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H: trigonometric representation of a complex number.
Let $z=e^{it}+1$ where $0\leq t\leq \pi$, Find the trigonometric representation of $z^2+z+1$.
(The trigonometric representation should be in the form of : $r(\cos \theta +i \sin \theta)$, where $r,\theta \in \mathbb{R}$ and $r>0$.
What I have done : $$z^2+z+1=e^{2i... |
H: Cramer-Rao lower bound for the variance of the unbiased estimator of $\tau(\theta)$
Given the pdf $f(x;\theta)=\frac{1}{\pi[1+(x-\theta)^2]}$ ; $-\inf < x<\inf$, $-\inf < \theta<\inf$
Show that the Cramer-Rao lower bound is 2/n
where n is the sample size.
AI: 1- you have the assumption that the samples are independ... |
H: Question about uniqueness function series
Please pardon me if this is elementary, but I've looked hard for an answer to this and am very surprised I have yet to find a good one. My question is simple: under what condition(s) does $$\sum_{n=1}^{\infty}f_n(x) = \sum_{n=1}^{\infty}g_n(x)$$ imply that $$f_n(x) = g_n(x)... |
H: Integration question verifying piecewise
I have the following question: from direct integration show
$\displaystyle \int \limits_{-L}^{L} \cos({m πx\over L})\cos({nπx\over L}) \ dx = \begin{cases}0 & m \neq n \\ L & m = n \\ \end{cases} $
I use a trigonometric identity and evaluate individually to produce:
$\displ... |
H: Find the range of: $y=\sqrt{\sin(\log_e\frac{x^2+e}{x^2+1})}+\sqrt{\cos(\log_e\frac{x^2+e}{x^2+1})}$
Find the range of:
$$y=\sqrt{\sin(\log_e\frac{x^2+e}{x^2+1})}+\sqrt{\cos(\log_e\frac{x^2+e}{x^2+1})}$$
What I tried:
Let:$$\log_e\frac{x^2+e}{x^2+1}=X,$$
then $$y=\sqrt {\sin X}+\sqrt{\cos X}$$
$$y_{max}at X=\pi/4$$... |
H: How to show that $\frac{1}{(1-\frac{1}{4}z^{-1})(1-\frac{1}{4}z)} = \frac{-4z^{-1}}{(1-\frac{1}{4}z^-1)(1-4z^{-1})}$
Can anyone help me clarify what rule is used in this rewriting of this fraction?
$$\frac{1}{\left(1-\dfrac{1}{4}z^{-1}\right)\left(1-\dfrac{1}{4}z\right)} = \frac{-4z^{-1}}{\left(1-\dfrac{1}{4}z^{-1}... |
H: Finding the value of distribution function of a converging random variable
There is this example in a note that I think this is supposed to be a simple problem, but I still find it not as straightforward.
Consider a sequence of random variables $X_n\equiv1/n,X\equiv0$. Then $F_{X_n}(t)\to F_X(t)$ at all $t\not=0$ (... |
H: Integrating $ a\,f(x) +b\,y(x)=\frac{dy}{dx}$
Can somebody put me on the right track for integrating the following equation? How do I separate the variables?
$$ a\,f(x) +b\,y(x)=\frac{dy}{dx}$$
AI: Hint: Write down a differential equation solved by $z(x)=\mathrm e^{-bx}y(x)$. |
H: inclusion of sets -transitive?
show that if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$
Can I do it with using injective functions?
$A\subseteq B$ means there exists an injective fcn $f:A\to B$
$B\subseteq C$ means there exists an injective fcn $g:B\to C$
then the composition $g\circ f:A\to C$ is also ... |
H: Bring close 3D point to another 3D point equivalence
Having 2 point in 3D field - $\text{p1(x1,y1,z1) , p2(x2,y2,z2)}$ .
How could I generate the equivalence , $f$ which take parameter $t\in[0,1]$ and bring $p1$ closer to $p2$ as well as $t$ increase such that $f(1)=p2$ and $f(0)=p1$ ?
AI: Use a convex combinat... |
H: Does $\sum_{k=1}^{\infty}k(p^{\frac{(k-1)k}{2}}-p^{\frac{(k+1)k}{2}})$ converge?
Does the sum:
$$\sum_{k=1}^{\infty}k(p^{\frac{(k-1)k}{2}}-p^{\frac{(k+1)k}{2}})$$
$$ p\in\mathbb{R}|0{\leq}p<1$$
converse, and if so, to what function?
AI: Let's simplify your expression to get Carl Najafi's expression :
\begin{align}
... |
H: $C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$
Consider $C([0, 1])$, the linear space of continuous complex-valued functions
on the interval $[0, 1]$, with the norm
$$\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx.$$
I have to show that ... |
H: Show that $\frac{-4z^{-1}}{(1-\frac{1}{4}z^{-1})(1-4z^{-1})} = \frac{16}{15}\frac{1}{(1-\frac{1}{4}z^{-1})}-\frac{16}{15}\frac{1}{(1-4z^{-1})}$
Can anyone help me clarify how this rewriting is done?
$$\frac{-4z^{-1}}{(1-\frac{1}{4}z^{-1})(1-4z^{-1})} = \frac{16}{15}\frac{1}{(1-\frac{1}{4}z^{-1})}-\frac{16}{15}\frac... |
H: How to establish an isomorphism between these two tensor products?
Let $V_1, ..., V_k$ be finite-dimensional vector spaces. A tensor product is defined to be the quotient space $U/I$, where $U$ denotes the free vector space generated by elements of $V_1 \times ... \times V_k$, and $I$ the subspace of $U$ generated ... |
H: number of positive real root of $f(x)=x^4+2x^3-2 , x\in \Bbb{R}$,
$f(x)=x^4+2x^3-2 , x\in \Bbb{R}$,
(A) has two roots in $[0,\infty)$
(B) has three roots in $[0,\infty)$
(C) has no roots in $[0,\infty)$
(D) has a unique roots in $[0,\infty)$
How can I do this?I am stuck on it.
AI: By Descartes rule of sign it has... |
H: How to prove $G_2$ is a normal subgroup of $G_1\times G_2$?
I am studying abstract algebra and in one of my previous midterms there was a question that I can not solve. Now I have final exam tomorrow and want to learn the answer to that question. Here it is:
Can anyone help me with the solution of this question? M... |
H: Row of $A = BC$ as a linear combination of the rows of $C$
How to prove that any row of a matrix $A = BC$ can be written as a linear combination of the rows of $C$. I am able to visualize it with an example, but finding it difficult to prove with notations of vector space.
AI: By definition of matrix multiplication... |
H: Determine the number of digits in $(2^{120})(5^{125})$.
Determine the number of digits in $(2^{120})(5^{125})$.
This is a bonus question I found in a Grade 12 math textbook, and I'm curious on how to solve it. What I find strange is that this question is in the "Power Functions" section of "Polynomial Functions"; ... |
H: Why do we believe the Church-Turing Thesis?
The Church-Turing Thesis, which says that the Turing Machine model is at least as powerful as any computer that can be built in practice, seems to be pretty unquestioningly accepted in my exposure to computer science. Why? Do we have any more evidence for its truth than... |
H: Maximizing sum of strictly non-negative functions?
In an answer to this question, it was stated that if a function is the sum of two functions $f(x)=g(x)+h(x)$, and ${(\forall{x})(g(x)\ge0)}$ and $(\forall{x})(h(x)\ge0)$, the relative maxima of $f(x)$ can be found by finding the relative maxima of $g(x)$ and $h(x)$... |
H: Measure Algebra Question
Background:
Let $G$ be a locally compact group, and define $M(G)$ to be the vector space of all regular complex measures defined on $G$, normed by total variation.
For $\mu,\nu\in M(G)$, define the product $\mu * \nu$ as follows:
Define $\varphi\in C_{0}(X)^{*}$ by $$\varphi(f) = \int_{G}\i... |
H: Image of the idele class group and its subgroup of idelic norm 1
[Sorry if the title isn't specific, it was too long.]
My question is: Why does $J_{K}/J_{K}^{1}\cong
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ imply that $J_{K}/K^{\ast }$ and $J_{K}^{1}/K^{\ast }$ have the same image
under any... |
H: Natural number solutions to $\frac{xy}{x+y}=n$ (equivalent to $\frac 1x+\frac 1y=\frac 1n$)
I have a question about the following problem from a Putnam review:
Let $n\in \mathbb{N}$. Find how many pairs of natural numbers $(x, y)\in \mathbb{N}\times \mathbb{N}$ solve
$$
\frac{xy}{x+y}=n.
$$
I have found some soluti... |
H: Inverse Laplace transformation using reside method of transfer function that contains time delay
I'm having a problem trying to inverse laplace transform the following equation
$$
h_0 = K_p * \frac{1 - T s}{1 + T s} e ^ { - \tau s}
$$
I've tried to solve this equation using the residue method and got the followin... |
H: Is the third order term of a Taylor approximation a covariant tensor of rank 3 restricted to (h,h,h)?
Recently reading Peter Lax's linear algebra text, he had a very concise way of writing the Taylor approximation up to second order: $f(x+h) = f(x) + l(h) + \frac{1}{2}q(h) + \|h\|^2 \epsilon(\|h\|)$, where $l$ is a... |
H: Example of vector space
In almost all the literature that I have seen, one of the examples of vector space is as follows:
Set of all real-valued functions $f(x)$ defined on the real line
what confuses me here is that the word "linear functions" should be used instead of just "functions" because I think we also ha... |
H: A an nxn matrix. P a permutation matrix that permutes columns of A. How many operations does P*A involve?
Essentially, I am supposed to count how many operations a particular computational algorithm involves, and I've gotten stuck on this one part. My understanding is that for two nxn matrices, matrix multiplicatio... |
H: Proving the product of two real numbers is maximum when the numbers are equal given their sum is constant
Let us consider two real numbers $x$ and $y$. How can we prove value of $xy$ is greatest when $x=y$ given the condition $x+y=$ constant?
I have already found a proof, but I am not entirely happy with it yet.
AI... |
H: Cyclic vectors in a real vector space
Let $V$ be an n-dimensional vector space over $\mathbb{R}$ and $T:V \rightarrow V$ be linear. Call a vector $v \in V$ cyclic if $V$ is spanned by $\{v, \ Tv, \ T^2v,..\}$.
Question:
Show that $v$ is cyclic for $T$ then there is some $k \geq 1$ such that
$$B=\{v, \ Tv,..T^{k-1}... |
H: finite length is stronger than finiteness of a module
Let $A$ be a commutative ring and $M$ an $A$-module.
I realized recently that the property of $M$ having finite length is stronger than $M$ being finitely generated.
Here is my reasoning: Suppose $M$ has finite length but it is not finitely-generated. Let $\lef... |
H: If $(1,1)$ is an eigenvector of $A=\begin{pmatrix}2 &5\\3&k\end{pmatrix}$,then one of the eigenvalues of $A$ is :-
If $(1,1)$ is an eigenvector of $A=\begin{pmatrix}2 &5\\3&k\end{pmatrix}$,then one of the eigenvalues of $A$ is :-
$0,-1,1,2$
Can I get some hints please.
AI: $(1,1)$ is an eigen vector.let $\lambda$ ... |
H: Prove that the tangent space of a hyperplane is itself
I know this might sound really stupid: I was trying to show that the tangent space of a hyperplane is itself.
I started by parametrising the hyperplane locally at $x$ with a diffeomorphism $\phi : U \rightarrow X$, where $U\subset R^n$ and $X\subset R^m$ is the... |
H: $\text{Inn}(G)$ cannot be nontrivial cyclic group
Let $G$ be any group, and let $Z$ be its center.
(a) Show that $G/Z\cong \text{Inn}(G)$.
(b) Conclude that $\text{Inn}(G)$ cannot be a nontrivial cyclic group.
I've already gotten part (a) by considering the mapping $\pi:G\rightarrow\text{Inn}(G)$ such that $\pi(g... |
H: Closure of a subspace with respect to a inner product
I just have a question in general. If we are trying to show that a subspace of a vector space is a closed subspace, I know we need to prove that all convergent sequences in that subspace converge to a limit in that set. But if the subspace is defined in terms of... |
H: Definition of a primitive polynomial
I understand there are already some questions (A, B) on primitive polynomials. But none of these clears my confusion.
In page 84 of Handbook of Applied Cryptography, primitive polynomial has been defined as,
Now, if I try to understand the definition by dissecting the parts,
... |
H: Characteristic polynomial, and a coprime polynomial
I'm a bit stuck on this following question in algebra, I hope someone can show me to the right direction.
Let $A\in M_n(F)$ be a matrix and let $f\in F[X]$ be a polyonimal such that $\gcd(f,P_a)=1$, with $P_a$ being the characteristic polynomial of $A$. Show that... |
H: Given $H\triangleleft G, K\le G\ $, $\;K\nsubseteq H,\;$ and $(G:H) = p$, $\;p$ prime, prove $HK=G$
Suppose that $H\triangleleft G, K\le G\ $ and $K\nsubseteq H$. How we can prove that $HK=G?$
Also $(G:H)=p$ where p is prime.
AI: With your modification, I give you a hint. By the index formula
$$p=(G:H)=(G:HK)(HK:H... |
H: elements of sets, subsets relations
give an example (if possible) such that:
a)$x\in y$ and $x\not\subseteq y$
here I take $x=\{1,2\}$ and $y=\mathcal{P(x)}=\{\emptyset, \{1\},\{2\},\{1,2\}\}$ then $x\in y$ but $x\not\subseteq y$ as e.q $1\not\in y$
b)$x\subseteq y$ and $x\not\in y$ here I cannot find any counter e... |
H: $G/Z$ cannot be isomorphic to quaternion group
Let $Q_8$ be the quaternion group. Show that $G/Z$ can never be isomorphic to $Q_8$, where $Z$ is the center of $G$.
Hint: if $G/Z\cong Q_8$, show that $G$ has two abelian subgroups of index $2$.
I'm trying to prove the hint by looking at the canonical homomorphism $... |
H: Prove $\log x!$ is $\Omega (xlogx)$
Find a positive real number $C$ and a nonnegative real number $x_o$ such that
$Cx$$\log x$ $\leq$ $\log x!$ for all real numbers $x > x_o$.
I tried to expand $\log x!$ into $\log 1 + \log2 +\log3 +....\log x$. But how do I choose $C$ and $x_o$ so the above inequality hold?
Any hi... |
H: Closure of operators
Let $X$ and $Y$ Banach. We say that the linear operator $A:\mathcal{D}(A)\subseteq X\rightarrow Y$ admits a closure if there's a linear operator $B:\mathcal{D}(B)\subseteq X\rightarrow Y$ such that $\mathcal{D}(A)\subseteq \mathcal{D}(B)$, $B|_{\mathcal{D}(A)} = A$
and $\mathcal{G}(B)= \overlin... |
H: integrating a vector over a sphere
I have the following triple integral in spherical coordinates $(r,\theta,\phi)$: $$\int_0^{2\pi}\int_0^\pi\int_0^RCr^3\hat\theta\cdot r^2dr\sin{\theta}d\theta d\phi$$
How do I handle the $\hat\theta$? If I ignore it, I get $\frac{2}{3}\pi CR^6$. So is my answer the vector $\frac{2... |
H: Integral over a ball
Let $a=(1,2)\in\mathbb{R}^{2}$ and $B(a,3)$ denote a ball in $\mathbb{R}^{2}$ centered at $a$ and of radius equal to $3$.
Evaluate the following integral:
$$\int_{B(a,3)}y^{3}-3x^{2}y \ dx dy$$
Should I use polar coordinates? Or is there any tricky solution to this?
AI: Note that $\Delta(y^3-3x... |
H: Cardinality summation
How do I prove that, for set $X$,
$$\sum_{S\subseteq X, S\neq \emptyset}\frac{(-1)^{|S|}}{|X|+|S|} = \frac{|X|!(|X|-1)!}{(2|X|)!}$$
I have been around this exercise all day and would much appreciate your help.
AI: Let us denote $n = |X|$. Then
\begin{align*}
\sum_{S\subset X} \frac{(-1)^{|S|}}... |
H: Is the following polynomial reducible over Q?
I am looking at an exercise saying that "Demonstrate that x4-22x2+1 is reducible over Q. I have the solution manual and it solves like the following:
If x4-22x2+1 is reducible over Z, then it factors in Z[x], and must therefore
either have a linear factor in Z[x] or fac... |
H: If $I\pi ^{-k}\subseteq D$ for all $k\leq m$ but $I\pi ^{-m-1}\nsubseteq D$, then $I\pi ^{-m}\nsubseteq (\pi )$
I'm working through a proof and I've got stuck on one detail, it seems like it is supposed to be totally obvious, but need help to figure out why.
Let $D$ be a Noetherian integrally closed domain with u... |
H: name of theorem for infinite order polynomial limit for small x
is there a name for this property?
$\mathop{\operatorname{lim}}_{x\to 0} \sum_{n=1}^\infty x^n = \frac{x}{1-x}$
I have seen it in a derivation but want to know where it comes from.
AI: Firstly, the formula as written is false. On the left hand side yo... |
H: Delta System Lemma: Kunen’s proof.
I'm trying to understand Kunen's Delta System Lemma proof (Set Theory: Introduction to Independence Proofs, Chapter II, Theorem 1.6).
I'm heaving issues on understanding the last line:
"Since $|\alpha_0^{ < k}|<\theta$ there is an $r \subset \alpha_0$ and $B\subset A_2$ with $|B|=... |
H: Question about improper integral
Does $\forall \epsilon \in (0,a]: \int _{\epsilon}^{a} f(x)dx =0$ imply that $\lim_{h \rightarrow 0} \int _h^a f(x)dx=0$. I guess it is somehow the definition of it, but I need to know this exactly.
AI: It's even stronger. The limit only says that the integral goes to zero when $h$ ... |
H: Morse Function Definition: Does it implies Morse function is $C^2$?
In my understanding, Morse function just means the determinant of Hessian matrix is nonsingular at critical points.
So my claims are:
the function itself should be continuous
the reference to Hessian matrix in the definition implies Morse function... |
H: $|\phi(G):\phi(H)|$ divides $|G:H|$
Let $\phi$ be a homomorphism defined on a finite group $G$, and let $H\subseteq G$. Show that $|\phi(G):\phi(H)|$ divides $|G:H|$.
Not quite sure where to start on this one. We have a theorem saying that $\phi(H)$ is a subgroup of $\phi(G)$, but what is $|\phi(H)|$? If $H$ cont... |
H: Definition of localization of rings
I'm trying to understand this definition of Hungerford's book:
The definition is simple, I think I understood what the author means, but...
What is $P_P$? because we will have $P_P=S^{-1}P$, with $S=P-P=\varnothing$.
What is $S^{^-1}P$, with $S=\varnothing$?
I'm sure it should ... |
H: What is the range of the following function?
I have difficulties in understanding the concept of range. Let $f:\mathbb Z_{12}\to \mathbb Z_3$, $f(x)=x$. What is the range of it? Here is what i think: Range of $f$ is the set $\{a \mid a\equiv x \bmod 3 \text{ and } x=0,1,2\}$. Is that right?
Thanks
AI: The range is ... |
H: Finding all Laurent expansions of $f=\frac{1}{z}$
I have the following homework problem I need help with:
Let $G=\{z\in\mathbb{C}:\, z\neq0\}$ and define $f:\, G\to G$ by
$f(z)=\frac{1}{z}$.
Find all possible Laurent expansions of $f$ which are not Taylor
expansions and for each such expansion specify in what ... |
H: Conditional Probability Example using permutations
You are dealt three cards. The events of interest concern the number of face cards that you are dealt (0, 1, 2, or 3). What is the conditional probability that you are dealt at least 2 face cards given that the last card dealt to you was a face card?
There 12 f... |
H: Proving that sequence $a_n = \sqrt{x \sqrt{x \sqrt{x \sqrt{\cdots}}}} = x^{1-2^{-n}}$
Let $x>0$.
For sequence $a_n$, such that $n$ denotes the $n$th term:
$$\begin{align} a_1&= \sqrt{x}\\
a_2&= \sqrt{x \sqrt{x}}\\
a_3&= \sqrt{x \sqrt{x \sqrt{x}}}\\
a_4&= \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}}\\
&\vdots\\
a_{n-1}&= \s... |
H: Finding the values of $z$ s.t. $\sum_{n=0}^{\infty} \left( \frac{z}{1+z} \right)^n$ is convergent
I manipulated the series to $\sum_{n=0}^{\infty} \left( \frac{1}{1-(-1/z)} \right)^n$, which converges for $|-1/z|<1$ by geometric series. Then solving for $z$, I obtained $z>(1/\bar{z})$.
Is this correct? I was expect... |
H: Differentiating $ f(x)= \frac{x + \sin x}{x - \cos x}$
Can someone help me? I'm having some trouble with this:
how can I differentiate $$ f(x)= \frac{x + \sin x}{x - \cos x} \quad ?$$
P.S. Is there any trick or something to derive this kind of limit?
Thanks!
AI: To find $f'(x)$, use the quotient rule:
$$f'(x) = \... |
H: Extracting the coefficient from a generating function
Consider (for fixed $r$) the following function:
$$f(z) = \frac{1}{1-z-z^2-\cdots-z^r} = \frac{1}{1-z\frac{1-z^r}{1-z}}=\sum_{j=0}^\infty\left(z\frac{1-z^r}{1-z}\right)^j$$
(Assume everything is ok with regards to convergence.)
The text I am reading claims that ... |
H: To which group is the $\mathbb Z_{20}^*$ isomorphic?
I have a question saying that to which group is $\mathbb{Z}_{20}^{*}$ is isomorphic, where $\mathbb{Z}_{20}^{*}$ is the set of the not zero divisors of $\mathbb{Z}_{20}$.
Here is what i think: $\mathbb{Z}_{20}^{*}=\{1,3,7,9,11,13,17,19\}$. It has 8 elements. Then... |
H: What are some examples of proof by contrapositive?
Applying the Modus Tollens argument to Fermat's Little Theorem really helped me to understand logical implication. I never knew that FLT was actually a compositality test.
Theorem (FLT): given integers $a>1$ and $n>1$, if $n$ is prime, then $a^n$ is congruent to $a... |
H: Random variable with density proportional to a function and finite in some points
Let $X$ be a random variable on $[-1,3]$ with density $f(x) = k x^2$ (with $k \in \mathbb{R} $ to be determined) on $[-1,3]$ apart from some points s.t. $p(X=-1) = p(X=3) = \dfrac{1}{4} $ and $p(X=0) = \dfrac{1}{3}$.
What is the cumu... |
H: Continuity of a function defined differently on $\mathbb Q,\mathbb R\setminus \mathbb Q$
Let's say I define the function $f(x)=2^x$ for rational $x$, and $f(x)=1$ for irrational $x$.
My question is: is this function continuous everywhere? I think it's not, because for any $2$ irrational numbers you can find a ratio... |
H: Differentiating $ y= xe^{1\over x} $
Can someone please help me? I'm trying, but I really can't find the second derivative of $y= xe^{1/x}$. Thanks!
AI: By the product rule:
$$y'=x'e^{1/x}+x\left(e^{1/x}\right)'=e^{1/x}-\frac{1}{x}e^{1/x}$$
I used: $y=e^{f(x)}$, then $y'=f'(x)e^{f(x)}$. Then:
$$y''=-\frac{1}{x^2}e^... |
H: Monic polynomial with certain properties is the characteristic polynomial?
I'm trying to determine whether this statement is true or false:
Assume that $dim V = n$ and let $T \in L(V)$. Let $f(z) \in P_n(F)$ be a
monic polynomial of degree
$n$ such that $f(T) = 0$. Then $f(z)$ is the characteristic polynomial of $T... |
H: How to prove statistical hypothesis?
I developed a caching method. I took 100 experiments and got that hit ratio is not less than 75%. Now, I want to prove that my method with some probability gives hit ratio not less than 75%. How should I make this?
AI: You can calculate the chance of a type I or type II error in... |
H: Square root of $8^3$
I'm only in intermediate algebra. I know that $\sqrt{8^3}$ is equal to $16\sqrt{2}$ but could you simply explain the process on how to get to that?
AI: Using the standard rules of algebra, we compute:
$$\sqrt{8^{3}} = \sqrt{8^2 \cdot8} = \sqrt{8^2}\cdot\sqrt{8} = 8\cdot\sqrt{4\cdot2} = 8\cdot\s... |
H: Determinant Expression
How I will be able to found any expression for the determinant of the matrix $R^{N\times N}$ wiht entries belong $\mathbb{R} $, if $R_{ij}=\dfrac{2}{N}-\delta_{ij}$?
AI: I assume that $N$ is the size of the matrix, if it is not, the solution can be adapted easily.
Your matrix is
$$R= \begin{... |
H: $X$ random uniform variable, what is the cumulative distribution of $|X|$?
Let $X$ be a random uniform distribution on $[-2,1]$. What is the cumulative distribution of $|X|$, i.e. $G(x) = p(|X| \leq x)$ ?
The cumulative distribution of $X$ is $F(x) = \dfrac{x+2}{3} $.
$$G(x) = p(|X| \leq x) = p (-x \leq X \leq x) =... |
H: Continuity of Infimum function on $\mathbb{R}^n$
Suppose $F$ is a non-empty closed subset of $\mathbb{R}^n$ and let $f(x)= \inf_{y \in F}|x-y|$, where $|x-y|$ is the usual Euclidean distance in $\mathbb{R}^n$.
Prove that $f$ is continuous and that $f(x)=0$ iff $x \in F$.
Thoughts:
We want to show that given some $ ... |
H: Solving summation $2n+2^2(n-1)+2^3(n-2)+....+2^n$
Can anyone help me with this summation? I tried to use the geometric series on this, but I can't use that.
$$2n+2^2(n-1)+2^3(n-2)+....+2^n$$
I am trying to do this for studying algorithms. Can we get a closed form for this ?
AI: All you have to find out is how to su... |
H: Doubt in $\varepsilon$-$\delta$ proof of continuity of $x^2$
I have one elementary doubt on the proof that $f(x)=x^2$ is continuous for every $a \in \Bbb R$. The initial steps usually presented are: to deduce which $\delta$ will work we write:
$$|x^2-a^2|=|x-a||x+a|$$
Now, we must find some bound for $|x+a|$, so we... |
H: Prove that this function define a norm on $\mathbb{R}^2$.
Let $\| \;\|:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that
$ \| (x,y)\|=\sqrt{|x|^2+|y|^2}$ for all $(x,y)\in\mathbb{R}^2$.
I need to show that $\|\;\|$ defines a norm. I would like a hint in order to prove that $\parallel \; \parallel$ satisfi... |
H: What's the probability of a set of only three digits appearing in a random 9 digit set?
I'd like to know the method for correctly calculating the probability of a random sequence of $9$ numbers only containing $3$ unique, different numbers. For the purpose of this question: there are 10 numbers $0,1,2,3,4,5,6,7,8,9... |
H: Giving an exact description for $E_{R}:=\{\cos(z):\, R<|z|<\infty\}$
I have the following homework problem:
For each $R>0$ prove $$A(0,0,\infty)=\{e^{z}:\, z\in A(0,R,\infty)\}$$
and give an description of the set $$E_{R}:=\{\cos(z):\, z\in
A(0,R,\infty)\}$$
Where $$A(z_{0},r_{0},r_{1}):=\{z\in\mathbb{C}:\, r... |
H: Is There A Function Of Constant Area?
If I take a point $(x,y)$ and multiply the coordinates $x\times y$ to find the area $(A)$ defined by the rectangle formed with the axes, then is there a function $f(x)$ so that $xy = A$, regardless of what value of $x$ is chosen?
AI: The curve you are after is the rectangular h... |
H: If $f$ is a Morse function, then so is $f \circ \phi^{-1}$, where $\phi: U \rightarrow \mathbb{R}^k$ is the coordinate chart.
I am trying to show: if when $f^\prime = 0$, then $f^{\prime\prime} \neq 0 \Leftrightarrow (f \circ \phi^{-1})^\prime = 0$, $(f \circ \phi^{-1})^{\prime\prime} \neq 0$.
But the problem is, b... |
H: The area between two curves
Hi there's one problem on my study guide that my teacher didn't go over and I don't know how to approach/solve it. Here's the problem:
Find the area between the curves on the given interval. Draw a graph of the functions and the region.
$$y=x^4 , y=x-1, -2 \le x \le 0.$$
AI: Hint: determ... |
H: What is the contragredient representation?
Let $V=M_2(\Bbb C)$ be the set of all $2$x$2$-matrices. Let $G=B$x$B$ where $B$ is the group of $2$x$2$ lower triangular matrices with non-zero diagonal entries. Then G acts on $V$ by $\rho (g,h)x=gx^th$ for $x \in V$ and $(g,h) \in G$. What is the representation $\rho^*$ ... |
H: Differentiate $g(t)= {e^t - e^{-t} \over e^t + e^{-t}}$
I'm having some trouble trying to differentiate the function $g(t)= \dfrac{e^t - e^{-t}}{e^t + e^{-t}}$
Can someone help me? Thanks a lot!
AI: $g(t) = \dfrac{a(t)}{b(t)}$
$g'(t)= \dfrac{a'(t)b(t)-a(t)b'(t)}{b^2(t)}$
$g'(t)= \dfrac{(e^t+e^{-t})(e^t+e^{-t})-(e^t... |
H: Find the equation of the sine function graphed below.
Find the equation of the sine function graphed below.
Write a cosine function for the graph below. Assume the least possible phase shift.
AI: Hint: Your answer should be of the form $f(x)=A \sin (kx+\phi)$. What is the amplitude of a sine wave before you mult... |
H: Continuity and limits. Please check epsilon delta
Suppose $f$ is continuous at $a$ and $f(a) = 0$. Prove that if $\alpha \neq 0 $, then $f+\alpha$ is nonzero in some open interval containing $a$.
Since $f$ is continuous, we take $\epsilon = |\alpha|$; then we have $|x - a| < \delta \implies -|\alpha| < f(x) < |\a... |
H: What are some examples of functions that are continous from $[a,b]$, differentiable $(a,b)$ but not at $a$ and $b$.
I am studying the MVT and Rolle's Theorem. I would like some examples of functions that are continuous from $[a,b]$, differentiable from $(a,b)$ but not differentiable at $a$ and $b$. I am aware that ... |
H: A theorem in Linear Algebra; linear dependence - Axler
I really am having trouble understanding the statement and the proof. Why does the theorem pick $v_1 \neq 0 $? Why not $v_2$? Also in proving (a), why do we consider the largest $j$?
I do not understand the statement Note all of $a_2, a_3, \dots, a_m$ can be... |
H: Other log solutions?
I am evaluating the expression:
$\ln(1)$
And I know the trivial solution is $0$.
Are there other solutions to this equation? I feel there should be, my logic is as follows:
if:
$\ln(1) = x \implies 1 = e^x \implies 1 = 1 + x + x^2/2! + x^3/3!... $
$\qquad\qquad\qquad\qquad\qquad\implies 0 = x ... |
H: Integrability of a function (Darboux)
Question:
Let $f: [a,b] \to \mathbb{R}$ and assume $0 \leq f(x) \leq B$ for $x\in [a,b]$. Show that
$$U(f^2,P) -L(f^2,P) \leq 2B\ ( U(f,P) - L(f,P) )$$
for all partitions $P$ of $[a,b]$.
Show that if $f$ is integrable on $[a,b]$, then so is $f^2$ [no positivity as... |
H: Given $x,y\in\mathbb{C}^n$ s.t. $f(x,y)=\sup_{\theta,\phi}\{\|e^{i\theta}x-e^{i\phi}y\|^2,\theta,\phi\in\mathbb{R}\}$
Given$$x,y\in\mathbb{C}^n,\quad f(x,y)=\sup_{\theta,\phi}\{\|e^{i\theta}x-e^{i\phi}y\|^2,\theta,\phi\in\mathbb{R}\}$$
Then which is/are the following are true?
$1.\ f(x,y)\le \|x\|^2+\|y\|^2-2Re|\la... |
H: Injectivity of $A-\lambda I$
I'm reading a paper on determinants and on one point the author states that:
A complex number $\lambda$ is called an eigenvalue of matrix $A$ if $A-\lambda I$ is not injective.
Why is this? Could someone clarify :)
Thank you! =)
AI: Let's view $A$ as a linear operator from vector spaces... |
H: $f(1/n)=\frac{2n}{3n+1}$, Then
Given that $f:D\rightarrow \mathbb{C}$ is analytic, $D=\{z:|z|<1\}$, analytic at $0$ and satisfies $f(1/n)=\dfrac{2n}{3n+1}$, Then
$f(0)=2/3$
$f$ has a simple pole at $z=-3$
$f(3)=1/3$
No such $f$ exists.
considered $g(z)=f(z)-\dfrac{2}{3+z}$ and zero set of $g$ is $\{\dfrac{1}{n}\}$... |
H: evaluating $\int_0^{\infty}\frac{e^{-t-\frac{x}{t}}}{t} dt$
I got to this integral, while proving some theorem in statistics:
$$\int_0^\infty \frac{e^{-t-\frac{x}{t}}}{t} \mathop{dt}$$
I have trouble evaluating it. I tried partial integration, tried substitution with some polynomial and some trigonometric functions... |
H: Proving $\prod _{k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_j}\!\!,\;\;1\le j\!<\!n$
Let $p_n$ denote the $n$th prime number.
How could one prove that:
$$\prod \limits_ {k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_j}\!\!,\;\;1\le j\!<\!n$$
Examples:
$n=3812,\;j=81\qquad\implies\quad\large{\prod \limits _{k=81... |
H: A Calculus Question on onto functions with a specified range.
The following question was from a mock test of a competitive exam.
Suppose $f:\mathbb{R} \to [-8,8]$ is an onto function and $f(x) = \dfrac{bx}{(a-3)x^3 + x^2 + 4}$ where $a,b \in \mathbb{R}^+$. If the set of all values of $m$ for which the equation $f... |
H: Unitary diagonalization and eigenspace dimensions
I was trying to diagonalize the matrix:
$\left(\begin{array}{ccc}
0 & 0 & i\\
0 & i & 0\\
i & 0 & 0
\end{array}\right)$
I got two eigenvalues, $\lambda_{1}=i$ and $\lambda_{2}=-i$, and found the eigenspaces:
$V_{\lambda_{1}=i}=\mathrm{span}\left\{ \left(\begin{ar... |
H: maximal antichain
I don't understand the definition of Jech (set theory) for "maximal antichain". Let $B$ a boolean algebra and $A$ a subalgebra of $B$.
$W\subseteq A^+$ is a maximal antichain if $\sum W=1$ and $W$ antichain. As $A^+\subseteq B^+$ and $1\in A^+\cap B^+$, $W$ is also a maximal antichain in $B$ (is ... |
H: Eigenvalues of outer product matrix of two N-dimensional vectors
I have a vector $\textbf{a}=(a_1, a_2, ....)$, and the outer product $M_{ij}=a_i a_j$. What are the eigenvalues of this matrix? and what can you say about the co-ordinate system in which $M$ is diagonal? I have proved that the only eigenvalue of the m... |
H: Why is $ \|T(x) \|\le \| T\| \|x \|$?
Let $X$ and $Y$ be normed linear spaces and let $T$ be a bounded linear operator from $X$ to $Y$. The norm of $T$ is defined as
$$\|T \|=\sup\{\|T(x) \|\;:\;\|x \|\le 1\}. $$
From the definition of the norm, we can say that $\|T(x) \|\le \|T \|\;\forall \;x,$ and that $\|T \|\|... |
H: Hodge Star Operator and definition of divergence operator on riemannian manifold
From my lecture notes, on a general Riemannian manifold $(M, g)$, the divergence operator $\operatorname{div}: T^\infty(TM)\rightarrow C^\infty(M)$ is defined as $\operatorname{div}X := \star^{-1}d(x\neg\operatorname{Vol})$ , where $\s... |
H: Prove $a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1$ without use of log properties
$a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1 $
Hint $u_{2n}$ = $u_{n}^2$
I have totally no idea how to prove this, this looks obvious but i found out proof is really hard...
I am do... |
H: Minimizing an expression (over the integers)
In the context of Hurwitz groups and manifolds, one comes by what wikipedia defines as a "remarkable" fact, that $1-\dfrac1 a -\dfrac 1 b - \dfrac1 c > 0$ has a minimal value of $1/42$ if $a,b,c \in \mathbb{Z}$ and $a<b<c$ (I'm not sure this last part is needed, but let'... |
H: Derivative of order 16 - is there a method to do so?
I have the following exercise:
Find the $16^{\text{th}}$ derivative of $y$, (i.e. $y^{(16)}$), for $y = \sin x$.
Is there any method to do so, or I simply have to differentiate the function $16$ times?
AI: Hint:
$y'=\cos x$
$y''= -\sin x=-y$
$y'''=- \cos x=-y'$... |
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