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H: Question about continuous function in terms of limits of sequences
I am reading about continuous function, in this site http://en.wikipedia.org/wiki/Continuous_function specifically the section "Definition in terms of limits of sequences". My question is, let be $c\in \mathbb{R}$ an arbitrary element belonging to d... |
H: Graph $y=1/\sin x$
Graph $y=\dfrac{1}{\sin x}$
Now, I looked at the graph on google and got this
Which I thought that $y=\dfrac{1}{\sin x}$ would be $y=\sin^{-1}x$ But it's apparently not. So if anyone can shed some light on this. It's not just about finding a graph and copying it. I would like a better understa... |
H: Inscribing a rhombus within a convex quadrilateral
I was wondering if it is possible to inscribe a rhombus within any arbitrary convex quadrilateral using only compass and ruler? If possible, could you describe the method?
If not could you give an example of a convex quadrilateral which one can not inscribe a rhom... |
H: What are vector states?
I am reading some papers from the 70s on operator theory. I come across the term 'vector state' of a $C^*$-algebra quite often. It is a little bit confusing.
Wikipedia redirects to quantum state vector which I found irrelevant.
So could someone give me a definition of a 'vector state'?
In p... |
H: Proving a number with $3^n$ equal digits is divisible by $3^n$
Prove a number with $3^n$ equal digits is divisible by $3^n$.
My thoughts about the problem are: a number with $3^n$ equal digits $d$ is equal to $d\frac {10^{3^n} - 1} {9}$. We use Lifting The Exponent lemma, or plain induction.
AI: This follows from ... |
H: Showing that the closure of the closure of a set is its closure
I have the feeling I'm missing something obvious, but here it goes...
I'm trying to prove that for a subset $A$ of a topological space $X$, $\overline{\overline{A}}=\overline{A}$. The inclusion $\overline{\overline{A}} \subseteq \overline{A}$ I can do... |
H: Find a continuous solution of the initial-value problem
This question is from DE book by Braun(Pg no 10, Q no 17),
Find a continuous solution of the initial-value problem $y'+y= g(t), y(0)= 0$ where $g(t)=\begin{cases}2, &0 \leq t\leq 1, \\0, &t > 1\end{cases} $
since the intial condition is given at (0,0), ther... |
H: Why are compact operators 'small'?
I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it.
I know that compact operators map bounded sets to totally bounded ones, that the perturbation of a compact operator does not change the index, and that the ca... |
H: Solving a fraction expression without a calculator using properties
I am preparing for the CAT and am not allowed to use a calculator to solve questions like these
If k is an integer and $\frac{35^2-1}{k}$ is also an integer then k could be any of the following except a)8 b)9 c)12 d)16 e)17 (Ans=D)
Are there a... |
H: How to select an field of study?
As my question implies, I am a young, developing mathematician, with concerns about my future math career. In particular, I'd like to know how to select a future field of study. Through my courses and readings, I've started to entertain the possibility of studying Lie theory. I t... |
H: Find all solutions, other than $2$ for $12x^3-23x^2-3x+2=0$
Find all solutions, other than $2$ for $12x^3-23x^2-3x+2=0$
I started off by taking out an $x$ and got $$x(12x^2-23x-3)+2=0$$
I do not know if this is the correct first step, if it is, then am I able to use the quadratic formula or complete the square ... |
H: Homeomorphisms of X form a topological group
So I'm just learning about the compact-open topology and am trying to show that for a compact, Hausdorff space ,$X$, the group of homeomorphisms of $X$, $H(X)$, is a topological group with the compact open topology. This topology has a subbasis of sets $\{f\in H(X):f(C)... |
H: Orthogonal complement of a vector bundle
Let $E \rightarrow X$ be a vector bundle with an inner product. If $F$ is a sub-bundle, we can define an orthogonal complement bundle $F^\perp$ (see http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf for the construction, and the source of the problem). I am trying to show tha... |
H: Solve for $x; 12x^3+8x^2-x-1=0$
Solve for $x$. $12x^3+8x^2-x-1=0$ all solutions are rational and between $\pm 1$
As mentioned in my previous answers, I'm guessing I have to use the Rational Root Theorem. But I've done my research and I do not understand what to plug in or anything about it at all. Can someone ple... |
H: Still another diophantine equation
Can any of you guys provide a hint for thew following exercise?
Exercise. There is no $3$-tuple $(x,y,z) \in \mathbb{Z}^{3}$ such that $x^{10}+y^{10} = z^{10}+23$.
Thanks a lot for your insightful replies.
AI: The first thing to do is to check for local obstructions, that is some ... |
H: Prove that for $n \in \mathbb{N}, \sum\limits_{k=1}^{n} (2k+1) = n^{2} + 2n $
I'm learning the basics of proof by induction and wanted to see if I took the right steps with the following proof:
Theorem: for $n \in \mathbb{N}, \sum\limits_{k=1}^{n} (2k+1) = n^{2} + 2n $
Base Case:
let $$ n = 1 $$ Therefore $$2*1... |
H: Solve $\, \mathrm dy/\, \mathrm dx = e^{x^2}$
I want to solve $$\frac{\, \mathrm dy}{\, \mathrm dx}=e^{x^{2}}.$$ i using variable separable method to solve this but after some stage i stuck with the integration of $\int e^{x^{2}}\, \mathrm dx$. i dont know what is the integration of $\int e^{x^{2}}\, \mathrm dx$.... |
H: How to prove that $2^\omega=\mathfrak{c}$?
Let $\mathfrak{c}$ denote the continuum.
My textbook says that $2^\omega=\mathfrak{c}$. How can one prove this equality?
Thanks ahead:)
AI: Clearly $2^\omega$ has cardinality equal to $|P(N)|$. Think of a subset of the naturals as a sequence of zero's and one's. (A one i... |
H: Solve the following inequality $x^2+x+1\gt 0$
Solve the following inequality $x^2+x+1\gt 0$
I understand how to solve inequalities and what the graphs look like. Usually the first step is to set this as in equation and then find the zeros. But for this one when I used the quadratic formula my two answers were:
$... |
H: Question about proof of chain conditions
Here is a proof from Atiyah-Macdonald:
For i) $\implies$ ii) could one not write "If $(x_n)$ is such that $x_m = x_{m+1} = \dots$ then obviously $x_m$ is a maximal element"?
I am asking because the book has been getting terser with proofs and has reached super-terse by now ... |
H: How to transform a differential equation to easier one using change of variables?
I am currently studying analysis of Algorithms and have come across a paper about Median of 3 partition for quick select. The authors have solved the recurrence using the generating functions and came up to a partial differential equa... |
H: Product of Riemannian manifolds?
Given two Riemannian manifolds $(M,g^M)$ and $(N,g^N)$ is there a natural way to combine them to be a Riemannian manifold? Some kind of $(M \times N, g^{M \times N})$.
AI: Yes. Using the natural isomorphism $T(M \times N) \cong TM \times TN$, define the metric on $T(M \times N)$ as... |
H: Showing that a homogenous ideal is prime.
I'm trying to read a proof of the following proposition:
Let $S$ be a graded ring, $T \subseteq S$ a multiplicatively closed set. Then a homogeneous ideal maximal among the homogeneous ideals not meeting $T$ is prime.
In this proof, it says
"it suffices to show that if $a,... |
H: What can be the possible value of $a+b+c$ in the following case?
What can be the possible value of $a+b+c$ in the following case?
$$a^{2}-bc=3$$
$$b^{2}-ca=4$$
$$c^{2}-ab=5$$
$0, 1, -1$ or $1/2$?
After doing $II-I$, $III-I$ and $III-II$, I got,
$$(a+b+c)(b-a)=1$$
$$(a+b+c)(c-a)=2$$
$$(a+b+c)(c-b)=1$$
I'm unable ... |
H: The neighborhood of each points of a special topological space
Let $R$ is the real line and $R^* = R \cup \{x^*\}$, where $x^*$ is not in $R$. For any subset $A$ of $R^*$, we define the $cl_{R^*}(A)$as following:
if $A$ is finite, then $cl_{R^*}(A)=A$; if $A$ is infinite, then $cl_{R^*}(A)=cl_R(A-\{x^*\}) \cup \{x^... |
H: For what value of k, $x^{2} + 2(k-1)x + k+5$ has at least one positive root?
For what value of k, $x^{2} + 2(k-1)x + k+5$ has at least one positive root?
Approach: Case I : Only $1$ positive root, this implies $0$ lies between the roots, so $$f(0)<0$$ and $$D > 0$$
Case II: Both roots positive. It implies $0$ lies... |
H: A simple question about the open mapping theorem
$X, Y $ : Banach space, $T : X \to Y$ : linear bounded operator, onto. I'm studying open mapping theorem, but how can I prove this?
If $B_Y (0, \epsilon_1 ) \subset \overline{T(B_X (0, \epsilon_2 ))}$ then $B_Y ( 0, 2 \epsilon_1 ) \subset \overline{T(B_X (0, 2 \eps... |
H: Showing that the last digit of $a$ and $a^{13}$ are the same
For $a \in \mathbb N$, show that the last digit of $a$ and $a^{13}$
are the same.
For example: $2^{13} = 8,192$
$7^{13} = 96,889,010,407$
AI: To rephrase the question, you want to show that $a^{13}\equiv a\pmod{10}$, or both $a^{13}\equiv ... |
H: Find the limit of $f(x)$ involving a sum of logarithms.
I need to find
$\lim_{x\rightarrow0}f(x)$ for the following function:
$f:(0,+\infty)$
$f(x)=[1+\ln(1+x)+\ln(1+2x)+\dots+\ln(1+nx)]^\frac{1}{x}$
I tried writing the logarithms as products:
$\lim_{x\rightarrow0}[1+\ln(1+x)(1+2x)\dots(1+nx)]^\frac{1}{x}$
and as ... |
H: What does this notation, $x>x_0(\epsilon)$, mean?
What does this notation, $x>x_0(\epsilon)$, mean?
I have seen this in several proofs and haven't quite figured it out.
AI: $x_0(\varepsilon)$ means that $x_0$ depend on $\varepsilon$. For example
$$\forall\varepsilon>0, \exists x_0(\varepsilon)>0, \forall x>x_0(\va... |
H: Example of Artinian module that is not Noetherian
I've just learned the definitions of Artinian and Noetherian module and I'm now trying to think of examples. Can you tell me if the following example is correct:
An example of a $\mathbb Z$-module $M$ that is not Noetherian: Let $G_{1/2}$ be the additive subgroup of... |
H: Notation for countable products of sets
Let $X$ be some set and $\Omega = \prod\limits_{i=0}^\infty X$, so that any element in $\Omega$ can be written as
$$
\omega= (\omega_0,\omega_1,\dots)
$$
where $\omega_i\in X$, $i \geq 0$. Given $A\subset X$ I wonder what is the right notation for the set
$$
A_0 = \{\ome... |
H: Proof of $M$ Noetherian if and only if all submodules are finitely generated
Is my proof of proposition 6.2 on page 75 correct? (it's different from what they do in the book)
Proposition 6.2.: $M$ is a Noetherian $A$ module $\iff$ every submodule of $M$ is finitely generated.
My proof:
$\implies$ Assume $M$ has a s... |
H: Inverse problem from pdes
A linear inverse problem is given by:
$\ \mathbf{d}=\mathbf{A}\mathbf{m}+\mathbf{e}$
where d: observed data, A: theory operator, m: unknown model and e: error.
To minimize the effect of the noise; a Least Square Error (LSE) model estimate is commonly used:
$\ \mathbf{\tilde{m}}=(\mathbf{A^... |
H: Codimension of the complement of a quasi-affine open subset of a variety
It is a known fact from Algebraic Geometry that the complement of an affine open subset of a variety is of pure codimension one. Does the same hold for the complement of a quasi-affine open subset of a variety? I don't know much about codimens... |
H: Finding the divisor of an unknown
I am trying to solve this problem
A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?
I have established two relations here but dont know how to procee... |
H: Finding a number when its remainder is given.
I am trying to solve this problem
W is a positive integer when divided by 5 gives remainder 1 and when divided by 7 gives remainder 5. Find W.
I am referring back to an earlier post I made. Now I am attempting to solve it that way.
We know that
$$w\equiv1(mod~5)$$
... |
H: A question about a proof of Noetherian modules and exact sequences
I proved part (i) of the following:
Proposition 6.3. Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then
i) $M$ is Noetherian $\iff$ $M'$ and $M'"$ are Noetherian;
ii) $M$ is Artinian $\iff$ ... |
H: The row- and column-sums of a nonengative matrix with spectral radius less than $1$
Is it true that if a matrix has nonnegative elements and spectral radius less than $1$, than the sum of its elements on each row (and column) is less than $1$?
Edit: What if the matrix has positive elements?
AI: $A=\begin{bmatrix}0 ... |
H: If $K/F$ is Galois and $E$ is a subextension then $E$ is generated by roots of a polynomial over $F$?
Let $K/F$ be finite Galois field extension, then $K$ is the splitting
field of a separable polynomial $p$ over $F$, i.e. $K=F(a_{1},..a_{n})$
where $p=(x-a_{1})...(x-a_{n})$.
My question is: is it true that if $E$ ... |
H: To show $a_{n}\log n\to 0$ as $n\to\infty$.
Suppose i have $a_{n}\downarrow 0$ and $\displaystyle \sum_{n=1}^{\infty}\Delta a_{n}\log n<+\infty$, where $\Delta a_{k}=a_{k}-a_{k+1}$ and $a_{n}\downarrow 0$ means $a_{n}$ is decreasing and convergent to $0$.
I want to show that $a_{n}\log n \to 0$ as $n\to\infty.$ Cle... |
H: Moment generating function for the uniform distribution
Attempting to calculate the moment generating function for the uniform distrobution I run into ah non-convergent integral.
Building of the definition of the Moment Generating Function
$
M(t) = E[ e^{tx}] = \left\{ \begin{array}{l l}
\sum\limits_x e^{tx} p(x) &... |
H: does the uniform continuity of $f$ implies uniform continuity of $f^2$ on $\mathbb{R}$?
my question is if $f:\mathbb{R}\rightarrow\mathbb{R}$ is uniformly continuous, does it implies that $f^2$ is so?and in general even or odd power of that function?
AI: No. For example, $f(x)=x$ is uniformly continuous, but $f(x)=... |
H: right-running waves
Can you help me please? The problem is:
1.Solve the wave equation in finite interval with Dirichlet boundary condition at he right and Neumann boundary condition at the left .
2.Choose the initial conditions for right-running waves.
3.Show the phase difference of wave reflection at the boundarie... |
H: Maximum area of a triangle in a square
For a given square, consider 3 points on the perimeter to form a triangle. How to prove that:
The maximum area of triangle is half the square's.
The maximum area of triangle occurs if and only if the chosen points are vertexes of the square.
AI: First, recall that the area of... |
H: Antiderivative of a function that is decreasing is concave.
I'm a little rusty on calculus and would like to know how one goes about showing this without having to rely on differentiating the antiderivative twice.
Also, is there an extension to this in the multivariate case (i.e., what condition is needed to ensur... |
H: Cardinality of set of all fixed points of a function
3.10 Let $f\in\mathcal{C}^1[-1,1]$ such that $|f(t)|\leq 1$ and $|f'(t)|\leq\frac{1}{2}$ for all $t\in[-1,1]$. Let $$A=\{t\in[-1,1]\colon f(t)=t\}.$$
Is $A$ nonempty? If the answer is 'yes', what is its cardinality?
Well, I was trying like suppose $\exists t_... |
H: continuous map from $S^1\rightarrow \mathbb{R}^1$
well, so far I know there exist no injective map from $S^n\rightarrow R^n$(due to Borsuk-Ulam), so in the case of $3.8$ they are asking are there different point on $S^1$ whic maps to same point in $\mathbb{R}$? so by Borsuk Ulam theorem I can say "Yes", If the $f... |
H: Difference between population, sample and sample value.
I was going through a book and reached a point where the author is comparing a Population, Sample and Sample Values. I don't seem to understand the difference at all.
(Caps are Random Variables, small font are values/data points)
What is the role of Random V... |
H: Fourier series on $\mathbb T$ and $S^1$
From my lecture notes: "The notation $\mathbb T$ will be used for the additive circle and $S^1$ for the multiplicative circle."
What I understand: As a topological group, $S^1$ has the subspace topology of $\mathbb R^2$ and multiplication is defined as $(e^{ia}, e^{ib}) \map... |
H: What exactly is a Haar measure
I've come across at least 3 definitions, for example:
Taken from here where $\Gamma$ is a topological group. Apparently, this definition doesn't require the Haar measure to be finite on compact sets.
Or from Wikipedia:
"... In this article, the $\sigma$-algebra generated by all compa... |
H: Even numbers have more factors than odd numbers...
This was an exercise to show that, in a sense, the even numbers have more prime factors than the odds, but--if it's right-- I still have a question.
As an heuristic calculation, we could take a large interval (1, 2N) on which the average number of prime factors wit... |
H: Compute the trigonometric integrals
How would you compute the following integrals?
$$ I_{n} = \int_0^\pi \frac{1-\cos nx}{1- \cos x} dx $$
$$ J_{n,m} = \int_0^\pi \frac{x^m(1-\cos nx)}{1- \cos x} dx $$
For instance, i noticed that the first integral is convergent for any value of $n$ since $\lim_{x\to0} \frac{1- ... |
H: Find center, radius and a tangent to $x^2+y^2+6x-4y+3=0$
For the circle $x^2+y^2+6x-4y+3=0$ find
a) The center and radius
b) The equation of the tangent line at the point $(-2,5)$
Now, I solved a) and got the equation
$$(x+3)^2+(y-2)^2=10$$ with center $=(-3,2)$ and radius $=\sqrt{10}$
Now, I've never lear... |
H: elementary measure theory problem.
I am trying to show that, a set $E$ in $\left( 0,1\right) $ is such that, if $\left( \alpha,\beta\right) $ is any interval, then $$\mu\left(E \cap \left( \alpha ,\beta \right) \right) \ge \delta \left( \beta -\alpha \right) $$ where $\delta > 0 $ then the $\mu\left(E\right)=1$.
W... |
H: Calculating maximum of function
I want to determine the value of a constant $a > 0$ which causes the highest possible value of $f(x) = ax(1-x-a)$.
I have tried deriving the function to find a relation between $x$ and $a$ when $f'(x) = 0$, and found $x = \frac{1-a}{2}$.
I then insert it into the original function: ... |
H: If $K,E$ are subfields of $\Omega/F$ then $KE/F$ is a finite Galois imply $K/K\cap E$ is Galois?
Let $\Omega/F$ be a field extension and $K,E$ be two subfields of
$\Omega/F$. Assume that $KE/F$ is a finite Galois.
I have a theorem in my lecture notes that claim $\text{Gal}(KE/E)\cong \text{Gal}(K/K\cap E)$,
while i... |
H: A notation question: $|\langle x,y\rangle|$
Someone please explain what is the meaning of the words in shade
From Proof from the Book, 4th edition page 96:
Let $q$ be a prime power, set $n=4q-2$, and let
$$Q = \{x \in \{+1,-1\}^n: x_1 = 1, \#\{i:x_i=-1\} \text{ is even}\}.$$
This $Q$ is a set of $2^{n-2}$ vectors ... |
H: Notation for function that returns exponent of primes in factorisation?
Consider the function $f(n, i)$ which returns the exponent of the $p_i$ in the factorisation of $n$, where $p_i$ is the $i$-th prime.
Question: is there a standard label for $f$?
Context: In the first edition of my Gödel book, I (thoughtlessly... |
H: What does diameter mean in the sentence of Borsuk's conjecture?
What does diameter mean in the following sentence of Borsuk's conjecture?
Sentence: Can every set $S \subseteq \Bbb R^d$ of bounded diameter $\operatorname{diam}(S)>0$ be partitioned into at most $d+1$ sets of smaller diameter?
AI: If $m:\Bbb R^d\times... |
H: Intersection of Simply-Connected Sets
Let $U,V$ be two simply connected subsets of a topological space.
Prove or disprove:
$U \cap V$ is simply connected.
AI: Let $S^1$ be the circle in $\mathbb R^2$, $U=\{(x,y)\in S^1: x\geq 0\}$ and $V=\{(x,y)\in S^1: x\leq 0\}$. Then $U$ is the right half of a circle and $V$ is ... |
H: Loopspace of Eilenberg Mac Lane space
Is the loop space of the Eilenberg-MacLane space $K(G,1)$ dependent only on the cardinality of $G$? For instance, is the loop space of $K(\mathbb{Z}_4, 1)$ homotopy equivalent to that of $K(\mathbb{Z}_2 \times \mathbb{Z}_2, 1)$?
AI: Yes. Note that for a based space $(X, x_0)$, ... |
H: What is $f(t)=X_{t+1}$, if $X_{t+1}=(1-p)(1-X_{t})+pX_{t}$ and $X_{0},p \in [0,1]$?
What is $f(t)=X_{t+1}$, if $X_{t+1}=(1-p)(1-X_{t})+pX_{t}$ and $X_{0},p \in [0,1]$?
And what are general methods for finding functions defined by such recurrent equations?
AI: I will assume that $t$ ranges over, say, the non-negat... |
H: Bounded ration of functions
If $f(x)$ is a continuous function on $\mathbb R$, and $f$ is doesn't vanish on $\mathbb R$, does this imply that the function $\frac{f\,'(x)}{f(x)}$ be bounded on $\mathbb R$?
The function $1/f(x)$ will be bounded because $f$ doesn't vanish, and I guess that the derivative will reduce t... |
H: Book suggestion for linear algebra "2"
I am almost finishing Gilbert Strang's book "An introduction to linear algebra" (plus video lectures at MIT OCW). First and foremost, I would like to suggest this course for everyone. It has been incredibly illuminating.
I would like to continue studying linear algebra, with ... |
H: Is there an algorithm to compute this matrix at $\frac{2}{3}n^3 + O(n^2)$ flops?
If $A$ is an $n\times n$ nonsingular matrix, and $b$ , $c$ are column vectors of length $n$. Is there an algorithm I can use to compute the matrix $W = bc^\top A^{-1}$ in $\frac{2}{3}n^3 + O(n^2)$ flops?
AI: Well,
Decompose $\mathbf A... |
H: show that if $a | c$ and $b | c$, then $ab | c$ when $a$ is coprime to $b$.
Given two numbers $a$ and $b$, where $a$ is co-prime to $b$,
Show that for any number $c$, if $a|c$ and $b | c$ then $ab| c$.
Is the reverse also true? In other words, if $ab |c$ and $a$ is co-prime to $b$, then do we have $a | c$ as wel... |
H: If $f$ is a polynomial of degree $n$, then $f(x) \equiv 0\pmod p$ has at most $n$ solutions.
I know that I have to prove this by induction on $n$ when we let $f(x)=a_{n}x^n + a_{n-1}x^{n-1} +\cdots + a_0$. I have two books in front of me with the complete proof but I don't see how after they assume that the theorem... |
H: Simple group of order $660$ is isomorphic to a subgroup of $A_{12}$
Prove that the simple group of order $660$ is isomorphic to a subgroup of the alternating group of degree $12$.
I have managed to show that it must be isomorphic to a subgroup of $S_{12}$ (through a group action on the set of Sylow $11$-subgroup... |
H: In a certain year, January had exactly $4$ Mondays and $4$ Fridays. What was the day on $2^{\text{nd}}$ October the previous year?
In a certain year, the month of January had exactly $4$ Mondays and $4$ Fridays. What was the day on Gandhi Jayanti $(2^{\text{nd}}$ October $)$ the previous year?
I am not sure how ... |
H: Solving Congruences and CRT
I have never really directly dealt with congruences until I was introduced to the Chinese Remainder Theorem. Although there are tons of different versions of this theorem out there, currently I am more interested in solving congruences.
I had some questions regarding steps that were used... |
H: Showing that the universal enveloping algebra of some $\mathfrak{g}$ is isomorphic to $\mathbb{C}[x_i,\partial/\partial x_i]$
The universal enveloping algebra $U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ is defined to be
$$
\dfrac{\mathbb{C}\oplus\mathfrak{g}\oplus ( \mathfrak{g}\otimes \mat... |
H: The minimum value of $(\frac{1}{x}-1)(\frac{1}{y}-1)(\frac{1}{z}-1)$ if $x+y+z=1$
$x, y, z$ are three distinct positive reals such that $x+y+z=1$, then the minimum possible value of $(\frac{1}{x}-1) (\frac{1}{y}-1) (\frac{1}{z}-1)$ is ?
The options are: $1,4,8$ or $16$
Approach: $$\begin{align*}
\left(\frac{1}{x}... |
H: Solving for coefficients on a Laurent series
I am having an issue with the following complex analysis problem. I am suppose to
find the coefficients of $z^{-1}$, $z^{-2}$ and $z^{-3}$ in the Laurent series for
$\displaystyle \frac{1}{\sin z}$ around $z_0 = 0$ which is valid for $2\pi < |z| < 3\pi$.
One way I tho... |
H: simple integration question
I've tried but I cannot tell whether the following is true or not. Let $f:[0,1]\rightarrow \mathbb{R}$ be a nondecreasing and continuous function. Is it true that I can find a Lebesgue integrable function $h$ such that
$$
f(x)=f(0)+\int_{0}^{x}h(x)dx
$$
such that $f'=h$ almost everywher... |
H: Identifying $SL(2,\mathbb{C})/H$ with $\mathbb{C}^2\setminus \{ 0\}$
Let $G=SL(2,\mathbb{C})$ and let $H$ be the set of unipotent matrices
$$
\left\{ \left[ \begin{array}{cc}
1 & b \\
0 & 1 \\
\end{array}
\right] : b\in \mathbb{C}\right\}.
$$
I am trying to work out the details to show that
$G/H$ can be i... |
H: Is periodic extension of Lipschitz function Lipschitz?
Let $f: [0,T] \rightarrow \mathbb{R}$, where $T>0$, be a Lipschitz with constant $K$ and $f(0)=f(T)$.
Let us define $g(x)=f(x)$ for $x \in [0,T]$ and $g(x+T)=g(x)$ for $x \in \mathbb{R}$.
Does $g$ satisfies $$|g(x)-g(y)| \leq K |x-y|$$ for $x,y \in \mathbb{R}... |
H: Unprovable statements in ZF
Possible Duplicate:
Advantage of accepting the axiom of choice
Advantage of accepting non-measurable sets
As you all know, Banach-Tarski paradox is solely a consequence of Axiom of Choice, and I think it is just absurd.
I'm trying to take ZF as my axiomatic model rather than ZFC. I wo... |
H: Irrational equation for a maximization problem
I have the following maximization problem
$ \max_h m_1 + 10 (h)^{1/4} + h + m_2 - 2 (h)^{1/4} + m_3 - (h)^{1/4} $
where $m_1, m_2, m_3$ are three fixed values. The FOC for a maximum is
$ \dfrac {10}{4} h^{-3/4}+1-\dfrac{1}{2}h^{-3/4}-\dfrac{1}{4}h^{-3/4}=0$
Rearranging... |
H: Are homeomorphisms order-isomorphisms?
Is every homeomorphism between topological spaces an order isomorphism (for orders of inclusion $\subseteq$ of sets)?
AI: Every bijection $f \colon X\to Y$ induces an order-isomorphism between $(\mathcal P(X),\subseteq)$ and $(\mathcal P(Y),\subseteq)$.
This follows easily fro... |
H: How the dimension of a subspace related to the differential operator
I am wondering the link as the title implies. The Spring 87 problem in Berkeley Problems in Mathematics is as follows:
Let $V$ be a finite dimensional linear subspace of $C^{\infty}(\mathbb{R})$. Assume that $V$ is closed under differentiation. Pr... |
H: A question on second order ODE
I want to ask for a hint in solving the following problem from Berkeley Problems in Mathematics:
Let $h>0$ be given. Consider the linear difference equation $$\frac{y((n+2)h)-2y((n+1)h)+y(nh)}{h^{2}}=-y(nh),n\in \mathbb{Z}^{*}$$
1) Find the general solution of the equation by trying s... |
H: Polynomial ring over $\mathbb{Z}_2$
As $f(x)$ is an irreducible over $\mathbb{Z}_2[x]$ so $R/(f)$ is an infinite field. Am I right?
AI: No the quotient ring is a finite field of order 4. The reason is every element in the quotient ring by the division algorithm is a linear polynomial. There are 2 choices for the c... |
H: Group of order $60$
[NBHM_2006_PhD screening test_Algebra]
Let $G$ be a group of order $60$, pick out the true statements:
a. $G$ is abelian
b. $G$ has a subgroup of order $30$.
c. $G$ has subgroups of order $2$, $3$, and $5$.
d. $G$ has subgroups of order $6$, $10$, and $15$.
My Attempt:
a is false because $... |
H: Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$
Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$.
My textbook only provides two simple examples.
Really have no idea ... |
H: Question about whether axiom of choice is needed in this proof
Do I need axiom of choice in this proof here?
I think not: at each step we choose one element from a set $N - \langle g_1, \dots, g_k \rangle $. So while there is indeed a countable number of sets involved from which we choose elements, I could also thi... |
H: Is every order isomorphism of sets induced by a bijection?
Let $f$ is an order isomorphism $\mathscr{P}A \rightarrow \mathscr{P}B$ (where $A$ and $B$ are some sets, the order is set-inclusion $\subseteq$).
Is it true that it always exist a bijection $F: A \rightarrow B$ such that $f(X) = F[X]$ for every $X\in \math... |
H: length of the tangent
A Circle is inscribed in a triangle $ABC$, where $AB=10 cm$, $BC=9 cm$ and $AC=7 cm$ . $X$, $Y$, $Z$ are points of contact of the sides $AC$, $BC$ and $AB$ with the circle respectively. $BZ=?$
In the first look, question looked simple to me. But I am not able to get to the answer. I found the ... |
H: How to prove the function $y = \sin x$ is not a closed function?
I came across a question:
Suppose that $f(x) = \sin x$ is a function from $\mathbb R$ to $[-1,1]$. How do I prove the function $f(x) = \sin x$ is not a closed function?
By "closed function", I mean a function such that the image of any closed set... |
H: Sufficient condition for convergence of a real sequence
Let $(x_n)$ be a sequence of real numbers.
Prove that if there exists $x$ such that every subsequence $(x_{n_k})$ of $(x_n)$ has a convergent (sub-)subsequence $(x_{n_{k_l}})$ to $x$, then the original sequence $(x_n)$ itself converges to $x$ .
Tha... |
H: Prove that this function is measurable
I cannot write a neat proof of this result, so I would like to see how to be precise in these kinds of arguments.. Here is the problem
Let $I=[0,1]$ and let $f\colon I\times\mathbb R\to \mathbb R$ be a function such that
i) $f(\cdot,x)$ is measurable for all $x\in\mathbb R$;
... |
H: Uniform boundedness principle statement
Consider the uniform boundedness principle:
UBP. Let $E$ and $F$ be two Banach spaces and let $(T_i)_{i \in I}$ be a family (not necessarily countable) of continuous linear operators from $E$ into $F$. Assume that $\sup_{i \in I} \|T_ix \| < \infty$ for all $x \in E$. Then $\... |
H: Compute integral $\int_{-6}^6 \! \frac{(4e^{2x} + 2)^2}{e^{2x}} \, \mathrm{d} x$
I want to solve $\int_{-6}^6 \! \frac{(4e^{2x} + 2)^2}{e^{2x}} \, \mathrm{d} x$ but I get the wrong results:
$$
\int_{-6}^6 \! \frac{(4e^{2x} + 2)^2}{e^{2x}} \, \mathrm{d} x =
\int_{-6}^6 \! \frac{16e^{4x} + 16e^{2x} + 4}{e^{2x}} \, \... |
H: Endomorphisms preserve Haar measure
I am having trouble following the argument in page 21 of P. Walters, Intro. to ergodic theory, of the following statement:
Any continuous endomorphism on a compact group preserves Haar measure.
Obviously, this is not true as stated, as the trivial homomorphism does not preserv... |
H: How to find back from generating function $\sum(Q(x)*z^m/m!)$ to $Q(x)$?
How to find back from generating function $\sum_{m=0}^{\infty}(Q(x)\frac{z^m}{m!})$ to $Q$?
In other words, find $Q$ from $\sum_{m=0}^{\infty}(Q(x)\frac{z^m}{m!})$.
finally is generating function depends on $x$ and $z$
Update
The reason for th... |
H: How to prove that a matrix is positive definite?
Let $L$ be a Laplacian matrix of a strong connected and balanced directed
graph. Define
$$
L^{s}=\frac{1}{2}\left( L+L^{T}\right) .$$
Let $D$ be a diagonal matrix with
$$
D=\begin{bmatrix}
d_{1} & & & \\
& d_{2} & & \\
& & \ddots & \\
& & & d_{n}%
\end{bmatr... |
H: Example of Artinian ring
By definition a ring $R$ is Artinian if it is Artinian as $R$-module. I think the following is an example of an Artinian ring: $\mathbb Q / \mathbb Z$.
Ideals in it are of the form $(\frac1n)$ (since $(\frac{1}{n_1}, \dots , \frac{1}{n_k}) = (\frac{1}{\text{lcm}_i(n_i)})$).
Since we have $... |
H: Properties for a matrix being invariant under rotation?
Consider a 2D case.
Let $R$ be a rotation matrix with angle $\theta$
$$R = \begin{bmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{bmatrix}.$$
Is it possible for a matrix $A$ to satisfy the following identity for any $\theta$
$$A = R A R^T?$$
... |
H: Putnam Problem: Partitioning integers with generating functions
We were given the following A-1 problem from the 2003 Putnam Competition:
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, $$ n= a_1+a_2+ \cdots + a_k$$
With $k$ an arbitrary positive integer, and... |
H: Poset of idempotents
Let $A$ be an associative algebra and consider the poset $I$ of idempotents in $A$, where as usual if $e,f \in A$ we say $e \leq f$ provided that $e = fef$. Clearly $0 \in I$ is minimal, but in general I can't say anything else about this poset.
This leads me to ask: are there any other general... |
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