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H: Prove a language is NP-Complete
$A$ is NP-complete.
$B$ is P.
$A \cap B = \emptyset $
$A \cup B \neq \sum^{*}$
Prove that $A \cup B $ is NP-complete.
How can I prove this ? I think if anything can be P-reducible to A then it can also be reducible to $A \cup B$, but don't know how to formally say this.
Thanks.
AI: ... |
H: Dividing rational expressions
How come I am able to divide the following:
$$\frac{2}{2} = 1$$
Yet I am not allowed to divide the $2x$'s in the following:
$$x^2 + \frac{2x}{2x} = x^2 + 1$$
Why can't I divide the $2x$ in numerator with $2x$ in denominator to get $1$? They are both equal parts, and just like in the fi... |
H: A first order linear inhomogenous ODE
Consider this ODE without any initial conditions.
$$ax'(t) + bx(t) + c = 0$$ where $a,b,c$ is a nonzero constant.
What method could be used to solve this ODE? I am seeking a technique that does not require differentiating both sides and turning it into a second-order ODE.
AI: H... |
H: Integrate $f(x)=x^2\ln\left(2\sqrt{\frac{a^2-x^2}{a^2+4x^2}}+\sqrt{\frac{5a^2}{a^2+4x^2}}\right)$
I am trying to find the integral of
$$f(x)=x^2\ln\left(2\sqrt{\frac{a^2-x^2}{a^2+4x^2}}+\sqrt{\frac{5a^2}{a^2+4x^2}}\right)$$
And I am having no luck with it. Does anyone have any ideas? Is it even possible?
I also ne... |
H: Proof that $\lim{a_n}=L$ when $n$ goes to infinity, then $\{a_n\}$ its a Cauchy Sequence
Proof that $\lim{a_n}=L$ when $n$ goes to infinity, then $\{a_n\}$ its a Cauchy Sequence
I start with this hypothesis
$\displaystyle\lim_{n \to\infty}{a_n}=L \Leftrightarrow{\forall{\epsilon}>0}$ $\exists{N}\in{\mathbb{N}}$ su... |
H: Congruence question with divisibility
I have this question and I have proved that a/d is congruent to b/d mod(m/d)
However, I don't know how to go forward to prove a/k is congruent to b/k mod(m/d)
Can anyone help me out? THX
AI: We have $a\equiv b\pmod m\iff a=b+c\cdot m$ where $c$ is some integer
Let $\displaysty... |
H: regular expression for a number
It is a regular expression for a number. I have several questions about it.
(0U1U2U3U4U5U6U7U8U9)* Does it means a set containing a number from 0 to 9 and then concatenate itself n times, or a set containing all of those 10 numbers and then concatenate itself n times.
Is the fir... |
H: Cardinality of a vector space versus the cardinality of its basis
Let $V$ an infinite dimensional vector space.
How to show that the cardinality of $V$ is the same of a basis of $V$?
I saw this argument here link in the main answer (MathOverFlow).
AI: You need the additional assumption that $\dim_F V \ge |F|$ (whe... |
H: Tensor Projection
I'm currently reading "Vector and Tensor Analysis with Applications" by A.I. Borisenko and I.E. Tarapov, and I'm having trouble following a particular mathematical step in where the author projects the moment of inertia tensor onto a set of axes, K. This occurs on page 68 in section 2.4.3. Below i... |
H: If $G \cong H/K$, does it follow that $H \cong G \times K$?
It is very tempting to perform this step, but I feel like it is not true. I couldn't come up with a counterexample though.
AI: No: $H = \mathbb{Z}_4$, $K = \langle 2 \rangle$.
More generally: $H = $ your favorite cyclic group, $K = $ your favorite non-tri... |
H: Finite abelian groups
Find all finite abelian groups (up to isomorphism) of order 320.
So I found the prime factorization to be $2^6 \times 5$. I found the 11 groups to be
$\mathbb{Z}_{64} \times \mathbb{Z}_5$,
$\mathbb{Z}_{32} \times \mathbb{Z}_2 \times \mathbb{Z}_5$,
$\mathbb{Z}_{16} \times \mathbb{Z}_4 \times \m... |
H: If integral is zero and function is continuous and non negative, then what about the function?
If $f$ is continuous on $[a,b]$, $f(x)≥0$ on $[a,b]$ and $$\int_{a}^{b} f(x) =0$$
then prove that $f(x)=0$ for all $x \in [a,b]$.
I tried with Riemann's definite integral definition but couldn't proceed
AI: Hint: Suppos... |
H: A single transferrable vote question
I'm preparing for the Oxford TSA, and am using past papers as a way to practice. One of the questions, I thought I had right, but turned out incorrect. Would love it if you were to analyze my reasoning here -->
Question:
In the elections for the mayor of Bitton, the single tra... |
H: Propositional Logic - Is my answer correct?
I have a question relating to Propositional Logic. Any help will be greatly appreciated.
Without changing the meaning of the following formulæ, which
rely on operator precedence to be interpreted correctly, introduce
brackets in each so that no precedence information is r... |
H: Fast algebraic expansion
Is there an algebraic trick to expand the following expression without multiplying each term with another, expanding the standard way it gives $6+6+6=18$ terms and then cancelling the same terms with opposite signs to get the final result, is there any shortcut?
$(b+c-a)(y+z)+(c+a-b)(z+x)+(... |
H: Reduced nonintegral domain contains at least two minimal primes
Let R be a ring such that R is reduced and R is not an integral domain. Show that R contains at least 2 minimal prime ideals.
AI: This is immediate once you know that the nilradical of $R$ is the intersection of its minimal primes. |
H: Minimum number of hemispheres covering a sphere
Here is a question which seems easy but seems to have many pitfalls. If I give you an arbitrary covering of the sphere by $N$ closed hemispheres. You can pick any of the hemispheres to keep. What is the minimum number you can keep while still covering the sphere? We s... |
H: Relations and Functions - Is my answer correct?
Could someone please advise if my answer is correct or incorrect? Any help will be greatly appreciated.
Given the sets $A = \{1, 2, 3\}$, $B = \{−1, 0, 1, 2\}$ and $C = \{3, 4, 5, 6\}$, indicate the members (pairs) in the following relations. Let $\Bbb N$ denote the n... |
H: $\mathbb{E}[e^{ t \sum_{i=1}^n X_i^2 }] = \Pi_{i=1}^n \mathbb{E}[e^{tX_i^2}]$
$$\mathbb{E}[e^{ t \sum_{i=1}^n X_i^2 }] = \Pi_{i=1}^n \mathbb{E}[e^{tX_i^2}]$$
How do I get this?
It seems like in here, $\mathbb{E}[y_1+y_2+...] = \mathbb{E}[y_1]\mathbb{E}[y_2]...$
AI: Note that
$$e^{t\sum\limits_{i=1}^n X_i^2}=e^{t... |
H: What is the limit of $\sum^{i=n}_{i=0} \left(-\frac{2}{3}\right)^i$?
What is the limit of $$\sum^{i=n}_{i=0} \left(-\frac{2}{3}\right)^i$$ ?
I am currently taking a introductory course in real analysis. Hints will be appreciated!
As a soft question, can I ask what should I search for in the search bar to find my a... |
H: What is the domain of $x^x$ as a real valued function?
Consider the function $f(x) = x^x$.
Wolfram alpha tells me that this function's domain is $x : x>0$, $x \in \mathbb{R}$. I can't see why it cannot be defined for a number like $(-2)$. I mean $(-2)^{-2}=0.25$, the same Wolfram Alpha told me. I realize that fract... |
H: Evaluation of $\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}$
Evaluate
$$\lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n}.$$
$\underline{\bf{My\;\;Try}}::$ Let $\displaystyle y = \lim_{n\rightarrow \infty}\frac{1}{2n}\cdot \ln \binom{2n}{n} = \lim_{n\rightarrow \infty}\frac{1}{2n}\c... |
H: Geometry of Complex Numbers
Write down in the form ${Z}\rightarrow{AZ+B}$ the following transformations of the complex plane:
(a) translation in the direction $(2,-3)$
(b) rotation about (0,1) through $\pi/4$
I know from my student answer key that the answer for
part (b) is ${Z}\rightarrow{AZ-iA+i}$
where $A=\frac... |
H: Hexadecimal to Octal and Vice Versa
Convert Hexadecimal number to Octal - $(FD56.52A)_{16}$ to octal
My answer - $(176526.2452)_8$
Convert Octal to Hexadecimal $(37.27)_8$
My answer - $(1F.5C)_{16}$.
Correct or incorrect?
Please help.
AI: First of all, use $ {inline formula} $ and $$ {paragraph formula} $$ to type... |
H: If a functor preserves finite limits, does it preserve subobjects?
I have just started learning category theory. My question really appears on the title. In other words, can a subobject be seen as some kind of limit?
AI: Let $f : X \to Y$ be a morphism. Then $f : X \to Y$ is a monomorphism if and only if the diagra... |
H: Existence of an unbounded positive sequence $\{x_n\}$ with $f'(x_{n})
Let $a>1$ and let $f:(0,+\infty)\longrightarrow (0,+\infty)$ be differentiable.
Show that there exists a positive sequence $\{x_{n}\}$ with $\lim_{n\to\infty}x_{n}=+\infty$, such that
$$f'(x_{n})<f(ax_{n}),\ \forall n\in \Bbb N^{+}.$$
My try: I ... |
H: Why not $f(z)=z^2$ conformal at $z=0$?
$$f(z)=z^2$$ is not conformal at $z=0$
Why?
Conformal definition:
$f$ is conformal at z if f preserves angles there.
AI: The angle between the rays $t$ and $it$ with $t\in[0,\infty)$ is $90^\circ$.
The angle between thier images $t^2$ and $-t^2$ is $180^\circ$ |
H: $B(R,R)$ is not closed in the topology of compact convergence
I'm doing this exercise in Munkres book, and got no clue to solve this problem. Help someone can help me.
Let $B(R,R)$ be the set of bounded functions $f: R \rightarrow R$. Prove that $B(R,R)$ is not closed in the topology of compact convergence
Than... |
H: $E[e^{X_1^2 t}] = \frac{1}{\sqrt{1-2t}}$ then $M(t) = \frac{1}{\sqrt[\color{red}n]{1-2t}}$
I have $E[e^{X_1^2 t}] = \frac{1}{\sqrt{1-2t}}$. How do I get to $M(t) = \frac{1}{\sqrt[\color{red}n]{1-2t}}$. Where did the $n$-th root come from?
See last line of image
Proposition $\bf 2.4.38.\;$ Let $X_1,\ldots,X_n$ be ... |
H: Epi-Mono factorization in presentable categories
If $\mathcal{C}$ is a locally presentable category, then it seems to be well-known that (Strong Epi, Mono) is a factorization system on $\mathcal{C}$. Where can I find a proof of this fact? Actually I only would like to see a proof that every morphism can be factored... |
H: Question about $a \equiv b \pmod{mn} \Leftrightarrow a \equiv b \pmod{m} \wedge a \equiv b \pmod{n}$
So Knuth's 'Discrete Mathematics' states that:
$a \equiv b \pmod{mn} \Leftrightarrow a \equiv b \pmod{m} \wedge a \equiv b \pmod{n}$ if $m$ and $n$ are relatively prime.
But being a curious human being that I am a ... |
H: Finding a basis of eigenvectors
For a linear operator $T$ on $V$ find the eigenvalues of $T$ and an ordered basis $\beta$ for $V$ such that $[T]_\beta$ is a diagonal matrix:
$V$=$R^3$, $T(a,b,c)$= $(7a-4b+10c,4a-3b+8c,-2a+b-2c)$.
I solved this question, and got that, the eigenvalues are $-1,1,2$ and
the basis $\be... |
H: How does a left group action on the fiber of a principal bundle induce a right action on the total space?
Suppose I define a "principal $G$-bundle" as follows:
A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, together with a... |
H: Lebesgue measure vs. Borel measure
Wikipedia states that
the Lebesgue measure \lambda is an extension of "the" Borel measure which possesses the crucial property that it is a complete measure (unlike the Borel measure).
However I have read that for every Lebesgue-measurable set a subset can be found, which is no... |
H: Conformal map example $ f(z)=e^z$
I an studying the example-1.
I understand $f(z)=e^z$ has a nonzero derivative at all points, hence it is everywhere conformal and locally $1-1$.
But I dont understand th part I underlined with yellow pencil. Please explain it. Thank you so much:)
AI: $z=x+iy$, $f(z)=e^z$, so $e^{... |
H: Problem with a set without accumulation points
Let $S$ be a nonempty subset in $\mathbb R^m$ without accumulation points in $\mathbb R^m$.
Is then
$$
\inf \{ \|x-y\|: x,y \in S, x\neq y \} >0 \textrm{ ? }
$$
AI: Counterexample : $A=\lbrace k; k+\frac{1}{k} | k\geq 1\rbrace$. |
H: why 64 is equal to 65 here?
how is this possible? I know there is some trick, should someone please explain?!
AI: This is a very well known optical illusion. Count the number of squares in each triangle (or at least in each non-vertical or non-horizontal line) and you'll see that they don't have the same slope. The... |
H: Does my logic statement make sense?
I'm trying to convert this sentence to logic notation.
"there is an integer less than or equal to all other integers greater than 0".
"An integer exists that is less than or equal to all other integers greater than 0".
So far I have -
L(x,y): x is less than or equal to y
∃y ∈... |
H: Computing $\sum\limits_{n=0}^{\infty}{\frac{(-1)^n}{\alpha n +1}}$ where $\alpha>0$
I'm trying to calculate $$\sum_{n=0}^{\infty}{\frac{(-1)^n}{\alpha n +1}};\;\; \alpha>0$$
using the fact that
$$\frac{1}{1-a}=\sum_{k=0}^{n-1}{a^k}+\frac{a^n}{1-a}$$
So taking $a=-t^\alpha$ we get
$$\int_0^1\frac{1}{1+t^\alpha}dt=\... |
H: the power series converges in compact convergence topology
Consider the sequence of functions $f_{n}: (-1,1) \rightarrow R$ defined by:$$f_{n}(x) = \sum_{k=1}^{n}{kx^{k}}$$
a) Prove that $(f_{n})$ converges in the topology of compact convergence, conclude that the limit function is continuous
b)Show that ... |
H: Evaluating the limit $\lim_{n \rightarrow +\infty} \frac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}}$
How would you solve the following limit? It's $\frac \infty \infty$ and L'Hospital doesn't seem to help:
$$\lim_{n \rightarrow +\infty} \frac{e^n+e^{-n}}{e^{n+1}+e^{-n-1}}$$
AI: Hint: Multiply the numerator and denominator by $e... |
H: sequence of elements in $\ell_2$
Consider the sequence in $\displaystyle{ \ell _2 (\mathbb N ) }$
$$ (x^k)_k = \sum_{j=1}^{k} \frac{1}{i} e_i $$
where $\displaystyle{ e_i \in \ell_2 }$ is the $i-\text{th}$ unit vector in $\ell_2$ ,i.e. $\displaystyle{ (e_i)_j = \delta_{ij} }$
Does this sequence converge in $\ell_2... |
H: Probability distributing ice cream satisfying the taste of each person.
Distributing randomly 5 vanilla ice-creams and 5 chocolate ice-creams to 10 people among which 3 prefer vanilla, 2 prefer chocolate and the others do not have preference, what is the probability that everyone has an ice-cream he likes?
AI: Fix ... |
H: Findin the most general harmonic polynomial of the form $ax^2 + bxy + cy^2$
The question says to find the most general harmonic form of $ax^2 + bxy + cy^2$. And I've seen one or two answered questions here on this topic but I couldn't understand $why$ certain steps were took and didn't see how this was applicable t... |
H: Matrix topology
Let $X$ be the space of all real $n \times n$ matrices. We define the function $f \colon X \rightarrow \mathbb{R}$ as the rank of a matrix. We induce the standard Euclidean topology on $X$. Show that for all $x$ in $X$ and for all $a$ in $\mathbb{R}$ the set $\{x \in X; f(x) \leq a \}$ is closed in ... |
H: How to Prove the $ {L}_{\infty} $ Ball Is Convex?
I want to prove that a ball for infinity norm is convex:
$$
B_\infty=\{x\in\mathbb R^n : \|x\|_\infty\le1\}
$$
I came up with this proof and appreciate it if someone can help to verify if this is correct:
\begin{align}
\|x\|_\infty&=\|(1-\lambda)x+\lambda y\|_\infty... |
H: Transformations Difference
On what basis we depend when we choose the tool to transform to frequency domain,
I can't distinguish in what case we use one of these transformations ( trig. fourier series, complex fourier series, fourier transform, laplace transform, z-Transform)
AI: These transformations are generally... |
H: How come when $2^{k} | (x-1)(x+1)$ one of the terms is divisible by $2$ and not by $4$ when $k \in \mathbb{N} $ and $3 \leq k$
So I'm reading Knuth's 'Discrete Mathematics' at the moment and there's a paragraph detailing how many solutions are there for $x^{2} \equiv 1 \pmod{p}$.
So other cases (when $p$ is an odd ... |
H: Sum of a set normalize by total items in set
This might be a be simple but I just want to make sure I didn't use the wrong notation.
If I have a set of weighted terms, ${w_1, w_2, \dots, w_n}$ and the score is the sum of $w_1$ to $w_n$ normalize by count of $w$. Count of $w$ is just the total number of terms, i.e.... |
H: Can I approximate a complex number by its imaginary part, if real part is small compared to imaginary part?
I have the following doubt.
How do you explain this?
Here $j$ means $\sqrt{-1}$.
AI: Assuming $\;j=i:=\sqrt{-1}\;$ :
$$ix+\frac{x^2}{a+ix}=\frac{-x^2+iax+x^2}{a+ix}=\frac{ax}{a+ix}i\xrightarrow[x\t... |
H: Lebesgue Measure as a Countable Sum of Probability Measures
Show that Lebesgue measure can be expressed as a countable sum of probability measures.
I'm trying to do something with the countable additivity property in order to show this, but so far nothing is working. I don't think this is supposed to be difficult, ... |
H: Find points on perpendicular line
I have $P_1=(x_1,y_1)$ and $P_2=(x_2,y_2)$ points and they are specified.
Also I have $h$ distance from $P_1$ point that is also known.
And I want to find $P_3$ and $P_4$ that located on perpendicular line and $h$ is distance to them.
How can i found those
$P_4=?$
$P_3=?$
AI: Hint... |
H: Is $(m \Leftrightarrow m) \Leftrightarrow (m \Rightarrow m)$ a tautology, contradiction or contingent?
Is this a Tautology, contradiction or contingent?
$(m \Leftrightarrow m) \Leftrightarrow (m \Rightarrow m)$
My answer is that It is a tautology. But what is yours?
Can someone please explain with a truth table?
... |
H: Leaky integrator in novice terms
I have read few article on the leaky integrator including the Wikipedia. They all give the same equation and the graph and say it is applicable in areas such as neuroscience etc.
But I still cannot understand how this works in practice.
So I would like to know an example scenario h... |
H: An inverse question of uniformly convergence
{Edit: since I made some mistake on the pointwise limit and the uniformly continuous.}
A classical results in elementary analysis state that if a sequence of continuous function $f_n(x)$ on $[0,1]$ is uniformly convergence to $f$, then $f$ is continuous on $[0,1]$ too.
I... |
H: Has the property $\int\Phi =1$ (of the fundamental solution $\Phi$ of the heat equation) a physical interpretation?
For each $t>0$ the fundamental solution of the heat equation is given by $$\Phi(x,t)=\frac{1}{(4\pi t)^{n/2}}\exp\left(-\frac{|x|^2}{4t} \right )$$ and satisfies $$\int_{\mathbb{R}^n}\Phi(x,t)\,dx=1.$... |
H: Reflection of the origin in the x,y,z plane?
The equation of the plane is x-2y-2z=27. It can be written in the form where d gives the distance of the origin from the plane, which I worked out to be 9 (can someone verify?). How do I work out the point which is the reflection of the origin? The answer is (6, -12, -1... |
H: injectivity of the function $f: \quad \mathbb{R}^2_+ \longrightarrow \mathbb{R}^2 \quad : (x,y)\quad \longmapsto \ (x^2-y^2, 2xy)$
Consider the function:
$$f: \quad \mathbb{R}^2_+ \ \longrightarrow \ \mathbb{R}^2 \quad : \quad (x,y) \ \longmapsto \ (x^2-y^2, 2xy)$$
I had to show that this function is injective. Ac... |
H: What is the remainder when $3^{1264}$ is divided by 549?
Please explain in detail.
I tried a lot by applying normal remainder theorem but I am not able to get anywhere.
AI: Here is something to get you started. $549=9\cdot61$, and $9$ and $61$ are mutually prime. So you can compute $3^{1264}\bmod 9$ and $3^{1264}\b... |
H: Question regarding compound Poisson random variable
I have a question regarding compound Poisson random variable.
Let $X:=\sum_{i=1}^{N}{\alpha_i}$ to be a compound Poisson random variable, where $N$ is a Poisson random variable with mean $1$, and $(\alpha_i)_{i \in \mathbb{N}}$ is a family of i.i.d random variable... |
H: Recurrence relation, generating function
I am trying to solve this recurrence relation using generating functions $$x_{n+2}+x_{n+1}+x_n=0$$
$$x_0 = x_1=1$$
I have got this generating function $f_a(x)=\frac{2x+1}{x^2+x+1}$. Since the denominator has complex roots I can't factor it. Is there a way to get the series w... |
H: In how many ways can $7^{13}$ be represented as product of $3$ natural numbers?
How i solved it:
all possible non-distinct groups $(a,b,c)$ are,
$a = 0 \Rightarrow (b,c) = (0,13)(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)$
$a = 1 \Rightarrow (b,c) = (1,11)(2,10)(3,9)(4,8)(5,7)(6,6)$
$a = 2 \Rightarrow (b,c) = (2,9)(3,8)(4,7)... |
H: find value (-2)^-(2)^(-2)
Find the value of $(-2)^{-(2)^{(-2)}}$. Is it 16/8/-8/none?
My attempt: $a^{-x}=\frac1{a^x}$, so, $(-2)^{-(2)^{(-2)}}=(-2)^{\frac{-1}{2^2}}=\frac{1}{(-2)^{\frac14}}$. That is, I would pick 'none of the given options' as my answer.
But it is given that answer is 16. And the comments below... |
H: What is the kernel of the tensor product of two maps?
Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product $f_1\otimes f_2\colon V_1\otimes V_2\to W_1\otimes W_2$ is defined, and ... |
H: Limit with e number, explain me this...
Please explain me how somebody got this, step by step what's done here...
AI: I think that there is a mistake: it is $\displaystyle \frac{\log(1+x)}{x} = \log(1+x)^{\frac{1}{x}}$.
So, remembering that $\displaystyle \lim_{x \to +\infty} \left(1 + \frac{1}{x} \right)^{x}=e$ ... |
H: Minimal polynomial of the inverse
Given $\mathbf{J}_t(\lambda)=\begin{pmatrix}\lambda&1&&\\&\lambda&1&{\LARGE\mathbf{0}}\\&{\LARGE\mathbf{0}}&\ddots&\ddots\\&&&\lambda\end{pmatrix}\in\mathcal{M}_{t\times t}(\mathbb{F})$, where $\lambda\in\mathbb{F}$. Suppose $\lambda\ne0$, show that $m(x)=\left(x-\lambda^{-1}\righ... |
H: The MLE of a $N(\theta, 1)$ distribution
I am trying to find the Maximum Likelihood Estimator of an i.i.d. sample $X_1, \ldots, X_n$ arising from the model $N(\theta, 1)$, where $\theta \in [0,\infty)$. I have done this problem previously where the mean was not restricted to be non-negative, and found the MLE to be... |
H: Is every theorem of PA true in the standard model of number theory $N$?
My understanding is that every theorem $\phi$ of $PA$ is true in $N$ because
$N$ is a model for $PA$, $N\models PA$.
By completeness of first order logic, "$PA\vdash\phi$" implies that "if $N\models PA$ then $N\models \phi$".
Hence $\phi$ is ... |
H: $ \lim_{x \to 0^{-}} 1 + e^{1/x}$ (Definition)
Using the definition of limit, I need to show that $\lim_{x \to 0^-} 1 + e^{1/x} =1$.
Using the definition, I have that $x > \frac{1}{\ln \epsilon}(|e^{1/x}|=e^{1/x} < \epsilon)$, but this is not helping since I need a positive delta. Also, I coudn't deal with the case... |
H: Existence of a prime between $ap$ and $(a+1)p$ - generalization of Bertrand's postulate
Conjecture:
There exists at least one prime number $p_{m}$ : $ap_{n} < p_{m} < (a+1)p_{n}$, $\forall$ $a \in \mathbb{N}$ and $\forall$ $p_{n}$ $\in \mathbb{P} $ if $(a+1)p_{n} < p_{n+1}^2$ .
Is there a name for this conjectur... |
H: Solving a problem with L'Hospital's rule
$$\lim _{x \to 2 \pi} (\cos x) ^{1/\sin ^22x} $$
I know that I should get it into $\frac{f(x)}{g(x)}$ form. How should I do it?
AI: Hint: $$\ln \left((\cos x)^{1/\sin^2 2x}\right) = \dfrac{\ln \cos x}{\sin^2 2x}.$$ |
H: Algebraic representation of an absolute value.
I tried several ways, but i could not come up with any way to have an equation as such:
|n| = ...
without using the absolute value signs on the right side of the equation.
I do not know if there is any way...
I tried using some form of n * i^(an + b) but those efforts... |
H: How to prove this just by using Natural Deduction?
I need your help to prove this by using Natural Deduction:
$$(\exists x)(p(x) \implies q) \dashv\vdash (\forall x)(p(x) \implies q).$$
I want to show the proof for both sides. It is a bit easy for me to get the first side from the second side. How can I get the ... |
H: Multiply a matrix by its transpose
I'm using some material found on the internet for learn how to use R.
One exercise is asking to return the result of a multiplication of a $15\times 3$ matrix by its transpose. Being the transpose a $3\times 15$ matrix, I would assume the result is a matrix of dimension $15\times ... |
H: First Order Differential Equations
I am having trouble isolating the $x$ and $y$ to separate side in the differential equations below. Could someone give me a hint as to how to to this.
Equation 1: $$\frac{dy}{dx} - \frac{x}{y} = \frac{1}{x}$$
Equation 2: $$xy\frac{dy}{dx} = y^2$$
AI: The first is not separabl... |
H: How to find angle $v$ in a rectangular diagonal
I am unable to find a way to find the angle $v$ (in degrees) in a rectangular diagonal.
Here's what I have:
Opposite: $15.1$ m
Hypotenuse: $23.5$ m
Adjacent: $x$
$v$ = measure (in degrees) of the angle opposite to the 'Opposite' side
Thanks a lot in advance!
AI: H... |
H: There are only two types of groups of order $6.$
There are only two types of groups of order $6.$
Could anyone advise on how to prove a/m claim? Here is my attempt but I'm stuck:
If $\exists g\in G$ such that $o(g) =6,$ then $G = \left \langle {g}\right \rangle.$
If not, let $G = \{g_1,g_2,g_3,g_4,g_5,e\},$ wher... |
H: Why $p \leftrightarrow q$ is equivalent to $(p \wedge q) \vee (\neg p \wedge \neg q)$? Without using the truth table
I want to know why $p \leftrightarrow q$ is equivalent to $(p \wedge q) \vee (\neg p \wedge \neg q)$? Without using the truth table.
Thanks all
AI: Just think about the statement.
$p \leftrightarrow... |
H: Express lattice axioms using implication and universal quantification
I'd like to ask for some help with homework. My task is to express lattice axioms in signature $(\leq, =, \sup, \inf)$ using only implication and universal quantification.
Here are these axioms in $(\leq, =)$ signature:
$\forall x (x \leq x)$ - ... |
H: Proving the cotangent function is uniformly bounded on the complex plane
I'm trying to prove that the function $\cot\left(z\right)=i\frac{e^{iz}+e^{-iz}}{e^{iz}-e^{-iz}}$
is uniformly bounded in the complex plane outside $\varepsilon$
neighborhoods of the poles (with the bound depending on $\varepsilon$).
The... |
H: Does statement 1 imply statement 2?
1) (For some $t, P(t).) \implies Q$.
2) For all $t, (P(t) \implies Q).$
I think so, and my reasoning is this: for Q to be true, we just need P to be true for some t. Therefore, over the range of all possible t's, once P is true, Q is immediately true.
AI: Yes.
(1) "If (there e... |
H: What am I doing wrong?
I am trying to prove the integral test for series, but got a strange result.
Assume that $f$ is decreasing and positive.
Because the series can be imagined as the area-sum of $1$-wide rectangles of height $f_n$ and each of those rectangles can be expressed as a constant integral on a $1$-wide... |
H: How do I calculate the area of a rectangular diagonal
I need to find a way to calculate the Area of a rectangular diagonal, this is what I have so far:
Opposite: 15.1 m
Hypotenuse: 23.5 m
Adjacent: x
EDIT: The picture was wrong.
AI: Given your link to the picture of a "rectangular diagonal," it looks like you ar... |
H: Arranging Prime Factors to form Integer Solutions
I have a problem as such:
How many solutions in positive integers are there to the equation $x_1 \cdot x_2 \cdot x_3 \cdot x_4 = 2^{20} \cdot 13^{13}$? Let $x_1,\ldots,x_4$ all be distinguishable, so $x_1=a,x_2=b,x_3=c,x_4=d$ is distinct from $x_1=d,x_2=c,x_3=b,x_4... |
H: Free abelian group has a subgroup of index n
A nonzero free abelian group has a subgroup of index n for every positive integer n.
Proof: Consider the free abelian group F(S), where S is the set of generators. Let x
be some element of S, and let xn be some element of S'. Then F(S') is a subgroup of F(S), and F(S)/F... |
H: How to derive this second derivative using the quotient rule?
If a given first derivative is: $\ {dy \over dx} = {-48x \over (x^2+12)^2} $
What are the steps using the quotient rule to derive the second derivative: $\ {d^2y \over dx^2} = {-144(4-x^2) \over (x^2+12)^3} $
My Steps:
\begin{align*}
{d^2y \over dx^2}
&=... |
H: Prove Heine-Borel Theorem
Prove Heine-Borel Theorem: "A subset $S$ of $\mathbb{R}$ is compact if and only if every open cover for $S$ has a finite subcover."
Suggestions: Let $S \subset \mathbb{R}$. If every open cover for $S$ has a finite subcover, then $S$ must be compact. Why? Now, assume that $S$ is compact and... |
H: Find $a\in\mathbb{N}$ such that $n^4+a$ is not prime $\forall n\in\mathbb{N}$
How would I go about finding such an $a$?
I've been thinking it is something to do with modular arithmetic, but don't know what base to consider.
AI: $n^4 + b^4 = (n^2-\sqrt{2}nb+b^2)(n^2+\sqrt{2}nb+b^2)$. Set $b = 2\sqrt{2}$. Then, $n^4 ... |
H: Random variable of twice another random variable
If Y is a random variable with a mean $\mu$ and a standard deviation, $\sigma$, how do I calculate W if W = 2Y? Is it just $2\mu$ and $2\sigma$?
AI: E(X) = mu
E(W) = c*E(X) = c*mu
VAR(W) = VAR(c*X) = c^2VAR(X)
SD(W) = c*SD(X) |
H: How can I get the limit of $(-1)^{2n} $ when $n$ goes to infinity
How can I get the limit of $(-1)^{2n}$ when $n$ goes to infinity? I checked it with wolframalpha but the result is $e^{2i}$, $0$ to $\pi$, why $i$???
AI: Note that $(-1)^{2n}=((-1)^2)^n=(1)^n$
As $n$ tends toward infinity, what happens now? |
H: Using Weyl sequences to prove relation between quadratic form and spectral radius
I know that the formula $$\lVert A\lVert=\sup_{\lVert x\lVert=1} \langle x,Ax\rangle$$
holds true for self adjoint operators. While reading Teschl's book I saw a comment that on can prove this formula for normal operators using Weyl s... |
H: Isomorphic Gaussian Integers
Can we show that the ring of Gaussian integers $$\mathbb{Z}[\sqrt{17}]:=\{a+b\sqrt{17}:a,b\in\mathbb{Z}\}$$
$$\mathbb{Z}[\sqrt{11}]:=\{a+b\sqrt{11}:a,b\in\mathbb{Z}\}$$
equipped with standard addition and multiplication are not isomorphic?
AI: An isomorphism of rings would map $1$ to $1... |
H: Find number of solutions of the equation $ x_{1}+x_{2}+x_{3} = 41$, where $x_{1}, x_{2}\ \text{and}\ x_{3}$ are odd and non negative integers
There are two constraints to this problem:
$x_{1}, x_{2}\ \text{and}\ x_{3}$ are non negative integers
$x_{1}, x_{2}\ \text{and}\ x_{3}$ are odd
If there had been just th... |
H: Where am I going wrong on this second derivative?
If a given first derivative is: $\ {dy \over dx} = {-48x \over (x^2+12)^2} $
What are the steps using the quotient rule to derive the second derivative: $\ {d^2y \over dx^2} = {-144(4-x^2) \over (x^2+12)^3} $
My Steps:
$$ {{d^2y \over dx^2} = {-48(x^2 +12)^2 - 2(x^2... |
H: Maximum N that will hold this true
Find the largest positive integer $N$ such that
$$\sqrt{64 + 32^{403} + 4^{N+3}}$$
is an integer
Is $N = 1003$?
AI: Note that with $N=2008$ we have $ (2^{N+3}+8)^2=4^{N+3}+2\cdot 8\cdot 2^{N+3}+64=4^{N+3}+2^{2015}+64=64+32^{403}+4^N,$ so we conjecture that the maximal value is $20... |
H: Prime gaps with respect to the squared primes
Conjecture
If we have two consecutive prime numbers $p_{a}$ and $p_{a+1}$, and two other consecutive primes $p_n$ and $p_{n+1}$,
so that $p_{a} < p_{a+1} < p^2_{n+1}$,
then $p_{a+1} - p_{a} < 2p_{n} $.
Are there any known counter examples and are there any known s... |
H: If $n^2+10$ is odd then $n$ is odd.
I have been having a little trouble with proofs, and would like to ask for a hint for this statement. I've already started it, however I'm unsure what I should be doing next.
Prove that if $n^2 + 10$ is odd then $n$ is odd.
My answer so far: Suppose that $n$ is an odd integer, ... |
H: Counting ways to partition a set into fixed number of subsets
Suppose we have a finite set $S$ of cardinality $n$. In how many ways can we partition it into $k$-many non empty subsets?
Example: There is precisely one way to partition such a set into $n$-many subsets.
and there is one way to partition into a single ... |
H: Let $f(x) \in F[x]$ and assume that $f(x)|g(x)$ for every nonconstant $g(x) \in F[x]$. Show that $f(x)$ is constant
because $f(x)|g(x)$ $f(x)$ must share at least one root with $g(x)$ and because $g(x)$ could be any degree polynomial that don't share the same roots (ex: $x+1$, $x-2$) $f(x)$ must be a constant value... |
H: In the proof that in a PID, every non-zero non-unit is the product of irreducibles...
In proving that all non-zero non-units of a PID are a product of irreducibles, theres:
"We now show that $a$ is a product of irreducibles. If $a$ is irreducible,
we are done. Otherwise let $p_1$ be an irreducible such that $p_1... |
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