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H: complex analysis -bounded but not holomorphic function
What are some examples of functions that are bounded but not holomorphic?
I know that some the not holomorphic functions are
Absolute value function, Piecewise functions (not analytic where the pieces meet)
AI: Well, you can take any bounded discontinuous funct... |
H: Number of combinations
You are given K prime numbers, bigger than 6, find the number of different number that can be made of those prime numbers(using 1 number, 2 numbers ..., k numbers).
Obviously you need to get the number of combiantions that you'll get using only 1 digit, 2 digits and so on and at the end add t... |
H: Simple probability with marbles
I have completely forgotten my statistics knowledge, and I have now found myself in dire need of this without having access to my books.
While I'm pursuing another context, I would like to ask my questions using marbles. Say I've got $100$ marbles of two colors: $6$ black marbles and... |
H: Does $x^T(M+M^T)x \geq 0 \implies x^TMx \geq 0$ hold in only one direction?
I know this is true for the "if" part, but what about the "only if"?
Can you give me one example when the "only if" part does not hold? I am not quite sure about this.
I forgot to tell you that $M$ is real and $x$ arbitrary.
AI: $M=\frac{M+... |
H: Proof of the discrete Doob inequality
$\mathbf{Theorem:}$ Let $n \in \mathbb{N}, a,b, \in \mathbb{R}, a < b$ and $S_i$ a submartingale for $i=1,2,\dots,n$. Define $$H_n(a,b;S_1,\dots,S_n) = \text{card} \Bigg( \{ (i,j) \in \mathbb{N}^2: 1 \le i <j \le n, S_i \le a, S_j \ge b, a<S_k<b, \forall k = i+1, i+2,\dots, j-1... |
H: easy homework: Equivalence classes, how do they look?
Let's say that I got a
set = { Arnold, Harrison }
and I want to display the equivalence class of [ Harrison ]
The actual condition for the relation doesn't matter in this case so let's just say that {Arnold} is the only relation to Harrison.
This would be di... |
H: Riemann or Lebesgue integrable
Lets consider the following two functions:
$$f(x)=\begin{cases} x &,x\in[0,1]\setminus\Bbb Q \\ 0 &,x\in[0,1]\cap\Bbb Q\end{cases}$$
$$g(x)=\frac{(-1)^{[x]}}{[x]}$$
where $[x]$ is the integer part of $x$. I am trying to determine which of these integrals are Riemann or/and Lebesgue in... |
H: How can partial derivatives feature in the definition of a function?
I have a map $f(t,g,h)$ where $f:[0,1]\times C^1 \times C^1 \to \mathbb{R}.$
I want to define $$F(t,g,h) = \frac{d}{dt}f(t,g,h)$$
where $g$ and $h$ have no $t$-dependence in them. So $g(x) = t^2x$ would not be admissible if you want to calculate w... |
H: Why does the (topology given by) Hausdorff metric depend only on the topology?
If I have a compact metric space $(X,d)$, I can define the Hausdorff metric on the set $K(X)$ of all non-empty compact (equivalently, closed) subsets of $X$ as $$d_H(A,B) = \max ( \sup_{x \in A} \inf_{y \in B} d(x,y), \sup_{y \in A} \in... |
H: Properties of determinants.
Is this property of a determinant true?
$$|A^3| = |A|^3.$$
I haven't studied about this but while working out on a sum, wondered if this could be true, I'll check out on other sums too if this works.
AI: Hint: use $\det(AB)=\det(A)\det(B)$. |
H: Extending a function beyond the completion/closure of its domain
In analysis there are certain theorems that tell under which conditions you can continuously extend a continuous functions to the closure/completion of its domain (which actually give the same set, since when talking about completions we have to be (... |
H: Does $X\times Y$ have countable chain condition?
Let $X$ and $Y$ have countable chain condition. Does $X\times Y$ have countable chain condition?
Thanks for your help.
AI: Consistently not necessarily. A consistent counterexample, under, for example, the assumption $\mathbf{V} = \mathbf{L}$ or just $\diamondsuit$,... |
H: How to evaluate the following limit?
How to evaluate this limit?
$$\underset{n\to \infty }{\mathop{\lim }}\,\left( {{n}^{-2}}\sum\limits_{i=1}^{n}{\sum\limits_{j=1}^{{{n}^{2}}}{\frac{1}{\sqrt{{{n}^{2}}+ni+j}}}} \right)$$
Thanks!
AI: The idea here is to rearrange in terms of a Riemann sum that leads to a double in... |
H: Show $x+\frac{\lambda}x \geq 2\sqrt{\lambda}$ all $x,\lambda>0$
For $\lambda>0 $ and $x > 0$,
$$x+\frac{\lambda}x \geq 2\sqrt{\lambda}$$
I tried to let function $g(x) =$ the difference of them and then find $g'(x) = 0$. With the given $x$, I can get the min point in $g(x)$ and find out $g(x)$ is greater than z... |
H: Tails of Fourier Transformed family of functions
I am reading a thesis where on page 39, Definition 4, $\epsilon$-oscillatory is defined as a property for a family of functions $\{f_{\epsilon}\}_{0<\epsilon<1}$ in $L^2(\mathbb{R}^d)$ to have if $$\lim_{R\to\infty}\limsup_{\epsilon\to 0}\int_{\{\left|\xi\right|>R\}}... |
H: Unicity of solution of pde
Let the pde $$\dfrac{\partial^2 u}{\partial t^2} - \dfrac{\partial^2 u}{\partial x^2}=f(x)$$
The question is:
Find the limit condition such that this pde admit a unique solution in $[a,b] \times [0,T].$
For this, I suppose the existence of to sulutions $u_1$ and $u_2$ and we put $v = u_... |
H: How to calculate module $a\cdot7 \equiv1\pmod8$?
How should i calculate following module ?
$a\cdot7 \equiv1\pmod8$
is value of $a = 1$ ?
Thanks.
AI: For every natural number $n$ we have $(-1)*(n-1) \equiv (-1)^2 = 1 \bmod n$. So in your example $a \equiv -1 \equiv 7 \bmod 8$. |
H: Is $\gcd(a,bc)=\gcd(a,b)\gcd({a\over\gcd(a,b)}, c)$?
Is it true that $\gcd(a,bc)=\gcd(a,b)\gcd({a\over\gcd(a,b)}, c)$?
It is true in quite a few examples that I came up with, e.g.
$a = 18, b = 21, c = 33$
$\gcd(18,21)\gcd({18\over\gcd(18,21)}, 33) = 3 \gcd({18 \over 3}, 33) = 3 \times 3 = 9$
$\gcd(18,693) = 9$
He... |
H: Golden ratio rectangles
I'm designing a layout and I would like to use four golden ratio rectangles. The total width of the layout is 960px. How do I find the height (x)? Below is a diagram of the layout.
AI: Finding the required height amounts to expressing $a$ and $c$ in terms of the height $h = a+b$. I will do t... |
H: Product of two compact topological spaces is compact.
Proof:
I am following this. But, I feel, I am missing something.
Consider two compact spaces $X_1$ and $X_2$ and some cover $U$ of their product space. Consider an element $x\in X_1$. The sets $A_{x,y}$ within $U$ contain $(x,y)$ for each $y\in X_2$.
Now, defin... |
H: Is the adjoint operation WOT-WOT continuous?
This is a well-known property of the Hilbert-space adjoint operator that it is WOT continuous. Is a similar true for Banach spaces? That is, for a given Banach space $X$ is the operation
$\varphi\colon B(X)\to B(X^*)$, $\varphi(T)=T^*$
continuous with respect to WOT-top... |
H: Approximating continuous functions $S^n \to S^n$
I'm trying to check that every continuous function $f:S^n \to S^n$ can be approximated by differentiable ones. Well, by Stone-Weierstrass I can approximate the coordinate functions $f_i:S^n \to \Bbb R$ by differentiable ${\tilde f_i}:S^n \to \Bbb R$. The problem is, ... |
H: Is the composite of an uniformly continuous sequence of functions with a bounded continuous function again uniformly continuous?
Let $\{f_n\}$ be a sequence of functions $f_n: J\to \mathbb{R}$ that converges uniformly to $f:J\to \mathbb{R}$ where $J\subseteq \mathbb{R}$ is an interval.
It is clear that for a unifor... |
H: we need to know at $z=0$ ,$f(z)={e^z+1\over e^z-1}$ has a
we need to know at $z=0$ ,$f(z)={e^z+1\over e^z-1}$ has a
removable singularity
a pole
essential singularity
residue is $2$
for removable singularity we need $\lim_{z\to 0}zf(z)=0$ but here this is not the case so $1$ is false, clearly $2$ is true, also $... |
H: Show $\frac{x_1}{x_n} + \frac{x_2}{x_{n-1}} + \frac{x_3}{x_{n-2}} + \dots + \frac{x_n}{x_1} \geq n$
I was recently asked this question which stumped me.
How can you show $\dfrac{x_1}{x_n} + \dfrac{x_2}{x_{n-1}} + \dfrac{x_3}{x_{n-2}} + \dots + \dfrac{x_n}{x_1} \geq n$ for any positive reals $x_1, x_2, \dots, x_n$?
... |
H: Why isn't the derivative of $e^x$ equal to $xe^{(x-1)}$?
When we take a derivative of a function where the power rule applies, e.g. $x^3$, we multiply the function by the exponent and subtract the current exponent by one, receiving $3x^2$. Using this method, why is it that the derivative for $e^x$ equal to itself a... |
H: Uniform integrability of RV's
$\mathbf{Theorem}$: Let $Y \in \mathbb{L}_1$, then the RV $(\mathbb{E}[Y \mid \mathcal{F}], \mathcal{F} \subset \mathcal{A} \space \sigma\text{-algebra})$ are uniformly integrable.
$\mathbf{Proof}$: Choose $K \in (0,\infty)$:
\begin{align}
\mathbb{E}[\mathbb{E}[Y \mid \mathcal{F}] \ma... |
H: Evaluating $\int_{0}^{1} \frac{\ln^{n} x}{(1-x)^{m}} \, \mathrm dx$
On another site, someone asked about proving that
$$ \int_{0}^{1} \frac{\ln^{n}x}{(1-x)^{m}} \, dx = (-1)^{n+m-1} \frac{n!}{(m-1)!} \sum_{j=1}^{m-1} (-1)^{j} s (m-1,j) \zeta(n+1-j), \tag{1} $$
where $n, m \in \mathbb{N}$, $n \ge m$, $m \ge 2$, and ... |
H: Localization and a particular exact sequence
Look at this proposition:
Let $R$ be a commutative ring with unity, and let $f_1,\ldots,f_n\in R$
generate the unit ideal in $R$. Then the following sequence is exact:
$$0\longrightarrow R\xrightarrow{\alpha} \bigoplus_{i=1}^n
R_{f_i}\xrightarrow{\beta} \bigoplus_{i,j... |
H: Gram determinant
How to prove that
$$\sqrt{\Gamma(\vec{a},\vec{b},\vec{c})}=|(\vec{a},\vec{b},\vec{c})|,$$
where $\Gamma(\vec{a},\vec{b},\vec{c})= \left | \begin{array} {ccc} \vec{a} \cdot \vec{a} & \vec{b} \cdot \vec{a} & \vec{c} \cdot \vec{a} \\
\vec{a} \cdot \vec{b} & \vec{b} \cdot \vec{b} & \vec{c} \cdot \vec{b... |
H: Estimating $\hat{p}$
let $X\sim Bin(n,p)$ and $\hat{p} =\frac{X}{n}$
a) Find a constant c such that $E[c\hat{p}(1-\hat{p})]=p(1-p)$
My work:
$$
\begin{align}
cE[\hat{p}(1-\hat{p})] &=E[\frac{X}{n}]-E[\frac{X^2}{n^2}]\\
&= \frac{1}{n}E[X]-\frac{1}{n^2}E[X^2] \\
\end{align}
$$
And I continue with $E[X] = p$ and ... |
H: $x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?
Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]:
Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in
F$. Then $x^p - c$ is irreducible in $F[x]$ if and on... |
H: What is the distribution of primes modulo $n$?
Let $n\geq 2$ and let $k$ be "considerably larger" than $n$ (like some large multiple of $n$). Then for each $i$ such that $0<i<n$ and $\gcd(i,n)=1$ let's define
$$c_i=\left|\{p_j\;|\; p_j\equiv i \mod n,\;\mbox{where $p_j$ is the $j$-th prime, $1\leq j\leq k$}\}\right... |
H: Geometric Tangent Vectors - looking for and understanding of and what the point is.
The problem that I am having is that I am having quite a hard time understanding the ideas of Geometric Tangent vectors and why they are even needed - I mean one already has the usual "Tangent" of a curve, why more definitions?
We ... |
H: Differentiable function and polynomials: Proof of $\phi (x)=ce^x + q(x)$ unclear to me.
I came across a proof which I can't quite understand:
If $p:\mathbb{R}\to\mathbb{R}$ is a polynomial of degree $n$ and $\phi:\mathbb{R}\to\mathbb{R}$ is a differentiable function with $\phi'=\phi+p$, there exists a polynomial $q... |
H: Prove that $C^{0}$ and $\mathbb{R}$ have equal cardinality
How to Prove that $C^{0}$ and $\mathbb{R}$ have equal cardinality ?
$C^{0}$ denote the set of all Continuous function $\mathbb{R} \rightarrow \mathbb{R}$
AI: Hint: A continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ is completely determind by its valu... |
H: Is $K=\{f_\lambda(x)=e^{\lambda x}\mid \lambda \in [0,a], x\in [0,b]\}$ equcontinuous?
I am trying prove that the following set is equcontinuous.
$K=\{f_\lambda(x)=e^{\lambda x}\mid \lambda \in [0,a], x\in [0,b]\}$
I read two prove of equcontinuty in two theorem and twice of theme have the hypothesis of compactne... |
H: The integral over a subset is smaller?
In a previous question I had $A \subset \bigcup_{k=1}^\infty R_k$ where $R_k$ in $\Bbb{R}^n$ are rectangles I then proceeded to use the following inequality $\left|\int_A f\right| \le \left|\int_{\bigcup_{k=1}^\infty R_k} f \right|$ which I am not really certain of. Does anyon... |
H: Quotient of Cayley graph of the free group on two generators by a subgroup.
If $F=F(\{a,b\})$ is the free group on two generators $a$ and $b$ and $G$ is the subgroup $$G=\:\langle b^n a b^{-n}|\: n\in \mathbb{N}\rangle \leq F$$
I am trying to work out what the quotient graph $\Delta / G$ looks like, where $\Delta =... |
H: Isometry fixing two points of a geodesic line
Let $H$ be a hyperbolic space, and let $\Gamma \subset H$ be a geodesic line, i.e., the image of an isometry from $\mathbb{R}$ to $H$. If $f$ is an isometry of $H$ that fixes two distinct points of $\Gamma$, is it true that $f(\Gamma) = \Gamma$?
AI: The answer is yes. T... |
H: $H$ is a subgroup of $G$ and every coset of $H$ in $G$ is a subgroup of $G$.Then which of the following is true?
$H$ is a subgroup of $G$ and every coset of $H$ in $G$ is a subgroup of $G$.Then which of the following is true?
(A) $H=${$e$}
(B) $H=G$
(C) $G$ must have prime order.
(D) $H$ must have prime order.
AI: ... |
H: Lagrange Multiplier Problem - Distance from point With a Circle Constraint
Given a point $P:(x_0,y_0)$ in $\mathbb{R}^2$, and a constraint function
$$x^2+y^2=R^2$$where $R$ is the radius of the circle. The distance from $P$ to any point on the circle is to be minimized using the method of Lagrange Multiplier. The d... |
H: Does weak*-convergence imply convergence of the operator norms?
Let $\mathcal A$ be a unital C*-algebra with topological dual $\mathcal A^*$ and denote the unit ball as $B_1^*:=\{\phi \in \mathcal A^* : \vert\vert \phi \vert\vert_{sup}\leq 1\}$.
If $\phi_n \rightarrow \phi$ is a weak*-convergent sequence with $\ver... |
H: Let N1 and N2 are normal subgroups in the finite group G. Is it true that if N1≃N1 then G∖N1≃G∖N2.?
Let $N_1$ and $N_2$ are normal subgroups in the finite group $G$. Is it true that if $N_1 \simeq N_2$ then $G/ N_1 \simeq G/ N_2$?
AI: No, take $G=\mathbb{Z}_2\oplus\mathbb{Z}_4$, with $H=\langle (1,0)\rangle$ and $K... |
H: Prove $\lim_{n\to \infty}\frac{n}{n+1} = 1$ using epsilon delta
$\lim_{n\to \infty}\frac{n}{n+1} = 1$
Prove using epsilon delta.
AI: By your notation I believe you're talking about the sequence $(a_n)$ of elements of $\mathbb{R}$ defined by:
$$a_n =\frac{n}{n+1}$$
Now, limit for sequences has the following definiti... |
H: Finding radius of convergence for this power summation $\sum_{n=0}^\infty \left(\int_0^n \frac{\sin^2t}{\sqrt[3]{t^7+1}} dt\right) x^n$
I have been given this tough power summation that its' general $c_n$ has an integral.
I am asked to find the radius of convergence $R$
$$\sum_{n=0}^\infty \left(\int_0^n \frac{\sin... |
H: When computing the Taylor series of $(\cos x)^2$ how does the slide jump to concluding it is $1-(\sin x)^2$?
In the following slide it shows how the taylor series of $(\cos x)^2$ is computed:
On the first line they simply take the taylor series of cosx and write it out twice, which makes sense. However, the orde... |
H: Reduction modulo $p$ in number fields
For every prime number $p$, there exist a map
$$f:\mathbb{P}^n(\mathbb{Q})\to\mathbb{P}^n(\mathbb{F}_p)$$ defined
by: for $P\in \mathbb{P}^n(\mathbb{Q})$, we can find a unique tuple
$(x_1,\dots,x_n)\in\mathbb{Z}^n$ of coprime integers such that
$P=[x_1,\dots,x_n]$. The... |
H: Difficult Continuity Multivariable Question
WIll you help me understand the following?
$ f(x,y)=\begin{cases}
\sin(y-x) & \text{for} & y>|x| \\ \\
0 & \text{for} & y=|x| \\ \\
\frac{x-y}{\sqrt{x^2 + y^2}} & \text{for} & y<|x|
\end{cases}
$
I need to check differentiability and continuity.
I tried substituting... |
H: Logarithmic Equations
How does one go about solving:
$(5x+2)^{\frac{4}{3}} = 16$
I'm confused as how to parse through the equation to solve it using logs.
AI: \begin{align}
&(5x+2)^{\frac{4}{3}} = 16\\\implies
&5x+2 = (16)^{\frac{3}{4}}\\\implies
&5x+2 = (2^4)^{\frac{3}{4}}\\\implies
&5x+2 = 2^3\\\implies
&5x+2 = ... |
H: Explanation of where this trig identity comes from
I'm working on a problem but it's been a while since I last saw trig identities so I'd love some help or being pointed in the right direction.
Basically, I'd like to understand where this identity comes from;
$$\tan(2t) = \dfrac{2\tan(t)}{1 - \tan^2(t)}$$
Thanks fo... |
H: how do determine the distribution of outcomes for a given probability?
For a game I generate various block types given certain odds. Say, there's a $0.001$ chance for the karma block. If a typical game has $600$ blocks, what's the distribution of games that have $0$ karma blocks, $1$ karma blocks, $2$ karma blocks,... |
H: Why $s/(1-s) = 1$ at $s=1$ in bode plot?
Wolfram plot of $\frac{s}{1-s}$ is $\pm\infty$ at $s=1$. But, bode plot of $\frac{s}{1-s}$ results in $1$ at $s=1$. Obviously, this is wrong. Why?
AI: The x-axis in the Bode plot isn't $s$, but $\omega$. Remember that $s=j\omega$, so what you're seeing in the Bode plot is as... |
H: Modulo question with negative
$60-88 \equiv \,\,? \pmod 5$
$60-88 = -28$
Then what do I do?
Please tell me how to answer this question. Thanks.
AI: When you have a negative number, like $-28 \pmod 5$, all you really need to do keep adding the modulus, or an integer multiple of the modulus $m$, to the negative numb... |
H: How many epsilon numbers $<\omega_1$ are there?
An epsilon number is an ordinal $\epsilon$ such that $\epsilon=\omega^\epsilon.$ What is the cardinality of the set of all epsilon numbers less than $\omega_1$?
I'm asking this because of a proof I've just read that seems to presuppose that there are countably many s... |
H: The 'compactness cardinal' of a space
I'm looking for references (and a name!) for the following invariant of a topological space $X$:
The least (infinite) cardinal $\kappa$ such that any open cover of $X$ has a subcover of cardinality less than $\kappa$.
For compact spaces, for example, this cardinal is $\aleph_0$... |
H: Isometries of a hyperbolic quadratic form
I am reading an article that says "The group of isometries (of a hyperbolic space) of a hyperbolic quadratic form in two variables is isomorphic to the semi-direct product $\mathbb{R} \rtimes \mathbb{Z}/2\mathbb{Z}$".
Could someone help me to understand that fact? First, it... |
H: Proof that $1/\sqrt{x}$ is itself its sine and cosine transform
As far as I understand, I have to calculate integrals
$$\int_{0}^{\infty} \frac{1}{\sqrt{x}}\cos \omega x \operatorname{d}\!x$$
and
$$\int_{0}^{\infty} \frac{1}{\sqrt{x}}\sin \omega x \operatorname{d}\!x$$
Am I right? If yes, could you please help me ... |
H: Showing a simple property in statistics
I know this is very elementary but I cannot remember how to show $\hat\alpha$ as below.
AI: You minimize the squared error
$$\epsilon^T\epsilon=(y-X\alpha)^T(y-X\alpha)=y^Ty-2\alpha^TX^Ty+\alpha^TX^TX\alpha$$
This expression can be minimized by setting its derivative w.r.t. $... |
H: $n^5-n$ is divisible by $10$?
I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that the respective last digits match. Another approach I tried is this: I factored $n^5-n$... |
H: Integrate function $\int_{0}^{\infty}\lambda^{x}e^{-2\lambda}d\lambda$
How can I integrate this function? It's originated by an exponential prior and a poisson likelihood.
$\int_{0}^{\infty}\lambda^{x}e^{-2\lambda}d\lambda$
AI: This is related to the gamma function:
$$\Gamma(z) = \int_0^{\infty} dt \, t^{z-1} e^{-t... |
H: What makes $5$ and $6$ so special that taking powers doesn't change the last digit?
Why are $5$ and $6$ (and numbers ending with these respective last digits) the only (nonzero, non-one) numbers such that no matter what (integer) power you raise them to, the last digit stays the same? (by the way please avoid modul... |
H: If we restrict the Heaviside step function to $\mathbb{R}\setminus\{0\},$ does it suddenly become continuous?
The Heaviside step function is discontinuous, despite that its continuous at every point except $0$.
Supposing we restricted it to $\mathbb{R}\setminus\{0\},$ does it suddenly become continuous?
I think 'ye... |
H: How to convert a geometric series so that exponent matches index of sum?
I need to convert the following series into a form that works for the equation $$\frac{a}{1-r}$$ so that I can calculate its sum. But the relevant laws of exponents are eluding me right now.
$$\sum_{n=1}^{\infty}\left(\frac{4}{10}\right)^{3n-1... |
H: Prove that there is a unique $A\in\mathscr{P}(U)$ such that for every $B\in\mathscr{P}(U), A\cup B = B$
$U$ can be any set.
For the existence element of this proof, I have $A = \varnothing$
But it's for the uniqueness element of this proof where I am having trouble. So far I have:
$\forall(C\in\mathscr{P}(U))(\for... |
H: Determine whether this relation is reflexive, symmetric...
Determine whether this relation $R$ on the set of all integers is reflexive, symmetric, anti-symmetric and/or transitive where $x\,R\,y$ iff
$x = y + 1$ or $x = y-1$
It is not reflexive:
Let $x = 2$: $2\neq 2 + 1$ and $2 \neq 2 - 1$.
It is symmetric:
I... |
H: Proof of Real Number Property
In his introduction to Calculus, Apostol gives a foundation for the properties
of real numbers. After laying down the field and order axioms but before stating the least-upper-bound axiom, the author uses the set $T = \{x \ : \ 0 \leq x < 1\}$ as an example of a set which has no maxim... |
H: Properties of limits when dealing with functions and parentheses .
My calculus instructor recently mentioned some odd properties of limits that I don't recall ever seeing, and seem alien to me. He says that the following statements are allowed:
$$\lim_{n\to\infty}\sin\left(\frac{1}{n}\right)=\sin\left(\lim_{n\to\in... |
H: Why is the bridge index of the trefoil equal to 2?
It seems to me, all three 3 bridges are needed?
AI: The bridge index of a knot $K$ is the minimum number of local maxima (or equivalently, minimum number of local minima) of the height function on all possible knot diagrams of $K$. So from the standard picture of t... |
H: Show $z^4 + e^z = 0$ has a solution in $\{z \in \mathbb{C} : |z| \leq 2\}$
Show $z^4 + e^z = 0$ has a solution in $\{z \in \mathbb{C} : |z| \leq 2\}$.
I would like if in the proof the tools of algebraic topology were preferred over the other tools of analysis, complex analysis, algebra etc.
AI: The tools are the s... |
H: Master Theorem change of variables with root other than 2
I'm working on this:
$$T(n) = 12T(n^{1/3}) + \log(n)^{2}$$
Using change of variables, and substituting $m = \log n$, I get as far as:
$$S(m) = 12S(m/3) + m^{2}$$
I see how a square root would work but with a cube root I'm not sure that $\Theta(m \log m)$ mak... |
H: If $\lim_{x \to \infty} \frac{f'(x)}{x}=2$ does it follow that $\lim_{x \to \infty} \frac{f(x)}{x^2}=1$?
I need to show that the following statement is true or false. $$\displaystyle\lim_{x \to \infty} \frac{f'(x)}{x}=2 \Rightarrow \displaystyle\lim_{x \to \infty} \frac{f(x)}{x^2}=1$$
I considered $f(x)=x^2$, and i... |
H: Solving the equation $10^{-x} = 5^{2x}$ with logarithms
$$10^{-x} = 5^{2x}$$
I'm having trouble isolating $x$. I get both logs on one side and then I'm stuck because I have nothing to divide with on the other side, and I can't factor it.
Thanks
AI: If $10^{-x} = 5^{2x}$, then $-x\log(10) = 2x\log(5)$ thus either $x... |
H: Simplified form of $\left(6-\frac{2}{x}\right)\div\left(9-\frac{1}{x^2}\right)$.
Tried this one a couple of times but can't seem to figure it out.
I am trying to simplify the expression:
$$\left(6-\frac{2}{x}\right)\div\left(9-\frac{1}{x^2}\right)$$
So my attempt at this is:
$$=\bigg(\dfrac{6x}{x}-\dfrac{2}{x}\bigg... |
H: Definition of topological group via neighborhood base -- weird difference condition
It's been over 2 years since I've seriously done point-set topology, so my apologies if this is simple. I'm working out of Liu's Algebraic Geometry and Arithmetic Curves. My question concerns a definition on page 16.
Liu defines a ... |
H: Find triple integral over a tetrahedron constructed by 3 planes
The question is as follows:
Find triple integral of f(x,y,z) = xy + z by dxdydz
Over the tetrahedron D created by the following coordinates: $(0,0,0),
(1,0,0), (0,1,0)$ and $(0,0,1)$
My answer doesn't agree with the book's answer. I got $\frac{3}{2... |
H: Evaluating $\sum_{k>\frac{N}{2}}\frac {1}{N}\cdot \frac{N-k}{k}$
Assuming $N$ is even, how can I evaluate the following sum:
$$\sum_{k>\frac{N}{2}}\frac{\binom{N}{k}(k-1)!(N-k)!}{N!}\cdot\frac{N-k}{N}=
\sum_{k>\frac{N}{2}}\frac {1}{N}\cdot \frac{N-k}{k}$$
I really don't know how to do it...
Thanks!
(Not HW BTW)
AI:... |
H: Given an semi-ellipse inscribed about a square, how do I find the equation of the ellipse?
Given the following diagram:
Where:
W = (-1, 0)
X = (-1, 2)
Y = (1, 2)
Z = (1, 0)
How can I find M?
The ellipse can be assumed to be a semi-ellipse with one of the foci on $\bar{XY}$. I'm guessing that this means that on... |
H: Convergence of series and disk or convergence
I applied the Ratio Test and I got
$$x + 2y < 1$$
Shouldn't this give me a half plane? The answer says it is (D). The only reason why I think it could be (D) is because $y$ could be positive or negative?
AI: What you want is $|x+2y|<1$ or $x+2y=-1$: this last option g... |
H: What is the relationship between $(u\times v)\times w$ and $u\times(v\times w)$?
Given three vectors $u$, $v$, and $w$, $(u\times v)\times w\neq u\times(v\times w)$. This has been a stated fact in my recent class. But what is the ultimate relationship between them? I would presume that one is a scalar multiple of t... |
H: A question regarding exponential distribution
Charlie and Bella and their friends Mark and Leonard each have a toy. Each toy breaks at a time that is exponentially distributed with expectation $24$ hours. Assume the toys are independent of each other.
What is the cumulative distribution function of the time until ... |
H: Prove that $V$ is the direct sum of $W_1, W_2 ,\dots , W_k$ if and only if $\dim(V) = \sum_{i=1}^k \dim W_i$
Let $W_1,\dots,W_k$ be a subspace of a finite dimensional vector space $V$ such that the $\sum_{i=1}^k W_i = V$.
Prove: $V$ is the direct sum of $W_1, W_2 , \dots, W_k$ if and only $\dim(V) = \sum_{i=1}^k \... |
H: $\mathbb{Z}:= \mathbb{N} \cup (-\mathbb{N}) \cup \{0\}$, $a,b \in \mathbb{Z}$, when $a \le b$?
Let $\mathbb{Z}:= \mathbb{N} \cup (-\mathbb{N}) \cup \{0\}$, and $a,b \in \mathbb{Z}$, when $a \le b$?
Thanks in advance!!
AI: $$a\le b$$ if and only if $$b-a\in \mathbb{N}\cup \{0\}$$ |
H: Working backwards with Determinants.
The determinant of A is -2. Find $\det(3A^TA^3)$. You may leave your answer as a product of integers.
What I did was this:
$$\det(3A^TA^3)$$
$$\det(3A^T)\det(A^3)$$
$$3\det(A^T)(\det(A))^3$$
This is the part where I became confused;
$$3\det(A^T)(-2)^3 $$
That's as far as I got, ... |
H: Elementary Row Matrices
Let $A$ =
$$
\begin{align}
\begin{bmatrix}
-4 & 3\\
1 & 0
\end{bmatrix}
\end{align}
$$
Find $2 \times 2$ elementary matrices $E_1$,$E_2$,$E_3$ such that $A$ = $E_1 E_2 E_3$
I figured out the operations which need to be performed which are;
$E_1$ = $R_2 \leftrightarrow R_1$
$E_2$ = $R_2$ = $... |
H: Cramer's Rule, 2x2 Matrix
Solve the following system using Cramer's Rule.
$$2x + y = 1$$
$$x - 4y = 14$$
I haven't done Cramer's rule for 2x2 matrices, but I figured that the same rules applied as in a 3x3, here's what I did;
$$\det(A) = -8-1 = -9$$
$D_x$ =
$
\begin{align}
\begin{bmatrix}
4 & 1\\
14 & -4
\end{bmatr... |
H: How can I determine $\lim_{x\rightarrow 2} \frac{(x^3-5x^2+8x-4)}{x^4-5x-6}$?
This is the limit:
$$\lim_{x\to2}\frac{x^3-5x^2+8x-4}{x^4-5x-6}$$
Thank you.
AI: As I'm guessing you found out, the numerator and denominator both evaluate to $0$ at $x = 2$ $(\dagger)$. But the limit may nonetheless exist as $x$ approac... |
H: Is there a divergent sequence such that $(x_{k+1}-x_k)\rightarrow 0$?
Is there a divergent sequence such that $\lim_{k\rightarrow\infty}(x_{k+1}-x_k)=0$?
AI: Let $f:(0,\infty)\to\mathbb R$ be any function such that $\lim\limits_{x\to\infty}f(x)=\infty$ and $\lim\limits_{x\to\infty}f'(x)=0$. Then $(f(1),f(2),f(3),\... |
H: Question about circles.
Find the value of x. If necessary, round your answer to the nearest tenth. The figure is not drawn to scale.
AB = 19, BC = 10, and CD = 5
A)23
B)53
C)38
D)58
What theorem should I use? And how do I use that theorem?
AI: AC = AB + BC = 19 + 10 = 29
if AC*BC=CD*(CD+X), then 29*10 = 5*(5+x)
2... |
H: Continuous Function and Open Subsets in $\mathbb R$
Let $E$ be a subset in $\mathbb R$, $f$ a real-value function on $E$.
Prove that $f$ is continuous on $E\iff$ for every open subset $V$ of $\mathbb R$, $f^{-1}(V)$ in open relative to $E$.
My question is about the ($\Rightarrow$) direction only.
Let $f$ be a conti... |
H: $\sum_{j=1}^{n} f\left(\frac{j}{n}\right) \cdot \chi _{\left[\frac{j-1}{n},\frac{j}{n}\right)} \longrightarrow f$ converges uniformly
Let $f:[0,1]\longrightarrow \mathbb{R}$ be a continuous function
and $\displaystyle f_n=\sum_{j=1}^{n} f\left(\frac{j}{n}\right) \cdot \chi _{\left[\frac{j-1}{n},\frac{j}{n}\right)... |
H: $f$ can be extended iff $\partial f = 0$
If
$0\rightarrow{A'}\rightarrow{A}\rightarrow{A''}\rightarrow{0}$
is an exact sequence of modules, then there exists an exact secuence
$0\rightarrow{}Hom(A'',B)\rightarrow{}Hom(A,B)\rightarrow{}Hom(A',B)\xrightarrow \partial{Ext}^1(A'',B)\rightarrow ...$
Suppose $A'\subsete... |
H: Span of an empty list
Its a simple question . But still I thought of asking it coz I don't see the logic behind it. Recently I read a statement "the Span of empty list ( ) equals {0}" .What we know is an empty list has no elements in it,so how can we even talk of a span of it as span is a linear combination of all ... |
H: What is "exclusive neighborhoods"?
I am trying to read a proof but don't understand what does this mean. Thank you very much!
So here is the question I am trying to prove: "Show that for a Lefschetz map f on a compact manifold X, there can be only finitely many fixed points." And the proof is: "The graph of f is tran... |
H: Riemann zeta function at zero
Can the value of Riemann zeta function at 0, $\zeta(0)=-1/2$, be deduced from the identity $E(z)=E(1-z)$, where
$$E(z)=\pi^{-z/2}\Gamma(z/2)\zeta(z)?$$
AI: The functional equation under consideration yields:
$$
\zeta(s)=\frac{\pi^s}{\sqrt{\pi}}\frac{\Gamma\left(\frac{1-s}{2}\right)}{\G... |
H: If $X_n\to X$ a.e. it does not follow that $\mu_n(P)\to \mu(P)$
If $X_n\to X$ a.e. and $\mu_n$ and $\mu$ are the p.m.'s of $X_n$ and $X$, it does not follow that $\mu_n(P)\to \mu(P)$ even for all intervals $P$.
I am having trouble coming up with an example that illustrates this. Why does Egoroff's theorem not guara... |
H: Proving that ${n}\choose{k}$ $=$ ${n}\choose{n-k}$
I'm reading Lang's Undergraduate Analysis:
Let ${n}\choose{k}$ denote the binomial coefficient,
$${n\choose k}=\frac{n!}{k!(n-k)!}$$
where $n,k$ are integers $\geq0,0\leq k\leq n$, and $0!$ is defined to be $1$. Prove the following assertion:
$${n\choose k}={n\cho... |
H: A tricky logarithms problem?
$ \log_{4n} 40 \sqrt{3} \ = \ \log_{3n} 45$. Find $n^3$.
Any hints? Thanks!
AI: By elementary arithmetic operations (after / describing next action):
$$\log_{4n}40\sqrt{3}=\log_{3n}45\ \ \ \mbox{ / definition of logarithm}$$
$$(4n)^{\log_{3n}45}=40\sqrt{3}\ \ \ \mbox { / } 4=\frac{4}{3... |
H: LU decomposition steps
I've been looking at some LU Decomposition problems and I understand that making a matrix A reduced to the form A=LU , where L is a lower triangular matrix and U is a upper triangular matrix, however I am having trouble understanding the steps to get to these matrices. Could someone please ex... |
H: When does $e^{f(x)}$ have an antiderivative?
today I tried to integrate $x^x$ by applying a reverse chain rule which turned out to be false. I was told $\int e^{f(x)}\,dx$ can be done when $f(x)$ is linear. This made me wonder what conditions we can find so that $\int e^{f(x)}\,dx$ can be expressed in terms of elem... |
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