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H: Is a probability distribution a measure on $( \mathbb{R}, {\cal R} )$?
Let $(\Omega, {\cal B}, P )$ be a probability space. Let $( \mathbb{R}, {\cal R})$ be the usual
measurable space of reals and its Borel $\sigma$- algebra. Let $X: \Omega \rightarrow \mathbb{R}$ be a random variable.
I am wondering if the followi... |
H: "Any"; universal or existential quantifier?
For any integers $m$ and $n$, if $7m+5n=147$, then $m$ is odd or $n$ is odd.
$$Q(m,n) \equiv 7m+5n=147$$
$$∀m∀n: Q(m,n) → \bigl(m \not\equiv 0 \!\!\pmod 2 \lor n \not\equiv 0 \!\! \pmod 2\bigr)$$
Am I right in assuming $\forall$ means "any" in this case? It doesn't seem... |
H: Monoidal Category - Equalizer
We have a category $\mathbb C$ with finite products and terminal object $1$. Further $\mathsf{Mon}(\mathbb C)$ is the category of monoids in $\mathbb C$ where a monoid is a triple $(M,m:M\times M \rightarrow M,e:1 \rightarrow M)$ which fulfills certian associativity and unit axioms. I ... |
H: Finding eigenvalues.
I'm working on the following problem:
Define $T \in L(F^n)$ (T an operator) by
$T(x_1,...,x_n) = (x_1+...+x_n,...,x_1+...+x_n)$
Find all eigenvalues and eigenvectors of $T$.
I've found that the eigenvalues of $T$ are $\lambda = 0$ and $\lambda = n$. Is there an easy way to prove that these are ... |
H: How to prove this max absolute value equation?
How to prove this equation?
$$\max(|x_1-x_2|,|y_1-y_2|) = \frac{\left|x_1+y_1-x_2-y_2\right|+\left|x_1-y_1-(x_2-y_2)\right|}{2}$$
AI: Let $a = x_1-x_2$ and $b = y_1-y_2$ to simplify the problem to proving $$\max(|a|,|b|)=\frac{|a+b|+|a-b|}{2}$$
Now let $c=|a|, d=|b|$ a... |
H: Math question from the GMATprep
If $xy=1$ what is the value of: $2^{(x+y)^2}/2^{(x-y)^2}$
A 1
B 2
C 4
D 16
E 19
$(x+y)^2/(x-y)^2$ because $2$ just cancels out from numerator and denominator, right?
AI: This isn't as hard as you think.
xy = 1*1 or -1*-1
If xy = 1*1 then:
2^(x+y)^2/2^(x-y)^2 = (2^(2)^2)/(2^0^2) =... |
H: Is it possible to extract some expressions from modulus brckets?
I have an expression:
$$\left| x-y-a\cdot \left( \frac{1}{x^2}+\frac{1}{y^2} \right) \right|$$
where $a>0$, $x>0$ and $y>0$. Can I make something with it to get:
$$\left| x-y \right| \cdot SomethingElse$$
AI: Of course:
$$\left| x-y-a\cdot \left( \fra... |
H: Partial fraction of 1 over (x^2+1)^2
Its been years since I solved PF. Now I am having hard time solving this partial fraction
$$
F = \frac{1}{\left( x^2+1\right)^2}
$$
I proceeded with(Is this right ?)
$$ \frac{1}{\left( x^2+1\right)^2} = \frac{A}{\left( x+\iota\right)} + \frac{B}{\left( x+\iota\right)^2} + \frac{... |
H: Finding Taylor approximation for $x^4e^{-x^3}$
I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$
I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just wondering is there a trick for this one or do I just have to use brut... |
H: Constructing invariant subspaces from scratch. An algorithm
So, basically i am trying to prove that there exist a basis w.r.t which there exists an upper triangular matrix in a complex field.
Most of the books which i read incorporate induction as a method which i find sadly non intuitive. Please have a look at the... |
H: Count of 3-digit numbers with at least one digit as 9
Find the number of $3$ digit numbers (repetitions allowed) such that at least one of the digit is $9.$
I've posted my answer below. If there is a better way to solve this question, I would be glad to learn about that.
AI: Suppose that 'three-digit' means $abc... |
H: How to find the representation of Lie algebra
I read a book about the Lie algebra, but I really don't understand the calculation of $ad(X)$. For example, we have a Lie algebra of bases:
$$e_1=\left[\begin{array}{cc}1 & 0\\0 & -1\end{array}\right], e_2=\left[\begin{array}{cc}0 & 1\\0 & 0\end{array}\right], e_3=\lef... |
H: Identical complex functions.
Uniqueness principle theorem :
If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$.
Now my question is this:
If $f$ and g are analytic on a domain $D$, and if $f(z)=g(z)... |
H: Unit Ball with p-norm
I am having trouble understanding the definition of p-norm unit ball. What I know is that for infinity (maximum norm), then it will shape as a square. I need a "click" to understand this, can someone be so kind to explain this to me in simple words?
If the norm is the distance of the vector, ... |
H: Multiple choice question about the dimension of a space of $10 \times 10$ complex matrices
Problem: Let $A \in M_{10}(\Bbb C)$ ,the vector space of $10 \times 10$ matrices with entries in $\Bbb C$.
Let $W_A$ be subspace of $M_{10}(C)$ spanned by $\{\,A^n : n\geq 0\,\}$
Then which of the following correct ?
1)... |
H: A question related to uniqueness principle theorem.
We know that the equation $ \sin^2z+ \cos^2z=1$ which holds $\forall z \in\Bbb R$, also holds $\forall z \in\Bbb C$.
This is obvious under the shadow of following theorem:
Uniqueness principle theorem :If $f$ and $g$ are analytic functions on a domain $D$, and i... |
H: Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$
Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$.
I tried
$$x\equiv5\pmod6\\x\equiv4\pmod5\\x\equiv3\pmod... |
H: For what values of m are the roots of $x^2 +2x+3 = m(2x+1)$ real and positive
I am only able to show that to be real, $m <-1$ or $m\geq2$
Don't know how to finish solution
Answer is $2 \leq m < 3$
So far:
After expanding and factorising,
$x^2 + 2(1-m)x + (3-m) = 0 $
Roots are real if discriminant $\ge 0 $
i.e. $... |
H: What will happen to the limits of this integration?
This is a joint probability distribution function (PDF)
f(x,y) = $C(x^2 + y)$ and the limits are $x>0$ and $0 < y < (1-x^2)$
We have to calculate the marginal PDF of Y, i.e. f(y).
I get that we will integrate the joint function with the limits of x. But can't fig... |
H: Question regarding diameter of subsets of a metric space
The question is :
Find a condition on a metric space$(X,d)$ that ensures that there exist subsets $A$ and $B$ of $X$ with $A \subset B$ such that $diam(A)$ = $diam(B)$.
I know that if $X$ is a metric space and $A$ and $B$ are subsets of $X$ with $A \subset B$... |
H: how can I sort a matrix
I have coordinates of some points$(x,y)$, that I want to sort them,at the first by $y$
coordinates and second sort $x$ coordinates between those points that have equal $y$ coordinate.
Is there anyone know any function in matlab about it?
I examined it by $SORT$ function, but this function j... |
H: Determine relationship between vectors.
Using the product i would like to understand if it is possible to determine relationship between two vectors,let us consider the following problem:
Vectors $u$ and $v$ have length $1$. Which of the assumptions $(a)-(g)$ below imply that vectors $u$ and $v$ are: $$\begin{arr... |
H: Problem with definition of regular surface in classical differential geometry
I am reading Do Carmo's differential geometry book and the definition of a regular surface in the second chapter is given to be this:
I have few doubts about this definition:
1) Why we need to find a neighbourhood of point $p$? Is it bec... |
H: When a pole lies outside the circle of integration, what does Cauchy integral formula state?
I have the following complex line integral:
$$ \int_{|z| = 2} \frac{z}{z - 3} $$
My prof said it is $0$, but did not explain. He just said that the point $3+0i$ lies
outside the circle.
But the Cauchy integral theorem does ... |
H: Find the Fourier series S(t) of the period 2$\pi$
Find the Fourier series S(t) of the period $2\pi$ function
$f(t)=\begin{cases}
-1& \text{if −$\pi$ < t < 0;}\\
\;\;1& \text{if $\:$0 < t < $\pi$;}\\
\;\;0&\text{if $t = −\pi, 0, or \;\pi$ }
\end{cases}$
Use MATHEMATICA to graph partial sums $S_N(t)$ of the Fourier s... |
H: Harmonic function (PDE) - Orthogonal matrix
Let $u\in C^2(\mathbb{R}^n)$ be harmonic in $\mathbb{R}^n$, i.e.
$$
\Delta u:=\sum\limits_{k=1}^{n}\frac{\partial^2 u}{\partial x_k^2}=0\mbox{ in }\mathbb{R}^n.
$$
Let $Q\in\mathbb{R}^{n^2}$ be an orthogonal matrix (i.e. $QQ^T=E_n$). Show that then the function... |
H: How does my professor go from this logarithm to the following one of a different base?
I don't understand on the second last line how my professor goes from $2^{log_3(n)} = n^{log_3(2)}$ how is that relation formed?
AI: Note that $\log_b a = \frac{\log b}{\log a}$, so
$$2^{\log_3 n} = 2^{\frac{\log n}{\log 3}} = \... |
H: Just a question regarding continuous differentiability
$ f: [0,1] \to [0,1] $ be a MONOTONE & CONTINUOUS function. Does it always imply that: $ f(x) $ is continuously differentiable??
AI: No: Take, for example, $f(x) = \frac{1}{2} x$ for $x < \frac{1}{2}$, and $f(x) = \frac{3}{2}x - \frac{1}{2}$ for $x \ge \frac{1}... |
H: Second order implicit derivative
The equation
$\begin{equation}
x^3 \ln x + y^3 \ln y = 2z^3 \ln z
\end{equation}$
defines $z$ as a differentiable function of $x$ and $y$ in a neighbourhood of the point $(x,y,z) = (e, e ,e)$. Calculate $z_1 (e,e)$ and $z_{11}(e, e)$.
Attempt at solution
Define $F(x,y,z) = x^3 \ln x... |
H: Intersection of two congruent spirals
Let $S_1$ and $S_2$ be two congruent circular spirals in $\mathbb{R}^3$,
both with their axes passing through the origin. They are congruent in that their
radii are equal, as are their winding frequencies; but aside from being constrained
to have their axes through the origin, ... |
H: Checking of continuity
While compactness & connectedness are preserved under continuous maps, this question comes to my mind:
$f : \mathbb R \to \mathbb R$ is strictly monotone increasing function such that {$ f(x) : x \in \mathbb R$} is dense in $\mathbb R$ , then prove or disprove, that: $f$ is continuous on $\ma... |
H: Is ∑ with negative value solvable?
Is it possible to have a negative value in sigma?
e.g.
$y = \Sigma_{k=0}^{k=-2} k \times 10$
Will this give the result $(0 \times 10) + (-1 \times 10) + (-2 \times 10) = -30 $?
Or will it be $\infty$ because $k$ will be increased with $1$ until it equals $-2$ (which is never).
... |
H: An exercise in Rudin's RCA
Would you please give me some help on the following problem?
Suppose $1 \leq p \leq \infty$, and $q$ is the exponent conjugate to $p$. Suppose $\mu$ is a positive $\sigma$-finite measure and $g$ is a measurable function such that $fg\in L^1(\mu)$ for every $f\in L^p(\mu)$. Prove that then... |
H: cohomology of Eilenberg-Maclane space
In line 5, Page 394 of Allen Hatcher's book Algebraic Topology, it is claimed that $H^n(K(G,n);G)=Hom(H_n(K(G,n),\mathbb{Z});G)$ for any abelian group $G$. How to get it?
I have tried but cannot continue: by universal coefficient theorem, we need $Ext(H_{n-1}(K(G,n);\mathbb{Z})... |
H: find maximum length of sum of two vector
let us consider following problem
so let us introduce vector with $2$ coordinates ,namely
$v=(v_1,v_2)$ with length $12$,which means that
$v_1^2+v_2^2=144$
and vector $s=(s_1,s_2)$ with length $10$,so it means
$s_1^2+s_2^2=100$
now let us consider sum of two vect... |
H: Calculation of $\frac {1-q^{n+1}}{1-q} \times \frac {1-q^{n+2}}{1-q}$?
I am trying to solve this equation:
$$\sum^{n}_{i=0} q^i \times \frac {1-q^{n+2}}{1-q} = \frac {1-q^{n+1}}{1-q} \times \frac {1-q^{n+2}}{1-q} = \frac {1-q^{n}q^1}{1-q} \times \frac {1-q^{n}q^2}{1-q} = ...?$$
where
$$\sum^{n}_{i=0} q^i = \frac ... |
H: Find Greatest Common Divisor and Least Common Multiple
Find GCD (320, 112) and LCM[320, 112].
Solve the equation 320x + 112y = a in the following situations:
(i) a = 32
(ii) a = 10.
Using Euclids Algorithm to find the GCD I have the following:
320 = 112*2 + 96
112 = 96*1 + 16
96 = 16*6 + 0
GCD(320, 112) = 16... |
H: Calculate Var(XY)
I was practicing and came across this, which I couldn't solve.
Calculate Var(XY) where X ~ Uniform(0,1) and Y ~ Normal(0,1)
AI: Let $W=XY$. We want to calculate $E(W^2)-(E(W))^2$.
On the (unstated) assumption that $X$ and $Y$ are independent, all we need is $E(X^2)$ and $E(Y^2)$, since $E(Y)=0$.
... |
H: Show that there exists a $k>o$ such that solutions of this system of differential equations never cross the line $y = kx$.
For the system:
$\frac{d x}{d t} = -y$
$\frac{d y}{d t} = x(1-x) - Ay$
Where $A \geq 2$.
I want to show that there exists a $k > 0$ such that $(x(t), y(t))$ cannot cross through the line $\{ ... |
H: Inverse Laplace Transform Problem
I have this problem $\frac{1}{(s^2+1)^3}$. I have to find its Inverse Laplace Tranformation.
I already try using partial fraction but it didn't work because I found it will back to the problem form.
Any other way for solutions?
AI: You may use the residue theorem. The ILT is
$$\... |
H: combined reliability
I thought I knew the answer to this question, but further reflection is showing some holes in my knowledge; my college math is twenty-five years old and google isn't helping today.
Let's say you have a computer with five disk drives. If any one of the drives crashes, the computer is down. For... |
H: On a sum involving fractional part of an integer
I was interested in estimating the sum of the form
$$
\sum_{j=1}^{N} \{ \sqrt{j} \}.
$$
I was wondering if there is a reference or maybe some one could
help me figure out what to do.
Thanks!
$\{ \alpha\}$ denotes the fractional part of the real number $\alpha$.
AI: T... |
H: Example function $f:(0,1) \to (0,1)$ such that $f^{-1}(y)$ is uncountable for all $y$.
Intuitively the interval $(0,1)$ can be an uncountable union of disjoint uncountable sets, but I'm wondering if there is a nice function to show this easily. Namely, what is a simple example of $f:(0,1) \to (0,1)$ such that $f^{-... |
H: How would one prove that $\# (E \cup G) = \#(E) + \#(G)$?
For a set $E$ and $n \in \mathbb{N}$ we say $\# (E) = n$ if there is a bijection from $I_n$ to $E$, where
$$I_n \overset{\text{def}}= \{k\in \mathbb{N}: k \leq n\}$$
Suppose that $\# (E) = n$, $\#(G) = k$ and $E \cap G = \varnothing$. Show that
$$\#(E \cup G... |
H: Divisibility of a prime number
I need help with the following:
Show that:
If $p$ is prime such that $p$ divides $a^n$
Then $p^n$ divides $a^n$
I know that if $p$ is a prime and divides a square number $a$ then $p$ also divides $a$ but I'm not sure how to apply this to the given problem.
AI: Notice that if $p$ is ... |
H: P p-sylow with $ P ⊂ Z(G) $
For me this problem is hard .
If $ P⊂ Z(G)$ is a $p$-sylow of $G$ then there is a $N$ normal subgroup of $G$ such that $P ∩ N = 1$ and $G = PN$.
I try use the Schur-Zassenhaus: All normal subgroup cop rime has a complement and all conjugates are conjugate, but i stuck in this problem ... |
H: Ask a question about the definition of trace norm.
Suppose $X\in \mathbb{R}^{M\times N}$
$\|X\|_*=\mathrm{trace}(\sqrt{X^*X})=\sum_i^{\min{M,N}}\sigma_i$
where $\sigma_i$ is the singular values of $X$.
I know that $\mathrm{trace}({X^*X})=\sum_i^{\min{M,N}}\sigma_i^2$. Thus, the following equation holds.
$\sqrt{\mat... |
H: If $p$ is prime and any integer $k>1$, then $p^{\frac 1k} $ is irrational. Prove this by assuming $p^{\frac 1k}$ rational
I've tried setting $p^{\dfrac 1k}= \dfrac a b$, and then raising $p^{\dfrac 1k}$ to the $k^\text{th}$ power, but I'm stuck.
AI: let $p^{\frac1k}$ be rational then $p^{\frac1k}=\frac ab$ with $(a... |
H: Verify the identiy, (cos(x+h) - sin x)/h = cos x * ((cos h - 1)/h)- sin x * (sin h /h)
Verify the identity:
$$\frac{(\cos(x+h) - \cos x)}{h} = \cos x \left(\frac{\cos h - 1}{h}\right)- \sin x \left(\frac{\sin h }{h}\right)$$
=(Cosxcosh - sin x sin h -cos x)/h.
I can't think of where to go from here.
Thanks
AI: I... |
H: Prove that $x$ is any positive real number greater than $0$, $x>0$, then exists $N$ in the natural numbers such that $\frac{1}{N^3}
Prove that $x$ is any positive real number greater than $0$, $x>0$, then exists $N$ in the natural numbers such that $\frac{1}{N^3}<x$
My steps:
Well I begin with $N\in\mathbb{N}$ and ... |
H: Linearly independent vectors and matrix
If $\{v_{1},v_{2},\cdots,v_{n}\}$ is $n$ linearly independent vectors
in $\mathbb{R}^{n}$, what would be necessary and sufficient condition of $A$ ($n\times n$ matrix) $A$ so that the vectors $Av_{1}$,
$Av_{2}$, $\cdots$, $Av_{n}$ are linearly independent?
AI: Hint:
The vect... |
H: Semigroup isomorphism
Does there exist an isomorphism between the semigroups $S(4)$ and $\mathbb{Z}_{256}$?
$S(4)$ is the set of all maps from the set $X$ to itself and $X =\{1, 2, 3, 4\}$,
$S(4)$ is a semigroup under the composition of mappings, and $\mathbb{Z}_{256}=\{0, 1, 2, \dots , 255\}$ is t... |
H: Approximation to Lattice points inside circle
In the Wikipedia here, there is an approximation that $N(r)=\pi r^2+E(r)$ and then it says that Gauss managed to prove that $E(r)\leq2 \sqrt2 \pi r$ can anyone prove the original formula and tell me how Gauss got his approximation.
AI: Around each lattice point consider... |
H: How to calculate expected value?
Expected numbers of 'yes' for following data
A = [1,2,3,5]
B = [5,6,7,8]
A 'yes' is when a^b > b^a where a is value randomly chosen from A and b is value randomly chosen from B . Also numbers a and b are discarded from A and B after randomly chosen.
Please explain how to solve it.
... |
H: Understanding the theorem about the almost disjoint sets partial order
I have read the Kunen's set theory. But I get a struck to understand the proof of a theorem.
Let $\mathcal{A}\subset\mathcal{P}(\omega)$ be a family of pairwise almost disjoint sets, then we define $(\Bbb{P}_\mathcal{A},\le)$ as $\Bbb{P}_\mathc... |
H: Conditional probability, Bayes' rule and chain rule
I'm reading the following paper A Rational Account of the Perceptual Magnet Effect and I'm puzzled by equation (3) on page 2:
$$p(T|S,c) \propto p(S|T)p(T|c)$$
where $T$, $S$ and $c$ are random variables. Apparently it should be trivial, but I tried to play with c... |
H: How do I caclulate the probability of hash collisions?
I have a 10Gb file and the entire file is overwritten with random data every day.
Afterwards, I divide the data into blocks and hash each block to generate a fingerprint.
I am trying to choose an appropriate1 hash size to make it unlikely that that will be a co... |
H: Prove that a module is projective or not
Let $R=\left(\begin{array}{cc}\mathbb{Q}&\mathbb{Q}\\0&\mathbb{Q}\end{array}\right), J=\left(\begin{array}{cc}0&\mathbb{Q}\\0&0\end{array}\right)$. Prove that $R/J$ is not a projective $R$-module.
I'm really misunderstand about $R/J$, about it's equivalent class. I can u... |
H: Why is it sufficient for a normal to only be orthogonal to 2 vectors on a plane instead of 3?
The following is an excerpt from my textbook:
It is clear geometrically that there is a unique plane containing any 3 points $A,B$ and $C$ that are not all on a line. In determining the equation of this plane, the problem... |
H: Question concerning finite rings
Let $R$ be a finite ring. Is it possible that $R$ has an element $a\in R$ such that $a$ is a left divisor of zero and $a$ is not right divisor of zero?
Thanks.
AI: Let $S$ be a finite semigroup of left zeroes ($ab=a$ for all $a,b\in S$), $F$ a finite field, $FS$ the semigroup ring ... |
H: Finding centre of ellipse using a tangent line?
I need to determine the centre coordinates (a, b) of the ellipse given by the equation:
$$\dfrac{(x-a)^2}{9} + \dfrac{(y-b)^2}{16} = 1$$
A tangent with the equation $y = 1 - x$ passes by the point (0, 1) on the ellipse's circumference.
I'm guessing I have to find th... |
H: What is the simplification of sum of $i(i+1)$?
I am trying to simplify $$\frac{\sum_{i=1}^n i(i+1)}{n(n-1)}$$.
I am not sure how to simplify ${\sum_{i=1}^n i(i+1)}$ part.
How can I simplify it?
AI: You may know the wellknown results $$ \sum_{i=1}^n i=\frac{n(n+1)}{2}$$
and
$$ \sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}{6}... |
H: Unsolved uniform continuity
Prove or disprove that: the function $ f : \mathbb R \to \mathbb R $ defined by $ f(x) := x^{1/3} \log (1+|x|)$ is uniform continuous!
I have tried all possible (I THINK) to prove it to be uniform continuous but failed. I think it would be NOT uniform continuous!Now, here I am still unab... |
H: Relationship Little '$\mathcal{o}$' and Big '$\mathcal{O}$'
I'm learning about asymptotic analysis and, as a starting point, big and little o definitions.
On the Wikipedia page, http://en.wikipedia.org/wiki/Big_O_notation
further down under the heading for little-$\mathcal{o}$ notation it states
"In this way little... |
H: Polynomials in nature
What polynomials occur in "nature"? I am interested in polynomials of degree three and higher. I am aware of Stefan Boltzmann Law and Chemical Equilibrium Examples.
Edit:
There are some formulas under beam deflection category.
I am particularly interested in full polynomials, as opposed to pu... |
H: Semigroups isomorphism
Does there exist an isomorphism between the semigroups $S(4)$ and $\mathbf Z_{256}$.
$S(4)$ is the set of all maps from the set $X$ to itself and $X = \{1, 2, 3, 4\}$. $S(4)$ is a semigroup under the composition of mappings and $\mathbf Z_{256} = {0, 1, 2, … , 255}$ is t... |
H: Need help with detail on proof regarding intermediate values of a derivative.
This is Theorem 5.12 in Rudin's Principles of Mathematical Analysis.
Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f^{'}(a) < \lambda < f^{'}(b)$. Then there is a point $x \in (a,b)$ such that $f^{'}(x)=\lambda$.
P... |
H: Apollonius circle, its radius and center
I've got the following set:
$\{|z-a|=k|z-b|\}$, where z is a complex number, a an b are fixed, and $k>0$,$k \ne 1$.
I need to prove that this is a circle (called Apollonius circle).
I also have to prove that this circle's radius is equal to $k|a-b||1-k^2|^{-1}$ and it's cent... |
H: Show that two logical expressions are equivalent without using the truth table.
I would like to show that the expression (S => P) AND (NOT S => Q) and the expression (S AND P) OR (NOT S AND Q) are equivalent.
I am not interested in any solution using truth tables or exhaustive trying of truth assignments, I find t... |
H: Existence of some linear mappinng's problem
Asume that $X$ is a linear finite dimensional vector space over $\mathbb R$ and $W$ is its subspace.
Let $f$, $h$ be linear mappings from $X$ to $X$ such that $f(X)=W$ and $h(X) \subset W$.
Is there a linear mapping $g:X \rightarrow X$ such that
$$
h=f\circ g
$$
AI: H... |
H: Equivalent definition of locally compact when $X$ is Hausdorff. How did they get $\overline{V} \cap C$ is empty?
The theorem in Munkres' Topology is
Theorem 29.2. Let $X$ be a Hausdorff space. Then $X$ is locally compact if and only if given $x \in X$, and given any neighborhood $U$ of $x$, there is a neighbordh... |
H: Conditional Probability using Venn Diagrams
In class we have been told (for now) to use Venn diagrams to solve probability questions, we were also set this question for homework. Instead of answers, a tip, or a nudge in the right direction would be help me the most.
Two events $A$ and $B$ are such that $P(A) = \fr... |
H: Evaluation of the integral.
$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$
The integration is taken along the imaginary axis,
an integer $n\gt 0$ and
$\epsilon \in \left[-1, +1\right]$
How to evaluate $I\left(n, \epsilon\right)$... |
H: $1_{\limsup A_n} = \limsup 1_{A_n} $
Do you have some hints on how to prove the following relation? I think it should be quite straightforward, but I cannot see it.
$$
1_{\limsup A_n} = \limsup 1_{A_n} $$
AI: Each side is a way of writing the function
$$ x\mapsto\begin{cases}1 & \text{if }x\in A_n\text{ for infinit... |
H: A difficulty solving a limit
Could anyone help me with this limit?
$$\lim_{x \to \infty} (\log_2^2 x - \sqrt{x})$$
All my usual method fail with this one, Wolfram Alpha looks at me funny, and I know the answers is $\ - \infty$, but have not idea whatsoever as to how to come to this conclusion.
AI: Hint:
$$
\log_2^2... |
H: Using change of variables, solve the integral and show the domain obtained by the change.
I need to solve the following integral using change of variables:
$$\int\int_D\frac{\sqrt[3]{y-x}}{1+x+y}dA$$
where D is the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$.
I tried to change the variables to $u=y-x$ and $... |
H: Solving for x in a equation involving natural logarithms
How would I solve for x in this equation here:
$$\ln(x)+\ln(1/x+1)=3$$
I realize that the answer is $e^3-1$, but I am not sure as to how to get it.
Any input is appreciated.
AI: $$ \ln x + \ln \left( \frac{1}{x} + 1 \right) = \ln \left( x(1/x + 1) \right) = 3... |
H: limit 0 times infinity, rewrite to find the limit
I need some help with: $\lim_{x\to 0+} x^3\cdot e^{1/x}$. How to start?
I've tried substitution $(y=1/x)$ without any luck.
I would prefer not to use L'Hopitals rule and apologizes for a bad title line.
AI: You can carry on with your substitution. In the case $y = 1... |
H: Adding Logarithms
Studying for my midterm.
Solve the following algebraically:
$$\log_2x+\log_2(x+4)=5$$
So I know that $\log_b(mn)=\log_b(n)+\log_b(m)$ therefore:
$$5=\log_2(x(x+4))$$
$$\text{or}$$
$$5=\log_2(x^2+4x)$$
$$5=2^{x^2+4x}$$
Now I don't know how to solve for x at this point. I'm stuck. Please let me know... |
H: Monic polynomials with integer coefficients
We have $\Pi_{j=1}^n (z-z_j)$ a polynomial with integer coefficients. Is also $\Pi_{j=1}^n (z-z_j^k)$ for k=1,2,3,... a polynomial with integer coefficients?
AI: Yes.
The coefficients of the latter are symmetric polynomials in the $z_j$, hence are (integer!) polynomials i... |
H: Characterization of $\mathrm{Im} f=\ker f$
Let $E$ a finite dimensional space over a field $\mathbb F$ and $f\in\mathcal{L}(E)$. The question is to prove the equivalence of:
$\mathrm{Im} f=\ker f$
$f^2=0$ and there's $h \in\mathcal{L}(E)$ such that $hf+fh=\mathrm{id}$
I proved that $2.\Rightarrow 1.$ and $1.\Righ... |
H: Showing $\lor$ in terms of $\to$ and $\lnot$
I was looking at a question a user posted yesterday (link below). And one of the answers mentioned one could express $∨$ in terms of $\to$ and $\lnot$.
In terms of $\land$ he stated it can be expressed like such:
\begin{align*}
p \land q &\equiv \neg\neg p \land \neg\n... |
H: Proving that $n^2-l$, $n^2$ and $n^2+l$ can't all be perfect squares
I tried to write a proof and used the argument that if $n^2$ is a perfect square, $n^2-l$ and $n^2+l$ can't both be perfect squares. However, I can't find a proof for this statement. Can you help me with this?
What I have tried:
Suppose that $n^2-... |
H: Number of ways to fill a grid with balls
Find the number of ways to fill a $3\times 3$ grid (with corners defined as $a,b,c,d$) if you have 3 black and 6 white marbles.
Note: This question was asked in an e-litmus exam and is not an assignment question.
AI: area of the grid =$9$
No. of ways of arranging 3 black mar... |
H: Find number no of ways to fill a grid with balls[Part 2]
Find the number of ways to fill a 3*3 grid (with corners indistinguishable) if you have 3 black and 6 white marbles.
Approach till now[Incorrect]:
No. of ways of arranging 3 black marbles or 6 white marbles = $(^9C_3)$ = 84.
Since there are 4 different ways t... |
H: Speed of object towards a point not in the object's trajectory?
Trying to study for my mid-term, but I'm having slight difficulties understanding what I'm supposed to do in this one problem:
A batter starts running towards first base at a constant speed of 6 m/s. The distance between each adjacent plate is 27.5 m. ... |
H: Interesting trigonometric equation
Find all $x\in \mathbb{R}$ such that
$\arccos x = 2\arctan \sqrt{\frac{1-x}{1+x}}$
Now, here is my approach, please state anything that is not correct/fully motivated.
$\cos 2\theta =2\cos ^2\theta -1$
$\cos ^2\theta =\frac{\cos ^2\theta}{1}=\frac{\cos ^2\theta}{\cos ^2\theta+\sin... |
H: Continuity of a Function with complex analysis
Problem:
Let $f$ be defined $$f(z)=\frac{{{\rm Re}(z^{2})}^2}{\left \| z^2 \right \|}$$ if $z\neq 0$ and $f(0)=0$.
Prove that $f$ is continuous at $0$.
Does any one have any idea on how to solve it?
AI: Recall that $|\Re z|\leqslant |z|$ so...? |
H: Rate of Change of Cylindrical Roll's Volume as it Unrolls
This is purely a "for-fun" question. I was minding my own business in the washroom this morning when I began to unroll some toilet paper from the roll, and in typical Breaking Bad fashion (sorry if you don't understand the reference) I had a serious moment w... |
H: Probability question: given $P(A|B)$ and $P(B)$ how do I find $P(A)$?
I have a probability distribution for some quantity $A$ given a fixed $B$, i.e. $P(A|B)$. I also have a prior distribution $P(B)$ for $B$. I'm trying to find the distribution $P(A)$.
I had thought about using Bayes' theorem which implies that:
$P... |
H: What am I doing wrong when trying to find a determinant of this 4x4
I have to find the determinant of this 4x4 matrix:
$
\begin{bmatrix}
5 & -7 & 2 & 2 \\
0 & 3 & 0 & -4 \\
-5 & -8 & 0 & 3 \\
0 & -5 & 0 & -6 \\
\end{bmatrix}
$
Here is my working which seems wrong according to the solutions. What am i doing wro... |
H: Example of a commutative ring with identity with two ideals whose product is not equal to their intersection
I need a specific example of a commutative ring with identity, and two ideals in the ring whose product is not equal to their intersection.
I know that for two such ideals I and J, IJ = I ∩ J if I + J = R. I... |
H: Show uniform convergence (Gamma function)
In order to exchange limit and integral of
$\displaystyle\lim_{n\to\infty}\int_{0}^{n}x^{s-1}\left(1-\frac{x}{n}\right)^n dx$,
I need to show uniform convergence of $f_n(x)=x^{s-1}(1-\frac{x}{n})^n$. Am i right?
It converges pointwise to $f(x)=x^{s-1}e^{-x}$, so $|f(x)-f_n(... |
H: Elementary number theory, sums of two squares
Suppose x can be written as a sum of two squares of integers, and y can be written as a sum of two squares of integers. Show that xy can also be written as a sum of two squares of integers.
AI: $$(a^2+b^2)(c^2+d^2)=(ac-bd)^2 +(ad+bc)^2.$$
Remark: let $u=a+ib$ and $v=c+i... |
H: Is $0^\infty$ indeterminate?
Is a constant raised to the power of infinity indeterminate? I am just curious. Say, for instance, is $0^\infty$ indeterminate? Or is it only 1 raised to the infinity that is?
AI: No, it is zero.
Consider the function $f(x,y) = x^y$ and consider any sequences $\{(x_0, y_0), (x_1, y_1), ... |
H: Derive the equation of the locus of all points (exact question in description)
Derive the equation of the locus of all points such that the line joining a point in the locus to the point (6,2) and the line joining the same point to the point (2,6) are perpendicular.
I need help with this proof for my calculus clas... |
H: Equivalence class help
I have a question that goes as follows:
Let d be a positive integer. Define the relation Rho on the integers Z as follows: for all m,n element of the integers.
m rho n if and only if d|(m-n)
Prove that rho is an equivalence relation. Then list its equivalence classes.
Now the first d that com... |
H: Finding the permutation that shows two permutations are conjugates method?
Problem:
Given $\sigma=(12)(34)$ and $\gamma=(56)(13)$ find $\tau\in S_6$ with
$\tau^{-1}\sigma\tau=\gamma$
Attempt: I'm kind of new to this but from what I understanding find $\tau$ that satisfies this will show that $\sigma$~$\gamma$ righ... |
H: Closed under intersections
I read this definition: "A collection C of subsets of E is said to be closed under intersections if A ∩ B belongs to C whenever A and B belong to C."
How could the intersection of ANY A and B belonging to C ever NOT belong to C?? Whats the point of this definition?
AI: Let $E=\{1, 2, 3\}$... |
H: Proving supremum for non-empty, bounded subsets of Q iff supremum in R is rational
Let E be a nonempty bounded subset of ℚ. Prove that E has a supremum in ℚ if and only if its supremum in ℝ is rational and that in this case, the two are equal.
This seems intuitive enough, and I know that not all nonempty, bounded s... |
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