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H: Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)
I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate:
$$\lim_{x\to-3}\frac{x!}{(2x)!}=-120$$
Neither $x!$, nor $(2x)... |
H: Smallest genus example of a non planar curve
A curve is a smooth projective connected curve over an algebraically closed field.
Every curve of genus 2 is planar.
Also, every curve of genus 3 is planar.
But what about curves of genus 4?
What is the dimension of the subvariety defined by planar curves in the moduli s... |
H: A linearly independent, countable dense subset of $l^2(\mathbb{N})$
Possible Duplicate:
Does there exist a linear independent and dense subset?
I am looking for an example of a countable dense subset of the Hilbert space $l^2(\mathbb{N})$ consisting of linearly independent vectors
AI: Based on Davide's answer. ... |
H: How to find the identity element in $(\mathbb Z_{40}, \odot)$
Let $R = \mathbb Z_{40}$ and let $\odot$ be defined on $R$ as follows:
$$\begin{aligned} a \odot b = a + 25b-10 \end{aligned}$$
I need to check if this structure has an identity element, so:
$$\begin{aligned} a \odot \mathbb 1_{R} = a \end{aligned}$$
$$... |
H: linear algebra linear transformation eigenvector and eigenvalues
i would be very thankful if someone could help me on this question, i know how to do the first bit but the last two questions confuse me a little. thanks in advance
$M = \left( \begin{smallmatrix} 8&40&-30\\ 25&98&-75\\ 35&140&-107 \end{smallmatrix} \... |
H: Is there a way to define the "size" of an infinite set that takes into account "intuitive" differences between sets?
The usual way to define the "size" of an infinite set is through cardinality, so that e.g. the sets $\{1, 2, 3, 4, \ldots\}$ and $\{0, 1, 2, 3, 4, \ldots\}$ have the same cardinality. However, is thi... |
H: From Presheaf to Sheaf
In Hartshorne's Algebraic Geometry is written that "A sheaf is roughly speaking a presheaf whose sections (i.e. elements of $\mathcal{F}(U)$ for open subset $U$) are determined by local data". What does it means? What is the local data?
After this remark Robin Hartshorne gave a definition of ... |
H: formal expression of $\tilde{z}$ is the nearest from z
$h = distance(z, \tilde{z})$, where $\tilde{z}$ is the element that is nearest from $z$ (that is, distance(z, $\tilde{z}$) is smaller than distance(z, any_other_z)).
Is it possible to expression this formally, instead of saying "where $\tilde{z}$ is the element... |
H: proving : $a_{1}=a_{2}=\cdots=a_{n}$ when satisfied relation
Suppose $k_1,k_2,k_3,\ldots,k_n$ are non-negative integer numbers such that sum $k_1+k_2+\cdots+k_n$ is an odd number. Let $a_1,a_2,\ldots,a_n$ be arbitrary numbers satisfied:
$$\frac {|a_1-a_2|}{k_1}=\frac {|a_2-a_3|}{k_2}=\cdots=\frac {|a_{n-1}-a_n|}{k_... |
H: Truth Table for If P then Q
Possible Duplicate:
In classical logic, why is (p -> q) True if both p and q are False?
The Logic table for If P then Q is as follows:
P Q If P then Q
T T T
T F F
F T T
F F T
What I don't understand is, How can there be a truth table for this?
As far as ... |
H: The Dynamics of Contrapositive Proofs
The Wikipedia Link for contrapositive proofs states that proving if p then q is the same as proving if not q then not p. I don't completely follow why.
Is there any way to understand what's happening without involving logic tables ? If logic tables are inevitable, can anyone gi... |
H: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
I want to prove that if $p$ is a prime and $x,y \in \mathbb{Z}$, then $$(x+y)^p \equiv x^p+y^p \pmod{p}$$
So far I know that
$$(x+y)^p = \sum_{k=0}^{p} \dbinom{p}k x^{p-k} y^k$$
A part of the above equation is supposed to cancel, I t... |
H: Fourier Transforms and the Laplacian
I need your help in the following question:
"Use Fourier transforms to prove that the domain of $H_0 := - \bar{\Delta} $ on $ L^2 (\mathbb{R}^3)$ consists entirely of continous bounded functions. If $V$ is a non-negative potential which is not $L^2 $ when restricted to any non-e... |
H: Pullbacks of categories
Let $\mathfrak{Cat}$ be the 2-category of small categories, functors, and natural transformations. Consider the following diagram in $\mathfrak{Cat}$:
$$\mathbb{D} \stackrel{F}{\longrightarrow} \mathbb{C} \stackrel{G}{\longleftarrow} \mathbb{E}$$
There are several notions of pullback one cou... |
H: Generalized PNT in limit as numbers get large
If $\pi_k(n)$ is the cardinality of numbers with k prime factors (repetitions included) less than or equal n, the generalized Prime Number Theorem (GPNT) is:
$$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln n)^{k-1}}{(k-1)!}.$$
The qualitative appearance of the actual dist... |
H: How to show that a map is an isometry
I'm having a difficulty understanding how to go on proving a certain map is an isometry. It should be really basic and simple, but for some reason I can't understand how to do this..
The situation is this: I have 2 manifolds, $D,M$:
$D$ is the Poincare disk $\{x\in\mathbb{R}... |
H: Subgroup Test with Conditions
Let $G$ be a group and let $A$ be a non empty subset of $G$. Let $H$ be a set defined by
$$H = \{ x \in G \mid \text{For all }a \in A,\text{ we have }xa \in A\text{ and }x^{-1}a \in A \}$$
Show that $H$ is a subgroup of $G$.
AI: Note that $H$ contains $e$, since for all $a\in A$, $ea =... |
H: Does existence of anti-derivative imply integrability?
If $f$ has an anti-derivative in $[a,b]$ does it imply that $f$ is Riemann integrable in $[a,b]$?
AI: Take $f(x)=\begin{cases} x^2\sin (1/x^2), &x\ne 0, \\ 0, &x=0. \end{cases}\quad$ Then $g=f'$ exists everywhere but is unbounded over $[-1,1]$. $g$ thus has a p... |
H: Solving for unknown inside square root
Sorry if this is a very primitive question, but I really not sure if I am right about this kind of situations. Imagine the following equation where $a$ , $b$ and $c$ are known numbers and $x$ is the unknown variable:
$$a\sqrt{bx}=c$$
Is it ok in this case to do it like
$$a^2bx... |
H: Fastest way to compare fractions
Which is the fastest method to compare the below fractions with minimum calculation possible and finding which is greatest and which the smallest??
$$\frac{26}{686},\quad \frac{48}{874},\quad \frac{80}{892},\quad \frac{27}{865}$$
AI: The denominators of the last three fractions are ... |
H: A consequence of Runge's theorem
I'd like to have a reference for the proof of the following fact of complex analysis. I think it follows from Runge's theorem, but I don't know how to prove it.
Fact. Let $U \subseteq V \subseteq \mathbb{C}$ be open sets such that no one connected component of $V \setminus U$ is com... |
H: Best books in the genre "______ for Mathematicians"
I once heard someone (perhaps from someone famous -- anyone have a citation?) say that there ought to be a series of books called "__ for Mathematicians," each one of which would explain a different topic or discipline using the tools of mathematics. (The idea is ... |
H: What is the preferred symbol to indicate the least positive number to start a sequence?
I need a least positive number and I am considering $\delta$, $\epsilon$ and $\theta$. Which one would be best to start a sequence? Are there any others I should also consider?
Edit:
$a(0)\text{:=}\theta$
$a(n)\text{:=}\left ... |
H: Constructing $\sqrt{a}$ for a constructable $0\leq a\in\mathbb{R}$ - Compass and straightedge constructions
Possible Duplicate:
Compass-and-straightedge construction of the square root of a given line?
I wish to understand how to construct $\sqrt{a}$ for a constructable
$0\leq a\in\mathbb{R}$ , the book Abstract... |
H: About Square Rooting
I have read that "every positive number $a$ has two square roots, positive and negative".
For that reason I have always (as far as I could remember) unconsciously done the following for such expressions
$$ x^2 = 4 \implies x= \pm 2 $$
What I wanted to know was that, in order to cancel the sq... |
H: What is an operator in mathematics?
Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
AI: Based on your comment it sounds like you're actually asking about operations, not operators. A binary operation on a set $S$ ... |
H: normalizer of a p-Sylow on $S_p$
Let $P$ be a group of order p, on $S_p$ , How can I prove that the cardinality of normalizer of $P$ it's $p(p-1)$ ?
If I compute that the number of conjugates of the group P, it's $
\frac{{n!}}
{{p\left( {p - 1} \right)}}
$ then I'm done, since equals to the index of the normalizer.... |
H: Difference Identity problem
I have a homework problem that I don't know what to do with. We were just introduced to sum and difference identities. We've always been provided values in degrees both in class and in homework until this problem. I checked the book to see if a similar problem had been worked out; it ... |
H: Let $x_n$ be a an unbounded sequence of non-zero real numbers
Let $x_n$ be a an unbounded sequence of non-zero real numbers. Then
it must have a convergent subsequence.
it can not have a convergent subsequence.
$\frac{1}{x_n}$ must have a convergent subsequence.
$\frac{1}{x_n}$ can not have a convergent subsequenc... |
H: sequence of function $f_n(x)= \sin(n\pi x)$
$f_n(x):[0,1]\rightarrow \mathbb{R}$ defined by
$$f_n(x)= \sin(n\pi x)$$ if $x\in [0,1/n]$, and $$f_n(x)=0$$ if $x\in (1/n,1]$ Then
It does not converge pointwise.
It converges pointwise but the limit is not continous.
It converges pointwise but not uniformly.
It co... |
H: Determining the existence of a solution to an additive equation
During my research in combinatorial geometry, I have encountered the following elementary question which I am hoping to have some help on.
Let $\zeta = 27.22236\ldots$. Does there exist a set of $\gamma_i$'s such that $\sum\limits_{i=1}^{8} \gamma_{i}... |
H: Finding a point near other points
Let $p_1, \ldots, p_k$ be $k$ points in $\mathbb{R}^n$ so that $$\max_{i,j}\|p_i - p_j\| = \epsilon$$ where we are employing the standard Euclidean norm.
What is the smallest $r > 0$ so that there exists some $x \in \mathbb{R}^n$ with $\|x - p_i\| \leq r$ for all $1 \leq i \leq n... |
H: will there be pairwise disjoint open sets in $\mathbb{R}^2$
Suppose $S$ is a collection of pairwise disjoint open sets in $\mathbb{R}^2$
$S$ can not be finite
S can not be countably infinite.
S can not be uncountably infinite
S is empty.
1 is wrong I can take any finite no of disjoint open sets by housdorff prope... |
H: Is $K\subset\mathbb{R}^2$ homeomorphic to an interval if $K$ is connected but $K\setminus\{x\}$ is not for any $x\in K$? Must it have empty interior?
Given that $K$ is a connected subset of $\mathbb{R}^2$ such that $\forall x\in K, K\setminus\{x\}$ is not connected, then
K must be homeomorphic to an interval of... |
H: Subgroup of order $9$ of $S_6$
Consider the permutation group $S_6$ and let $H\subseteq S_6$ be a subgroup of $9$ elements
It is abelian but not cyclic
It is cyclic
It is not abelian
If H is abelian then it is cyclic.
Wel I know a general result that group of order $p^2$ is abelian where $p$ is a prime number, he... |
H: Is there an explicit way to determine $\mathrm{Mat}_n(R[X_1,\dots,X_m])\simeq\mathrm{Mat}_n(R)[X_1,\dots,X_m]$?
For a commutative ring $R$, let $\mathrm{Mat}_n(R[X_1,\dots,X_m])$ denotes the matrix ring with entries from $R[X_1,\dots,X_m]$, and let $\mathrm{Mat}_n(R)[X_1,\dots,X_m]$ denotes the polynomial ring with... |
H: The Number of symmetric,PD, $8\times 8$ matrices
The Number of symmetric,Positive Definite, $8\times 8$ matrices having trace$=8$ and determinant$=1$ is
$0$
$1$.
$>1$ but finite.
$\infty$
I am not able to do this one.
AI: If $A$ is pos. def. then its eigenvalues $\lambda_i$ are real and positive. Besides, we kno... |
H: Show inequality generalization $\sum (x_i-1)(x_i-3)/(x_i^2+3)\ge 0$
Let $f(x)=\dfrac{(x-1)(x-3)}{x^2+3}$.
It seems to be that:
If $x_1,x_2,\ldots,x_n$ are positive real numbers with $\prod_{i=1}^n x_i=1$ then $\sum_{i=1}^n f(x_i)\ge 0$.
For $n>2$ a simple algebraic approach gets messy. This would lead to a genera... |
H: If $Q \in \mathbb{R}^{n \times n}$ is both upper triangular and orthogonal, then $\textbf{q}_j = \pm \textbf{e}_j, j = 1,\ldots, n$
I can get this far:
If $n = 1$, then the only matrices that are both upper triangular and orthogonal are $[1]$ and $[-1]$, so $\textbf{q}_j = \pm\textbf{e}_j, j = 1$ is true. Then if... |
H: Why solving $\dfrac{\partial u}{\partial x}=\dfrac{\partial^2u}{\partial y^2}$ like this is wrong?
Try let $v=x+y$ , $w=x-y$ ,
Then $\dfrac{\partial u}{\partial x}=\dfrac{\partial u}{\partial w}\dfrac{\partial w}{\partial x}=\dfrac{\partial u}{\partial w}$
$\dfrac{\partial u}{\partial y}=\dfrac{\partial u}{\partial... |
H: Two different characterization of "differentiable function"
In a calculus class we were given the following definition of "differentiable function" (working with 2 variables):
Definition: Let $A \in \mathbb{R^2}$, and $f : A \to \mathbb{R}$. We say that $f$ is differentiable in $(x_0, y_0) \in A$ if the graph of $f... |
H: Weak categoricity in first order logic
In a certain sense, only finite structures are definable up to isomorphism in first order logic. But if we rely on a metatheory containing a sufficient strong set theory (like required for second order logic), would it be possible to also define certain infinite structures up ... |
H: Does ODE initial value problem produce beat or resonance phenomenon?
$$x''+9x=\sin(3t),$$
$$x(0)=x'(0)=0.$$
This question was asked on a test. We are allowed to solve differential equations with TI-89.
My steps:
Solve with TI-89, solution
$$x(t) = \frac{1}{18} (\sin(3 t)-3 t \cos(3 t)) .$$
Plot the solution, and ... |
H: If $a\ge 0$ and $b\ge 0$, then $\sigma(ab)\subset\mathbb{R}^+$.
This is an exercise in Murphy's book:
Let $A$ be a unital $C^*$-algebra and $a,b$ are positive elements in $A$. Then $\sigma(ab)\subset\mathbb{R}^+$.
The problem would be trivial if the algebra is abelian. On the other hand I do not have a clue for t... |
H: Can we solve $2a(x^2-y^2)/(x-y)=b$ for $a$ without multiplying $b$ by $x-y$?
I would like to know if its possible to pull $a$ out of the following equation without multiplying $b$ by $(x-y)$
$$ \frac{ 2a(x^2 - y^2)}{x - y} = b $$
Its part of a more complex problem I'm stuck on.
Cheers
AI: Yes indeed, you have the ... |
H: Asymptotic behaviour of $\sum_{p\leq x} \frac{1}{p^2}$
As the title suggests, I want to find the asymptotic behaviour of this sum as $x\rightarrow \infty$, I tried by summation by parts but didn't succeed I also tried using the asymptotic behvaiour of the sum
$$\sum_{p\leq x} \frac{1}{p} \sim_{x \to \infty} \log \... |
H: Find the area of the regular polygon described given the side-length
I am being asked to calculate the area of a triangle with a side-length of 15.5 inches.
The formula for calculating a regular polygon's area is 1/2Pa
Where P is the perimeter of the polygon, and a is the apothem. I am completely lost.
AI: Hint:... |
H: Proving the suprema of $\{b^r\mid x\geq r\in\mathbb{Q}\}$ and $\{b^r\mid x\gt r\in\mathbb{Q}\}$ are equal if $b\gt 1$
Please help me with the proof that $$\sup\{b^r\in \mathbb{R}\mid x\geq r\in \mathbb{Q}\} = \sup\{b^r\in \mathbb{R}\mid x\gt r\in \mathbb{Q}\}$$
where $1<b\in \mathbb{R}$ and $x\in \mathbb{R}$.
AI: I... |
H: How many $3\times 3$ binary matrices $X$ are there with determinant $0$ and $X^2=X^T$?
How many $3 \times 3$ binary matrices $X$ are there with determinant as $0$ that also satisfy $X^2 = X^T$?
AI: There are $2^9=512$ binary $3\times 3$ matrices. Of these, $7\times 6\times 4 = 168$ are invertible, so there are $344... |
H: Simplifying a fraction?
$$\frac {n-2}{n} \cdot \frac {n-3}{n-1} \cdot \frac {n-4}{n-2} \cdots \frac{2}{4} \cdot \frac{1}{3} = \frac {1}{n(n-1)}$$
Why is this true? Notice the denominators and numerators cancel out, but since they "aren't in sync" the first two denominators and the last two numerators will not be... |
H: Help find hard integrals that evaluate to $59$?
My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$.
I want to try and make a definite integral that equals $59$. So far I can only think of ones that are easy to evaluate. I was wonder... |
H: Improper integral about exp appeared in Titchmarsh's book on the zeta function
May I ask how to do the following integration?
$$\int_0^\infty \frac{e^{-(\pi n^{2}/x) -(\pi t^2 x)}}{\sqrt{x}} dx $$
where $t>0$, $n$ a positive integer.
This came up on page 32 (image) of Titchmarsh's book, The Theory of the Riemann Ze... |
H: Quadratic System of Equations
I'm trying to define a quadratic that can pass through any 3 points. I've obviously done something wrong but can't figure out where. Any help would be appreciated.
$$ ax_1^2 + bx_1 + c = y_1 $$
$$ ax_2^2 + bx_2 + c = y_2 $$
$$ ax_3^2 + bx_3 + c = y_3 $$
Solve for C using the first equa... |
H: Area of a regular octagon with a side-length of 10 km
I am asked to calculate the area of a regular octagon given the side-length of 10 km.
I saw some examples saying that I should start by splitting the octagon into eight isosceles triangles, and that the length of the base would be 10 km, since we're given that... |
H: Find an ideal in $K[x,y]$ that is maximal but not principal.
Let $K$ be a field. Find an ideal of $K[x,y]$ that is maximal but not principal. Prove your claims.(Here we are working in a commutative ring with $1\neq 0.$)
My idea:
Choose $K=\mathbb{Q}.$ Then we claim that an ideal $I\subset K[x,y]$ which is maxima... |
H: Proving equipotency between sets.
Prove that the set $[2,5[$ is equipotent with the set $[3,4[$
According to my book, I have to find the linear function that passes by the points $(2,4)$ and $(5,3)$. How do you do that? This is a particular case that doesn't seem to be explained in my book.
... and why a linear fun... |
H: Multivariable Limits
Can someone help me calculate the following limits?
1) $ \displaystyle\lim _ {x \to 0 , y \to 0 } \frac{\sin(xy)}{\sqrt{x^2+y^2}} $ (it should equal zero, but I can't figure out how to compute it ) .
2) $\displaystyle\lim_ {(x,y)\to (0,\frac{\pi}{2} )} (1-\cos(x+y) ) ^{\tan(x+y)} $ (it should ... |
H: Solving heterogeneous successions
I know how to get the explicit formula for homogeneous successions, kinda. What I do is get the characteristic equation, get the solutions and then solve a system to obtain the values of A,B,C... constants to build the explicit formula.
... But what if the succession is heterogeneo... |
H: Orthogonal Trajectories
I am asked to show that the given families of curves are orthogonal trajectories of each other.
$$x^2+y^2=ax$$
$$x^2+y^2=by$$
I know that two functions are called orthogonal if at every point their tangents lines are perpendicular to each other. If I differentiate both of these functions, a... |
H: Convergence for expectation
$X_n$ converges to $X$ in $L^1$, then $\limsup_H|EX_n1_H-EX1_H|=0$.
I want to prove it, is the following proof right?
$$\lim|EX_n1_H-EX1_H|=\lim|E(X_n-X)1_H|=|\lim E(X_n-X)1_H|\\=|E\lim(X_n-X)1_H|=0$$
It's true for all $H$.
So, $\limsup|EX_n1_H-EX1_H|=0$
And I also confuse how I can get ... |
H: Seminorm exercise
Can you tell me if my answer is correct? It's another exercise suggested in my lecture notes.
Exercise: Consider $C[-1,1]$ with the sup norm $\|\cdot\|_\infty$. Let
$$ W = \{f \in C[-1,1] \mid \int_0^1 f d\mu = \int_{-1}^0 f d \mu = 0 \}$$
Show that $W$ is a closed subspace. Let $f(x) = x$ and ca... |
H: What functions maintain inequality?
In my calculus book it mentions that increasing functions maintain inequality relations and that's the reason you can apply $\exp$ and $\ln$ to two sides of an inequality to solve them. Is there some general classification for the types of functions that maintain inequality? Fo... |
H: Orthogonal projection to closed, convex subset in a Hilbert space
I don't understand one step in the proof of the following lemma (Projektionssatz):
Let $X$ a Hilbert space with scalar product $(\cdot)_X$ and let $A\subset X$ be convex and closed. Then there is a unique map $P:X\rightarrow A$ that satisfies: $\|x... |
H: Two Lie algebras associated to $GL(n,\mathbb{C})$
I have elementary questions about Lie groups and their associated Lie algebras.
Let $G=GL(n,\mathbb{C})$. Then associated to this Lie group is the Lie algebra $M_n(\mathbb{C})$ with the commutator relation $[x,y]=xy-yx$ or we can define its Lie algebra to be $M_n(\m... |
H: finding all integer $n$ such that $ n\mid2^{n!}-1$
how to find all integer $n$ such that $ n\mid2^{n!}-1$
I find:
Of course $2 \nmid n$. We prove that, if $2 \nmid n$ then $n \mid 2^{n!}-1$. $2 \nmid n \iff n = 2k+1 , k \ge 0$, we'll prove:
$2^{(2k+1)!} \equiv 1\pmod{2k+1}$
Let $n = p_1^{a_1}\cdot p_2^{a_2} \cdot ... |
H: Fourier transformation of sin, cos, sinh and cosh
I am trying to solve the following exercise
Use $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$ to calculate the Fourier-Transformation of $\sin x$, $\cos x$, $\sinh x$ and $\cosh x$
Now I am a little bit confused, because the fourier transformation of $\sin x$ is simply... |
H: To show $f$ is continuous
Let $f:[0,1]\rightarrow \mathbb{R}$ is such that for every sequence $x_n\in [0,1]$, whenever both $x_n$ and $f(x_n)$ converges , we have $$\lim_{n\rightarrow\infty} f(x_n)=f(\lim_{n\rightarrow\infty}x_n),$$ we need to prove $f$ is continuous
well, I take $x_n$ and $y_n$ in $[0,1]$ such tha... |
H: What can we say about transport equation?
If we have a transport equation, i.e.
$$u_t + \vec{b} \cdot D_x u=0,$$
is it true that at some point the particular directional derivative of $u$ becomes $0$, i didn't understand why? As Evans says that this property can be exploited to get the solution, but I didn't get h... |
H: to show $f(t)=g(t)$ for some $t\in [0,1]$
Let $f,g:[0,1] \rightarrow \mathbb{R}$ be non-negative, continuous functions so that
$$\sup_{x \in [0,1]} f(x)= \sup_{x \in [0,1]} g(x).$$
We need To show $f(t)=g(t)$ for some $t\in [0,1].$
Thank you for help.
AI: If $f$ is nowhere equal to $g$, then by
continuity $f-g$ has... |
H: Security of a particular cryptosystem
I recently came across this problem, and while I'm fairly certain the solution is not too 'conceptually-challenging', I've been stumped at finding the right trick/manipulation to make any solution work.
Alice chooses two large primes $p,q$ and denotes $N=pq$; then she also cho... |
H: Can $a^2+b^2+2ac$ be a perfect square if $c\neq \pm b$?
Can $a^2+b^2+2ac$ be a perfect square if $c\neq \pm b$?
$a,b,c \in \mathbb{Z}$.
I have tried some manipulations but still came up with nothing. Please help.
Actual context of the question is:
Let say I have an quadratic equation $x^2+2xf(y)+25$ that I hav... |
H: $ \lim_{ k \rightarrow \infty } { \frac{ \lambda^k }{k}} = \infty$ when $1 < |\lambda| \in \mathbb{C} $.
Can someone show why $ \lim_{ k \rightarrow \infty } { \frac{ \lambda^k }{k}} = \infty$ when $1 < |\lambda| \in \mathbb{C} $.
AI: Just to make things a little easier to follow let $|\lambda|=1+x$ with $x >0$.
... |
H: NP-completeness and NP problems
Suppose that someone found a polynomial algorithm for a NP-complete decision problem. Would this mean that we can modify the algorithm a bit and use it for solving the problems that are in NP, but not in NP-complete? Or would this just shows the availability of a polynomial algorithm... |
H: Possible combinations of items in a certain number of sets
How many ways are there of arranging n elements into k sets given that all elements must be used in each arrangement? No set can be empty and order doesn't matter (i.e. {a, b, c} is the same as {c, b, a}). So for example, say n is 5 and k is three, there wo... |
H: What does a "convention" mean in mathematics?
We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that we can't prove with mathematical logic, or is it just intuitions, or s... |
H: Boundaries on Probability of Independent Events
Given an integer n, and an event n that happens with $P(\frac{1}{n})$, is the probability that e will happen in n trials bounded by any constant?
For example, if I had an n-sided fair die and a target value t, can I say with certainty that regardless of the value of n... |
H: Is there an easy way to see associativity or non-associativity from an operation's table?
Most properties of a single binary operation can be easily read of from the operation's table. For example, given
$$\begin{array}{c|ccccc}
\cdot & a & b & c & d & e\\\hline
a & e & d & b & a & c\\
b & d & c & e & b & a... |
H: Weak*-convergence of regular measures
Let $K$ be a compact Hausdorff space. Denote by $ca_r(K)$ the set of all countably additive, signed Borel measures which are regular and of bounded variation. Let $(\mu_n)_{n\in\mathbb{N}}\subset ca_r(K)$ be a bounded sequence satisfying $\mu_n\geq 0$ for all $n\in\mathbb{N}$.... |
H: How to prove the uniqueness of the solution of $ax+b=0$?
I have no background in mathematical analysis or the like, but I am interested to know how to prove the uniqueness of the solution of $ax+b=0$? Perhaps your answers will help me to prove other uniqueness problems.
AI: A standard way of showing that a certain ... |
H: Local minimum example
Help me please with this question.
Let's $\Delta u>0$ in connected domain in $\mathbb{R}^{n}$.
Is it possible that function $u$ have local minimum? Can you show an example?
Thanks!!
AI: $f(x,y)=x^2 + y^2$ on all of $\mathbb{R}^2$ |
H: Documentaries about mathematics and mathematicians
Possible Duplicate:
List of Interesting Math Videos/ Documentaries
I have watched "Fermat's last theorem" a documentary about Andrew Wiles proof of the theorem, it was a great show. and i am asking if there is some other good documentaries about mathematics and... |
H: Relationship Between Basis For Vector Space And Basis For Dual Space
There exist the famous theorem about a basis for dual space
Let $\mathbb V$ be finite dimensional vector space over $F$ and $\mathcal{B} = \{\alpha_1, \ldots ,\alpha_n\}$ is basis for vector space $\mathbb V$ then
$\mathcal{B^*} = \{f_1, \ldots ,... |
H: About positive semidefiniteness of one matrix
It is not clear how to prove that the matrix $(\min(i,j))_{i,j=1,\dots,n}$ is (or is not) positive semidefinite. There are some facts from Horn and Johnson's book Matrix Analysis: if $A \in M_n$ is positive semidefinite, then $a_{ii}a_{jj} \ge a_{ij}^2, i,j=1,2,\dots, n... |
H: Weak-* sequential compactness and separability
Let $X$ be a Banach space, and let $B$ be the closed unit ball of $X^*$, equipped with the weak-* topology. Alaoglu's theorem says that $B$ is compact. If $X$ is separable, then $B$ is metrizable, and in particular it is sequentially compact.
What about the converse... |
H: Typo in lecture notes?
The following is an example in my lecture notes:
"Let $X$ be a locally compact topological space (that is, a topological space in which every point has a
compact neighborhood). Then
$C_0(X)=\{f \in C_b(X)| \lim_{x \to \infty} f(x)=0 \}$
is a closed subspace of $C_b(X)$, the space of bounded c... |
H: Questions about the definition of group actions
A group $G$ is said to act on a set $X$ when there is a map $\phi : G\times X\rightarrow X$ such that the following conditions hold for all elements
$\phi(e,x)=x$ where $e$ is the identity element of $G$.
$\phi (g,\phi(h,x))=\phi(gh,x)$ for all $g,h\in G$.
This ... |
H: Too many topics taught in class? Frustrated student in need of advice and encouragement.
Location: New York CUNY (as education systems might be different in other places)
I started my life studying philosophy and psychology and then at 22 transitions to computer science. It took me a long time to understand the im... |
H: a question on symmetric group $S_5$[NBHM_2006_PhD Screening Test_algebra]
Given that $x=(1 2)(3 4 5)\in S_5$ so its order $6$, and its a product
of two cycle,I want to know whether $x$ commutes with all elements of
$S_5$ and is it conjugate to $(4 5)(2 3 1)$?
Thank you for the help.
AI: There is a rule for co... |
H: Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$
I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating $\lim\limits_{n\to\infty} \prod\limi... |
H: Finding gradual values
I'm writing some code for a pressure level sensor for propane tanks. The manual provides me with the following table with the caption:
"Best accuracy will be obtained using the calibration data in the table below:"
I assume:
0.000 - 0.318 is a E-stop
0.319 - 0.590 is 10 etc.
What I'd like ... |
H: How to think of the field $F(\alpha)$
The way I learned it was given a field extension $F \subset E$, and an element $\alpha \in E$
$$F(\alpha) := \{p(\alpha)/q(\alpha) : p(x), q(x) \in F[x] ,q(\alpha) \not = 0\} $$
Is there an easier way to think about the field $F(\alpha)$
AI: The definition you've given yields ... |
H: Name of this discrete stochastic process
Suppose we have $n$ blocks of wood. At each step, we choose one of these boxes uniformly at random and paint it red (so at later steps, we may be re-painting an already-red box). Let $X_t$ denote the percentage of the boxes painted red at time $t$.
In other words, take $X_... |
H: Question about $L^p$ spaces
Suppose $1<p<\infty$ and let $L^1$ and $L^p$ denote the usual Lebesgue spaces on $[0,1]$. Let $$A=\{f\in L^1:\|f\|_p\leq 1\}.$$ Show $A$ is closed in $L^1$.
I took a sequence $\{f_n\}$ in $A$ and assumed it converges to $f$ in $L^1$. I am having trouble showing $\|f\|_p\leq 1$.
AI: ... |
H: Two Representations of $\log \zeta$
I was looking for representations of $\log \zeta$ and found these two:
$ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted by me],
$ \displaystyle \log \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,d... |
H: if $ax+by = d$, then $a'x+b'y=d$ where $x>0$ and $0 \leq b' \leq x$
I've been trying this for a little while now,
if $ax+by = d, \ $ then $a'x+b'y=d$
where $x>0\ $ and $0 \leq b' \leq x\ $ and $a,b,a',b',x,y, \in \mathbb{Z}$
My first thought is:
$$ \begin{align}
ax+by &= d \\
ax &= d - by \\
\end{align} $$
Impl... |
H: $A=\{x\in \mathbb{R}\mid b^x < y\}$ is nonempty
Let $1<b\in \mathbb{R}$ and $y\in \mathbb{R}$. I have proved that $A=\{x\in \mathbb{R}\mid b^x < y\}$ is nonempty when $y > 1$. Please give me any hint how to show that $A$ is nonempty when $y\leq 1$.
AI: Consider the sequences $x_n=-n$ and $b_n = b^{x_n}$. Ask what t... |
H: How to design a convolutional error correcting code
I'm trying to understand how one would design a convolutional code, like the (2,1,2) code that is always used in examples (see here for an example: https://en.wikipedia.org/wiki/Convolutional_code#Convolutional_encoding)
It is clear to me how to decode an arbitrar... |
H: Free PDF for MV Calculus
I was looking for a free PDF from which I can review MV calculus.
Specifically:
MV Limits, Continuity, Differentiation.
Differentiation of vector and scalar fields
Surface/Multiple Integrals
A succinct book would be great, (coherent) course notes and presentations would do as well.
I ran ... |
H: Properties of a $3\times 3$ orthogonal matrix
[NBHM_2006_PhD Screening Test 2006_Algebra]
Let $A$ be an $3\times 3$ orthogonal matrices with real entries,Then
which are true
$\det A$ is rational number
$d(Ax,Ay)=d(x,y)$ for any two vector $x,y\in \mathbb{R}^3$ where $d$ is ussual eucledean distance.
All entri... |
H: Unique expression of a polynomial under quotient mapping?
I have a weird feeling about something I'm reading.
Suppose $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is a polynomial over a field $F$. Let $y=x+(f(x))$ be the image of $x$ in the quotient $F[x]/(f(x))$. Then every element of $F[y]$ can be uniquely expresse... |
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