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H: A property dealing with complete metric spaces
I came across a property in a textbook that caught my eye. The property is: If $X$ is a complete metric space, then the intersection of any two dense $G_{\delta}$-subsets of $X$ is dense in $X$.
This property seems simple, but I am having trouble figuring out the proof... |
H: Disk Method for Volume of Solid with negative exponent.
Use the disk method to find the volume of the solid generated when the region bounded by $y=(1-9x)^{-1/4}$, $y=0$, $x=0$, and $x=1/18$ is revolved about the x-axis.
I know that to set this problem up, I have to use the equation
$$V=\pi \int_0^{1/18} (1-9x)^{-... |
H: A condition for the image of a metrizable space to be metrizable
I noticed this problem on a previous exam that I completely missed, and I was wondering if anyone could help me out.
Suppose $f: Y \rightarrow X$ is a continuous mapping of a separable metric space $Y$ onto a compact Hausdorff space $X$. Show that $X... |
H: Why the weak * topology on the dual of a Banach space has the stronger meaning of locally compact
Let us say that for a Hausdorff topological space to be locally compact means that every point has a compact neighborhood. Why do locally compact have the property that if $x \in U$ and $U$ is open in $X$, then $x$ ha... |
H: Representing a countable field by $\Bbb N$ [transport of algebraic structure]
To make sure that we are talking about the same, I would like to post the relevant definitions I know first.
Definitions:
A pair $(G, +)$ where $G$ is a set and
$+: G \times G \rightarrow G$
is called a commutative group if it has all of ... |
H: Algebra help please, substitution method for these equations
Note: This isn't homework, I'm skipping ahead of class. Please answer all these equations, I'm deathly stuck on them.
Use the substitution method only please. (Find $x$ and $y$.)
\begin{align}
ax\left({\frac {1}{a-b}-\frac {1}{a+b}}\right)+by\left({\fra... |
H: Probability ( counting principles)
I just need an explaination to a question I saw in a statistics book. I understand the concepts but I don't really understand what the question is asking. This question asks '' Find the probability of being dealt at random and without replacement a thirteen card bridge hand consis... |
H: Can anyone find any rhyme or reason to these sequences?
In the following sequence of fractions,
$$21/8,273/32,1333/72,4161/128,10101/200,20881/288$$
the denominators are simply $8n^2$, but can't figure out what the numerator is. It's not in OEIS.
Also need help with this one if possible:
$$1/2,73/18,601/50,2353/98... |
H: Question about a defined function in D.Marker Model Theory book.
In page 162 of D.Marker Model theory book he proved that the set $S_n(F,T)$ of all $F$-types realized by some $n$-tuple in some countable model of $T$ is analytic (this is with any $F$:= $countable$ $fragment$ from an infinitary lenguage, and a comple... |
H: how can I solve for $x$?
I am having a bit of trouble solving for x when trying to find $f^{-1}$. I have $$y=\frac{x+5}{x-4}$$ How can I get x on one side? I tried multiplying both sides by the denominator but I am still left with an $x$ on both sides...
AI: You just need to expand out the brackets and re-factor:
$... |
H: How to prove $a = c$ if $a = b$ and $b = c$?
It seems like a simple question, but is actually quite hard to prove. You don't just simply say:
$a = c$ because $a = b$ and $b = c$.
Like I actually want a proof behind it, something like using formulae, arithmetic, etc.
How is this possible?
AI: In any part of mathem... |
H: How to prove that the exterior angle sum of an n-sided polygon is 360 degrees?
Just wondering about the smartest and best way to prove such a question. I know of many ways, and also I don't want to use anything related to $180(n - 2)$.
AI: I don't know exactly what are you looking for, but perhaps try this:
You can... |
H: Geometric interpretation of frechet derivative in one variable
Given a function $ f:(a,b) \to E $ where $E$ is a Banach space, then clearly when $f$ is differentiable, the derivative $ f^\prime(x) $ can be seen as the tangent vector to the curve in $E$ graphed by $f$, at the point $f(x)$.
But the book I use says t... |
H: Motivation behind definitions of the Integral without reference to Derivatives
If a (definite) Integral can simply be calculated as the difference of two Antiderivatives and Antiderivatives are simply the "reverse process" of Differentiation. Then it seems to me that the Integral can be entirely defined in terms of... |
H: Particular solution to a 3rd order ode (method of undetermined coefficients not working)
I am trying to get the particular solution to the equation -
$y''' + 4y'' + 5y' + 2y = e^{-t} $
We are taught the method of undetermined coefficients to solve such equations. However, one of the solutions of the homogenous equ... |
H: $p$ is prime and $p^2\large\mid\normalsize|G|$
Hints needed:
Let $p$ be a prime and $G$ a finite group such that $p^2\large\mid\normalsize|G|$ then $p\large\mid\normalsize|\text{Aut}(G)|$.
AI: We know there is a sylow $p$-group with order $p^{n}$, $n\ge 2$. Name this group to be $H$. Consider the inner automorphis... |
H: In how many different ways can we fully parenthesize the matrix product?
We have a finite number of matrices that we wish to compute the product of . Say we wish to compute a product of n matrices and we have the subroutine to compute a pair of matrices . We also know that matrix products are associative ie :
$\lef... |
H: Automorphisms of projective line
how does one find the set of Automorphisms of the complex projective line?
PS: no scheme theory is assumed.
AI: 0) I'll use coordinates $(t:z)$ on the projective line $\mathbb P^1(\mathbb C)$, with the embedding $\mathbb C\to \mathbb P^1(\mathbb C)$ given by $z\mapsto (1:z)$ and wi... |
H: Perturbation of an eigenvalue
Suppose $A$ is a square matrix over complex numbers and $u$ is an eigenvector with eigenvalue $\alpha$. Consider perturbing $A$ using $u$ to get $B = A + uv^H$ for some vector $v$.
Then we would like to relate the characteristic polynomial of $A$ with that of $B$ as
$(\lambda - \alpha)... |
H: How was the quadratic formula found and proven?
Possible Duplicate:
Why can ALL quadratic equations be solved by the quadratic formula?
History of Quadratic Formula
How was the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ found and proven?
AI: It seems that Alkharazmi was the first person who found the for... |
H: Find $\lim\limits_{n\to+\infty}(u_n\sqrt{n})$
Let ${u_n}$ be a sequence defined by $u_o=a \in [0,2), u_n=\frac{u_{n-1}^2-1}{n} $ for all $n \in \mathbb N^*$
Find $\lim\limits_{n\to+\infty}{(u_n\sqrt{n})}$
I try with Cesaro, find $\lim\limits_{n\to+\infty}(\frac{1}{u_n^2}-\frac{1}{u_{n-1}^2})$ then we get $\lim\limi... |
H: Is such a function of class $C^1$
Let $X$ be a real Banach space and $f:\mathbb{R} \rightarrow X$ be continuous on $\mathbb{R}$, of class $C^1$ on $\mathbb{R}\setminus \{0\}$ and there is a limit $$g=\lim_{x \rightarrow 0} f'(x).$$ Is then $f$ of class $C^1$ on $\mathbb{R}$?
AI: Rudin in his "Principles of mathemat... |
H: How to prove that the Torus and a Ball have the same Cardinality
How to prove that the Torus and a Ball have the same Cardinality ?
The proof is using the Cantor Berenstein Theorem.
I now that they are subsets of $\mathbb{R}^{3}$ so I can write $\leq \aleph$ but I do not know how to prove $\geqslant \aleph$.
Thank... |
H: Find $n$ value satisfying equations in $\mathbb Z_n$
Determine for which $n \in \mathbb N^*$ values the following equations in $\mathbb Z_n$ are satisfied:
$[3] \cdot [5] = [3] + [5]$
$[3] \cdot [5] = [27]$ and $[3] + [5] = [11]$
I can't come up with anything leading to a solution, so every piece of advice is ver... |
H: A basis of Galois extention $L/K(R)$
Ler $R$ be a normal domain with its quotient field $K$. Let $L$ be a finite Galois extention of $K$. Let $T$ be the integral closure of $R$ in $L$.
Then we can take a basis $t_1,\cdots t_m$ of vector space $L$ over $K$ so that each $t_i\in T$.
It would be very appreciated if you... |
H: Limit of a definite integral $\frac{\sin(x)}{x}$
Given the integral:
$$S=\int_{-a}^{a}\frac{\sin(ax)}{ax}$$
I get from MAPLE this result:
$$\lim_{a\to \infty}S=0$$
My question is: how can I prove this result?
Thanks.
AI: By a change of variables, you can rewrite your integral above as
$$S= \int_{-a^2}^{a^2} \frac{... |
H: complex projective space with some identification with groups
Could any one tell me how to show :
$\mathbb{C}P^n ≅ SU ( n + 1)/ U ( n)$?
$\mathbb{C}P^n ≅ S^{2 n +1}/ U (1)$ ?
what is transitive group of $\mathbb{C}P^n$?
AI: This really depends on how you define $\mathbb{C} P^n$. For example if we define $\mathbb{C}... |
H: Little help with some algebra
Note: This isn't homework, I'm skipping ahead of class. Please answer all these equations, I'm deathly stuck on them.
Use the substitution method only please. (Find $x$ and $y$.)
\begin{align}
ax\left({\frac {1}{a-b}-\frac {1}{a+b}}\right)+by\left({\frac {1}{b-a}-\frac {1}{b+a}}\righ... |
H: What is the number of distinct combinations for choosing a pair of numbers between a sorted sequence?
What is the total number of ways in which we can choose pairs of $a_i$ and $a_j$ between a sequence of increasing sorted numbers numbers $a_1,a_2,a_3,\ldots,a_n$ such that $a_1\le a_i\le a_j\le a_n$.
AI: In fact, a... |
H: Confidence interval for uniform
A random variable is uniformly distributed over $(0,\theta)$. The maximum of a random sample of $n$, call $y_n$ is sufficient for $\theta$ and it is also the maximum likelihood estimator. Show also that a $100\gamma\%$ confidence interval for $\theta$ is $(y_n, y_n /(1 − \gamma )^{1/n... |
H: Three questions on chapter 7 of Jech's Set Theory
In the proof of Pospisil's Theorem (theorem 7.6) that there are $2^{2^\kappa}$ uniform ultrafilters on $\kappa \geq \omega$, the author writes : Let $\mathcal{A}$ be an independent family of subsets of $\kappa$. For every function $f : \mathcal{A} \to \{0,1\}$, con... |
H: Image set of a function.
I am revising some old exam question that i have now reflecting back on it.. the question is to find the image set of the function.. I forgot how to do this.. can some one tell me the steps not just the answer...cause if I remember correctly we need to have a limit right?
$$
f(x) = 3x^2 + 1... |
H: A double inequality by Ramanujan
Ramanujan presented this double inequality without showing a proof, namely:
$$\sqrt\pi \left(\frac{x}{e}\right)^{x}\left(8x^3+4x^2+x+\frac{1}{100} \right)^{1/6}<\Gamma(x+1)<\sqrt\pi \left(\frac{x}{e}\right)^{x}\left(8x^3+4x^2+x+\frac{1}{30} \right)^{1/6}, \space x \ge 0$$
I wonder ... |
H: Power curve calculation
I have a power curve relating to a turbo trainer I use on my bike.
I save my sessions on a website called strava using speed, cadence, heart rate and time.
Using the power curve I've been given I wish to calculate the gradient of the curve, providing me with a power calculation for a given s... |
H: A question about non-compact metric spaces.
I noticed the following statement on a comprehensive exam, and I am having trouble proving it. Can anyone help?
If $X$ is a non-compact metric space, then there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f(X)$ is dense in $\mathbb{R}$.
Thanks ... |
H: Is it true that every normal countable topological space is metrizable?
I've been reading about and working on various proofs about metrizabililty. I'm having trouble answering the following question: Is it true that every normal countable topological space is metrizable? Then it asks to explain your reasoning.
Any... |
H: Find the basis for the space of $2\times 2$ matrices.
Question: Find a basis for $M_{2\times 2}$, the space of $2 \times 2$ matrices.
My query: Can I say that all of the following are valid answers(out of many other possible answers),
$\bigl[ \begin{smallmatrix} 1 & 1\\ 1 & 1
\end{smallmatrix} \bigr]$
$\big< ... |
H: How long should I insist?
While studying math, I usually get stuck on some topics, how long should I insist trying to understand it alone? What are the strategies for when this happens?
AI: This is an answer that is based on my opinion and not necessarily fact.
It really does depend on what the topic/problem is. In... |
H: If $G$ is a finite group and $g \in G$, then $O(\langle g\rangle)$ is a divisor of $O(G)$
Does this result mean:
Given any finite group, if we are able to find a cyclic group out of it (subgroup), then the order of the cyclic group will be a divisor of the original group.
If I am right in interpreting it, can one... |
H: Is $T$ singular or nonsingular for $T(A)=AB - BA$?
Let $B$ be a complex $n\times n$ matrix. Prove or disprove: The linear operator $T$ on the space of all $n\times n$ matrices defined by $ T(A) = AB - BA $ is singular.
AI: $T(I)=0$, where $I$ is the identity. So.. the map has a nontrivial kernal. |
H: $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}\iff x \equiv a\pmod{pq},\ $ i.e. CCRT = Constant case CRT [Chinese Remainder Theorem]
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$ for $p\neq q$ distinct primes.
Where can I start with this proof?
It looks similar to the Chinese Rema... |
H: Method of Least Squares-Why is it preferred?
Possible Duplicate:
Why do we use a Least Squares fit?
To find the normal equations for derivation of the regression line we use the method of least squares, We want to make the error smallest for each individual,so we may take summation of mod of the error term for e... |
H: Do all representations of finite groups have one-dimensional subrepresentations?
Let V be a representation of a finite group G, and $v\in V$ - a nonzero vector. Put $$u = \sum_{g\in G} gv.$$
Then for any $g\in G$ we have $gu = u$ and therefore $<u>$ is a subrepresentation of V.
I know there is an error here since t... |
H: Induction proof: $S$ contains powers of 2 and predecessors implies $S={\bf N}$
Let $S$ be a subset of $\mathbb{N}$ such that
$$2^k\in S\quad\forall\ k\in\mathbb N$$
and,
$$k\in S\Rightarrow k-1\in S\quad\forall\ k\ge2,k\in\mathbb N$$
Show by induction that $S=\mathbb N$.
AI: Hint $\ $ If one views this as a sort of... |
H: Distinct-degree factorization in finite fields
In $\mathbb{Z}_3$
$$ x^9 : x^4+x^3+x^2+2x+1 = x^5+2x^4+2x^2+2x$$
with remainder of $x$.
In $\mathbb{Z}_7$
$$x^7 : x^4+5x^3+x+5 = x^3+2x^2+4x$$
with remainder of $x$.
Is this random? Or is there some kind of trick which I do not see to avoid polynomial long-divison by h... |
H: Bayesian estimation
I am a bit confused with this example.
Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean λ is known to be either 1 or 1.5. Suppose the prior mass function for λ is $π_λ(1)$ = 0.4, $π_λ(1.5)$ = 0.6 .
A collection of 5 rolls of t... |
H: Can you take Dedekind Cuts of the real numbers?
My professor defined Dedekind cuts in the following way:
"Given two nonempty sets E,F $\subset \mathbb{R}$, we say that the pair (E,F) is a Dedekind cut of $\mathbb{R}$ if
$E \cap F = \emptyset $
$E \cup F = \mathbb{R}$
$x < y$ for all $x \in E$ and for all $y \i... |
H: group order $p^2q^2$ will be abelian
let $(G,*)$ a group order $p^2q^2$ such that $q\nmid p^2 -1 $ y $p\nmid q^2 -1$ then $G$ is abelian.
for Sylow theorem $n_p\equiv 1\mod (p)$ then $n_p = 1, q, q^2 $ but $n_p\neq p^2$ the same form $n_q \neq q^2$ but what next??
AI: You can show that your group $G$ is as $H\time... |
H: Irreducible polynomials on $\mathbb Z_2$ and $\mathbb Z_k$
I had to solve many exercises to take my algebra exam lately and lots of them are on polynomials, whereas I am asked to look for roots and verify if the given polynomials are reducible in a specific $F[x]$ field or not.
I have noticed something while doing... |
H: Given $p$, $e$, and $m^e$ mod $p$, compute $m$ mod $p$?
Is it possible to compute $m$ mod $p$, if you are given $p$, $e$, and $m^e$ mod $p$?
$p$ is prime.
$(p-1)$ and $e$ are relatively prime.
AI: compute $d$ such that $ed\equiv 1\pmod {p-1}$, then $(m^e)^d\equiv m^{ed}\equiv m^{k(p-1)+1}\equiv (m^{p-1})^k.m\equiv ... |
H: When does $\lfloor (n-1)x \rfloor + \lfloor x \rfloor = \lfloor nx \rfloor$?
I am trying to find the conditions under which $\lfloor(n-1)x\rfloor + \lfloor x \rfloor = \lfloor nx \rfloor$. The trivial case is whenever $x \in \mathbb{Z}$. If $n = 2$, then $x - \lfloor x \rfloor \lt \frac{1}{2}$.
As $n$ increases how... |
H: Connection between derivatives in zero of some functions
Let's assume that we have two functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ of class $C^\infty$ such that $f(x)=g(x^p)$ for $x \in \mathbb{R}$, where $p \in \mathbb{N}$ is fixed.
I wish to calculate $g^{(n})(0)$ for $n \in \mathbb{N}$.
In particular case... |
H: Question about the Souslin property
I've been working on problems dealing with the Souslin property. A topological space $X$ has the Souslin property if every pairwise disjoint family of non-empty open subsets of $X$ is countable.
I came across this problem, and I am having trouble with formulating a proof.
A metr... |
H: On cardinality of equivalence classes
Say $R$ is the set of all Riemann integrable functions $f:[a,b]\rightarrow \mathbb{R}$. We define an equivalence relation in $R$ as follows: $f$ and $g$ are said to be integrally equivalent iff they differ only on a (Lebesgue) measure zero set. Prove that the collection of thes... |
H: What morphisms of schemes fulfill this property?
Let $f:Y\to X$ and $g:Z\to Y$ be morphisms of schemes and take the product $Z\times_X Y$ with respect to the composition $g\circ f:Z\to X$ and $f$.
What property of a scheme morphism does $f$ have to fulfill such that the projection $Z\times_X Y\to Z$ is an isomorphi... |
H: Every point closed $\stackrel{?}{\Rightarrow}$ space is Hausdorff
If a topological space is Hausdorff, then every point is closed.
Is the converse true?
Edited: Let $G$ be a topological group and $H$ the intersection of all
neighborhoods of zero. Since every coset of $H$ is closed, every point
of $G/H$ will be clos... |
H: Existence of NFA for this language
I'm given a task to find (and prove) such language $L$ in the alphabet $\Sigma = \{a,b\}$ with all words less than $1000$ in length, for which any DFA/NFA will have more than $10^{10}$ of states. For the first part of the task, with DFA, I've found such language (see below), but c... |
H: In Group theory proofs what is meant by "well defined"
What is exactly meant or required for a mapping to be well defined? I was reading the First Isomorphism theorem (link), and the first thing the proof does is define a map and find out if it's well defined.
Intuitively it makes sense, but what are the requiremen... |
H: How to prove that an element is zero in $G/\langle M\rangle$
Suppose $(G,+,0)$ is an abelian group and $(M,+,0)$ is an abelian submonoid, so $M\subseteq G$, the operation of the monoid coincides with the restricted operation of the group, the units coincide and $a,b\in M\Rightarrow a+b\in M$.
Let $\langle M\rangle$... |
H: Chebyshev's first $\vartheta(x)$ function question
This was an exercise using the first Chebyshev function, $\vartheta(x)= \sum_{p \leq x} \log p.$ The question is simply how to prove (2) below, the rest is my two thoughts on how to proceed. [Edit: trimmed after answer.]
$p_k$ is a number with k prime factors inclu... |
H: Evaluating $ \int_{1}^{2}\left ( x+3 \right )\ln x \:dx $
I would appreciate some help on evaluating this integral: $$ \int_{1}^{2}\left ( x+3 \right )\ln x \:dx $$
I know that I must use integration by parts, but I am stuck at the first step. I chose $u = \ln x$, but what should I do with $(x+3)$, because derivati... |
H: Constructing a discrete Markov chain
Klenke gives a construction for a discrete Markov chain (Section 17.2 "Discrete Markov Chains: Examples", pp. 353-354). I don't understand several points in this construction, as indicated below.
The construction
Let $\left(R_n\right)_{n\in\mathbb{N}_0}$ be an independent family... |
H: Number of Divisor
How to find the Number of divisors of a number 'n' that are also divisible by another number 'k' without looping through all the divisors of n?
I tried the following:
Stored powers of all prime factors of n in an associative array A and did similarly for k, stored the powers of all primes factors ... |
H: Find a plane $\Pi'$ given two points, a plane, and a prescribed distance.
I have to work this out:
Let $\Pi:2x-y+2z=4$, $P=(2,2,2)$ and $Q=(1,0,1)$. Find a plane $\Pi'$ containing $P$ and $Q$, and a point $R$ of $\Pi$ such that $d(P,R)=d(P,\Pi)$.
Now, observe that $Q$ is in $\Pi$, so we can find $d(P,\Pi)$ as fol... |
H: Showing the Weyl element is sent to its negative by a certain element of the Weyl group
Let $\mathfrak g$ be a complex semi-simple Lie algebra with a choice of positive root system. Let $\rho$ be the Weyl element, i.e. the sum of the fundamental weights (or half the sum of the positive roots). There is a unique e... |
H: how to prove this extended prime number theorem?
A Generalized Prime Number Theorem?
Conjecture
Let $n$ and $k$ be positive integers with $n - 50 > k^2 > 0$ and $n$ sufficiently large. Then for the odd primes we have, when $p$ is the biggest odd prime $\le n$,
$$
3^k + 5^k + 7^k + 11^k + ... + p^k \sim \frac{n^{k+... |
H: Proving the odd integer power function is injective over the reals
I am taking Abstract Algebra and in the first assignment we were asked to determine which values of $n$ make the following function injective.
$f: \mathbb{R}\rightarrow \mathbb{R}$
$x \mapsto x^n$ $| n \in \mathbb{N^+}$
Obviously the case where $n$... |
H: What is the basis for a proof?
What is the basis for a proof? I am trying to understand what a proof is, what is the most simple example of a proof? like can you have a 'proof' for $1+1=2$?
AI: What is a proof? Well, commonly in mathematics this either means "a rigorously convincing argument" or a "formal proof". I... |
H: The difference of meaning between X and E(X) or Y and E(Y) in covariance formula?
Is there some basic examples to show that there are difference meaning between X and E(X) in the covariance formula?
AI: Suppose $X$ and $Y$ are random variables which can take the following pairs of values with the following probabi... |
H: Two Lie groups which are isomorphic but not homeomorphic
I am looking for an example of two Lie groups which are isomorphic as groups but not homeomorphic as topological spaces. Or, even more interestingly, a proof that two such groups cannot exist. Does anyone have an example or a proof?
AI: $\mathbb{R}$ and $\mat... |
H: Graph theory: adjacency vs incident
Okay, so I think if 2 vertices are adjacent to each other, they are incident to each other....or do I have it wrong? Is this just different terminology. I thought I was totally clear on this for my class, but now I am doubting myself reading the book and looking at my notes. I ju... |
H: Method for determing least period length of sequence that is the sum of two modulo sequences.
How does one determine the length of the smallest period of the following sequence:
$$f[n] = \left< n \right>_N - \left< n \right>_M$$
where $\left<n\right>_N$ represents $n \pmod N$ and $\left<n\right>_M$ represents $n \p... |
H: Proving that matrix $B=C$
Let $A,B$ be $n×n$ matrices such that $BA = I_n$, where $I_n$ is the identity matrix.
$(1)$ Suppose that there exists $n×n$ matrix $C$ such that $AC = I_n$. Using properties of the matrix multiplication only (Theorem 2, sec. 2.1, p.113) show that $B = C$.
$(2)$ Show that if $A, B$ sat... |
H: Is thickness supposed to affect a box's dimensions?
If I'm told that a box's height is 10 and that the thickness is 5, should I
Assume that the box's real height is 15.
Assume that the box's content height is actually 5, and due to the thickness it is said that the total box height is 10
Edit: Think of it as a ca... |
H: Probability/Combinatorics Problem. A closet containing n pairs of shoes.
A closet contains n pairs of shoes. If 2r shoes are chosen at random, (where 2r < n), what is the probability that the chosen shoes will contain no matching pair?
I have tried thinking about this problem on my own and have not gotten much fart... |
H: Needing to find a variance from a data set below
I am trying to get the variance of the data set below. I get a negative number upon applying 1/2(2.5)^2 + 1(3)^2 + 0.5(4.5)^2 + 1(5)^2 + 1(6)^2 - (17.5)^2 = variance, but a negative and weird number.
Please use this data below from the link, the problem is as is and ... |
H: Show $ \int_0^\infty\left(1-x\sin\frac 1 x\right)dx = \frac\pi 4 $
How to show that
$$
\int_0^\infty\left(1-x\sin\frac{1}{x}\right)dx=\frac{\pi}{4}
$$
?
AI: Use
$$
\int \left(1-x \sin\left(\frac{1}{x}\right)\right) \mathrm{d} x = x - \int \sin\left(\frac{1}{x}\right) \mathrm{d} \frac{x^2}{2} = x - \frac{x^2}{2}\... |
H: Difference between Slope and Gradient
It has been a few years since studying contour maps.
Often I hear slope and gradient interchangeably in describing steepness.
Does anyone know any good definitions and analogies of slope and gradient.
Thanks,
Amanda
AI: Best Answer - Chosen by Voters
A gradient is a vector, a... |
H: Conditional probability of guessing the correct rank of a card drawn from a standard deck
Alice draws a card from a standard deck, doesn't look at it, and gives it to Bob and he looks at it and says "it's not a King" (a spoiler of this type will, henceforth, be referred to as a Bob-Spoiler). Alice wants the probabi... |
H: Explain the Stokes -theorem from differential from into Integral form
I want to understand the Stokes -theorems deeper. I am trying to understand the operation from
$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$
to
$$\oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = - \int_{... |
H: What is the reason to define the variance?
Possible Duplicate:
Usefulness of Variance
i know Standard deviation is equal to a constant times the shortest distance from the point $(x_1,x,_2,\ldots,x_n)$ to a line $(r_1,r_2,r_3,\ldots,r_n)$ where the r is equal provide that these objects is in n-dimensional euclid... |
H: Length of the least period of a "double" modulo sequence.
What is the length of the smallest period of the following sequence
$$f[n] = \left< \left< n \right>_N \right>_M$$
where $\left<n\right>_N$ represents $n \pmod N$. Is there a special term for performing nested modulo operations?
For $N$ even (say 8) and $... |
H: Closing 3 numbers
I have 3 numbers that physically must add up to zero. Unfortunately, each is obtained from a noisy measurement and they don't add up exactly. Assuming the noise is Gaussian and given 3 corresponding standard deviations (one for each number), what is the correct way to change the numbers so that ... |
H: Probability that k or more i.i.d. Bernoulli random variables are realized 'success'
Let $X_n\sim \operatorname{Bern}(p)$.
How does one calculate that there will be $k$ more more successes in a row? I can only think of taking the complement of the Geometric c.d.f..
$$1-\sum_{i=0}^{k-1} p^i(1-p)$$
But this can get t... |
H: What's the answer of (10+13) ≡?
As to Modulus operation I only have seen this form:
(x + y) mod z ≡ K
So I can't understand the question, by the way the answers are :
a) 8 (mod 12)
b) 9 (mod 12)
c) 10 (mod 12)
d) 11 (mod 12)
e) None of the above
AI: The OP may be taking a Computer Science course, in which if $b$ ... |
H: Is the set of all Heisenberg Matrices Connected?
The set of all $3\times 3$ matrices of the form
$$\begin{pmatrix}
1 & a & b\\
0 & 1 & c\\
0 & 0 & 1\\
\end{pmatrix}$$
forms a group surely, are they connected?
AI: Sure. Each is path connected to the identity, by a linear segment, replace $a,b,c$ by $at,bt,ct$ with ... |
H: Find whether or not an inverse exists algebraically
Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus?
I'm an undergrad engineer and can obviously solve this using basic calculus, but I'm having trouble finding a way to explain how to do this using o... |
H: Is there a picture of the tangent lines through a point inside a circle?
In Shafarevich's Basic Algebraic Geometry, he gives an example of why we use algebraically closed fields. He considers a circle $C$ and a point $P$ outside of it. He constructs the polar of $P$ as the line joining the points of contact of the ... |
H: Show $\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$
How to show that
$$
\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1
$$
where $\psi(x)$ is the digamma function.
AI: Simply use recurrence relations:
$$
\frac{\Gamma(x)}{\psi(x)} = \frac{x \cdot \Gamma(x)}{x \cdot \psi(x)} = \frac{\Gamma(x+1)}{x \cdot \psi(x+1) - 1}
$$
Sinc... |
H: Help solving CDF for transformation of $ \ge 2 $ random variables or if it's impossible.
Suppose independent random variables $U, T$. Let $U$ have continuous Uniform distribution over $(0, 2\pi)$. Let $T$ have Exponential distribution with $\lambda = 1$. Let random variable $Y$ be a function $Y=r(U, T) = \sqrt{2T... |
H: Prove: Every compact metric space is separable
How to prove that Every compact metric space is separable$?$
Thanks in advance!!
AI: Hint: Consider the countable family of coverings $\mathcal U_n=\left\{B\left(x,\frac1n\right)\mid x\in X\right\}$ for all $n$, use compactness to distill a countable family of points, ... |
H: Distribute distinct objects in identical boxes
Number of ways to distribute $6$ distinct objects to $3$ identical boxes such that each box should have atleast one?
$\mathbf {Is\ there\ any\ standard\ formula\ for\ these\ sums}$, as we have for $\langle identical\ object,\ distinct\ box\rangle$ pair as $$N+R-1\choos... |
H: Show that $X$ is independent of $\mathcal{G}$ given $E(X|\mathcal{G})$
Given that $\mathcal{G}$ is a sub-sigma field. $Z=\mathbb E(X|\mathcal{G})$, how can we show that $X$ is independent of $\mathcal{G}$ given $Z$?
I am struggling about the interpretation of this result.
By definition, we only need to show that gi... |
H: Can you help me to identify this unknown notation in his arm?
What's this in his arm? I've never seen such notation.
AI: This look like Frege's "Begriffsschrift". Try to translate it using the syntax desciption in the Wikipedia article I linked. |
H: What does it intuitively mean that the divergence of a vector field is 0?
I was going through an Electrodynamics textbook, and as a prerequisite it requires elemenets of Vector calculus and Multivariable calculus. They discussed divergence, and gave examples of fields with positive and negative divergence. And they... |
H: Where can I find a comprehensive guides to linear algebra and calculus?
I'm a software engineer with a keen interest in all sorts of artificial intelligence and machine learning applications, and also quantum computing.
Both areas require quite a bit of linear algebra and calculus, and I'm afraid mine is quite rust... |
H: Compute this limit of series: $\lim_{n\rightarrow \infty} \frac{1}{n^{3}}\sum_{k=1}^{n-1}k^2 $
Compute this limit of series:
$\displaystyle \lim_{n\rightarrow \infty} \frac{1}{n^{3}}\sum_{k=1}^{n-1}k^2 $
I used the definition of the definite integral
$\displaystyle \int_{a}^{b}f(x)dx=\lim_{n\rightarrow \infty} S_{n... |
H: Proving Conjugacy Class Theorem with Lemma and why is it necessary
The Theorem states that:
$\left | G \right |=\left | Z(G) \right |+\sum_{g'}\frac{\left | G \right |}{\left | Z(g) \right |}$
The author goes on to state a Lemma: If $S(a)$ is the conjugacy class containing $a \in G$ and $\gamma(a) =$ number of elem... |
H: Action of Groups over group quotient
Let $G$ a group, $H \le G$ and $A= G/H $. Then there exists an action $\phi: G\rightarrow S_A$ such that the kernel is the maximum subgroup normalized by $G$ and contained in $H$.
AI: the homomorfism is well defined for $g\in G$: $\phi_g (aH)=gaH $ for $ aH\in G/H$.
then
$$\begi... |
H: Finding $E(\bar{X^2}|\bar{X})$
Possible Duplicate:
Finding $E\Bigl(\overline{Y^2}\Bigm|\overline{Y\vphantom{Y^2}}\Bigr)$ by Basu's theorem?
Suppose $X_1,\ldots,X_n$ are a random sample of $N(\theta,1)$. if $\bar{X^2}=\displaystyle\frac{1}{n}\sum_{i=1}^n X_i^2$, how can I find $E(\bar{X^2}|\bar{X})$?
AI: It's lat... |
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