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H: Prove $\det(kA)=k^n\det A$
Let $A$ be an $n \times n$ matrix and $k$ be a scalar. Prove that $\det(kA)=k^n\det A$.
I really don't know where to start. Can someone give me a hint for this proof?
AI: First, let's recall what multiplication of a matrix by a scalar means:
When we multiply $\;kA$, where $A$ is an $n\t... |
H: Who realized $\int \frac 1x dx =\ln(x)+c$?
Who discovered the non-obvious $\int \frac 1x dx=\ln(x)+c$ ? Were power series involved? The series look similar on opposite sides of 1:
$$ \frac 1x =\sum_{n=0}^\infty (-1+x)^n \text{ for } |x-1|<1 $$
$$ \ln(x) = \ln(-1+x)-\sum_{k=1}^\infty \frac{(-1)^k}{k(-1+x)^k} \text{ ... |
H: Integrate $\int e^{-x} \cos x \,\mathrm{d}x$
I know that integration by parts leads to an infinite loops of sin and cos so what do I do?
I can't do $u$ substitution because I can't get rid of all the variables.
$$\int e^{-x} \cos x \,\mathrm{d}x$$
AI: That's the entire point - you want to go through a loop once an... |
H: Error calculating the limit $ \lim_{x \to 0}{\frac{x-\tan(x)}{x^2 \cdot \sin(x)}} $
Given this limit:
$$ \lim_{x \to 0}{\frac{x-\tan(x)}{x^2 \cdot \sin(x)}} $$
Wolfram says the result is $\frac{1}{3}$ , but I tried to solve it and I get 0:
$$ \lim_{x \to 0}{\frac{x \cdot (1-\frac{\tan(x)}{x})}{x \cdot (x \cdot ... |
H: Please help me to prove that $|f(x)| \le M \Vert x\Vert$ around $0$ when $|f(x)| \le \Vert x \Vert^\alpha$ around $0$
Question:
Suppose that $0<r<1$ and that $f\colon B_1(0) \to \Bbb R$ is continuously differentiable.
If there is an $\alpha>0$ such that $|f(x)| \le \Vert x \Vert^{\alpha}$ for all $x\in B_r(0)$, th... |
H: Mergelyan's theorem from Runge's theorem?
From Conway, A course in functional analysis, page 85. Corollary 8.5.
I want to ask for a hint how to deduce Mergelyan's theorem from Runge's theorem, assuming a functional analysis rhetoric proof. This is listed as a corollary after Runge's theorem with no proof given. I ... |
H: Next step in solving an equations...
I have managed so far to break down an the following equation:
$x^n+y^n=1$
to
$x^n=1-y^n$
but what is the next step to get $x$ on it's own?
I have hopped over here from StackOverflow where I am trying draw superellipse where a and b are always 1. So applogies for my lack of term... |
H: Poisson Estimators
Consider a simple random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$.
Let $T=\sum X_{i}$. Show that $\tilde{\theta}=[(n-1)/n]^{T}$ is an unbiased estimator of $\theta$.
AI: In this calculation, we assume that you are familiar with the moment generating fun... |
H: Integrate $\int e ^ \sqrt{x} \, dx$
$$\int e ^ \sqrt{x} \, dx$$
I don't even know how to begin. I tried u sub but obviously doesn't work, no variable to cancel. I tried the formula to memorize $$\int ae^ {au} \, du = \frac{1}{a^2} (au - 1)e^au$$ but that didn't work and I think maybe it isn't suppose to.
AI: HINT: ... |
H: Is the Fibonacci number-like function too trivial to investigate about?
To make some interesting recursive function, I generalized Fibonacci numbers to a function $f(x)$ such that satisfies the following condition:
Given a function $g(x)$, such that $g(0)=0$ and $g(1)=1$, defined on the interval $x\in [0,2)$, $f(x... |
H: Find the number of ways of picking the following cards from a standard, 52-card deck.
a) a king and a queen
b) a king or queen
c) king and a red card
d) king or a red card
For a), I can see that, since there are 4 kings and 4 queens in a deck, the number of ways of picking both would be $(4)(4) = 16$. This is the m... |
H: How to graph the trigonometric function when period is more than the range?
I have to graph this trigonometric equation for a given range,
$y = -\sin \left(\dfrac{x}{3}\right) - 2, ~-\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2}$
But the period is coming out to be $6 \pi$. So the question is how?
I would appreciate a... |
H: Uniformly continuous function on a disconnected domain
$f:A=\mathbb{Q}\cap (0,7)\to\mathbb{R}$ be an uniformly continuous function
can anyone tell me which of the following are correct?
$1. f \text{ is bounded}$
$2. f$ must be constant
$3. f$ is differentiable at all rational points
$4. f$ is differentiable in $(0,... |
H: derivative as a linear map
$f:\mathbb{R}^n\to\mathbb{R}$ is given by $f(x_1,\dots,x_n)=a_1x_1+\dots+a_nx_n$
where $a=(a_1,\dots,a_n)$ is a fixed nonzero vector, $Df(0)$ denote the derivative of $f$ at $0$ could anyone tell me which of the following are correct?
$1. Df(0)$ is a linear map from $\mathbb{R}^n\to\mathb... |
H: Learning to read mathese - how do I interpret $(c_j = a + (j - \frac{1}{2}) \Delta x$
$(c_j = a + (j - \frac{1}{2}) \Delta x$
I am trying to learn the midpoint rule for approximation of the area under a curve but I can't translate this into something I can work with.
The formula to memorize is $M_n = \Delta x ( f(c... |
H: (Qual Question) Example of a non-measurable function $a_{ij}:\mathbb{Z}\times\mathbb{Z}\to\mathbb{R}$
The title says it all. The question arises from a qualifying exam question in which it asks to provide an example in which we may have $A\neq B$ where
$A=\sum_{i\geq1,j\geq1}^{\infty}a_{ij}$ and $B=\sum_{j\geq1,i\... |
H: Partial Derivatives on Manifolds - Is this conclusion right?
I'm self-studying Differential Geometry and I've asked here about how to describe functions on a manifold, and now that I'm pretty sure that my conclusions about that are correct I've started to think on how do we compute partial derivatives. Well, Spivak... |
H: $x \sin x=2$ why is my proof that there no solutions wrong?
$\frac 12 x \sin x=1$ . Let's look at a right triangle with base $x$ and altitude $\sin x$ . Then our equation is for the area of this triangle. Let the sides of the triangle be $a=x$ , $b=\sqrt {x^2+sin^2 x}$ , and $c= \sin x$ . According to wikipedia, He... |
H: Interest Calculation
A university student receives his statement for his tuition and notices that he doesn't have enough money to pay it all off at once. The student inquires about interest rates at his university and is told the following information:
You will be subject to interest charges of one per cent month... |
H: Meaning of Problem in Evaluation
In Algebra, a good rule-of-thumb I saw was If solving an equation leads to a contradiction, there is no solution. And this makes sense to me, particularly in the following case:
$x-1 = x +1 \Rightarrow -1 = 1$
We are being asked to find a number whose predecessor is the same as its ... |
H: Calculation for absolute value pattern
I have a weird pattern I have to calculate and I don't quite know how to describe it, so my apologies if this is a duplicate somewhere..
I want to solve this pattern mathematically. When I have an array of numbers, I need to calculate a secondary sequence (position) based on t... |
H: Invariant subspace
Suppose that $v = v_1 + iv_2$, where $v_1$ and $v_2$ are real vectors. Show that if we view $A$ as defining a map $α$ of $\mathbb{R^3}$ into itself, then $α$ leaves the subspace spanned by $v_1$ and $v_2$ invariant.
I'm looking for hints of how I should approach this problem. I mean, do I need to... |
H: Find the number of bytes that begin with 10 or end with 01.
A sequence of digits where each digit is 0 or 1 is called a $binary\
\> number$. Each digit in a binary number of a component of the number. A
binary number with eight components is called a byte. Find the number
of bytes that begin with 10 or end wit... |
H: Partition problem
While doing some programming, I found that I needed to design or implement an algortihm to basically do the same as the Partition Problem,that consists in partitioning a set of integers in two groups with the most equitative sum, but with floating point numbers. Is there any "efficient" solution? ... |
H: Does $\lim_{t\to 0}\frac{x \sin(xt)}{1+x^2}=0$ uniformly over $x\in \mathbb{R}$?
Consider the following limit: $$\lim_{t\to 0}\frac{x \sin(xt)}{1+x^2}.$$The question is to determine whether it exists, and if it does, whether it is uniform, i.e. whether there is $\delta_\epsilon>0$ not depending on $x$ such that $\l... |
H: Norm of integral operator
Consider the operator $T(f(t)) = \int_0^t f(s)ds$, where $t \in [0,1]$, and $f(t) \in C[0,1]$.
To prove $$\|T^n\| = \frac{1}{n!}$$
Thanks for suggestions.
AI: HINT:
$$
\|T^n\| = \sup_{\|f\|=1} \|T^nf\| = \sup_{\|f\| =1}\sup_{s_{n+1}\in [0,1]} \left|\int^{s_n=s_{n+1}}_0\cdots\int^{s_2}_0 f(... |
H: Explanation/How to use the Lattice isomorphism theorem
I am having trouble understanding some of the wordings of the Lattice isomorphism theorem (Also known as 4th isomorphism theorem) in group theory. I quote here the theorem as in Dummit and Foote
Let $G$ be a group and let $N$ be a normal subgroup of $G$. Then ... |
H: I have to determine which of the following define a metric on $\Bbb R \,\,$?
I am stuck on the following problem:
Determine which of the following define a metric on $\Bbb R$:
$d(x,y)=\frac{|x-y|}{1+|x-y|}$
$d(x,y)=|x-2y|+|2y-x|$
$d(x,y)=|x^2-y^2|$
MY ATTEMPT:
In each of the aforementioned cases, $... |
H: $F(u)= \frac{2}{\pi}\int_{0}^\infty \frac{uf(x)}{u^2 + x^2}dx.$ Show that $\lim\limits_{u\downarrow0}F(u)=f(0)$.
This is an problem from p. 296 of Buck's Advanced Calculus: Let $f$ be continuous on the interval $0\leq x < \infty$ with $|f(x)|\leq M$. Set $$F(u)= \frac{2}{\pi}\int_{0}^\infty \frac{uf(x)}{u^2 + x^2}d... |
H: Where does $\sin 3° =3\sin 1° -4 \sin^3 1°$ come from?
Wikipedia makes the claim:
"Though a complex task, the analytical expression of $\sin 1°$ can be obtained by analytically solving the cubic equation $\sin 3° =3\sin 1° -4 \sin^3 1°$ from whose solution one can analytically derive trigonometric functions of al... |
H: Are there infinitely many rational outputs for sin(x) and cos(x)?
I know this may be a dumb question but I know that it is possible for $\sin(x)$ to take on rational values like $0$, $1$, and $\frac {1}{2}$ and so forth, but can it equal any other rational values? What about $\cos(x)$?
AI: There are infinitely many... |
H: Is the tensor product of 2 free Abelian groups free?
Ok, basically, I think that is true. Let's consider $A$, and $B$ are both free $\mathbb{Z}-$modules, i.e $A = \bigoplus_{i \in I} \mathbb{Z}$, and $B = \bigoplus_{j \in J} \mathbb{Z}$, so we'll have:
$$\begin{align}A \otimes B &= \left(\bigoplus_{i \in I} \mathbb... |
H: Taylor series expansion for $f(x)=\sqrt{x}$ for $a=1$
I seem to be stuck defining an alternating sequence of terms in this series because $f^{(0)}(x)=f(x)$ is positive, as well as $f'(x)$, but then every other term starting with $f''(x)$ is negative. How can I define $f^{(n)}(x)$ given this?
\begin{array}{ll}
f... |
H: A proof in circles.
I need help proving this problem:
$AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD = AB$. The tangent at B meets the line $AC$ produced at $E$. Prove that $ AE = 2AB $.
What I've got so far is this:
on extending the line $CD$ to the tangent at $B$ such that $CD$ and the... |
H: Heaviside step function squared
I have a question about the Heaviside step function $\theta(\xi)$, defined by
$$\theta(\xi):=\begin{cases}
1, & \xi\geq0\\
0, & \xi<0
\end{cases}$$
I need to evaluate the square of the Heaviside step function, i.e.
$$[\theta(\xi)]^2$$
So, my question is: does the relation
$$[\theta(\... |
H: How is a sequentially compact space not compact?
Let us assume the space $X$ is not compact. Then there exists a covering with no finite subcovering, such that every set contains at least one point that no other does.
Select a countable number of such points assuming the axiom of countable choice. We have an infin... |
H: Hyperbolic integration solving
$$ \therefore x-x_0 = \pm \int_{\phi(x_0)}^{\phi(x)} \frac{d \phi}{\sqrt\frac{\lambda}{2}\left( \phi^2-(\frac{m}{\sqrt \lambda})^2\right)} $$
How can we write the above equation to as,
$$
\phi(x) = \pm \frac{m}{\sqrt \lambda} \tanh\left[\frac{m}{ \sqrt 2} (x-x_0)\right]$$
Do I need... |
H: For which values of M vectors$A(m-4,2,2m-12),B(2,m-12,2)$ are orthogonal
I want to find for which values of M vectors$A(m-4,2,2m-12),B(2,m-12,2)$ are orthogonal.
what I did is to do $A*B=0$ and the result was $m=7$ then I inserted $7$ and tried to check if they are orthogonal but it didnt gave me $0.$
there is ... |
H: $x+y> \epsilon$ then $x>\frac{\epsilon}{2}$ or $y>\frac{\epsilon}{2}$
Since the statement is true,
$P(x+y>\epsilon) \le P(x>\frac{\epsilon}{2} \text{ or } y>\frac{\epsilon}{2})$
Why the inequality exists there?
AI: The statement
$$x+y > \epsilon \Longrightarrow x > \frac \epsilon 2 \ \text{ or } \ y > \frac \eps... |
H: Series expansion of $\sqrt{\log(1+x)}$ at $x=0$
Mathematica gives the following series expansion of $\sqrt{\log(1+x)}$ at $x=0$.
$$
x^{1/2}-\frac{1}{4}x^{3/2}+\frac{13}{96}x^{5/2}-\cdots
$$
You can find it from Wolfram alpha too.
How can I obtain the expansion?
Obviously Taylor expansion is impossible because $\sqr... |
H: Find $x$ such that $\frac{1}{x} > -1 $
How do I solve this type of inequalities analytically?
I know the answer is $ x<-1 $ and $ x> 0 $ but:
$$\frac{1}{x} > -1 $$
$$1>-x $$
$$ x>-1 $$
Wat I'm doing wrong?
AI: You can't multiply both sides of an inequality by an unknown (which is your first step). This is because w... |
H: Apparent equivalent notations for the axiom of infinity
I'm just begining to build the systems of numbers based on the axioms of set theory ($\mathsf{ZF}$). Accordingly the axiom of infinity is no more than assuming the existence of $\mathbb{N}$ (of course the axiom is formulated in terms of the existence of an ind... |
H: Name for $X^\infty=\bigcup\limits_{k=0}^\infty X^k$
I'm making structures associated with groups, rings and so on in OCaml and in order to do so I started by defining sets and a few operations (intersection, union, difference, carthesian product, carthesian power) and now I want to define a set $X^\infty=\bigcup\li... |
H: The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$?
The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$?
We know that minimum value of a quadratic is $-\cfrac{b}{2a}$.
We will get one condition from here and $-\cfrac{b}{2a}$ should be eq... |
H: Probability Question : Pick a ball.
Here's the problem statement.
Given a bag with three type of balls, i.e. winning ball, losing ball and try-again ball. If a person picks up a winning ball then he wins, if he picks losing ball then he loses and if he picks a try-again ball, he tries again and that try-again bal... |
H: Will a point moving on a sphere always at an angle x (0 deg. < x < 90 deg.) to the "equator" reach a "pole"?
Formulating my question seems to have given me the answer: that the point will continue getting closer to the pole but never reach it. Am I correct?
Edit in response to Martin Argerami:
I see your point.... |
H: Find the vector that meets the following criteria
I want to find the vector $X$ by the following lines:
$$(1,-3,5) \cdot X=49$$
$$(4,1,-1) \cdot X = 0$$
$$(2,0,-3)\cdot X=-9$$
I would like to get some advice how to find him.
Thanks!
AI: Hint: Assume the vector $X=(x,y,z)$ and then solve the system of equations. |
H: Complex logarithm vs real logarithm
Suppose we are given the function
$$\theta = \ln |z| \quad \text{defined on the upper half plane $\{ z \in \mathbb{C} \colon \Im( z) > 0\}$}$$
Naively, I would go and manipulate
$$
\ln | z| = \ln (z\bar z)^{\frac{1}{2}} = \frac{1}{2}(\ln z + \ln \bar z + 2\pi i k(z)) \quad \tex... |
H: Replacing Axiom of Extensionality with a logical formalism
Is it possible to replace the Axiom of Extensionality with a formalism from logic, namely the following one: $\forall a \forall b (a=b\Leftrightarrow \forall P (P (a)\Leftrightarrow P (b)))$ ($ P $ is obviously a predicate)?
AI: In general, the axiom
$$
(\f... |
H: matrix product with trace zero
$D$ is a positive definite matrix, $A$ and $B$ are both positive semidefinite matrices, $c$ is a postive integer. I want to know whether $trace\{(A+B+cI)^{-1}ABD\}=0$ implies that $AB=0$?
AI: No. Let
$$A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\!, \quad B = \begin{bmatrix} 1 \\ ... |
H: $f:\Bbb{R}^n\to\Bbb{R}^m$ is differentiable iff all coordinate functions $f_i:\Bbb{R}^n\to\Bbb{R}$ are differentiable
What I tried:
Since $f$ is differentiable, there is a linear function $L$ such that
$$f(x)=f(a)+L(x-a)+(\text{remainder})$$
Let $f_i$ be a coordinate function. Is it true that $\pi_1\circ L:\Bbb{R... |
H: Does smooth section of a quotient space $G/H$ define an immersion?
Question 1:
Let $G$ be a Lie group and $H<G$ a Lie subgroup of $G$, given the projection $p:G\to G/H$
and a smooth (local) section $s:G/H\to G$ s.t. $p\circ s=id$, then does this imply that $s$ is an immersion? So a section can in general be conside... |
H: Calculate the inner angles of the triangle $A(2,-3,5),B(0,1,4),C(-2,5,2)$
I want to calculate the inner angles of this triangle.
$$A(2,-3,5),B(0,1,4),C(-2,5,2)$$
I know that for calculate the angle I need to do the following thing:
$$\cos(\alpha)=\frac{A\cdot B}{|A||B|}$$
I need to calculate AB with BC and AC with ... |
H: Tangent map of the inclusion map of a submanifold
Let $M$ be the Minkowski spacetime, let $f\in C^{\infty}(M)$ be defined as $f(m)=x^{0}(m)$, with $\{x^{\mu}\}$ being a global Cartesian coordinates system, and let $M\supset F_{t}=f^{-1}(t)$ be the submanifold relative to a regular value $t\in\mathbb{R}$ of $f$. How... |
H: Need an easy CDF for Inverse transform sampling
I want to use inverse transform sampling to generate some random numbers, which all fall into a given interval $(0,x_{max})$. The numbers are not necessarily distributed evenly but can be "skewed". I do not know the true distribution, all I know is that there can be s... |
H: Is't a correct observation that No norm on $B[0,1]$ can be found to make $C[0,1]$ open in it?
There's a problem in my text which reads as:
Show that $C[0,1]$ is not an open subset of $(B[0,1],\|.\|_\infty).$
I've already shown in a previous example that for any open subspace $Y$ of a normed linear space $(X,\|.\|... |
H: Check if $u + v\sqrt 2 > u' + v'\sqrt 2$ without computing $\sqrt 2$
I'm building an algorithm that perform some computations on two inputs, m and n. These are numbers of the form $u + v\sqrt 2$, where $u$ and $v$ are integers.
I'm asking here because at a certain point the algorithm checks if m $>$ n, and, in orde... |
H: Are there algebraic structures with more than one neutral element and/or more than one inverse element?
I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than... |
H: Ordering Relation
Prove the following theorem:
Suppose $A$ is a set, $F \subseteq P (A)$, and $F \neq \varnothing.\;$ Then the least
upper bound of $F$ (in the subset partial order) is $\bigcup F.$
AI: Hint: Show that for every $B\in F$ we have $B\subseteq\bigcup F$, so it is an upper bound; and if $C$ is such that... |
H: Prove that $\sum\limits_{k=0}^{n-1}\dfrac{1}{\cos^2\frac{\pi k}{n}}=n^2$ for odd $n$
In old popular science magazine for school students I've seen problem
Prove that $\quad $
$\dfrac{1}{\cos^2 20^\circ} +
\dfrac{1}{\cos^2 40^\circ} +
\dfrac{1}{\cos^2 60^\circ} +
\dfrac{1}{\cos^2 80^\circ} = 40. $
How to prov... |
H: Eigenvalues of a specific $9\times9$ matrix - a simpler way?
Let $ρ$ be the permutation of $\{1, \dots , 9\}$ be given by $$ρ=\bigg(\begin{array} \\1 2 3 4 5 6 7 8 9 \\2 3 4 1 6 7 5 9 8 \end{array}\bigg)$$ and let $α : C^9 → C^9$ be the linear map defined
by by $α(e_j) = e_{ρ(j)}$, $j = 1, \dots , 9$. What are ... |
H: Eigenvectors of a $2 \times 2$ matrix when the eigenvalues are not integers
How can I calculate the eigenvectors of the following matrix?
$$\begin{bmatrix}1& 3\\3& 2\end{bmatrix}$$
I calculated the eigenvalues. I got
$$\lambda_1 = 4.541381265149109$$
$$\lambda_2 = -1.5413812651491097$$
But, now I don't know how to... |
H: Transverse a single point
I got very confused with understanding this theorem. So $\{y\}$ is a point, how could it be transversed by $f$?
Proof: Given any $y \in Y.$ alter $f$ homotopically to make it transversal to $\{y\}$.
Thank you for your help~~
AI: The tangent space to a point is trivial, so to say that $f$... |
H: MLE of Poisson Variable
Consider a random sample of size $n$ from a Poisson distribution with mean $\mu$. Let $\theta=P(X=0)$. Find the MLE of $\theta$ and show that it is a consistent estimator.
--We have $\theta=P(X=0)=e^{-\mu}$. To find the MLE, I took the log likelihood, $\ell(\mu,\mathbf{x})=-n\mu$, which has... |
H: A cardinality of a graph
If I have graph $G=(V,E)\\$
What is the meaning of $|G|$? (The cardinality of G).
I'd like to know few words about it.
Thank you!
AI: Generally, for a given graph $\,G=(V,E),\;$ the standard meaning of $|G|$ is simply $$|G| = |V|$$ |
H: Differentiation solving
According to the question , answered by martini , I am failing to evoke my memory that how can we write this $$\sum_i \partial_i^2\rho = \frac{D\rho^2 - \sum_i x_i^2}{\rho^3} = \frac{D-1}{\rho}$$
AI: Well, $\rho^2 = \sum_i x_i^2$ hence $D \rho^2-\sum_i x_i^2 = (D-1)\rho^2$ then the $2$ can... |
H: Does there exist such a function?
Can we find a function $ f: \mathbb{R} \to \mathbb{R} $ that is continuous only at the points $ 1,2,\ldots,100 $?
AI: Take $$f(x) = \prod_{i=1}^{100}(x-i)1_{\mathbb{Q}}(x)$$ where $1_{\mathbb{Q}}(x) = 1$ if $x$ is rational and $0$ otherwise. Can you show that it has the desired pro... |
H: A step in computing the cohomology ring of $\mathbb{C}P^n$
On page 250 of Hatcher's Algebraic Topology, he uses a certain corollary to compute the cohomology ring of $\mathbb{C}P^n$. The relevant section is below for convenience:
I understand the proof except for his statement that once can deduce $H^{2i}(\mathbb{... |
H: Irrational sum to integers?
Is it possible for $(a-b)k + bf$ to be an integer if $k,f$ are irrational numbers and $a,b$ are integers? What about $(a-b)k - bf$?
AI: $$(0-1)\sqrt2+1\cdot\sqrt2=0$$
$$(2-1)\sqrt2-1\cdot\sqrt2=0$$
In general this is possible, if and only if the set $\{1,k,f\}$ is linearly dependent ov... |
H: Larger circuit design for same boolean function?
I've designed this circuit with 4 logic gates, and did Karnaugh map's simplification and Quine McCluskey method. However I found out that actually my circuit design is already optimized and I can't really compare how the simplifications offer a less expensive circuit... |
H: How many quartic polynomials have single-digit integer coefficients?
Let $X$ be the set of all polynomials of degree 4 in a single variable $t$ such that every coefficient is a single-digit nonnegative integer. Find the
cardinality of $X$.
This is a question from Balakrishnan's Intro. Discrete Mathematics, and ... |
H: $\int_{0}^\infty e^{-xu}\sin x \, dx$: Where is my mistake?
Below find a a scan of a page out of Buck's Advanced Calculus. Right below the line marked (6-33), he seems to claim that $\int_{0}^\infty e^{-xu}\sin x \,dx = \frac{1}{1+u^2}$ (for $u>0$). I keep getting $\frac{u^2}{1+u^2}$, and I'm wondering if I'm makin... |
H: A sentence that has infinite models, finite model, but no finite model above certain cardinality
Let $T$ be a theory and $\sigma$ a sentence, such that
there exists infinite $\mathfrak{A} \models T + \sigma$.
there exists finite $\mathfrak{A} \models T + \sigma$.
there exists $n \in \mathbb{N}$, such that for all ... |
H: Real variable lemma
Someone can help with the following lemma?
Lemma: Let $\theta'(r)\geq0,\theta(r)>0$ and $\dfrac{\theta'(r)}{\theta(r)}$ decreasing for $r>0$. Then
$$\frac{\displaystyle\int^r_0\theta'(s)\,ds}{\displaystyle\int^r_0\theta(s)\,ds}$$
is decreasing.
Thanks!
AI: I'll assume $\theta(r)$ doesn't... |
H: Is this a vertical asymptote?
I have this function:
$$ f(x) = (x+1) \cdot e^{\frac{1}{x}} $$
I have the two side limits:
$$ \lim_{x \to 0^-} { (x+1) \cdot e^{\frac{1}{x}} } = 0 $$
$$ \lim_{x \to 0^+} { (x+1) \cdot e^{\frac{1}{x}} } = +\infty $$
So the right left side limit is not infinity. Is that still conside... |
H: To show an analytic function is one-to-one on the unit disk
Let $\displaystyle f(z) = \sum_{n=0}^\infty a_nz^n$ be analytic in the unit disk $D_1(0)$ with $f(0) = 0$ and $f'(0) = 1$. Prove that if $\displaystyle \sum_{n=2}^\infty n|a_n| \le 1$, then $f$ is one-to-one in $D_1(0)$.
I am able to show that $f$ has a ... |
H: Evaluate $\int_2^\infty{\frac{3x-2}{x^2(x-1)}}$
To be shown that $\int_2^\infty{\dfrac{3x-2}{x^2(x-1)}}=1-\ln2$
My thought: $\dfrac{3x-2}{x^2(x-1)}=\dfrac{3x}{x^2(x-1)}-\dfrac{2}{x^2(x-1)}$
• $\dfrac{3x}{x^2(x-1)}=\dfrac{3}{x(x-1)}=\ldots=-\dfrac{3}{x}+\dfrac{3}{x-1}$
• $\dfrac{2}{x^2(x-1)}=\ldots=-\dfrac{2}{x^2}+\... |
H: Are cross-references context-free?
First, a little background: In XML there is the ability for one part of an XML document to reference another part of the document (i.e., a cross-reference). Below is an example. The BookSigning element references a Book element:
<Library>
<BookCatalogue>
...
... |
H: Calculating Log-likelihood using Raphson and Jacobian matrices?
I am reading the following paper:
http://www.ntuzov.com/Nik_Site/Niks_files/Research/papers/stat_arb/Ahmed_2009.pdf
and in particular page 13. I want to try and calculate lambda_t(p) = exp^(Beta^T x_t) but I am having problems with the fact beta is der... |
H: (Revisted) Surjectivity: Examples for Compositions
I'm asked to give an example where $f$ is surjective, but $g\circ f$ is not. I suspect that $f(x)=x$ and $g(x)=\frac{1}{x}$ will do the trick, namely for $f,g : \mathbb{R}\rightarrow \mathbb{R}$, right?
Well, what about where $g$ is surjective...
AI: Since $\opera... |
H: Running time of adding $n$ items
I am trying to calculate how many binary additions it takes to add $n$ items. I see that with each iteration of binary addition, I am left with $n/2$ items so I see that it would take $\log_2 n$ iterations.
By writing out a few examples, I see that it is $n-1$ binary additions to ad... |
H: How to go from a sum to a product and a product to a sum?
I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?
AI: Consider the set $\mathbb R$ of all real numbers together with the operation of addition and con... |
H: How to study positivity of $x^3 -x -1$?
Studying this inequality:
$$x^3 -x -1 \ge 0 $$
Since I can't apply Ruffini's rule, I cannot recognize a method to study the function positivity. I could decompose it to:
$$ x \cdot (x+1) \cdot (x-1) -1 \ge 0$$
But the $-1$ is a problem, how do I go on?
AI: You can study i... |
H: Eigenvectors from different eigenspaces of an operator are orthogonal?
Let $V$ be a vector space with an inner product on $\mathbb C$ with a finite dimension, and $T : V \to V$ an operator (not necessarily Normal,) with eigenvectors $\{v_1,...,v_k\}$ for different eigenvalues $\{\lambda_1,...,\lambda_k\}$. Are $\{... |
H: Show that: $\lim\limits_{r\to\infty}\int\limits_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$
I would like to show $\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$.
Now, of course, the integrand does not converge uniformly to $0$ on $\theta\in [0, \pi/2]$, since it has value $1$ at $\theta ... |
H: Commutant of bounded linear operators on a Hilbert space
Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by
$$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall B \in \mathcal{B}(H)\}$$
the commutant of $\mathcal{B}(H)$. ... |
H: Finite set on compact manifolds
I feel blocked with this claim - it sounds intuitively true, just thinking as a jellyfish entering a real line, the intersection of her legs with the real line is certainly finite since the jellyfish is compact - but I stuck with why.
$X$ and $Z$ are closed submanifolds inside $Y$ w... |
H: Show that $f(x)=g(x)$ for all $x \in \mathbb R$
Let $f,g:\mathbb R \rightarrow \mathbb R$ be both continuous. Supose that $f(x)=g(x)$ for all $ x \in D$, where $D\subseteq\mathbb R$ is dense. Show that $f(x)=g(x)$, for all $x \in \mathbb R$. Id like a hint to solve this question.
AI: One more hint:
Define $h(x) = f... |
H: Why does $\lim\limits_{x\to0}\sin\left(\left|\frac{1}{x}\right|\right)$ not exist?
Can someone explain, in simple terms, why the following limit doesn't exist?
$$\lim \limits_{x\to0}\sin\left(\left|\frac{1}{x}\right|\right)$$
The function is even, so the left hand limit must equal the right hand limit. Why does thi... |
H: questions about total derivative
I am learning some stuff about the total derivative and got these two questions:
1) I was wondering if a linear map is totally differentiable. So let $A$ be linear, $A\in\mathcal L(\mathbb R^n,\mathbb R^m)$. Then
$$\lim_{h\rightarrow 0}\frac{A(x_0+h)-A(x_0)-A(h)}{\|h\|}=0$$ since $A... |
H: Indefinite integral $\int{\frac{dx}{x^2+2}}$
I cannot manage to solve this integral:
$$\int{\frac{dx}{x^2+2}}$$
The problem is the $2$ at denominator, I am trying to decompose it in something like $\int{\frac{dt}{t^2+1}}$:
$$t^2+1 = x^2 +2$$
$$\int{\frac{dt}{2 \cdot \sqrt{t^2-1} \cdot (t^2+1)}}$$
But it's even ... |
H: Glueing morphisms of sheaves together - can I just do this?
While trying to solve a certain exercise in Hartshorne I realized that I need to use the following result:
Let $X,Y$ be two ringed topological spaces. Suppose we have a covering $\{U_i\}$ of $X$ and morphisms $f_i :U_i \to Y$ such that $f_i|_{U_i \cap U_... |
H: Is $\{ a-b=y, a \oplus b=x \}$ solvable?
Is the system
$$ a \oplus b = x$$
$$ a-b = y $$
Where $a,b$ are variables and $x,y$ known, and $\oplus$ denotes bitwise xor, solvable?
I've tried to substitute $b=a\oplus x$ in the second equation but it didn't yield anything.
I need this in order to identify 2 lone items in... |
H: Confusion regarding transversal for a partition in Smith Introductory Mathematics: Algebra and Analysis
In Smith's Introductory Mathematics: Algebra and Analysis, I came across the definition of a transversal for a partition along with examples. Either I don't understand one of the examples, or it is simply wrong. ... |
H: Show $\int_{0}^\infty\frac{\sin^2(xu)}{u^2}du=\frac{\pi}{2}|x|$ for all $x\in \mathbb{R}$
I would like to show $$\int_{0}^\infty\frac{\sin^2(xu)}{u^2}du=\frac{\pi}{2}|x| \,\,\,\,\,\,(\forall x\in \mathbb{R}).$$If I ask whether the integral converges uniformly, of course $\int_{\epsilon}^{\infty}\frac{\sin^2(xu)}{u^... |
H: Zech's logarithms - Why are they called "Zech"?
Zech's logarithms are defined in here.
I couldn't find a reason why they are called "Zech". The only thing a dictionary suggests is that Zech is an abbreviation for Zechariah, which doesn't seem relevant, right?!
So, could you please shed light on this naming?
AI: App... |
H: Looking at the intermediate fields of $\Bbb{Q}_7 = \Bbb{Q}(\omega)/\Bbb{Q}$ where $\omega = e^{i2\pi/7}$ .
From Basic Abstract Algebra (Robert Ash):
The question that I'm concerned with is number 3, but I will write problems 1 and 2 as well, since they are all related...
We now do a detailed analysis of subgroups ... |
H: Show that every finite group of order n is isomorphic to a group of permutation matrices
Show that every finite group of order n is isomorphic to a group consisting of n x n permutation matrices under matrix multiplication.
(A permutation matrix is one that can be obtained from an identity matrix by reordering its ... |
H: Flaw with proving that $S$ is a basis of $V$ iff $\|v\|^2=\sum_{i=1}^{n}{c_i}^2$
Let $V$ be a vector space over $\mathbb{R}$ equipped with a positive-definite scalar product $(\cdot,\cdot)$. Let $S = \left\lbrace v_1,...,v_n \right\rbrace$ be a set of orthogonal vectors of unit length in $V$ and $c_i =(v,v_i), \qua... |
H: Show that $\int_{0}^\infty \frac{1-\cos x}{x^2}dx=\pi/2$.
I am trying to show that $$\int_{0}^\infty \frac{1-\cos x}{x^2}dx=\pi/2.$$The hint is "try simple substitution", and not incidentally, the previous problem has shown that $\int_0^\infty \frac{\sin^2(xu)}{u^2}du=\frac{\pi}{2}|x|$. This looks an awful lot like... |
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