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H: diagonal matrix with $k$ 1's and $n-k$ 0's
This is from an old qualifying exam.
Let $V$ be a vector space of dimension $n$ and let $P$ be a projection on $V$ with
$\dim(P(V))=k$. Prove that there is an ordered basis for $V$ such that the matrix representation of $P$ with respect to this basis is a diagonal matrix ... |
H: Confidence Interval
After taking 90 observations, you construct a 90% CI for μ . You are told that your interval is 3 times too wide.(i.e. your interval is 3 timers wider than what was required. You sample size should have been. (a)30 (b) 270 (c) 810 (d) 10
The correct answer is c. But why is C?
AI: Often the endpo... |
H: Evaluate $\int_{D} \frac{dw}{w \cdot (1-w)}$ where $D$ is the rectangle
How can I evaluate the below integral:: $$\int_{D} \frac{dw}{w \cdot (1-w)}$$ using the $\textbf{Cauchy Integral Formula}$, where $D$ is the rectangle with vertices at the points $3 \pm{i}$ and $-1 \pm{i}$.
I know that the Cauchy Integral form... |
H: Why does the XOR operator work?
In many coding problems, I see applying XOR to the set of values gives the result.
For example :
In the game of nim
Let n1, n2, … nk, be the sizes of the piles. It is a losing position
for the player whose turn it is if and only if n1 xor n2 xor .. xor nk
= 0.
Can somebody expl... |
H: Find the intersection (vector) of two 2D planes
Wall #1 is defined by [1,1,6]^T and [2,0,7]^T. Wall #2 is defined by [1,1,2]^T and [3,2,-1]^T. Perform the steps that a CAD program might do to find a vector which represents the corner created where these two walls intersect.
Hint: the left nullspace is perpendic... |
H: Statistic question
1) A student took a chemistry exam where the exam scores were mound-shaped with a mean score of 90 and a standard deviation of 64. She also took a statistics exam where the scores were mound-shaped, the mean score was 70 and the standard deviation was 16. If the student's grades were 102 on the c... |
H: $(a_n+a_{n+1})$ is convergent implies $(a_n/n)$ converges to $0$
Let $(a_n)$ be a sequence such that $(a_n+a_{n+1})$ is convergent. Prove that $(a_n/n)$ converges to $0$.
$(a_n+a_{n+1})$ convergent to $L$ means that for all $\epsilon$, there exists $N$ such that for all $n\geq N$, $|L-a_n-a_{n+1}|<\epsilon$, or i... |
H: recurrence relation for proportional division
Consider the following recurrence relation, for a function $D(x,n)$, where x is a positive real number and n is a positive integer:
$$ D(x,1) = x $$
$$ D(x,n) = \min_{k=1..n-1}{D(xk/n,k)} \ \ \ \ [n>1] $$
This formula can be interpreted as describing a process of divid... |
H: Clarification needed on finding last two digits of $9^{9^9}$
I stumbled across this problem here. In the answer given by the user Gone, I don't see how he makes use of the second line in the last line. Could someone explain why he calculated $9^{10}$ via binomial theorem and where it is applied?
AI: The idea is tha... |
H: Convergence of nth root iteration?
I'm on problem 18 of baby Rudin, which asks you to describe the behavior of the sequence defined by
$$x_{n+1}=\frac{p-1}{p}x_n+\frac{\alpha}{p} x_n^{1-p}$$
with $x_1>\sqrt{\alpha}$, for given $\alpha>1$ and positive integer $p$. From what I can tell it's a common formula for calcu... |
H: Find an orthogonal matrix $Q$ so that the matrix $QAQ^{-1} $ is diagonal.
The question is as follows:
$A=\left( \begin{array}{ccc}
1 &1& 1 \\
1 & 1 & 1 \\
1 & 1 & 1 \end{array} \right) $ Find an orthogonal matrix $Q$ so that the matrix $QAQ^{-1} $ is diagonal. Verify this by direct computation.
My friend knows ho... |
H: Why do Artinian rings have dimension 0 and not 1?
One of the properties of an Artinian ring $R$ is that every prime ideal is maximal. So, if $\mathfrak{m}$ is a nonzero prime ideal, $(0)\subseteq \mathfrak{m}$ is a length-$1$ chain of prime ideals, meaning that $\dim R\geq 1$. Since every prime ideal is maximal, no... |
H: What is continuity correction in statistics
Can someone please explain to me the idea behind continuity correction and when is it necessary to add or subtract $\dfrac{1}{2}$ from the desired number (how do we tell whether we need to add or subtract), how do we tell when we need to use continuity correction?
AI: The... |
H: A recursive formula for $a_n$ = $\int_0^{\pi/2} \sin^{2n}(x)dx$, namely $a_n = \frac{2n-1}{2n} a_{n-1}$
Where does the $\frac{2n-1}{2n}$ come from?
I've tried using integration by parts and got $\int \sin^{2n}(x)dx = \frac {\cos^3 x}{3} +\cos x +C$, which doesn't have any connection with $\frac{2n-1}{2n}$.
Here's ... |
H: Partial fractions on complex function for Laurent series
Whats the proper way to calculate the partial fractions for functions like these
$$ f(z)=\frac{z}{z^2(z+1)} \\
f(z)=\frac{z}{(2-z)^3z}
$$
before calculating the Laurent series?
AI: $$\frac z{z^2(z+1)}=\frac1{z(z+1)}=\frac1z-\frac1{z+1}\;--\text{almost no nee... |
H: Differential manipulation
Let $v \equiv F(y, z)$. The partial derivatives of $F$ are $$F_1 \equiv \frac{\partial F(y,z)}{\partial y} = \frac{H(v)}{H(y)},$$ $$F_2 \equiv \frac{\partial F(y,z)}{\partial z} = r\frac{H(v)}{H(z)},$$ where $H$ is an arbitrary function. This is taken from the book Probability Theory by E.... |
H: Sperner's Lemma in infinite-dimensional spaces?
I've been looking at Sperner's Lemma for a little while and have managed to come to grips with some of the combinatorial proofs. Some descriptions I have encountered claim to prove it for "simplices" and some for "$n$-simplices", and there didn't seem to be any partic... |
H: Simple question of maximum value a part can have?
We have to partition n chocolates among m children.
Children will be happy if max and min a child has got is less than 2.
What is the max a child can get??
For n=6 m=3 ,the partition will be 2 2 2
Maximum a child can get is 2
For n=7 m=3 ,the partition will be 2 3 2... |
H: Simple problem on restricted partition
When finding number of ways to partition n distinct chocolates among m children such that each child has at most
$$\left\{\begin{matrix}
\left \lfloor \frac{n}{m} \right \rfloor & \text{if} \ \ n\ \ \left (mod \ \ m \right )\equiv 0 \\
\\
\left \lceil \frac{n}{m} \right \rc... |
H: Analytic continuation of Riemann Zeta function.
I am reading about zeta function from book by Ingham. In that book the following theorem is given. I am unable to understand what does he mean by finite part of plane in the statement.
AI: By finite part of the plane he probably means the complex plane itself. The inf... |
H: Some questions on outer region and outer solution of a boundary problem
There is this problem below that I have some doubts and confusions, I will appreciate if anyone could provide some clarifications and explanations. I am new to the Boundary Layer Theory, this question involves the Boundary Layer Theory, it is a... |
H: In ZF, does there exist an ordinal of provably uncountable cofinality?
Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm cf}(\aleph_1)=\aleph_0$, and most conceivable limit ... |
H: Find $f(5)$ of a non-constant polynomial function $f(x)$
Suppose $f(x)$ is a non-constant polynomial such that $f(x^ 3) − f(x ^ 3 − 2) = f( x )\cdot f(x) + 12$ for all $x$. Find $f(5)$?
I find this problem on Quora just now, and I try to solve it but do not know where to start (every time I substitue a number $a$ ... |
H: Sandwich measurable set between Borel sets
If $B$ is Lebesgue measurable then we can sandwich it closely between Borel sets in the sense that there is Borel $A , C$ such that
$A \subset B \subset C$ and $m(C\setminus A) = 0$.
does anyone know a reference for this statement or how to go about a proof? this is somet... |
H: Are two independent events $A$ and $B$ also conditionally independent given the event $C$?
If we know that two events $A$ and $B$ are independent, can we say that $A$ and $B$ are also conditionally independent given an arbitrary event $C$?
$$P(A\cap B) = P(A)P(B) \overset{?}{\Rightarrow} P(A\cap B|C) = P(A|C)P(B|C)... |
H: Can there be a repeated edge in a path?
I was just brushing up on my discrete mathematics specifically graph theory and read the following definition of a walk in a graph
"A walk in a graph is an alternating sequence of vertices and edges from a start vertex to an end vertex where start and the end vertices are no... |
H: How to find $\lim_{n\rightarrow\infty} \int_{0}^{1} f(t)\phi(nt) dt$ for $1$-periodic $\phi$?
How to solve the following problem:
Let $\phi\in L^{\infty}(0,1)$ be a $1$-periodic function and $f\in L^{1}(0,1)$. Find $\lim_{n\rightarrow\infty} \int_{0}^{1} f(t)\phi(nt) dt$.
Thanks in advance.
AI: Claim: $$\lim_{n\to... |
H: Arc length of $y = x^{2/3}$
Find the arc length of $y=x^{2/3}$ from $x=-1$ to $x=8$.
I have tried applying the arc length formula but for some reason I keep getting $7.63$, but the answer is $10.51$.
AI: The complex outcome of Maazul seems to be caused by the definition of $x^{1/3}$ for $x < 0$. Many computer al... |
H: Composition $R \circ R$ of a partial ordering $R$ with itself is again a partial ordering
If $R$ is a partial ordering then $R\circ R$ is a partial ordering.
I cannot seem to prove this can anyone help ?
AI: Denote $R$ with $\le$, and $R \circ R$ with $\mathrel{\underline\ll}$. Expanding what reflexivity, transi... |
H: How to simplify summation $(1\cdot2) + (2\cdot3) + (3\cdot4) + (4\cdot5) + (5\cdot6) + ... + (N\cdot(N+1))$ in terms of N?
I have function f(n) like:
N=1 result = 2
N=2 result = 8
N=3 result = 20
N=4 result = 40
N=5 result = 70
N=6 result = 112
N=7 result = 168
N=8 result = 240
N=9 resul... |
H: How to find the minimum of the expression?
Let $a$, $b$, $c$ be three real positive numbersand $a^2 + b^2 + c^2 =3$. Find the minimum of the expression
$$P = \dfrac{a^2}{b + 2c} +\dfrac{b^2}{c + 2a}+ \dfrac{c^2}{a + 2b}.$$
I tried
$$\dfrac{a^2}{b + 2c} +\dfrac{b^2}{c + 2a}+ \dfrac{c^2}{a + 2b} \geqslant \dfrac{(a... |
H: Cyclotomic integers: Why do we have $x^n+y^n=(x+y)(x+\zeta y)\dots (x+\zeta ^{n-1}y)$?
Why do we have the factorization $$x^n+y^n=(x+y)(x+\zeta y)\dots (x+\zeta ^{n-1}y)$$ for $\zeta$ a primitive $n^{\text{th}}$ root of unity where $n$ is an odd prime?
AI: If $y = 0$, the factorisation reads $x^n = \underbrace{x\,.... |
H: Simple inequality with unknown in the exponent
Let $0<\alpha\ll1$ I have the following inequality:
$$
2\alpha^2x\geq \alpha^{2x}
$$
It looks trivial, but I wasn't able to find the $x$ that verify the condition. Anyone any clue?
AI: Here is a general approach.
Let $$f(x) = \alpha^{2x} \text{ and } g(x) = 2 \alpha^2 ... |
H: If $f^2$ is differentiable, how pathological can $f$ be?
Apologies for what's probably a dumb question from the perspective of someone who paid better attention in real analysis class.
Let $I \subseteq \mathbb{R}$ be an interval and $f : I \to \mathbb{R}$ be a continuous function such that $f^2$ is differentiable. ... |
H: How do I find the probability of these independent events
The following is taken from the ETS math review for the GRE:
Let A, B, C, and D be events for which P(A or B)=0.6, P(A)=0.2, P(C or D)=0.6, and P(C)=0.5 The events A and B are mutually exclusive, and the events C and D are independent.
Find P(D).
I'd apprec... |
H: Is each space filling curve everywhere self-intersecting?
Consider a continuous surjection $f:[0,1]\to [0,1]\times[0,1]$. Is
$$\{x:\exists(t_1\not=t_2) f(t_1)=f(t_2)=x\}=[0,1]\times[0,1]?$$
AI: No. Consider for instance the Hilbert space-filling curve, on which Brian Hayes wrote a nice popular article recently (HTM... |
H: What is a place?
In Specialization of Quadratic
and Symmetric Bilinear
Forms (page: 3) the author writes "Let also $\lambda: K \to L \cup \infty$ be a place, $\mathfrak o = \mathfrak o_\lambda$ the valuation ring associated to $K$ and $\mathfrak m$ be the maximal ideal of $\mathfrak o$."
What is a place and what is... |
H: $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant?
is it true if $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? I am not able to find out what property of holomorphic map I need to apply. please help.Thank you.
$f(z)=u(x,y)+iv(x,y)$, $\bar{f}(z)=u(x,y)-iv(x,y... |
H: Union of two partial orderings
Suppose S and R are partial orderings. Does is necessarily mean that $R \cup S$ (union) is a partial ordering? If not what conditions would have to be met for it to be a partial ordering?
AI: As exitingcorpse remarks, antisymmetry may fail.
Transitivity can be a problem too, for examp... |
H: On one representation of Green's function
The Green's function for heat equation on finite interval is well known (with Dirichlet conditions):
$$
G(x,x', t) = \frac{2}{l}\sum\limits_{n=1}^{\infty}
\exp\left(-\frac{\pi^2n^2}{l^2}at\right)\sin\left(\frac{x\pi n}{l}\right)
\sin\left(\frac{x'\pi n}{l}\right)
$$
But re... |
H: Probability Independence to reduce terms.
Is the following probability reduction correct?
p(A,B|C,D) = p(A,B|C), where (A,B) is independent of D
Is it because of the following prove:
p(A,B|C,D)
= p(A,B,C,D) / p(C,D)
= p(D)*p(A,B,C) / p(C)*p(D)
= p(A,B,C) / p(C)
= P(A,B|C)
If so, then isn't it (A,B,C) is independ... |
H: Is this graph coloring problem solved correctly?
On this Wikipedia page about Burnside's lemma, it is calculated that there are 57 rotationally distinct colorings of the faces of a cube with three colors. I'm confused by the way it is done.
They apply the lemma to the set of all $3^6$ functions assigning one color... |
H: limit with quaternion
Let $v\in \mathbb{H}$ and $q:t\in\mathbb{R}\rightarrow q(t)\in \mathbb{H}$. That $q(t) \neq 0$ for all $t\in \mathbb{R}$ and $q^{-1}(t) = \frac{1}{q(t)}$. So don't confuse $q^{-1}(t)$ with inverse function(as pointed by rschwieb).
Than we have limit:
$$\lim_{\Delta t \rightarrow 0} \frac{1-q(t... |
H: Properties of a one to one continuous function from $[0, 1]$ onto itself
Let $f$ be a one to one continuous function from $[0, 1]$ onto itself. Show that
(i) $f$ is a homeomorphism.
(ii) $f$ is strictly monotone on $[0, 1]$
(iii) Is it true that if $f$ is strictly monotone on $[0, 1]$ and onto $[0, 1]$, then $f$ is... |
H: antipodal map from sphere to projective space is immersion and submersion?
Define a map $\mathbb{S}^n \to \mathbb{RP}^n$ given by $x \mapsto \{ x,-x\}.$
Clearly this is not a diffeomorphism, but how can one show that it is an immersion and submersion?
AI: This is a local diffeomorphism, and the result follows. |
H: Representation problem from Serre's book
I asked this question yesterday on the setting of an exercise problem (Ex 2.8) from Serre's book "Linear representations of Finite Groups" (I'm teaching myself representation theory...)
Now that that is sorted, I'm still stuck on the actual problem: Let $\rho: G\to GL(V)$ be... |
H: Dropping the modulus sign in integrals of the form $\int 1/t ~dt$
In the process of solving a DE and imposing the initial condition I came up with the following question.
I've reached the stage that
$$\ln y + C = \int\left(\frac{2}{x+2}-\frac{1}{x+1}\right)dx$$
$$\Rightarrow \ln y +C=2\ln|x+2|-\ln|x+1|$$
$$\Rightar... |
H: $1500=P \times { (1 + 0.02) }^{ 24 }$, what is the value of $P$?
Hey guys could you please tell me what is the faster why to solve this equation. It's a compound interest equation and I'm stuck at the ${ (1 + 0.02) }^{ 24 }$ I really don't know how to proceed in this part. I thought about log, but how can I apply l... |
H: Showing $\varphi(t)\neq 0$ when $\varphi$ is a characteristic function of an infinitely divisible distribution
Let $\varphi$ be a characteristic function of an infinitely divisible random variable. Show that $\varphi(t) \neq 0$ for all $t$.
Sorry, I have no clue how to do it, because if the exponential is not rea... |
H: Is this integral improper? If yes - why?
Is this integral improper? If yes - why?
$$
\int\limits^2_0 \,\frac{1}{x-1} dx
$$
AI: Definition:
The integral $\int_a^b f(x)dx$ is called improper integral if:
$a=+\infty$ or $b=\infty$ or both.
$f(x)$ is unbounded at one or more points of $a\le x\le b$.
As @Git sugges... |
H: What is the ratio of the perimeter of $OPRQ$ to the perimeter of $OPSQ$?
Area of circle $O$ is $64\pi$. What is ratio of the perimeter of $OPRQ$ to that of $OPSQ$ ($\pi = 3$)?
Okay i have tried couple of things but seems its not working . Please suggest me proper solution of this example so that , i can solve si... |
H: Proving the origin is a saddle point.
I have the function $g(x,y) = x^6 -y^6x^2$ and want to prove that the origin is a saddle point.
I know that a critical point with an indefinite Hessian matrix is a saddle point, but this is only a sufficient condition.
$(0,0)$ is indeed a critical point of $g$, but the Hessian ... |
H: How to find $x^{2000}+x^{-2000}$ when $x + x^{-1} = \frac{1}{2}(1 + \sqrt{5})$
Let $x+x^{-1}=\dfrac{1+\sqrt{5}}{2}$. Find $x^{2000}+x^{-2000}$.
How many nice methods do you know for solving this problem? Thank you everyone.
My method: because $x+\dfrac{1}{x}=2\cos{\dfrac{2\pi}{5}}$, so $$x^{2000}+\dfrac{1}{x^{200... |
H: How to show $d(x,A)=0$ iff $x$ is in the closure of $A$?
This is a problem form Topology by Munkres:
Let $X$ be a metric space with metric $d$ and $A$ is a nonempty subset of $X$. Show that $d(x,A)=0$ if and only if $x$ is in the closure of $A$.
I think this problem is quite easy to understand emotionally but I d... |
H: Find coordinates of vertex of equilateral triangle
$ABC$ is an equilateral triangle , $AC = 2 $
What is the value of $p$ and $q$ ?
AI: HINT:
So, $C$ has to be $(2,0)$
Now, equating the squares of lengths of the sides $$(p-0)^2+(q-0)^2=(p-2)^2+(q-0)^2$$
Solve for $p$ and find $q$ from $p^2+q^2=2^2$ |
H: Contravariant Metric Tensor
Okay, so I have exactly ZERO experience with tensors and this project I am working on involves tensors. I have looked through a bunch of online resources, and attempted to look for textbooks (not available to me) and I am getting really confused. The extent to which I have picked up is... |
H: Let $f : \mathbb{R} \to \mathbb{ R}$ be such that $f' (x)$ exists
Let $f : \mathbb{R} \to \mathbb{ R}$ be such that $f' (x)$ exists for all non zero $x$ and $\lim_{x\to 0} f' (x) = 0$.
Then
(i) $f$ is continuous but not differentiable at $0$.
(ii) $f$ is differentiable at $0$ and $f' (0) = 0.$
(iii) $f$ has either ... |
H: How can i solve a differential equation like this one?
My Problem is: this given differential equation
$$x^3+y^3+x^2y-xy^2y^{\prime}=0$$ $$(x\neq 0,\ y\neq 0)$$
My Approach was: i had the idea to bring it in this form:
$$x^3+y^3+x^2y-xy^2y^{\prime}=0$$
$$x^3+y^3+x^2y=xy^2y^{\prime}$$
$$\frac{x^3}{xy^2}+\frac{y^3}{... |
H: Neyman-Pearson lemma proof
$\mathbf{Theorem}$ Let $\forall \alpha \in (0,1) \space \exists \space k$, such that for $W_0 = \{ x: p_1(x) \ge k p_0(x) \}$, $$\int_{W_0} p_0(x) d\mu(x)=\alpha$$
where $p_i(x)$ is the likelihood function under the hypothesis $i=0,1$. Then $\forall W$, such that
$$\int_{W_0} p_0(x) d\mu... |
H: Find the coefficient of $x^m$ in the expansion $(1 + ax + bx^2)^n$
Find the coefficient of $x^m$ in the expansion of $(1 + ax + bx^2)^n$
One approach would be:
Let $p = 1+ax$
and $q = bx^2$
Now expand $(p+q)^n$ and then expand $p$ and $q$ individually. But that is so clumsy. Can we derive a direct formula?
AI:... |
H: Solve the second order differential equation.
Find a general solution to the equation:
$u''-e^tu'-e^tu=1$.
AI: Note that you can rewrite the equation as
$$\frac{d}{dt} (u \,e^t) = u''-1$$
which is equivalent to
$$u'-e^t u = t+C$$
where $C$ is a constant of integration. This equation has an integrating factor of $e... |
H: Joint and individual probability independence
If (A,B,C) is independent of D, are the following true?
A then also independent of D
B then also independent of D
C then also independent of D
AI: Due to the symmetry of the argument, it is equivalent to ask whether independence of $A\cap B \cap C$ and $D$ implies t... |
H: Standard Approach to the Fundamental Counting Principal
I'm trying to teach myself combinatorics from a textbook. The last question of the first chapter is as follows:
If A is a finite set, its cardinality $o(A)$, is the number of elements in $A$. Compute
(a) $o(A)$ when $A$ is the set consisting of all five-d... |
H: $\bigcup_{n}V_n$ is dense in $V$ (Hilbert spaces)
I read:
$\bigcup_{n}V_n$ is dense in $V$ (Hilbert spaces)
Does this mean: for every $v \in V$, there is a sequence $\{v_n\}$ with $v_n \in V_n$ for each $n$ such that $|v_n - v|_V \to 0$?
I guess so. But then I also read
$V_n$ is dense in $V$
Does the author mea... |
H: How to show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$?
I really think I have no talents in topology. This is a part of a problem from Topology by Munkres:
Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$.
I always have the feeling that it is easy to understand the problem emotio... |
H: Every morphism in Set is regular
I am trying to prove that every morphism in the category Set is regular, that is, that for every set-function $f:A\to B$ there exists a function $g:B\to A$ such that $f\circ g\circ f=f$. The assumption is that $A\neq\varnothing $, because otherwise $B=\varnothing$.
Define an equival... |
H: Can a set of non self-intersection points of a space-filling curve contain an arc?
Consider a continuous surjection $f:[0,1]\to[0,1]\times[0,1]$.
It can be proved that set of self-intersection points must be dense.
In the Hilbert curve, the set of self-intersections are points (a,b) such that either a or b can be w... |
H: Is there a way to factor $uv-u-v-1$?
$uv-u-v-1$. I tried $(u+1)(v-1)$ and it's almost correct but I can't quite get it/ This is part of a limit problem I'm doing. Thanks!
AI: This polynomial in $u$ and $v$ is irreducible (not factorable). Observe that since it is degree 2, it can only factor nontrivially as the pro... |
H: Pumping Lemma Proof for $ww$
The proof of language $F = \{ww\mid w ∈ \{0,1\}^*\}$ is not a regular language using pumping lemma most of the solutions i found uses the string $0^p10^p1$. I understand the proof using that. But in Michael Sipser's Introduction to the Theory of Computation book he mention that using $... |
H: Some questions about $f(z) = \frac 1 {e^z -1}$
Let $f(z):= \frac 1 {e^z-1}$.
First question: Why has $f$ a pole of order $1$ in $z = 0$? Second question: How can we determine the radius of convergence of $\sum_{n} a_n z^n$ which is the Laurent-series of $f$ around $z=0$ ? I have computed that
$$
\forall n > 0: b_... |
H: Unitarily equivalent?
I'm confused about that notion.
In my textbook there are two examples.
(1) $A=\begin{pmatrix} 1&1&0\\0&2&3\\0&0&3\end{pmatrix}$ and $B=\begin{pmatrix} 1&0&0\\0&2&0\\0&0&3\end{pmatrix}$
They have same eigenvalues but not unitarily equivalent because one is symmetric and the other is not.
(2... |
H: Convergence of the series $\sqrt[n]n-1$
Let $a_n=\sqrt[n]n-1$. Does $\sum_{n=1}^\infty a_n$ converge?
AI: Hint: $$\sqrt[n]n = e^{\frac{\log n}{n}} > 1+\frac{\log n}{n}$$ |
H: lim sup and sup inequality
Is it true that for a sequence of functions $f_n$
$$\limsup_{n \rightarrow \infty }f_n \leq \sup_{n} f_n$$
I tried to search for this result, but I couldn't find, so maybe my understanding is wrong and this does not hold.
AI: The inequality
$$\limsup_{n\to\infty}a_n\leq\sup_{n\in\mathbb{N... |
H: Proof derivative using Cauchy-Riemann
Using the Cauchy–Riemann equations I have to prove that $f(z)=e^{iz}$ is analytic and its derivative is $ie^{iz}$
Using $z=x+iy$ this is what I have done:
$$e^{iz}=e^{i(x+iy)}=e^{ix-y}= \frac{e^{ix}}{e^{y}}=\frac{\cos x+i\sin x}{e^{y}}$$ so, $u(x,y) = \frac{\cos x}{e^y}$ and $v... |
H: Expand log function with two terms
HOw can I expand ln(1+2/(A-1))? I think I need to use taylor series but the 1 is messing me up.
Should I just ignore the 1?
AI: You can expand it iff $|\frac{1}{2(A-1)}|<1$.
The expansion is $\displaystyle\ln(1+w)=w-\frac{w^2}{2}+\frac{w^3}{3}\dots=\sum_{n=1}^{\infty}(-1)^{n+1}\fr... |
H: Mathematics Essence
I started reading History of Philosophy and readily noticed that the origins of our actual natural sciences were due to the proper use of inductive logic.Our Physics/Chemistry and Biology all are known to have started by the revolution of Thales and the Pre-socratic Thinkers, conscious human-bei... |
H: Does a lower bounded set always have an infimum?
Let $A$ be a partially ordered subset of $X$. If $A$ is bounded below, does $\inf(A)$ exist?
AI: Let us assume $A$ is nonempty to avoid pathologies.
The statement does not have to be true even in the nice case that the ambient space $X$ is totally ordered:
Let $\Bbb ... |
H: The Abelianization of $\langle x, a \mid a^2x=xa\rangle$
I wish to verify the following statement (which comes from Fox, "A Quick Trip Through Knot Theory", although that is probably not important).
"$\Gamma=\pi_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so the homology of $M$ is infinite cyclic."
So, I need to find t... |
H: Must an ideal contain the kernel for its image to be an ideal?
I'm trying to learn some basic abstract algebra from Pinter's A Book of Abstract Algebra and I find myself puzzled by the following simple question about ring homomorphisms:
Let $A$ and $B$ be rings. If $f : A \to B$ is a homomorphism from $A$ onto $B... |
H: Number of sides a regular polygon has.
The question is "Both tile A and B are regular polygons.
Work out the number of sides A has."
For this I put B is equilateral ∴ all angles are 60.
However, I have no idea where to go from this.
Could anyone give me any tips for solving this and similar questions? Thanks.
AI... |
H: FInd the number of pairs $(A,B)$
Let $n,r,s$ be given, where $n\geq 1$,$1\leq r\leq n$ and $1\leq s \leq n$.
a) determine the number of pairs $(A,B)$ with $A\subseteq N_n, |A|=r,B\subseteq N_n, \text{and} |B|=s $
Now my intuition says that these two could intersect. But dont have to. So I would define it as being... |
H: Confusion related to smoothness of a function
I just found this thing that $\operatorname{trace}(AB)$ where $A$ and $B$ are two matrices, it is a smooth function. I didn't understand how it is a smooth function. Any suggestions?
AI: A smooth function has derivatives of all orders. In this case, trace$(AB)$ is a ... |
H: Show that an algebraically closed field must be infinite.
Show that an algebraically closed field must be infinite.
Answer
If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F cannot be algebraically closed.
My Question
Could we not use th... |
H: show that $|\cup^n_{i=0} X_i| = \sum \binom{n}{r}\binom{r}{r-i}\binom{n-r}{s-i}$
Define $$
X_i = \{(D,E,F) : D \subseteq N_n, |D| = r,E\subseteq D, |E|=r-i,F\subseteq N_n - D, |F|=s-i\}
$$
Where $N_n$ denotes the set of all subsets.
$$|X_0 \cup X_1 \cup \ldots\cup X_n| = \sum_{i=0}^n \binom{n}{r}\binom{r}{r-i}\... |
H: what are necessary conditions for $\mathbb{E}[X_n] \to \mathbb{E}[X]$?
Say $\{X_n\}$ is a sequence of random variables. What type of convergence to $X$ (or additional conditions) is required to ensure convergence of the means?
I think convergence in probability is not enough in general. However, convergence in prob... |
H: How did we know to invent homological algebra?
Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought.
(Original question follows.)
I want to understand how anyone knew that the long exact sequence i... |
H: On the ramification index in Dedekind extensions
Citing from my textbook...
Definition If $R\subseteq R'$ are Dedekind domains and $\mathfrak{P}$ is a nonzero prime ideal of $R'$ and $\mathfrak{p}=\mathfrak{P}\cap R$ then the ramification index of $\mathfrak{P}$ over $R$ is the power of $\mathfrak{P}$ appearing in ... |
H: The system of equations $x^2 + y^2 - x - 2y = 0$ and $x + 2y = c$
I have
$(1.) \quad x^2 + y^2 - x - 2y = 0 \\
(2.) \quad x + 2y = c$
Solving for $y$ in $(2.)$ gives
$y = (c - x) / 2$
Is there a way to simplify equation $(1.)$?
Because at the end I arrive at
$c^2 - 2x - c = 0$
and can't proceed. Need to get typica... |
H: How to prove the properties derived from a matrix's signature
We've recently learned about metric signatures following the proof of Sylvester's law of inertia but we didn't quite say which properties does the signature of a given matrix $A\in \mathcal{M}_n\left(\mathbb{R}\right) $ have. Let's call it $(n_{_+} ,n_{_... |
H: questions about proof of reverse formula of Fourier transform
We had the following theorem in class for a fourier transform $\widehat f$:
Let $\widehat f$ be the restriction of a $\mathbb C$ definied meromorphic function $F$. Let $F$ have a finite number of poles and let $z\cdot F(z)$ be bounded for big $z$. Then ... |
H: the product of a matrix and a permutation matrix
Can a permutation matrix ($P$) be used to change the rank of another matrix ($M$)?
Is there any literature to this effect, or to the contrary?
I've tried a few small examples and the resulting matrix ($M_2$) seems to always have the same rank as the input matrix ($M$... |
H: Suppose that $T: \mathbb R^2 \to \mathbb R^2$ is the linear transformation that rotates a vector by 90°.
Suppose that $T: \mathbb R^2 \to \mathbb R^2$ is the linear transformation
that rotates a vector by 90°.
(a) What is the null space of T?
(b) Is T one-to-one?
(c) What is the range of T?
(d) Is T onto?
Well I'... |
H: If $f:I\to\mathbb{R}$ is $1{-}1$ and continuous, then $f$ is strictly monotone on $I$.
Suppose that $I\subseteq\mathbb{R}$ is nonempty. If $f:I\to\mathbb{R}$ is $1{-}1$ and continuous, then $f$ is strictly monotone on $I$.
The answer in the back of the book$^1$, which I found after writing the following proof, sa... |
H: Linearly independent subset under linear 1-to-1 transformation
Suppose that $T:\mathbb{R}^n \to \mathbb{R}^m$ is linear and one-to-one. Let $\{v_1, v_2, \ldots, v_k\}$ be a linearly independent subset of $\mathbb{R}^n$.
Prove that the set $\left\{T(v_1), T(v_2), \ldots, T(v_k)\right\}$ is a linearly
independent su... |
H: A planar graph problem
The problem is this:
Let $G$ be a connected, planar graph whose number of vertices is a multiple of $8$. $5/8$ of the vertices have degree $3$, $1/4$ have degree $4$, and $1/8$ have degree $5$. All faces of $G$ are triangles or quadrilaterals. Find the number of triangular faces, the number o... |
H: Is $G $ contained in $ G *_H K $ if $ H\rightarrow G $ and $ H\rightarrow K$ are injections?
Given injections $ H\rightarrow G $ and $ H\rightarrow K $, is the canonical morphism $ G\rightarrow G *_H K $ of G into the free product with amalgamation also injective?
AI: Yes, since every element $x\in G_1 *_H G_2$ ha... |
H: Let $F$ be a finite subset of the natural numbers and consider the sum
$$\sum(-1)^l\tag{1}$$
Define $F_{\text{even}} = \{n \in F : n\text{ is even}\}$ and $F_{\text{odd}} = \{n \in F : n\text{ is odd}\}$
(a) Suppose that $\#F$ is odd. Show that $\#F_{\text{even}}\ne \#F_{\text{odd}}$
(b) Suppose that $\#F$ is odd... |
H: Is there NO solution to this linear system of 3 equations, $3$ unknowns?
I have the following linear system:
$$\begin{align}
&x + y + 2z + 2 = 0 \\
&3x - y + 14z -6 = 0 \\
&x + 2y +5 = 0
\end{align}$$
I immediately noticed that there was no $z$ term in the last equation and thus determined that I will end with $0... |
H: Consequence of the Chinese Remainder Theorem
We want to prove the following:
For any $n+1$ distinct real numbers $a_0, a_1, ..., a_n$ and any $n+1$ real numbers $b_0, b_1, ..., b_n$, there exists a polynomial of degree at most $n$ taking the value $b_i$ at $a_i$ for all $i=0, 1, ..., n$.
This was discussed in a c... |
H: Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$
Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$?
With added assumptions on how $X_n$ behaves, you can prove that $E(X_T)=E(X_0)... |
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