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H: Cohomological definition of the Chow ring
Let $X$ be a smooth projective variety over a field $k$. One can define the Chow ring $A^\bullet(X)$ to be the free group generated by irreducible subvarieties, modulo rational equivalence. Multiplication comes from intersection. The problem is, verifying that everything is... |
H: Determinant of a nilpotent matrix
Let $A$ be a nilpotent matrix. Prove that $\det(I+A)=1$
Could someone at least give me a clue ?
AI: Since $A$ is nilpotent, we have
$A^m = 0 \tag{1}$
for some positive interger $m$. This implies every eigenvalue of $A$ vanishes, since the equation
$Av = \lambda v \tag{2}$
for non-... |
H: ENS is an abbreviation of?...
In CWM Mac Lane uses the term $\mathbf {ENS}$ for a category having as objects the subsets of a given set and as morphisms the functions from these sets to these sets. What is abbreviated by the letters ENS?
AI: I believe ENS stands for ensembles, which means sets in French. |
H: Calculation of an integral via residue.
$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$
How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked here. Thank you for help.
AI: I'm pretty sure this has been done on this site,... |
H: Hilbert's basis theorem for power series ring in many variables
My teacher Proved that if R is Noetherian then $R[x]$ and $R[[x]]$ are Noetherian , how can i prove that again R is Noetherian if and only if $R[[x_1,...,x_n]]$ is Noetherian ,thanks for your time and help.
AI: Hints:
A quotient ring of a Noetherian ri... |
H: Need help finding the Laurent Series
Been struggling with this for a while now. Just looking for a nudge in the right direction :
Find the Laurent series for $f(z) = \frac{z^2-2z+3}{z-2}$ about $z=2$.
The only thing I can think to do is
$$ \frac{z^2-2z+3}{z-2} = z + \frac{3}{z-2}$$
but I don't think this is corre... |
H: Computing $\displaystyle\int \frac{1}{x^{2}\sqrt{x^{2}+7}}$
Find the primitive function of $$\frac{1}{x^{2}\sqrt{x^{2}+7}}$$
Attempt. And my answer is
$$\arcsin \left( \frac{\sqrt{\left( x^{2}+7 \right)}}{\sqrt{7}} \right)$$
Why am I wrong? See link for whole calculation.
[Wolfram](http://www.wolframalpha.com/inp... |
H: Does the function $f: \mathbb R \to \mathbb R^2, t \mapsto (t^3, t^2)$ have a history?
I heard one mathematician briefly mentioning that the function $f: \mathbb R \to \mathbb R^2, t \mapsto (t^3, t^2)$ is very famous and has a history.
Do you know what was meant by that?
AI: “It seems that $f$ is the historic ea... |
H: Axiomatic definition of complex numbers
Trying to build axiomatically the set $\mathbb C$ of complex numbers, my first attempt was to define $\mathbb C$ with three structures: addition, multiplication and conjugate: $\langle\mathbb C,+,\times,{}^*\rangle$, with $\langle\mathbb C,+,\times\rangle$ forming a commutati... |
H: Countability in topology
I am looking at the following example: $\mathbb{R}$ is second countable because consider the open intervals $(a,b)$ with rational endpoints. This is countable base for the usual topology on real line $\mathbb{R}$.
How is the above base base (interval) with rational endpoints countable? Isn'... |
H: Completely baffled by this question involving putting matrices in matrices
This is homework, so only hints please.
Let $A\in M_{m\times m}(\mathbb{R})$
, $B\in M_{n\times n}(\mathbb{R})$
. Suppose there exist orthogonal matrices $P$
and $Q$
such that $P^{T}AP$
and $Q^{T}BQ$
are upper triangul... |
H: Show that f is a continuous function if and only if for every closed set C in Y, $f^{-1}(C)$ is closed in X.
Suppose X and Y are topological spaces with topology $T^x$ and $T^y$
Let $f: X \to Y$ be a function.
Show that $f$ is a continuous function if and only if for every closed set $C$ in $Y$, $f^{-1}(C)$ is cl... |
H: Result and proof on the conditional expectation of the product of two random variables
My problem is the following:
$X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algebra. Given that $X$ and $Y$ are independent, and that $X$ is independent of $\mathcal{F}$, can I affirm that
$$\mathbb{E}(XY \... |
H: Calculate $a^3+b^3+c^3+d^3$ for the real roots of $x^4+2x^3-3x^2-3x+2$
$a,b,c,d \in \mathbb{R}$ are the real roots of $x^4+2x^3-3x^2-3x+2$.
Calculate $a^3+b^3+c^3+d^3$.
With approximation i found out, that $a^3+b^3+c^3+d^3 = -17$, but how can I proof that without calculating the roots exactly?
Cheers
AI: Express $... |
H: what is the meaning of $\mathbb{Q}(\sqrt{2},i)$
I have learnt so far that:
$\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}|a,b\in\mathbb{Q}\}$
Also
$\mathbb{Q}[i]=\{a+bi|a,b\in\mathbb{Q}\}$
What is $\mathbb{Q}(\sqrt{2},i)$ ?
Is it $\mathbb{Q}+\mathbb{Q}\sqrt{2}+\mathbb{Q}[i]$ but then we include $\mathbb{Q}$ twice as $\mathb... |
H: Let $f: (a,b) \rightarrow \mathbb{R}$ a function, and $(a,b)$ contains the origin.
Let $f: (a,b) \rightarrow \mathbb{R}$ a function, and $(a,b)$ contains the origin.
Prove that if $f$ is monotone and $$\lim_{x\to 0} \frac {f(x)-f(-x)}{x}=0,$$ then f is differentiable at $0$.
I tried to add and subtract $ f (0) $ a... |
H: Sigma-finiteness of a point mass measure
Let $\mathcal{e}_{a}(x)$ denote point-mass at the point $a\in \mathbb{R}$. Let $a_{n}$ be a sequence of points in $\mathbb{R}$ and let $\nu = \sum_{n=1}^{\infty}\mathcal{e}_{a_{n}}$. What are properties of the sequence $(a_{n})_{n\geq 1}$ that render $\nu$ sigma-finite?
I wo... |
H: Product of quadratic residues mod p $\equiv 1$ mod p iff $p \equiv 3$ mod 4
Let $p$ be an odd prime number. Prove that the product of the quadratic residues modulo $p$ is congruent to $1$ modulo $p$ if and only if $p \equiv 3 \pmod 4$.
I've tried using the fact that any quadratic residue modulo $p$ must be one of t... |
H: What approach should I take to establish this logical proof?
I need to design a logical math proof:
Write a detailed structured proof to prove that if m and n are integers, then either 4 divides mn or else 4 does not divide n.
Hint: Think about the form of the statement.
I was thinking about first proving that ei... |
H: Quotient is principal
Let $R$ be a finite commutative ring, let $J$ be a maximal ideal of $R$ and $n$ some positive integer greater or equal than $2$. Is it always true that every ideal of the quotient $R/J^{n}$ is principal?
AI: No. For instance if $R$ is local then $J^n = 0$ for sufficiently large $n$ and then y... |
H: Show f is continuous if and only if for any x $\in$ X and any open set O$^y$ in Y containing f(x), ...
Suppose $X$ and $Y$ are topological spaces with topology $T^x$ and $T^y$
Let $f: X \rightarrow Y$ be a function.
Show $f$ is continuous if and only if for any $x \in X$ and any open set $O^y$ in $Y$ containing $f... |
H: Subset of A Regular Language
I need to show that a subset of a regular language is regular or not. I think it may not be regular but I could not find a counter example.
Do you have any simple example to prove that?
Thanks in advance.
AI: Let $N$ be some non-regular language over some alphabet $\Sigma$. You should ... |
H: Let $f:[0, \infty) \rightarrow \mathbb{R}$ a function of class $C^1$ in its domain, suppose $f'(x)$ is a non-decreasing function.
Let $f:[0, \infty) \rightarrow \mathbb{R}$ a function of class $C^1$ in its domain, suppose $f'(x)$ is a non-decreasing function.
Using the monotonicity of $f'$, prove that the function ... |
H: Incongruent Solutions Modulo $p$
Let $p$ be an odd prime and $k$ a positive integer such that $\gcd(p,k)=1$. Show $x^2\equiv k \bmod p$ has zero or two incongruent solutions.
I think we are supposed to assume that $x$ is a solution, and that $y$ is also a solution where $y$ does not equal $x$. Then show $y \equiv... |
H: Embarrassingly-basic fraction question
I'm trying to do a calculation at work to figure out what the average # of pages a visitor is viewing.
I am given:
47,000 visits
12% of visits do not bounce (that is, 12% navigate the site at least one link after stumbling upon it)
1.23 pages/visit is the average amount of pag... |
H: Why is $\frac{d}{dx}\exp(x) = \exp(x)$?
What is the explanation for
$$(e^x)'=e^x$$
I searched the SE, 'cause this can't be the first time this has been asked. But the question seems hard to formulate and search for here and on Google. Any help?
AI: Note that for $f(x) = e^x$ you have by definition
$$
f'(x) = \lim_{... |
H: Use a lemma to prove that $A_4$ has no subgroup of order $6$.
Use the following lemma to prove that $A_4$ has no subgroup of order $6$:
Lemma: If $H\le G$ has index $2$, i.e. $[G:H]=2$, then for any $a\in G$ we have $a^2\in H$.
The $12$ elements of $A_4$ are $(1), (12)(34), (13)(24), (14)(23), (123), (132), (124), ... |
H: Find a bijective function between two segments of R
In my homework in set theory i was asked to find a bijective function between (a,b) and (c,d) that are subsets of R. d>c and b>a.
I'm having problems thinking of a function that is surjective and injective.
i thought of the function for example f(x) = x+d-b, where... |
H: Growth of exponential functions with different base and/or exponent
Assuming that the following is trivial:
$\lim_{x \to +\infty} \frac{2^x}{2.1^x}=0$
$\lim_{x \to +\infty} \frac{2^x}{2^{x^2}}=0$
What is the most simple, intuitive way to show that:
$\lim_{x \to +\infty} \frac{2^{x^2}2.1^x}{2.1^{x^2}}=0$
AI: I th... |
H: convexity and the interior sphere condition
Consider $\Omega $ a open, convex bounded subset of $R^n$.
Let $x_0 \in \partial \Omega$. I believe that exists a open ball $B \subset \Omega$ such that $\partial B \cap \partial \Omega = \{ x_0 \}$. (this is the interior sphere condition in $x_0$) ? I have no idea how ... |
H: Can you split up expectation over multiplication?
I was wondering if the property exists where if you have $\ E[(Y- \mu)^3]$ you can write it as $\ E[(Y- \mu)^2] E[(Y- \mu)] $ ?
AI: Is $Y$ normally distributed by any chance? If it is then Stein's Lemma can be useful for calculating higher order moments.
Stein's Lem... |
H: α² is a cycle if and only if s is odd
let $\alpha$ be a cycle of length $s$, say $\alpha = (a_1, a_2, \ldots, a_s)$
Prove $\alpha^2$ is a cycle if and only if $s$ is odd.
Let me start off by saying I am in my 5th week of Group Theory. I often have trouble getting these problems started. This is my first proof bas... |
H: If a function $f:X \to Y$ is continuous on any compact subset $K \subseteq X$, is $f$ continuous?
I hope I phrased that right. If $X$ is not able to be covered by a family of compact spaces, how could we prove this? Am I just missing something simple?
(Let $f:X \to Y$ (EDIT: $X$ and $Y$ metric spaces) have the prop... |
H: Upper bound for $(1-1/x)^x$
I remember the bound $$\left(1-\frac1x\right)^x\leq e^{-1}$$ but I can't recall under which condition it holds, or how to prove it. Does it hold for all $x>0$?
AI: Starting from $e^x \geq 1+x$ for all $x \in \mathbb{R}$:
For all $x \in \mathbb{R}$
$$ e^{-x} \geq 1-x. $$
For all $x \neq 0... |
H: $ \lim_{x \to 0} a^x = 1 $ using the definition
Given $a > 1$, define $ f: \mathbb{Q} \to \mathbb{R}$ such that for all $ x \in \mathbb{Q}$ we have $ f(x) = a^x$ . Prove that $ \lim_{x \to 0} f(x) = 1$
I'm having issues to find my delta. I think we need to use $ \log_a$ of something in function of $ \epsilon$ but ... |
H: perfect squares possible?
If we let a, b, c, d, and x be integers is it possible that $$x^2+a^2 = (x+1)^2 + b^2 = (x+2)^2 + c^2 = (x+3)^2 + d^2$$
My initial thought is no way! I tried expanding and simplifying, getting $$a^2 = 2x+1 + b^2 = 4x+4 + c^2 = 6x+9 + d^2$$.
It seems that the difference between these perfec... |
H: Upper bound for product of exponents
From here we have the bound
$$\left(1-\frac1N\right)^N\leq e^{-1}$$
where $N$ is a positive integer.
Written another way, it is
$$\left(1-\frac1{N_1}\right)^{k_1}\left(1-\frac1{N_2}\right)^{k_2}\ldots \left(1-\frac1{N_r}\right)^{k_r}\leq e^{-1}$$ where $N_1=N_2=\ldots=N_r=N$ ... |
H: If $a < b$ and $b = \infty$, then $a < \infty$?
A very simple question, but I am not sure for this moment.
I have a strict inequality $a < b$. And I prove that $b = \infty$, say $b$ is an integral. Does this prove that $a < \infty$, that is, $a$ is finite? Question seems simple but things can easily be messed up wi... |
H: The conditions of applying L'Hospital's rule
I just finished calculus 1,2 last semester and I am learning calculus 3 now. I saw this question and I post a solution as follow:
Prove series convergent,consider the limit:
$$\lim_{n\rightarrow\infty}\frac{\tan{n}}{1.5^n} = \lim_{n\rightarrow\infty}\frac{\frac{1}{\cos^... |
H: Hypercube and dihedral group
Let $G_n$ denote the subgroup of the orthogonal group $O_n$ of elements that send the hypercube to itself, the group of symmetries $C_n$, including the orientation-reversing symmetries.
It would like to show that $G_2$ is isomorphic to the dihedral group $D_4$ of order 8
I determined a... |
H: Unique maximal ideal implies set of non-units is an ideal
This is not for homework, but I would just like a hint please. The question asks
If a commutative ring $R$ (with $1$) has a unique maximal ideal, then the set of non-units in $R$ is an ideal.
This is actually an 'if and only if', but I have shown one dire... |
H: question about cluster points of a set of cluster points
Let A be a set in a metric space X. A′ is the set of cluster points of A. Is it A′′ ⊆ A′ ? I think it is not. But I can not give a counterexample or proof. Could someone give me a clue? Thank you!
AI: HINT: Yes, it is always the case that $A''\subseteq A'$. I... |
H: Limit point of a bounded sequence
My textbook says that if $x_n$ is a bounded sequence ($\exists m,M: m \le x_n \le M$) and a is a limit point of this sequence, then $m \le a \le M$. Can somebody explain why is that so? How can I show that it is true?
AI: HINT: Suppose that $a>M$, and let $\epsilon=M-a>0$. The seq... |
H: What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$?
What is the easy way to calculate the roots of $z^4+4z^3+6z^2+4z$?
I know its answer: 0, -2, -1+i, -1-i.
But I dont know how to find? Please show me this. I know this is so trivial, but important for me. Thank you.
AI: Add 1 to both side to get:
$... |
H: Solving coupled langrangian derivatives
I have been told that the solution to
$$\frac{Du}{Dt}=2\Omega v, \frac{Dv}{Dt}=-2\Omega u$$
is
$$u(t) = u_0 \cos2\Omega t+ v_0 \sin 2\Omega t$$
$$v(t) = -u_0 \sin 2\Omega t + v_0 \cos 2\Omega t$$
But how would I arrive at this solution by analysis? Is there a method I can us... |
H: Mistake in excercise with differentiation
I have an excercise given where $f: \mathbb{R} \rightarrow \mathbb{R}$ and $a \in \mathbb{R}$ and we have that $x(t):=\int_0^t exp(a(t-s))f(s)ds$. Now we are supposed to show that $x$ solves the differential equation $d_t(x(t))= ax(t)+f(t)$.
I would say that this is imposs... |
H: How to expand of the plane wave in Legendre polynomials
There is the expression for the plane wave:
$$
e^{i(\mathbf k \cdot \mathbf r )} = e^{ikrcos(\theta )} = e^{\frac{kr}{2}\left( ie^{i \theta} - \frac{1}{ie^{i \theta }}\right)} = e^{\frac{t}{2}\left( \omega - \frac{1}{\omega}\right)}.
$$
This is the generating ... |
H: Scalar product in vector/coordinate form
As I know, $a*b = |a|*|b|*cos(a,b)$ in vector form
And $a*b = (a_1,a_2)*(b_1,b_2) = a_1*b_1+a_2*b_2$.
1)
$$a*b=?$$
$$a=2i-3j+5k$$
$$b=i+2j+8k$$
SOLUTION: $a*b = (2,-3,5)*(1,2,8) = 2*1+(-3)*2+5*8 = 36.$
I'm unsure about the angle between them? Do I have to do only multiplica... |
H: Heuristic approach to winding number
I'm working on problem 8.23 of Rudin's PMA, that is:
Let $\gamma:[a,b]\to\mathbb C$ be a closed curve, $\gamma \in C^1([a,b])$ and $\gamma(t) \neq 0 \ \forall t\in [a,b]$. Show that $$\text {Ind}(\gamma)=\frac{1}{2 i \pi} \int _a ^b \frac{\gamma '}{\gamma}$$
is an integer.
... |
H: proof by negation - there is no supremum
A = { x + 1/x : x > 0 }
How do you prove that there is no supremum for this set?
I think this is the inequality needed:
M - ε > x + 1/x
M - the supremum
How do you keep from this step?
thanks
AI: Suppose the set
$$A=\{x+1/x:x>0\}$$
has a supremum. Call it $M$. Then,
$$x+... |
H: Max value of Anti-symmetric Relation
Here is the question: Let A be a set with |A| = n and R be a relation on A that is anti-symmetric. What is the max value for |R|? How many antisymmetric relations can have this size?
So I'm not sure how to answer this. My first inclination is to just use the formula that tells y... |
H: Grandi's Series; tends to $1/2$, but why is this considered a valid sum?
Grandi's series,
$$1+1-1+1-1+1-1+...$$
can be expressed as the below:
$$\sum_{n=0}^\infty(-1)^n$$
Two valid sums that make sense to me are $1$, and $0$, depending on how you approach the series. $(1+1)-(1+1)-(1+1)-...=0$, and $1+(1-1)+(1-1)+(1... |
H: Assignment: Find the number of parameters in the general solution to a system of linear equations
This is a question given in an assignment I'm working on:
If the coefficient matrix $A$ in a homogeneous system of 33 equations with 28 unknowns is known to have rank 12, how many parameters are there in the general s... |
H: How can we derive the pseudo inverse of a matrix from its Singular value decomposition?
For a matrix $M$ with its singular value decomposition $UΣV^T$, the pseudo inverse of $M$, i.e., $M^+$ is $VΣ^+U^T$.
How can I derive the pseudo inverse(Moore–Penrose) $M^+$ from the singular value decomposition of a matrix $M... |
H: Why nonlinear programming problem (NLO) called "nonlinear"? What does "nonlinearity" actually mean? Is it "not linear" or something different?
My teacher in the course Mat-2.3139 presented the same definition as in Wikipedia for the nonlinear programming problem here
but he did not specify what the nonlinearity a... |
H: Group algebras, Maschke's lemma and direct sums of matrix algebras
Let $G=\{g_1,g_2,\dots,g_n\}$ be an arbitrary finite group. We consider its representations over $\mathbb{C}$. There is Maschke's theorem which states that each representation of $G$ is a direct sum of irreducible ones. I'm trying to link this resul... |
H: Computation of $n$-th order difference of falling factorial
I was reading a difference equation textbook and came across a problem. The question asks to compute ${\Delta}^nt^{\underline3}$ for $n=1,2,3,...$, where $t^{\underline3}$ is the falling factorial "$t$ to the $3$ falling" and ${\Delta}^n$ is the $n$-th ord... |
H: Why are two directions enough for the Cauchy-Riemann equations to imply differentiability?
If the complex function $f(z)$ is complex differentiable $\Rightarrow$ the Cauchy Riemann equations hold. $($This is because if $f'(z)$ is the same no matter in what direction $\delta z\rightarrow 0$. Choosing the special cas... |
H: prove that there is no supremum for mn/1+m+n
D = { mn/(1+m+n) } for m,n natural numbers.
To simplify the expression, I presumed m=n, which means:
D = { n^2 / (2n + 1) }
Now, I know by intuition there is no supremum, for this series convergencing to infinity, but I couldn't find a formal way to prove it.
I thought a... |
H: Integral $\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$
Is it possible to evaluate this integral in a closed form?
$$\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$$
AI: We first remark the following identity:
\begin{align*}
\int_{0}^{\infty} \frac{x... |
H: One question about topology.
Why is the set $ A=\{ (x ,x^{-1}):0<x\leqslant 1\}$ is closed in $\Bbb R^2$ but is not bounded?
Why is the set $ S=\{(x,\sin(x^{-1})) :0<x\leqslant 1\}$ is bounded in $\Bbb R^2$ but is not closed?
I am just a beginner in topology, so I hope someone can answer it in detail. Thanks.
AI... |
H: Prove that the set $H$ is a subspace of $\mathbf{P}_3$
The question is:
Consider the vector space $\mathbf{P}_3$ of all polynomials of degree at most 3 with real coefficients.
Prove that the set $H$ of all polynomials $p$ in $\mathbf{P}_3$ which vanish at $t = 2$ (meaning $p(2) = 0$) is a subspace of $\mathbf{P}_... |
H: Problem of understanding instructions on practice sheet
Can someone help me understand the scalar product of problem 23 in the attached practice sheet? I should show that this defines a nondegenerated symmetric scalar product on the vector space of trivariate polynomials.
My problem is that I don't understand the a... |
H: Proof that $\sin(x)$ don't have limit to infinity
I just used the Heine's definition.
Let $\alpha,\delta \in \mathbb{R}$ such that $\sin(\alpha)=a$ and $\sin(\delta)=b$. Let $(u_{n})=\alpha+2\pi n$ and $(v_{n})=\delta+2\pi n$ and $f(x)=\sin(x)$.
So one have,
$$\lim\limits_{n\rightarrow \infty} u_{n}=+\infty$$
$$\li... |
H: Combination with quantity
We need to ship 100 kits, each containing three beads of different colors. There are four colors: purple, blue, green, silver. We have the following quantities for each respective color: 53, 53, 85, 53. What are the fewest beads we will need to order, and what colors should we order to ... |
H: Find this limit without L'hopital Rule : $\lim_{x\rightarrow +\infty}\frac{x(1+\sin(x))}{x-\sqrt{(1+x^2)}}$
Find this limit without l'Hopital rule : $$\lim_{x\rightarrow +\infty}\frac{x(1+ \sin x)}{x-\sqrt{1+x^2}}$$
I tried much but can't get any progress!
AI: The limit does not exist. Multiply top and bottom by... |
H: Limit Comparison Test - ex. prob wrong?
I'm doing some practice problems on the Limit Comparison test from this site:
http://archives.math.utk.edu/visual.calculus/6/series.14/index.html
But I'm a bit confused on this problem: $\sum_{n=1}^\infty \sin (\frac 1n)$ .
The solution says to use the harmonic series for com... |
H: Determine if projection of 3D point onto plane is within a triangle
In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
AI: The question is a slight extension of the question given here: Ch... |
H: Example involving Uniform Continuity
Question: Could someone give an example of a sequence of uniformly continuous real-valued functions on the reals such that they converge point-wise to a function that is continuous but not uniform continuous.
My attempt so far: I managed to prove this is true in the case of uni... |
H: Prove f(x) <= x for all x>=0 if f ' (x) <= -2 for all x and f(0) = 0
The title basically states the whole question..I was trying to invoke the Mean Value Theorem on it but it hasn't worked..I was wondering if I'm supposed to solve it some other way. I just need hints, please. Thank you.
AI: If $f$ is continuously d... |
H: Can 'a family of sets' be an empty set?
1.Let $X$ be a set.
Let $\mathscr{A}$ be a family of subsets of $X$.
Here, what does 'a family' means precisely?
Does this mean $\{f(\alpha)\}_{\alpha\in A}$ for some $A\subset P(X)$ and $f:A\rightarrow P(X)$?
AI: Yes; the empty set is indeed a family of sets. |
H: Let $W$ be a vector space and let $U$ and $V$ be finite-dimensional subspaces.
a) Show that $U ∩ V$ is a subspace of $W$.
b) Show that $U + V = \{u + v : u \in U, v \in V\}$ is a subspace of $W$.
c) Show that $\dim(U+V) = \dim(U) + \dim(V) - \dim(U ∩ V)$.
I have no idea where to start. Any help?
AI: For part a) Us... |
H: Whats the difference between axiom and primitive concept?
I've read the definitions, but they are not very clear to me.
Looks like both are a premisse so evident to be accepted as true without controversy.
But, what about the axioms on the set theory?? Many of them are not evident, which contradicts the definition ... |
H: Calculating primitive roots
Wikipedia cleanly demonstrates that $3$ is a primitive root modulo $7$. Here
is the table, and my question is how do they calculate the 4th column? It appears that they take the exponent from the previous column and multiply it by $3$ but this pattern fails for the 4th row.
Is there so... |
H: Fermat's Little Theorem and Prime Moduli
I am given two distinct primes $p$ and $q$, where $$m = p*q$$ Also,
$$ \begin{cases} r\equiv 1\mod p-1\\ r\equiv 1\mod q-1 \end{cases} $$
I have to show that given an integer a, show that $$a^r \equiv a \mod (m)$$
I'm not sure how to get started. I know I can tie this in to... |
H: Starting Bisection Proof of Extreme Value Theorem
I am having difficulties beginning a proof for the following statement:
Use a proof strategy of bisection to prove that every function $f:[a,b] \to \mathbb{R}$ that is not bounded above is discontinuous at some point $c \in [a,b]$ (and discontinuous from the right o... |
H: Big-O evaluation:
I have the expression:
$$f_{k}(n,m) = (n - k)(m - k) + f_{k+1}(n,m)$$
which runs until k = n or m.
What is the big theta of this function in terms of n,m?
A naive approach is to assume that m does not vary at all since:
$f_k(n,m) < (n - k)m + f_{k+1}(n,m) $
Which gives the expression:
$$nm + (n-1... |
H: Assignment: determining sets are bases of $\mathbb{R}^3$
This is question from an assignment I'm working on:
Which two of the following three sets in $\mathbb{R}^3$ is a basis of $\mathbb{R}^3$?
\begin{align*}
B_1&=\{(1,0,1),(6,4,5),(-4,-4,7)\}\\
B_2&=\{(2,1,3),(3,1,-3),(1,1,9)\}\\
B_3&=\{(3,-1,2),(5,1,1),(1,1,1)\... |
H: If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every subordinate norm.
I need some guidance showing that:
If a matrix of the form I+B is singular, where I is the identity matrix, then for any subordinate norm $\|\cdot\|$, $\|B\|\geq1$.
AI: If the matrix is singular, then for some $v$ with $\|v\| = 1$... |
H: Sum of squares of the quadratic nonresidues modulo $p$ is divisible by $p$
Let $p$ be a prime number with $p > 5$. Prove that the sum of the squares of the quadratic nonresidues modulo $p$ is divisible by $p$.
My idea is to use the fact that any quadratic residue is congruent modulo $p$ to one integer in the set $\... |
H: Pigeonhole question and generalization
Let H be a regular hexagon with side length 1 unit.
(a) Show that if more than 6 points are specied inside H then the points of at least one pair of them are at most 1 unit apart.
(b) State and prove a generalization of the result in (a) to the situation where there are in exc... |
H: How many solution with the equation $f_{2013}(x)=\frac{x}{2013}$
let $f(x)=f_{1}(x)=\mid \cos{(2\pi x)}\mid,f_{2}(x)=f(f_{1}(x))=\mid \cos{(2\pi (\mid\cos{(2\pi x)}\mid)}\mid$
$f_{n}(x)=f(f_{n-1}(x))$,
Question:
How many solution with the equation
$$f_{2013}(x)=\dfrac{x}{2013}$$
This problem is from china s... |
H: If a sequence converges, then every subsequence converges to the same limit -- but how do I know a subsequence exists?
I have been reading the following post:
Prove: If a sequence converges, then every subsequence converges to the same limit.
I understand the idea, but I wonder, does this proof imply that such a su... |
H: In how many ways can one divide 10 people into 4 unequally sized groups?
Many questions on this site involve counting the number of ways one can divide a set of n people into equally-sized groups, but how would one do so for unequally-sized groups?
The answers for this question don't provide an explanation of how t... |
H: How do I prove the definition of a homomorphism?
The question is asking me to prove that $f(a \circ b) = f(a) \circ f(b)$.
This I believe is referencing our previous proof which tells us:
Assume $g:x \rightarrow y $ is a bijection and for an $a \in S(X)$ set $f(a)= g \circ a \circ g^{-1}$ and then I proved $f$ is... |
H: Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
Proving an Combination formula $\displaystyle \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
$\bf{My Try}$::$\displaystyle{\binom{n-1}{k}+\binom{n-1}{k-1}=}$
$\displaystyle{\frac{\left(n-1\right)!}{k!\left(n-k-1\right)!}+\frac{\left(n... |
H: Sheaf of differetials on $\mathbb{P}^n_A$
I am not clear how to display the sheaf of differetials $\Omega_{X/A}$ on $X=\mathbb{P}^n_A$ explicitly,
What is its gobal section $\Omega_{X/A}(X)$ and section on the complement of the hyperplane $T_0=0$, $\Omega_{X/A}(D_+(T_0))$, and its stalk ${\Omega_{X/A}}_x$?
I am r... |
H: Proof by Induction (discrete series)
Let P(n) denote the statement that n^p = n + kp
Base case - P(n) is true for n=1
Inductive step - Assume P(n) is true
- Show that P(n+1) is true
Show that (n+1)^p = (n+1) + cp
= RHS of P(n) + (c-k)p + 1
= LHS of P(n) + (c-k)p + ... |
H: Different real roots polynomial, roots of $P'+aP$
Let $P$ be a polynomial of degree $n$ with real roots $t_1<t_2\ldots<t_n$. Show that $P' + aP$, with $a\in\mathbb{R}$, has only real roots.
Is easy to conclude if $a=0$, by Rolle's theorem. But I can't see what to do if $a\neq0$. Any hint?
AI: Hint: Use the functi... |
H: The Derivative of $\cos(x-2)$
I would think that the solution would be $-\sin(x-2)$, but when i use WolframAlpha it says that the answer is $\sin(2-x)$. Are these $2$ answers equivalent or I am missing some fact here? Thanks in advance.
AI: Since $\sin(x)$ is an odd function, $\sin(-x)=-\sin(x)$.
In particular, $\f... |
H: How to efficiently compute $17^{23} (\mod 31)$ by hand?
I could use that $17^{2} \equiv 10 (\mod 31)$ and express $17^{23}$ as $17^{16}.17^{4}.17^{3} = (((17^2)^2)^2)^2.(17^2)^2.17^2.17$ and take advantage of the fact that I can more easily work with powers of ten ($17^2 \equiv 10 (\mod 31), (17^2)^2 \equiv 100 (\m... |
H: Prove: Sum and Difference of two distinct positive integers are both perfect squares.
I'm trying to prove that there exists two distinct positive integers whose sum and difference are both perfect squares. I cannot find any pattern or characteristic between the pairs of numbers that work i.e.
4, 5
6, 10
8, 17
1... |
H: $A_5$ has no subgroup of order 15 and 20
Show that $A_5$ has no subgroup of order 15 and 20.
I have been thinking about this problem for so much time but I'm still clueless. Can anyone tell me how to do this problem? Thanks.
I looked up online and saw some proofs with simple groups or Sylow Theorem. Can somebo... |
H: Symmetric matrices are a subspace of the space of $n\times n$ matrices
Hey, I'm trying to learn how to properly prove this problem. Any advice on how to go about this problem?
AI: It is easy to check that the sum of symmetric matrices is symmetric, and that any multiple of a symmetric matrix is symmetric. Hence i... |
H: Continuity implies Borel-measurability?
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. Is it necessary that $f$ is Borel-measurable?
I'm considering $A=f^{-1}((a,\infty))$ where $a\in\mathbb{R}$. Is $A$ necessarily a Borel set? It looks like it should be, but I'm not sure.
AI: By definition of co... |
H: how to prove mean value property for harmonic functions?
For a harmonic function $u(x)$, on domain $\Omega$ where $x \in \Omega \subset \Bbb R^n $, how to show that
$$ u(x) = \frac{1}{\omega_n R^{n-1}}\int_{\partial B_R(x)} u(\sigma) d\sigma$$
where $\omega_n$ is the area of the unit sphere $\partial B_1(x)$.
I a... |
H: About assumptions in the monotone convergence theorem
Why is the hypothesis that $\left\{f_n \right\}$ be an increasing sequence essential to the monotone convergence theorem? Could someone provide a nice, easy to understand counterexample if I were to assume otherwise? Thank you.
AI: Take for instance $f_n = \frac... |
H: Easy question about set notation
I have 2 sets and am wondering what B actually is.
A = {a, b} and B = {A}
Does B = {a, b} or B = {{a, b}}
AI: $B$ is a set that consists of another set $A$ therefore $B = \{A\} = \{\{a,b\}\}$ hence why there are two brackets instead of just one. |
H: integration using substitution: symbols $\frac{dy}{dx}$ are used as variables?
I'm learning integration using substitution and the symbols $\frac{dy}{dx}$ are used as variables which is confusing me as I thought they weren't normal variables. So if I'm integrating something and have to substitute $x^2$ so that $u =... |
H: How would I differentiate $\sin{x}^{\cos{x}}?$
How can I differentiate $\displaystyle \sin{x}^{\cos{x}}$? I know the power rule will not work in this case, but logarithmic differentiation would. I'm not sure how to start the problem though and I'm not too comfortable with logarithmic differentiation.
AI: Let $y = \... |
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