithm. My understanding is that
The domain of analyticity of any function $f(z) = \Log\left[g(z)\right]$, ... |
H: How can I prove that one of $n$, $n+2$, and $n+4$ must be divisible by three, for any $n\in\mathbb{N}$
Intuitively it's true, but I just can't think of how to say it "properly".
Take for example, my answer to the following question:
Let $p$ denote an odd prime. It is conjectured that there are infinitely many twin... |
H: Picking teams for a game
This was a question that came up at work, and we ended up with several different answers!
In the game, there are two teams of four, and each person gets a randomly assigned number from one to eight. This is done by picking the numbers out of a bag.
The question is: if the order of people wi... |
H: Is the Dirac delta function $L^1$ integrable?
The Riemann-Lebesgue lemma says that the Fourier transform of any $L^1$ integrable function on $\mathbb{R}^{d}$ satisfies:
$$\hat{f}(z):=\int_{\mathbb{R}^{d}} f(x) e^{-iz \cdot x}\,dx \rightarrow 0\text{ as } |z|\rightarrow \infty$$
This does not seem to be the case if ... |
H: Can't seem to figure out steps of factoring
I can't seem to figure out how to get from
$5(5\cdot 3^{n}-3\cdot 2^{n})-6(5\cdot3^{n-1}-3\cdot2^{n-1})$
to
$(5\cdot5\cdot3-6\cdot5)3^{n-1}-(5\cdot3\cdot2-6\cdot3)2^{n-1}$
by factoring out $3^{n-1}$ and $2^{n-1}$
AI: As @lab-bhattacharjee said, use the identity $a^{m+n}=a... |
H: Complex integration
I'm not sure how to solve this integral:
$$\int_\gamma {e^{|z|^2}}Re (z)$$ with $\gamma$ being the segment with vertices $0$ and $1+i$.
I started this by saying that $f(z)= {e^{|z|²}}Re (z) = xe^{x²}e^{y²} $
$\gamma (t) = t \; , \;t \; \epsilon \; [0,1] $
But what do I do know? Should I calculat... |
H: Arithmetical or geometrical sequence?
I have some problem in sequence, especially when I want to find the general formula of 'not arithmetic and also not geometric' sequences.
If I already knew that the sequence is arithmetic sequence, I can easily find the formula for arithmetic sequence, such that finding the gen... |
H: Show that $(a+b+c)^\alpha
For any positive real $ a, b, c$, show that $(a+b+c)^\alpha<a^\alpha+b^\alpha+c^\alpha, 0<\alpha<1$.
I can show that it works for special cases like $\alpha=1/p, p\in\mathbb{N}$... but I don't know how to generalize further...
AI: Calvin Lin's answer seems sufficient. However, here are a ... |
H: How do you take this limit algebraically (Not using the graphing calc)
$$\lim_{x\to0}{\frac{e^x-1}{x}}$$
I determined the limit by graphing this and seeing that the graph approaches 1 as x approaches 0. But, is there a way to algebraically determine this limit?
AI: No limit can really be computed by graphing; it'... |
H: Ideals in $Z_{24}$
The ideals in $Z_{24}$ are
$(\overline{0}), (\overline{12}), (\overline{8}), (\overline{6}), (\overline{4}), (\overline{3}), (\overline{2})$ and $Z_{24}$ itself.
Now why isn't, say, $(\overline{5})$, also an ideal in $Z_{24}$?
Ie. $(\overline{5})$ contains the elements $\overline{0}, \overline{5}... |
H: $\mathcal{M, N}$ are $\sigma$-algebras. Prove that $\mathcal{M \setminus N}$ also is a $\sigma$-algebra
Could you tell me why if $\mathcal{M, N}$ are $\sigma$-algebras, then $\mathcal{M \setminus N}$ also is a $\sigma$-algebra?
I've just started reading about measure theory, but I don't think it can be true, becaus... |
H: Prove that $\lfloor an \rfloor +\lfloor (1-a)n \rfloor = n-1 $
Given and irrational $a$ and a natural number $n$ prove that $\lfloor an \rfloor +\lfloor (1-a)n \rfloor = n-1 $.
Is this solution correct?
$\lfloor an \rfloor +\lfloor (1-a)n \rfloor = \lfloor an \rfloor +\lfloor n-na \rfloor =$ (we take out $ n $... |
H: "It can be checked locally that $Z$ is a closed subset"
Look at the following proposition/exercise:
A subspace $Z$ of a topological space $X$ is closed if and only if exists an open cover $\{U_\alpha\}$ of $X$ such that $Z\cap U_\alpha$ is closed in $U_\alpha$ for every $\alpha$.
Now the implication $(\Rightarro... |
H: Positive semi/definite matrix claim.
If $A$, $B$ is positive semidefinite (PSD) and $C$ is positive definite (PD), all are Hermitian, complex valued. I want to claim that $$(B+C)^{-1/2}A(B+C)^{-1/2}$$ is PD. (I am sure it is PSD but looking for PD).
My attempt: I know $B+C$ is PD so $(B+C)^{-1}$ is PD then I know ... |
H: How to prove the space of orbits is a Hausdorff space
Let $M$ be a connected smooth n-dimensional manifold and $G$ a lie group acting smoothly on $M$.
for $x\in M$, the orbit $G\cdot x=\{g(x)\mid g\in G\} $ is a sub-manifold of $M$ and if the action is proper, namely
the inverse image of every compact subset of $M\... |
H: Problem finding in simple algebra
It is given,
$$x= \sqrt{3}+\sqrt{2}$$
How to find out the value of $$x^4-\frac{1}{x^4}$$/
The answer is given $40 \sqrt{6}$ but my answer was not in a square-root form
I have done in thsi way:
$$x+ \frac{1}{x}= 2 \sqrt{3}$$
Then,
$$(x^2)^2-\left(\frac{1}{x^2}\right)^2= \left(x^2 + ... |
H: Evaluate $\lfloor \frac{x}{m} \rfloor + \lfloor \frac{x+1}{m} \rfloor + \dots + \lfloor \frac{x+m-1}{m} \rfloor $
For any $x \in \mathbb{R}$ and $m \in \mathbb{N} $ evaluate $\lfloor \frac{x}{m} \rfloor + \lfloor \frac{x+1}{m} \rfloor + \dots + \lfloor \frac{x+m-1}{m} \rfloor $.
Well if $x=m$ then we obviously ... |
H: Mathematical Induction Can't get past base step... Please help
The Question:
For all integers $n ≥ 1$ prove $1+2^1 +2^2 +\dots+2^n = 2^{n+1} −1$.
I am having a hard time with this. when I let $n=1$, my base step is false. What do I do now?
AI: If the base step is false, the proof fails as a whole.
However, for the ... |
H: 17539 decimal to binary not getting the same result
I'm trying to convert 17539 to binary. My math says its 110000010010001, but online calculators like this and this say it equals to 100010010000011. Who is making something wrong.
AI: \begin{align}
&&17539=2\cdot8769+1\\
&&8769=2\cdot4384+1\\
&&4384=2\cdot2192+0\\... |
H: How to calculate error size from division of two ratios
How one would calculate an error from division of two rations?
I am given 1/y and x/y as their decimal representation (numbers a and b).
Then I perform the calculation b / a, trying to find the value of x.
Both this representaitons have errors e[1] and e[2].
S... |
H: Verify a solution - Chain Rule
I've submitted a homework assignment online where it is then marked by a program and a score is instantly given back.
I was 100% correct in all solutions besides this one.
Use the Chain Rule to find $\displaystyle \frac{dw}{dt}$.
$$
w = xe^{\frac{y}{z}},\,\,\, x = t^5, \,\,\, y = ... |
H: Solution check for $z^2 + |z|^2 = 2+ 2i$ where $z^2=t$
I solved this equation,
$$z^2 + |z|^2 = 2+ 2i$$
placing $z^2=t$
at the end remains:
$$\begin{align}
t&=(2+2i)/2 \\
t&=1+i \\
z^2&=i+1 \\
(z+1)(z-1)&=i \\
z_1&=1+i \\
z_2&=-1+i
\end{align}$$
I checked with Wolfram|Alpha, but the result are:
$z=-1-i$
$z=1+i$
cou... |
H: Find $\lim\limits_{n→∞}\left(t+\frac{x}{n}\right)^n$
How would one go about finding the limit of $\displaystyle \left(|t|+\frac{x}{n}\right)^n$ as $n\rightarrow\infty$? ($t$ and $x$ are both positive.)
Of course $\displaystyle \left(1+\frac{x}{n}\right)^n$ has limit $e^x$, but how would this help?
AI: Use:
$$\lim\l... |
H: Proof of index number in complex analysis
Pictures above show the lemma and its proof in my complex analysis course. My question is how do we know we need to define $h$ to be in that form? Because when I try to prove the lemma on my own, I can't get anywhere.
AI: Note that by definition $$\int_\gamma\frac{\mathrm... |
H: Testing using training data
I've been trying to prove that estimates of a classifier's performance using training data is a bad thing.
Does "bad" mean it is biased?
This is part of a larger proof.
If somebody knows of previous work that proves this or a quick proof, any pointers would be much appreciated!
Thanks in... |
H: Double Coset Closed
Let $G$ be a locally compact group and $H$ a closed subgroup. Under what conditions can we say that the double cosets $H\cdot x \cdot H$ are closed? Is this always true? I am interested mainly in the case when $H$ is discrete.
AI: Closeness of cosets is definitely false in general even for discr... |
H: Solving systems of linear equations with an unknown 'a' using matrices and elementary row operations
Came across this one the other day... while I can narrow 'a' down I can't seem to find an exact/ optimised figure. For example 'a' cannot equal 1/3, 'a' must be less than 0.5...
Anyway, here's the problem. I've got ... |
H: Product of Lebesgue-measurable sets in $\mathbb{R}$ and $\mathbb{R}^2$
Let $M_1$ be the Lebesgue-measurable subsets of $\mathbb{R}$, and $M_2$ be the Lebesgue-measurable subsets of $\mathbb{R}^2$. Prove that $M_1\times M_1\neq M_2$, by considering a set $E\times\{0\}$, where $E\subseteq[0,1]$ and $E\not\in M_1$.
... |
H: Average distance to perimeter of a polygon?
Trying to calculate heat transfer which is a function of distance of each molecule to the closest wall for various container shapes. For example, a rectangular prism versus a cylinder.
So I think that a 'thin' rectangular prism of volume V average distance to wall can b... |
H: Rolles Theorem Simple and multiple zeros
I have this problem with Legendre polinomials
Use Rolle's Theorem to show that Pn cannot have multiple zeros in the open interval (-1, 1). In other words, any zeros of Pn which lie in (-1, 1) must be simple zeros.
The question is that Im a little bit confuse about what is mu... |
H: Fields and $\sigma$ - fields generated by given sets.
Could you check if my reasoning is correct?
The field of subsets of $\mathbb{N}$ (let's denote it by $\mathcal{M}$) generated by singletons is, I think, the set of all subsets of $\mathbb{N}$, because
$\emptyset, \mathbb{N} \in \mathcal{M}$ and
if $\{n\} \i... |
H: Induction and Countable Set
Ok well everytime ive seen induction being used, its been on the naturals for a statement we wish to prove. My question is would any countable set also work? Hence, doing induction on the rationals as they are countable. If this is possible, could someone please give some examples.
Than... |
H: Possible determinant relation for PSD matrices.
Is $$\det(I+ABC)=\det(I+ACB),$$ when $A,B,C$ are symmetric positive semi/definite and $I$ is the identity matrix. I am mostly interested in the case when the matrices are in complex field.
I know $\det(I+BC)=\det(I+CB),$ when $B,C$ is PSD.
Thanks a lot in advance.
AI... |
H: Taylor's Theorem Problem
This is from my engineering mathematics textbook. Is this version of taylor's theorem correct ?
Successive Differentiation, Maclaurin's and Taylor's Expansion of Function $-147$ TAYLOR'S THEOREM Let $f(x)$ be a function of $x$ and $h$ be small. If the function $f(x+h)$ is capable of bei... |
H: What is the symbol $\triangleq$?
I came across this new symbol while reading a document about writing proofs, and I have never seen it before.
AI: It’s is defined to be equal to; it’s my preferred symbol, but the most common one is $:=$, and I’ve also seen $\overset{\text{def}}=$.
For each $x\in X$ there is an op... |
H: Can one prove $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ if and only if $gH = Hg$ for all $g \in G$?
Let $G$ be a group and $H$ a subgroup of $G$.
I have proven in an exercise that $gH = Hg$ for all $g \in G$ implies $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$
where $(g_1H)(g_2H) = (xy | x\in g_1H, y\in g_2H)$.
... |
H: Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant.
Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant.
I do not know how to do it.
AI: Hint: Let $v(x)=f(x)-\int_0^1 f(t)\,dt$. |
H: Factoring Quadratics: Asterisk Method
I'm teaching my students about factoring quadratics. We've done GCF, difference of two squares, squared binomials, and grouping. One of my colleagues then found this asterisk method on line.
It's basically the grouping method but presented in a little different light. Has an... |
H: Dominated convergance of $\frac{1-e^{-xt^2}}{t^2}$
$$\begin{align}f(x, t)&=\frac{1-e^{-xt^2}}{t^2}\\
F(x)&=\int^\infty_0f(x,t)\ \mathrm{dt}\end{align}$$
I need to show that $F$ is continuous on $\Bbb R^+$. $F$ is defined everywhere and $f$ is continuous with respect to $x$, and now I need some uniform, integrable u... |
H: "Differential" of a measure
Let $\mu$ be a finite measure on $\mathbb{R}$. What is the definition of the operator $d$ in the expression: $d\mu$. For example, I have an exercise where at one point:
\begin{equation}
d\mu(x) = \frac{d x}{1+x^2}
\end{equation}
I would said this a "differential" of $\mu$, but I can not ... |
H: The ring $ℤ/nℤ$ is a field if and only if $n$ is prime
Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$
is prime.
Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any hints for this ?
Suppose $ℤ/nℤ$ is a field. Therefore: for every $\b... |
H: Is this epsilon-delta proof correct?
Consider the function $f:\mathbb{R}\rightarrow\mathbb{R}$ $$f(x)=\begin{cases}x,\ x\in\mathbb{Q} \\ -x,\ x \notin \mathbb{Q}.\end{cases}$$
I'm trying to prove that for all $a \neq 0$, $\lim_{x \to a}f(x)$ does not exist. I tried to do this by contradiction, so my first step was ... |
H: Finite subsets and the Definable Power Set Operation
I'm starting to read chapter VI of Kunen's "Set Theory: An I ntroduction to Independence Proofs". Lemma 1.2 says that for each formula $\phi(v_0, ..., v_{n-1}, x)$ with all free variables shown, $\forall A \forall v_0, ..., v_{n-1}\in A[\{x \in A: \phi^A(v_0, v_1... |
H: Perfect numbers less than 10 000
Im trying to find perfect numbers less that 10 000. I was told that the best way to do this is by using maple, but I don't know how to use latex except the basics like graphing. Can someone help me do this in maple? thanks.
AI: You can find much in wiki (http://en.wikipedia.org/wiki... |
H: Lebesgue integral of $x^{-3/4}$ is finite
Let $X=(0,1]$ and $f(x)=\dfrac{1}{x^{3/4}}$. Show that $A=\int_X f d\mu$ is finite, but $B=\int_X f^2 d\mu$ is infinite, where the integrals are Lebesgue integrals.
For $B$, I bound the integral below by simple functions, taking the values of $f^2$ at $x=1/n, 2/n, \ldots,... |
H: Perpendicular line intersection issues
Do not downvote questions for being 'simple' to you. What one might find trivial another may find helpful. It is not in the spirit of SE. That being said,...
I have a line with the equation $y = -2.08x - 44$, and I must find the perpendicular equation, which will be $y \app... |
H: Is zero irrational?
I think of the number zero as a whole number.
It can certainly be a ratio = $\frac{0}{x}, x \neq 0.$
Therefore it is rational.
But any ratio equaling zero involves zero, or
is irrational, e.g.$\frac{x}{\infty}, x \neq 0$ is not a ratio
of integers.
Can a rational number that is rational only wh... |
H: Multivariable local maximum proof
Suppose we have a twice differentiable function $f: \mathbb{R} ^n \to \mathbb{R}$, a point ${\bf x^0} = (x_1 ^0 , \ldots , x_n ^0)$ and we know that
$\nabla f({\bf x}^0) = 0$
$({\bf x - x^0})H({\bf x^0})({\bf x - x^0})^T <0 $, $(H({\bf x^0})$ is the Hessian matrix of $f$)
Prove ... |
H: $\lim_{x\to0^{+}} x \ln x$ without l'Hopital's rule
I have a midterm coming up and on the past exams the hard question(s) usually involve some form of $\lim_{x\to0^{+}} x \ln x$. However, we're not allowed to use l'Hopital's rule, on this year's exam anyways.
So how can I evaluate said limit without l'Hopital's rul... |
H: Bounding the density of finite coprime sets
I am currently running into a problem related to coprime numbers.
Consider a set of $d$-dimensional integer vectors, $z \subset \mathbb{Z}^d$ such that each component $z_i$ is bounded by another integer $K$. Let us denote this set as:
$$Z_K^d = \Big \{ z \in \mathbb{Z}^d ... |
H: Is there a name for the value $x$ such that $f(x)$ is maximum?
Obviously, $f(x)$ is called the "maximum value" or simply "maximum", but what is $x$ called? The maximizer?
Additionally, what if $f(x)$ is minimum or simply an extremum?
AI: It's often called the 'arg max' ('argument maximum'; similarly 'arg min'):
ht... |
H: Differentiating an integral using dominated convergence
Let's say we have
$$F(x)=\int^b_af(x,t)\ \mathrm{dt}$$
And we want to calculate $F'(x)$. Then:
$$F'(x)=\lim_{h\to0}\frac{\int_{a}^{b}f(x+h,t)\mathrm{dt}-\int_{a}^{b}f(x,t)}{h}\mathrm{dt}=\lim_{h\to0}\int_{a}^{b}\frac{f(x+h,t)-f(x,t)}{h}\mathrm{dt}\tag1$$
Now, ... |
H: How do we write a function F(x) into F(x,t) ??
Please see the link below:
Plane wave expansion
My question is if you notice equation (13) in the manual is written the function f(G) as F(G,lambda) using a transverse property?? My question is how is that possible???
AI: If you read the text below, you see that $h(G_i... |
H: Euler function and $\mathbb{Z}/n\mathbb{Z}$
I am trying to solve a very interesting problem about the ring $\mathbb{Z}/n\mathbb{Z}$ and Euler function $\phi (n)$, but i am not sure how to start, i have a few ideas, but none of them leads me to the end of the proof. So, here is the problem.
Let $n$ be a squarefree i... |
H: Using roots of irreducible polynomials to rewrite products.
Suppose $F$ is a field, and $p(x)\in F[x]$ is irreducible, of degree $n$, with a root $\alpha$.
"$F(\alpha)$ is closed under multiplication since $\alpha^n,\alpha^{n+1},\ldots $ can be written as combinations of $1,\alpha,\ldots, \alpha^{n-1}$ using the e... |
H: Prove that $\tau(n) \leq 2\sqrt{n}$
I'm looking at the following problem:
Prove that for a positive integer $n$, $$\tau(n) \leq 2 \sqrt{n}$$ where $\tau(n)$ is the number of divisors of $n$.
So my idea was to split the set of the number of divisors of $n$ into two subsets: One subset containing the divisors those l... |
H: Conjecture $\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$
$$\int_0^1\frac{\ln\left(\ln^2x+\arccos^2x\right)}{\sqrt{1-x^2}}dx\stackrel?=\pi\,\ln\ln2$$
Is it possible to prove this?
AI: Substitute $x \mapsto \cos x$ to obtain an equivalent formulation:
$$ \int_{0}^{\frac{\pi... |
H: Unit sphere parametrization
Consider the unit sphere $S^2 = \{(x,y,z): x^2+y^2+z^2 = 1\}.$
a. Given any point $(x,y,0)$ in the $xy$-plane, parameterized the line
that contains $(x,y,0)$ and $(0,0,1)$.
b. Show this line intersects $S^2$ in exactly two points $(0,0,1)$ and
another point.
How can I do this probl... |
H: Is there a nonstandard characterization of Lipschitz continuity?
Let $f: \mathbb R \to \mathbb R$ be Lipschitz continuous with finite constant $L$.
Then
$$
|f(x) - f(y) \le L |x-y|, \tag{1}
$$
and, by direct transfer, this property holds for $^*\!f$.
For continuity, there is the infinitesimal characterization
$$
x ... |
H: implicit differentiation $\large\frac{-x}{y}\frac{dx}{dt} = \frac{dy}{dt}$ to $\small(\cos{\theta})\large\frac{d\theta}{dt} = \frac{dy/dt}{13}$
Suppose $\frac{y}{13} = \sin{\theta}$.
Please show the steps to implicitly differentiate $\large\frac{-x}{y}\frac{dx}{dt} = \frac{dy}{dt}$ with respect to $t$ to reach $\sm... |
H: A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$
A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help.
Prove:
$$\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\rig... |
H: Primitive Root Modulo $m$
I need help with the following: Show that if $b$ is a primitive root modulo $m$, then $$\{b,b^2,b^3,...,b^m-1\}$$ is a complete set of units modulo $m$.
AI: If we define a primitive root as an element with order $\varphi(m)$ modulo $m$, then we can prove as follows. Each power $b^i$ is cop... |
H: How to evaluate $\pm$ operations
When finding the root of a number with an even exponent, $x^y$ becomes $\pm x$. How would this work in a situation such as $a = \sqrt{(5x + 12)^2 + m}$? I know that the result is not $a = \pm 5x + 12 + \sqrt m$ because the parentheses must be evaluated first.
Could anybody please ... |
H: Showing $F$ of Characteristic $p$ is Separable Provided $\phi(a) = a^p$ is Surjective on $F$
Let $F$ be a field of characteristic prime $p$. Let $\phi: F \rightarrow F$ be defined as $\phi(a) = a^p$ for all $a \in F$.
Goals:
(i) Show that $\phi$ is an injective homomorphism of $F$.
(ii) Show that $F$ is separable... |
H: Why does the vector Laplacian involve the double curl of the vector field?
The scalar Laplacian is defined as $\Delta A =\nabla\cdot\nabla A $. This makes conceptual sense to me as the divergence of the gradient... but I'm having trouble connecting this concept to a vector Laplacian because it introduces a double c... |
H: Can you transform any coordinate from any "space" to another "space" that's defined?
This question pertains to Matrix Transformations. So to provide an example, if I have 3D coordinates where $X = -1$ to $1$, $y = -1$ to $1$, $z = -1$ to $1$. They are "normalized" in my mind. Can I use a Matrix to transform these c... |
H: How to show that $\sum_{i=0}^{n-1}x^i=\frac{1}{1-x}-x^n=\frac{x^n-1}{x-1}$
This is not a homework assignment, but a problem that is bothering me nonetheless. I feel like I should be able to figure this out, but either I'm missing some detail, or I'm doing something completely wrong.
The question arises because I re... |
H: Gaussian elimination on matrix.. Is there a better way to extract the solution?
So, I used Gaussian elimination on this matrix
$$\left( \begin{array}{c} -1 & 3 & 5 & 13 \\ 3 & -2 & 2 & 16 \end{array}\right)$$
to turn it to this:
$$\left( \begin{array}{c} -1 & 3 & 5 & 13 \\ 0 & 7 & 17 & 55 \end{array} \right)$$
I do... |
H: Are there non-periodic continuous functions with this property?
Suppose $ f$ is a real-valued continuous non-constant function defined on all of $ \mathbb{R}$. Let $ A = \text{image} f $. Suppose also that there is a $L > 0$ such that for every half open interval $ I \subseteq \mathbb{R} $ with $| I | = L $, $\tex... |
H: Number of 3-digit numbers which do not contain more than 2 different digits.
[1] Total number of 3-digit numbers which do not contain more than 2 different digits.
[2] Total number of 5-digit numbers which do not contain more than 3 different digits.
$\underline{\bf{My\; Try}}::$ I have formed different cases.
$\... |
H: Further simplify this: $\sum\limits_{k=0}^{n}{n\choose k}\cdot a^k\cdot b^{n-k}\cdot \frac{1}{(k+1)^2}$
I know the simplified form of $\sum\limits_{k=0}^{n}{n\choose k}\cdot a^k\cdot b^k\cdot \frac{1}{k+1}$, which is $\frac{(a+b)^{n+1}-b^{n+1}}{a\cdot (n+1)}$, I am wandering if there exists the simplified form of $... |
H: $50^{th}$ digit from the left in the expansion of $(\sqrt{50}+7)^{50}$.
The $50^{th}$ digit from the left in the expansion of $(\sqrt{50}+7)^{50}$ after the decimal point.
$\underline{\bf{My\; Try}}::$ Let $\left(\sqrt{50}+7\right)^{50} = I+f$, where $I = $Integer part and $f = $ fractional part. and $0\leq f<1$
No... |
H: Directly showing $H_0(X, x_0) \cong \widetilde{H}_0(X)$?
I'm trying to show directly that
$$ H_0(X, x_0) \cong \widetilde{H}_0(X) $$
where $x_0$ is a point in the topological space $X$ and $\widetilde{H}_0$ denotes the zeroth reduced (singular) homology group.
What I have so far gives me a wrong answer, I'd be grat... |
H: Discrete math proof issue
This is a question from my discrete math quiz.
I was asked to prove there exists a Q(x). I used Disjunctive Syllogism to prove it. I was marked incorrectly because I used two different variables in the syllogism. My view is the variables can be implicitly the same, and I don't think this i... |
H: Find the number of cosets$ [G:H] $?
Assume that $G$ is a cyclic group of order $n$, that $G =\ <a> $, that $k|n$ , and that $H=<a^k>$.
Find $[G:H] $ the number of cosets to the subgroup H
I think that since $k|n$ $\Rightarrow$ $<a^k> = e$
then $[G:H] = 0 $
Am I right ?
AI: It can't ever be possible that $[G : H]... |
H: Proof of compactness theorem in first order logic - clarification
My question is about this question and the users's answer to it.
Here's the statement of the compactness theorem: If $T$ is a first order theory in some language $L$. The $T$ has a model if and only if every finite subset of $T$ has a model.
One dir... |
H: Derivatives of these two functions of $x$ containing sine and exponential functions
Can you help me with getting the derivatives of the following two functions please.
\begin{gather*}
f_1(x)=3^{\sin x}5^{\cos x} \\
f_2(x)=e^{x^2}+\sin^2 x.
\end{gather*}
It is too complicated for me. Could someone provide me with s... |
H: Which of the following statement is not necessarily true for the product of rings $R \times R$ when it is true for $R$?
$R$ is a ring. Which of the following statements is not necessarily true for the product of rings $R \times R$ when it is true for $R$?
A. There exists some generator whose order is finite.
B. $R$... |
H: Probability of A given $P(A) + P(B), P(A|B),$ and $P(B|A)$.
Given $P(A|B)= 0.5, P(B|A)=0.4,$ and $P(A) + P(B) = 0.9$ what is $P(A)=$ ?.
AI: Let $a=\Pr(A)$ and $b=\Pr(B)$. We are told that $a+b=0.9$.
We have $\Pr(A|B)\Pr(B)=\Pr(A\cap B)$. Thus $0.5b=\Pr(A\cap B)$.
Similarly, $0.4a=\Pr(A\cap B)$.
It follows that $0.5... |
H: How to calculate Bessel Function of the first kind fast?
I have wrote a C++ code to calculate the first kind of Bessel Functon by its infinite series definition. I took the sum of the first 20 series as the value of Bessel Function, which is same as MATLBA besselj()'s result by the accuracy of 1e-14.
However, in my... |
H: Constant limits of integral: ∭D dxdydz/(x+y+z+1)^3, where D={x>0,y>0,z>0,x+y+z<2}
What are the constant limits (obtained through change of variables) of the integral:
$$\iiint\limits_D \frac{dx \ dy \ dz}{(x+y+z+1)^3}, \quad \text{where} \; \; D=\left\{x>0,y>0,z>0,x+y+z<2 \right\}$$
I am supposed to use the 'tripl... |
H: Continuous functions in topology
I am trying to solve the following problem:
In which topologies (uniform, product, box) are the following functions from $\mathbb{R}$ to $\mathbb{R^\omega}$ continuous?
$f(t)=(t,2t,3t,....)$
$g(t)=(t,t,t,.......)$
$h(t)= (t,\frac{1}{2}t, \frac{1}{3}t,...)$
I dont get why do the fun... |
H: Left-isolated points in totally ordered set
Let $X$ be a totally ordered set. I say that a point $x$ is "left-isolated" if there is some $\alpha < x$ such that there is only one point $y$ which satisfies $\alpha < y \leq x$, that is, $y=x$. I assume that there is some countable subset $D \subset X$ such that any no... |
H: showing that some topology of $\mathbb{R}^2$ is not Hausdorff
Let $T$ be a topology on $\mathbb{R}^2$ generated by a basis consisting of sets that equal $\mathbb{R}^2$ subtracted with finitely many complete lines.
The question is that showing that $T$ is not Hausdorff.
In fact, I solve this problem using connectedn... |
H: Number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$?
The number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$ is
(A)0
(B)2
(C)3
(d)4
Actually im a 10 class student i don't know any of it,but my elder brother(IIT Coaching) cannot solve them,he told me post these questions on this... |
H: $A^TA=B^TB$ implies $\exists P \in M_{m\times m}(\Bbb{R})$. such that $A=PB$,where $P$ is an orthogonal matrix
Assume $A,B \in M_{n\times m}(\Bbb{R})$,and $A^TA=B^TB$,show that there exists an orthogonal matrix $P$, such that $A=PB$.
AI: Note that the condition implies $\langle x, A^TAy \rangle = \langle Ax, Ay \ra... |
H: Modular 2-adic Integers Question
I would like to know if the following statement is true in the 2-adic integers.
$\forall n( n=0 \lor Ex( (x \neq 0 \land x+x=0 \bmod n) \lor (x+x=1 \bmod n) ))$
I will define a modulo predicate as:
$M(x,n,r) := (n=0 \lor Ey(x=yn+r))$
$\forall n( (n=0 \lor Ex( x \neq 0 \land M(x,n,0)... |
H: Roots of the equation?
If $p,q,r$ are real numbers satisfying the condition $p + q + r =0$, then the roots of the quadratic equation $3px^2 +5qx +7r=0$ are
(A)Positive
(B)Negative
(C)Real and distinct
(d)Imaginary
Actually im a 10 class student i don't know any of it, but my elder brother (IIT Coaching) cannot solv... |
H: Why do these stationary subsets union to the entire set?
In proving the following theorem, I do not see why $S$ is the union of the pairwise disjoint stationary sets $S'_\eta$. It seems that for this to hold, you need every $\alpha_\xi$ to be equal to some $\gamma_\eta$?
Theorem. Suppose $\kappa$ is regular unco... |
H: Is $\omega = dU = sin(x+y)dx+cos(x+y)dy$ an exact form?
In my thermodynamics homework I should prove that $dU = sin(x+y)dx+cos(x+y)dy$ is a function of state. Which means it's integration over any path be constant or in other word $dU$ should be an exact form.
I used the Poincare Lemma and had the following calculu... |
H: Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?
Is a set of single element $\{x\}$ connected in a metric space $(X,d)$?
Definition: Suppose that $(X,d)$ is a metric space. A set $E \subseteq X$ is said to be disconnected if there exist two non-empty open sets $G_1$ and $G_2$ such that
$G_1 \... |
H: Supremum - why this holds?
Could you please explain to me why this holds?
$$\sup_{x \in (0,5)}\left|\frac{x}{n}\cdot \ln \frac{x}{n} \right| \neq \sup_{x \in \mathbb{}R^+}\left|\frac{x}{n}\cdot \ln \frac{x}{n} \right| $$
where $n \in \mathbb{N}$.
I think that both sup values are $\infty$ but it is incorrect.
AI:... |
H: Proof: If $r \in R$ is irreducible then $ur$ is irreducible where $u$ is a unit.
If $r \in R$ is irreducible then $r=ab, a,b \in R$ implies $a$ or $b$ is a unit.
How does one proof $ur$ is irreducible if $u$ is a unit.
I must proof: $ur = mn, m, n\in R$ then $m$ or $n$ is a unit.
Suppose $n$ is a unit, then we are ... |
H: ellipse boundary after rotation
Assume I have this vertical ellipse with a certain major axis $a$ and minor axis $b$.
If we take the center of the ellipse to be at $(0,0)$, then the top right small red circle will be at $(b,a)$.
Then I rotate it (say by an arbitrary angle $\theta$) about its center:
My question i... |
H: How to find the point on a parabola where x and y are equal?
On a parabola how could i find the point at which the y and x points are equal and meet on a point of the graph, algebraically?
AI: Substitute $x$ for $y$ in the parabola equation and solve for $x$. |
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