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Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$.
I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking:
\begin{align*}
ax + by &= 0\\
yb &= -ax\\
y &= -ax/b
\end{align*}
I then looked up the answer to check if I was right,... |
New to the site so excuse any clumsy writing:
I am attempting to write an equation that finds the surface area along a pyramid as you move up and down the height of it. I know the initial areas of the top and bottom and thought I could use the Frustum pyramid equation to find the volume:
\begin{align}
V=(\frac{h}{3})*(... |
for $I = [0,1]$, is $I \times I$ convex in $\mathbb{R} \times \mathbb{R}$?
The definition of convex seems to be that
$Y \subset X$ is convex in $X$ if $\forall a < b $ in $Y$ whole of $(a,b)$ in $X$ lies in $Y$.
For example, if I take $(1/3, 1/3) < (2/3,1/3)$ in $I \times I $ then, the rectangular strip in $\mathbb... |
I can prove that if $\gcd (a,n)=1$, then $as=1 \pmod{n}$ has a solution. However, I cannot prove that the solution $s$ is in the set $\{1, 2, ..., n-1\}$.
|
Let $k$ be a field and consider the rational function field $k(t)$. I was just reading that if $f(t),g(t),h(t)\in k(t)$ are such that $f(h(t)) = g(h(t))$, then "$f = g$ in $k(h(t))$." What does that mean? I know that $k(h(t))$ is the smallest subfield of $k(t)$ containing both $k$ and $h(t)$, but what does it mean that... |
I'm trying to work through the proof given here that the sample mean and sample variance of a random sample $X_1, X_2, ..., X_n \sim N(\mu,\sigma^2)$ are independent.
The part I can't seem to follow is the assertion that the Jacobian of the transformations...
$$
Y_1 = \bar{X} \\
Y_2 = X_2 - \bar{X} \\
\vdots \\
Y_n... |
Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$
where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values.
Is there a way of expressing the zeros of $f(z)$ in terms of the parameters ($\gamma,\beta_1,\beta_2$)?
|
There are $5$ points on the surface of a sphere. Is it always possible to divide the surface into $5$ connected congruent regions such that each region contains one of the $5$ points?
|
In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, $\frac{dV(x(t))}{dt}=\frac{dx^T}{dt}Qx+x^T Q\frac{dx}{dt}=x^T A^T Qx+x^T QAx$.
I don't see how the right-most side was obtained. Would appre... |
Prove that $||a|-|b||\leq |a-b|$ for all real numbers
I was thinking divide it into $a\geq b$ and $a<b$, but then I realized I need to include situations when they are greater than zero and less than zero, will that too comlicated? Is there a simpler way to show this?
|
I am trying to program the situation as show in figure below.
I have two circles, with centres at $(x_1,y_1)$ and $(x_2,y_2)$. The line segment connecting $(x_1,y_1)$ and $(x_2,y_2)$ makes an angle $\theta_2$ with the horizontal. I need to find the point $(h,k)$, such that the circle with this centre is tangent to t... |
Questions
Without computing, is $\int_0^1 t(t-1)(t-2)\,dt$ positive or negative? Give reasons for your answer.
Without computing, is $\int_0^2 t(t-1)(t-2)\,dt$ larger or smaller than $\int_0^1 t(t-1)(t-2)\,dt$? Give reasons for your answer.
Without computing, is $\int_0^{2.25} t(t-1)(t-2)\,dt$ larger or smaller than $... |
Hello I am a sophomore and I would like to know if Michael's other book named 'Calculus' is enough preparation for the Manifolds, if not please do tell what else should I be reading? I am taking an introductory course to topology next semester as well.
Thanks!
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I am asked to prove that $Y_t$ is a martingale where $Y_t=\exp\left(\int_0^tf(s)\,dW_s-1/2\int_0^tf(s)^2\,dt\right)$ using Ito's formula.
After applying Ito's formula (I hope I made no mistake) I get $dY_t= Y_t \, dM_t$ where $M_t=\int_0^tf(s) \, dW_s$
What to do next ?
|
Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$.
I am asked the following question:
Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ contained in the ideals $\mathfrak{a}$ and $\mathfrak{b}.$
Proof:
Let $x \in \mathfrak{a} \cap \mathfrak{b}$ $\implies x \in \... |
I am studying strongly convex functions and they mention if $f(x)$ is strongly convex with Lipschitz gradients $L$, which means $\parallel \nabla f(y) - \nabla f(x)\parallel \leq L\parallel x - y \parallel $ , then $ g(x):= f(x) - \frac{m}{2}\parallel{x}\parallel^2 $ has Lipschitz gradients $L-m$. I tried proving that ... |
I want to prove that if a group has no subgroups other than the identity and itself, then the order of the group is a prime number.
A hint would be appreciated. Is there any theorem on the relation of the order of the subgroup and that of the group?
Note that the finiteness is not given, unlike in the linked question.
|
Question: A particle is projected vertically upwards from the Earth's surface. Its distance $x$ from the centre of the Earth is connected with its upwards speed $v$ by the differential equation:
$$ v \frac{dv}{dx} = -g \frac{a^2}{x^2}$$ , where $a$ is the Earth's radius and g is a constant.
If the initial speed a... |
Suppose that $A \in M_{m\times n}$ & $B, C \in M_{n\times m}$ are matrices that satisfy $BA= I_n$ and $AC=I_m$. Prove that $B=C$.
In my mind, a good way to go about this proof is proving that $B$ is a subset of $C$ and $C$ is a subset of $B$, but I am not sure how to start.
|
For the differential equation $$\dot x = x(1-x), x(0)= \frac 12$$
Decide if the solution exists for all $t \ge 0$ or only on a finite time interval $0 \le t \lt T$.
By the theorem, for the maximal solution either
$T = \infty $,
${T \lt \infty }$ but $\lim_{t \to T} DNE$,
or ${T \lt \infty }$ and $\lim_{t \to T}$ exists... |
Sometimes it's easier to add numbers when you recognise that they're close to some round number, and then add the differences separately.
$$97+198$$
$$=(100-3)+(200-2)$$
$$=(100+200)+(-3-2)$$
$$=300-5$$
$$=295$$
What is this property of addition called? I have found "commutative" but I'm not sure that this is the right... |
Let $(\mathcal{C}_i)_{i\in I}$ be pairwise independent classes of the probability space $(\Omega, \mathcal{F},P)$, i.e. $\mathcal{C}_i,~\mathcal{C}_j$ are independent for every $i\neq j$ in $I$. True or false: $(\mathcal{C}_i)_{i\in I}$ are independent.
My thoughts: i strongly believe that this is false (we should ha... |
I am confused about ranks, nullities and bases of the kernel.
From what I understand the rank is the dimension of a vector space generated by a matrix.
How would I do the following examples?
Find the ranks and nullities of the following linear maps $T \colon U \to V$, and find bases of the kernel and image of $T$ in e... |
I have no idea how to start this problem. Any help would be greatly appreciated. Thanks
|
Show that $P_n = \{$Polynomials with real coefficients of degree $≤ n\}$ is an $(n+1)$-dimensional subspace of the infinite-dimensional vector space of all real polynomials.
|
In Lang's algebra on pp.559, he writes of the nilpotent part of a matrix $M$:
"We observe also that the only case when the matrix $N$ is $0$ is when all the roots of the minimal polynomial have multiplicity $1$. In this case, if $n = \dim E$, then the matrix $M$ is a diagonal matrix, with $n$ distinct elements on the ... |
I've run into a matrix dimension proof I'm having some trouble with:
Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems $A\underline{x}=\underline{0},B\underline{x}=\underline{0},AB\underline{x}=\underline{0}$ respectively.
Prove that if $rank(B)\leq rank(AB)$ ... |
Suppose that $X \geq 0$ is distributed according to some distribution $F$. What can be said about $E[e^{-r X}]$?
I.e. is there a way to express this expectation only in terms of some characteristics of the distribution $F$ (mean, variance, etc)?
It is a part of a project where I try to find a distribution $F$, but I a... |
I have an assignment where I need to count the number of models of a certain sort which satisfy a given sentence, and I keep finding that the number of models I count exceeds the total number of possible models, so I'm wondering where I'm going wrong.
I need models for languages having only predicates $F$ and $G$, an... |
I want to show that for the general wave equation $u_{tt} - \nabla \cdot (c^2\nabla u) + qu = 0, \quad u(x, 0) = \phi(x), \quad u_t(x, 0) = \phi(x)$
we have
$$ E(t) = \int_{|x-x_0| < R_0 - c_2t} (u_t^2 + c(x)^2|\nabla u|^2 + q(x)u^2) \, dx $$ is non-increasing when $c, q \geq 0$ and $c_1 \leq c(x) \leq c_2$ for some $c... |
The riddle:
Consider $N$ different bags $B_1$ to $B_N$. Each bag may be filled with numbers. Can you fill these bags with numbers from $1$ to $N$ so that the following conditions hold?
1) $n \in B_n$ (The $n$-th bag contains $n$.)
2) $B_i \cap B_j \neq \{\}$ (All bags must share numbers.)
3) Each number is element of ... |
For two sequences $a_n$ and $b_n$, “If $\{a_n\}$ and $\{b_n\}$ are increasing, then $\{a_nb_n\}$ is increasing.” Show this is false, make the hypothesis on $\{b_n\}$ stronger, and prove the amended statement.
I was thinking to let both $a_n$ and $b_n$ be positive, but it only let me change the hypothesis on $b_n$, ho... |
I have troubles proving that the power series
$\sum_{n=1}^\infty n^3\frac{x^n}{n!}$
represents the function
$f(x)=e^x(x^3+3x^2+x)$.
My idea was using the identity theorem for power series and the definition of the Taylor series of a function to somehow show that the only function whose $n$-th derivative at $x=0$ equals... |
Question 23: The string zyyzy belong to the language, right?
|
I am looking at the continuity of the following function $f(x) = \sin(1/|x|), f(0) = 0$
So this is $f(x) = \sin(1/|x|)$ filled in at $x = 0$
Clearly, $\lim\limits_{x \to 0} f(x) = 0 $ by squeeze theorem.
Is $f(x)$ continuous then?
In general, when is a function that is filled in continuous or discontinuous?
|
Assume that $N$ and $X_1, X_2, \ldots $ are all independent and identically distributed over $(0,1)$ with the density function:
$f (x) = cx^2 (1 − x)^2$.
An integer–valued random variable, $N$ specifies a random sum of first $(N + 1)$ variables,
$$\sum_{j=1}^{N+1}x_j$$
We assume, for integer values of $k$, that $N$ is ... |
Problem: Describe the set of points $z^2$ as $z$ varies over the second quadrant: $\{z = x + iy : x < 0 \text{ and } y > 0 \}$. Show this is an open connected set. (Hint: use the polar representation of $z$.)
The following is my attempt at a solution:
so $z = |z|(\cos\theta + i\sin\theta)$. Then $z^2 = |z|^2(\cos2\the... |
True or False: if $P(n)$ is the $n^{th}$ Taylor polynomial for $f$ at a point $x_0$, then the first $n$ derivatives of $P(n)$ and $f$ are equal at $x_0$.
|
If I want to prove a function $f(t)$ is continuous at $ t_o$. By def of continuity, I need to show that for $\epsilon >0 $, there exist $ \delta>0$ s.t
$ |t-t_o| < \delta$ implies $ |f(t)-f(t_o)|< \epsilon $ .
If in order to show this, I need other condition to satisfy along with a bound on $ |t-t_o|$ to control $|... |
If $||v|| = 5$ and $||w|| = 3$, what are the smallest and largest values
of $||v - w||$? What are the smallest and largest values of $v \cdot w$?
How can I solve these two problems? For $||v - w||$ how can you have different values when it is clear that the magnitude of v and w are given?
|
I know that there exists an elementary matrix that switches the rows in another matrix when they are multiplied, but how do you prove that this elementary matrix actually does this job?
|
Let $X_1, X_2, \cdots$ be i.i.d. such that $X_i > 0$ and $\mathbb E[X_i]=1$ and consider $\mathbb F = \{\mathcal F_n\}_{n\ge 1}$ to be the discrete filtration. Denote $Y_n = \prod\limits_{i=1}^n X_i$. I need to show that:
i. For any bounded $\tau \in \mathbb F$ we have $\mathbb E[Y_{\tau}]=1$.
and the for any $\tau \in... |
Mary has on her bookshelf 5 novels, 5 biographies, and 8 textbooks. Mary decides to take three novels and four non-fiction books with at least one of the non-fiction books a biography. How many ways are there to make such a selection?
I thought that the answer would be obtained using multiplication of several combina... |
Given a number, the answer is a power of $2$.
Given $1.000$ the answer is $16384$.
Given $5.000$ the answer is $131072$.
Can someone find a function, so given any number we can get the answer?
|
I take the next exercise from Apostol's book. I thought for a while , but nothing. Then I read the solution. I try to think: how did he get it , but nothing. I want to know how to get to the solution. So ...
exercise :
A periodic function with period $a$ satisfies $f(x + a) =f(x)$ for a11 $x$ in its domain. What can... |
Suppose $A$ and $B$ are events with $P(A)+P(B)>1$. Show that the
largest possible value of $P(A \cap B)$ is $ \min(P(A), P(B))$.
I suspect I'm supposed to use $P(A \cap B) = P(A)+P(B) -P(A \cup B)$ but I've no idea how. I also understand that from this we get $P(A\cap B)\leq P(A)$. However, I don't know how to use... |
In my algebraic structures textbook I have come across a tricky question that I am trying to solve which goes as follows:
suppose that $d|(a^n-1) $ and $d|(a^m-1)$ where $m,n$ are natural numbers and $a,d$ are integers, then show that $ d|(a^{gcd(m,n)}-1) $ . What I know is that clearly $a^m$ and $a^n$ are congruent t... |
I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question:
(i) Let $U, U'$ be subspaces of a vector space $V$ (both not equal to $V$). Prove that the union of $U$ and $U'$ does not equal $V$.
(ii) Find an example of $V$ and $U,U',U''$ containe... |
Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with respect to Uniform$(0,3)$' mean!
As I know Uniform$(1,5)$(I think in this question 'continuous' uniform has been meant, not s... |
I need to prove that ${S^n} \times {S^k}$ is homeomorphic to a subspace of ${\mathbb{R}^{n + k + 1}}$ by constructing an explicit map between the two.
I am unsure how to start this as I can't seem to think of any suitable maps. A spherical coordinate type map does not have a continuous inverse and a stereographic proj... |
I am trying to better understand vector spaces and dimensions.
I could prove (i) via induction and the definition of linear independence?
However how can I approach the questions (ii),(iii) which consider finite dimensions?
|
I have found three cases:
1) If $f$ and $g$ are the same function.
2) If $f$ and $g$ are mutually inverse.
3) If both are polynomials of degree $1$
Maybe there are more.
|
If $x,y \in \mathbb {R}$, find the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$.
This problem was inspired by a problem which asked if $x,y \in \mathbb {R}$ and $xy \neq 0$, find the minimum of $x^2+y^2$ when $xy(x^2-y^2)=x^2+y^2$.
By setting $x=a\sin\theta$, $y=a\cos\theta$, the equation can ... |
I have a question on finding probabilities of joint distributions, specifically two independent random variables that are Uniformly distributed. The question I wish to solve is this one:
We agree to try to meet between $12$ and $1$ for lunch at our favorite sandwich shop. Because of our busy schedules, neither of us i... |
Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?
I know that
\begin{align*}
\mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\
\mathbb{Q}(i) &= \{a+bi \mid a,b \in \mathbb{Q}\}
\end{align*}
I tend to believe it is, since every element in $\mathbb{Q}(\sqrt{2})$ belongs to $\math... |
my professor gave practice problems for an upcoming midterm, and one is to find a differentiable parametrized curve with trace $y = |x|$ with $-1 \leq x \leq 1$.
My thinking is that I should find a parametrization with $\alpha'(0) = (0,0)$ and of course $x = -y, x = y$ for negative and positive $x$ respectively, howeve... |
How would one evaluate $\sin(\frac {72\pi} {11})$?. The prime number in the bottom is getting me stuck. I couldn't see how to use it using the sum of two angles trig identity.
|
Can anyone help me with the following question:
Let $V$ be a vector space over field $F$, possibly not finite dimensional. Let $T \colon V \to F$ be a linear map. Prove that there is no subspace $U$ such that $U$ contains $\ker(T)$ but is strictly not equal to $\ker(T)$ or $V$.
|
There's a section in chapter 3 in Hatcher's Algebraic Topology where he talks about the long exact sequence of the pair $(Y \times I, Y \times \partial I)$, where $Y$ is any topological space. The claim is that the map
$$H^k(I \times Y, \partial I \times Y) \to H^k(I \times Y)$$
in the long exact sequence is zero for ... |
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$.
$f'(x)=f(x)$ is true when $f(x)=ke^x$.
$f(x)=0$ satisfies the condition. So $f(5)=0$ which is also the correct answer. But is there any... |
Let $(\mu_n)_{n \geq 1}$ be a sequence of real numbers and $(\sigma_n)_{n \geq 1}$ be a sequence of positive numbers, and let $\mu \in \mathbb{R}$ and $\sigma>0$. For $n \geq 1$, let $X_n\sim N(\mu_n, \sigma_n^2)$. Assume $\mu_n \rightarrow \mu$ and $\sigma_n \rightarrow \sigma$.
I need to prove that the $$X_n \rightar... |
I just started to learn about functional analysis and have started to learn about various spaces, like $L^{p}$, Banach, and Hilbert spaces. However, right now my understanding is rather mechanical. That is, my understanding of say the Hilbert space is that it is a vector space with an inner product such that the norm d... |
If $A+B+C=π$, prove that
$$\cos^2A+\cos^2B-\cos^2C=-2\cos A\cdot\cos B\cdot\cos C.$$
ATTEMPT:
Given $$A+B+C=π,$$
$$A+B=π-C$$
Taking "cos" on both sides
$$\cos(A+B)=-\cos C.$$
Now,
\begin{align*}
L.H.S&=\cos^2A+\cos^2B-\cos^2C\\
&=\frac{1+\cos 2A}{2}+\frac{1+\cos 2B}{2}-\cos^2 C\\
&=\frac{2+\cos 2A+\cos 2B}{2}-\co... |
Im having difficulty intuitively understanding how to solve this problem:
$x(t) = A\cos(\omega t + \phi)$
$A > 0$
$\phi\in(−\pi,\pi]$.
$x(t) ≥ 2.4$ for $18$% of each period
takes $0.123$ seconds for $x(t)$ to drop from $2.4$ to the next minimum
the first zero for $t>0$ occurs at $t = 0.040$ seconds.
How does one solve ... |
True or false (give a reason if true or a counterexample if false):
(a) If $u$ is perpendicular (in three dimensions) to $v$ and $w$, those vectors $v$ and $w$ are parallel.
(b) If $u$ is perpendicular to $v$ and $w$, then $u$ is perpendicular to $v + 2 w$,
(c) If $u$ and $v$ are perpendicular unit vectors then $\lVer... |
Continuing on my series of questions, for those following, is the following question:
Let $\alpha: I \to \mathbb{R}^3$ be a regular curve. Suppose that for some
$t \in I$ the distance from the origin, $|\alpha(t)|$, attains maximum. Prove that
$|\kappa(t)| \geq \frac1{|\alpha(t)|}$.
Intuitively this seems a straightfor... |
$f:[0,+\infty) \rightarrow \mathbb{R}$ twice differentiable. If $f''$is limited and there is $\lim\limits_{x\to \infty}f(x)$, show that $\lim\limits_{x\to \infty}f'(x) = 0$.
|
Candidate A has already garnered 80 votes.
Candidate B has already garnered 50 votes.
The number of votes a candidate must have in order to win the election is 115.
The votes of 5 states are still being counted.
Winning a state gives a candidate 20 votes. Losing a state gives them no votes.
Assuming that both candidat... |
We are given a fixed point on a circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0,1,2,\ldots$ from it we obtain points with abscissas $n = 0,1,2,\ldots$ respectively. How many points among them should we take to ensure that some two of them are les... |
As far as my understanding goes, the Wronskian $W(t)$ for a second order homogenous differential equation with continuous coefficients can help us govern whether the solutions will be linearly dependent or independent. Suppose we consider $t^2X''-3tX'+3X=0$ which has two solutions $X_1(t)=t^3$ and $X_2(t)=|t^3|$. Clear... |
A deer population grows logistically and is harvested at a rate proportional
to its population size. The dynamics of population growth is modeled by
$P' = rP (1-\frac{P}{K}) - hP$
where $r$ (the rate of growth), $K$ (capacity of the environment) and $h$ (harvesting rate) are constants
For $h > 0$, use a bifurcation dia... |
What is the best way to show that $S = \{(x,y) | x^2 + 3y^2 = 1\}$ is contained in the unit ball without graphing the set?
|
In $\mathbb{R}^3$ consider following 2-form given by $$\omega = xy \: dx \wedge dy + 2x \: dy \wedge dz + 2y \: dx \wedge dz$$ and $$A = \{(x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1, z\geq 0\}.$$
Show, that the Integral of $\omega$ over $A$ vanishes.
Using the map $$\phi\colon\mathbb{R}^2\rightarrow \mathbb{R}^3, \:... |
Find
$$\lim_{n\to\infty} \frac{1}{n}\left(1+\frac{1}{2}+\frac{1}{3}+\dotsb+\frac{1}{4n-1}+\frac{1}{4n}\right)^5.$$
I don't know how to calculate this limit. Please show me the how to solve this limit.
|
Determine the interior points, the boundary points, the cluster points, the limit points,
and the isolated points of each of the following subsets of $\mathbb R^2$. Also, classify each of
the sets as open, closed, neither or both.
{$(x, y)$ ∈ $\mathbb R^2$| $0 < x ≤ 1$}
Interior points should be all the points in the... |
For a parabola $y^2 = 4ax$ ,we can draw $2$ tangents from any point.If the point is outside of parabola then obviously we can draw $2$ tangents. If the point is on the parabola then the two tangents will coincide and it is easy to draw that picture. I know that if the point is inside the parabola then the tangents are ... |
$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$
for $t\in\Bbb{N}$
How might one represent $a,b$ in terms of $t$?
(Note that $a,b\in \Bbb{N}$)
If possible, could one also provide a proof please?
|
So I'm trying to figure out what the angle $θ$ would equal at $x=-2$ for the polar equation $r=θ+sin(2θ)$.
All I know is that $θ$ has a domain of $0\leθ\le \pi$ and $y < 1$ (pretty sure).
I think I either have to convert the equation into Cartesian form (but I can't figure out how to do that) or find the argument θ=ar... |
Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$.
\begin{equation}
A=I+L
\end{equation}
Define spectral norm (or the largest singular value) of a matrix $X\in R^{n\times n}$ as
\begin{equation}
\|X\|=\sup_{\|v\|_{\ell_2}=1} ... |
I'm having trouble figuring out how to show the general existence part of the following problem.
Suppose $n\in\{1,2,3...\}$ and $n\equiv 7\mod{10}$. Show that $\exists$ a prime divisor $p$ of $n$ s.t. $p\equiv 3\mod{10}$ or $p\equiv 7\mod{10}$.
|
If I have that $X$ is a random variable, for $\epsilon > 0$, and $r \geq 1$, is it true that:
$$\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r.$$?
If so, is there a reason why? Here, I have that $\mathbb{P}$ is the probability measure. I have tried to see it intuitively in terms of drawing a graph, but I... |
I'm trying to show that $U(n)^2$ is a proper subgroup of $U(n)$. Here
$$ U(n)^2 = \{x^2 \mid x \in U(n)\}$$
where $U(n)$ is the group of units modulo $n$.
My idea was to argue as follows:
Consider all elements of orders that are powers of $2$ in $U(n)$. Say these orders are $2^{k_1}, \dots, 2^{k_n}$ where $k_1 < \dots ... |
Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known whether in three dimensions, three generic ellipsoids will have a common point of intersection (unless one is contained in the... |
Let $f(z) = \frac{z}{|x|}$, with $z \not=0$
(a) Construct two sequences ${u_n}$ and ${v_n}$ such that
$\lim_{n \to \infty} u_n = 0$ and $\lim_{n \to \infty} v_n = 0$
$\lim_{n \to \infty} f(u_n)$ and $\lim_{n \to \infty} f(v_n)$ exist
$\lim_{n \to \infty} f(u_n)\not= \lim_{n \to \infty} f(v_n)$
(b) Is it possible to ... |
I am still confused about the difference between free module and finitely generated module. For example, $\Bbb Z/2\Bbb Z$ is finitely generated module, but why it is not finitely generated free module? What is difference between finitely generated module and finitely generated free module?I understand free module has ... |
Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$.
\begin{equation}
A=I+L
\end{equation}
All diagonal entries of $L$ are equal to $0$, so that, all diagonal entries of $A$ are equal to $1$.
Define spectral norm (or the largest singular value) of a matrix $X\in R^{n... |
I'm trying to figure out what happens to the graph of $f$ in the following to situations:
$f = f(|x|)$
$f = f(\frac{1}{x})$
For the first, I know if $f = |f(x)|$ then the points below the $x$ axis are reflected above the $x$ axis. However, I can't wrap by head around what happens when the absolute value bars are mov... |
I need to find the volume of a solid that is created by rotating the area within the following boundaries:
$y=x^3$
$y=8$
$x=0$
which is rotated over $x = 3$.
I thought I had the correct integral with
$$
\int_{0}^{3} (x-3)(8-x^3)dx
$$
but I do not believe that this is correct. How would I correctly get the integral to ... |
Prove that the function $f(z) = \frac{1}{1-z}$ is not uniformly
continuous on $(-1,1)$.
Partial proof : Suppose $f$ is uniformly continuous.
$\implies \forall \epsilon > 0, \exists \delta > 0, \forall z, w \in (-1,1) :$ $ |\frac{1}{1-z} - \frac{1}{1-w}| < \epsilon$ as long as $|z-w|<\delta$
Let $\epsilon = 1$. Try... |
In the image above, how does one know that $c=e$ and $c$ is not equal to $f$? and $e$ is not equal to $f$? How does one know that $b=d$?
|
I have this table of information:
(source: rit.edu)
Probabilities:
\begin{array}{c|c}
.919 & ????\\\hline
???? & .274
\end{array}
How do I find the probabilities of the question marks? I thought each row and each column was supposed to equal $1.0$, but clearly this won't work.
$1 - .919 = .081$, and $1 - .274 = .726... |
Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category of open sets on $X$ to the category of rings, assigning to each open set $D_{f} \subseteq X$ a local ring $\mathscr{O}_{... |
Do any mathematican still reserach about applied trigonometry? If so, what are the subject area called in the PhD level except fourier analysis?
In many area, you could see a lot of trig and hyperbolic function including finance, physics etc...
Please list some example of subject area of trigonometry that is alive or g... |
Suppose $V$ is an inner product space and $T$ is a linear operator that is orthogonally diagonalizable. Show that $T^*$ is also orthogonally diagonalizable.
Here, $T^*$ denotes the adjoint operator. The reason I ask is that I am trying to think of linear algebra separately from matrix/vector operations as I was taugh... |
I am looking for a series of video lectures on the subject of complex analysis which follow Conway's text Functions of Complex Variable I. Any recommendations?
|
Since I've worked my way through Spivak's Calculus book, I thought I'd give Courant's allegedly fantastic exposition of the subject a go as well. However, I've run into a problem. People in stackexchange threads always praise and suggest Courant's Calculus without specifying whether that's supposed to be his "Different... |
Find the projection of the line $x+y+z-3=0=2x+3y+4z-6$ on the plane $z=0$
The equation represents the line of intersection of two planes. Using augmented matrix
$$
\begin{bmatrix}
1 & 1 & 1 & 3 \\
2 & 3 & 4 & 6
\end{bmatrix}$$
$R_2\rightarrow R_2-2R_1$
$$
\begin{bmatrix}
1 & 1 & 1 & 3 \\
0 & 1 & 2 & 0
\end{bmatrix}... |
Let $X$ be a $T \times K$ matrix. $X'X$ positive definite means that for all $c \not = 0$ $c'X'X c >0$, so then $(Xc)' Xc > 0$ which implies $(X\cdot c)\cdot (X\cdot c)$ (I'm not sure what proper notation is here, sorry), and therefore $X\cdot c \not = 0$ for any $c$., so $X$ has full column rank, and $X'X$ has full ra... |
$$2a +4b +3c +5d +6e=37$$
$$4a +8b +7c +5d +2e=74$$
$$-2a -4b +3c +4d -5e=20$$
$$a +2b +2c -d +2e=26$$
$$5a -10b +4c +6d +4e=24$$
find $a,b,c,d,e$
I tried solving the system of equations above but ended up with many fractions and decimals that I could not solve it by hand. Is there a shortcut or trick to solving this e... |
Let $X$ denote a $T\times K$ matrix. I have seen the definition for full column rank as "There is no vector $c \not = 0$ with $X\cdot c = 0$. Would a definition for full row rank then be "There does not exist a $b\not = 0$ such that $ b\cdot X = 0$?
Where I use $c$ for a $K\times 1$ vector and $b$ for a $ 1 \times T$ v... |
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