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H: Significance of starting the Fibonacci sequence with 0, 1....
DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic.
QUESTION: What is the significance of starting the Fibonacci sequence with $0,1$ ?
For instance, if... |
H: stuck trying to find a matrix using lamda?
I'm stuck with the first step converting to a matrix to find a solution. I've got:
$$ 6y + (3-\lambda)x = 0 $$
$$ (4-\lambda)y + 5x = 0 $$
I need help finding two values of $\lambda$ so that they don't have unique answers?
AI: If you are looking for eigenvalues, we have t... |
H: Pointwise Convergence of Continuous, Real valued functions
Let $(f_n)$ be a sequence of continuous, real-valued functions on $[0,1]$ converging pointwise to $f$ .
Prove that there is some closed sub-interval of $[0, 1]$ on which $f$ is bounded.
I'm struggling with a proof for this, any help will be appreciated (I... |
H: Rearranging logarithmic equation
I've tried hard to rearrange the following equation to calculate the (AGE)
$$\mu=18.8144+(-1.8443\log(\text{age}))+(-1.4032\log(\text{SBP}))+(-0.3899\cdot\text{Smoke})+(-0.5390\log(\text{TC}/\text{HDL})).$$
For example if someone with
$\mu=3.13422$,
$\text{SBP}= 140$,
$\text{smoke}... |
H: binary division and remainder
Q=A/B , Q is a real number expressed as a pair of 8 bits:
most significant 8 bits for the integer part
least significant 8 bits for the fractional part
the number is unsigned
for example:
0 0 1 0 1 1 0 1 . 0 1 0 1 0 0 0 0
Can you find the remainder of division if you know B?
exampl... |
H: Problem 4.4 - Lie Algebras - Humphreys
I read the exercise 4.4 in the book Introduction to Lie algebras and representation theory of J. Humphreys, and I do not quite understand the sentence :
We start with $L\leq\mathfrak{gl}(p,F)$ as in Exercise 4.3, and let $M:=L+F^p$, the direct sum. We then make $M$ into a Lie... |
H: does a continous $f(x)$ limit is a necessary condition for uniform convergence?
Suppose I have function sequence $S = \{f_n(x)\}_{n = 1}^{\infty}$.
and $S$ converge to $f(x)$. (I mean $\lim_{n \to \infty}f_n(x) = f(x)$)
in order that $\{f_n(x) \}_{n =1}^\infty$ converge unfirmly(uniform convergence) to $f(x)$,
doe... |
H: finding sequence for e converging at some speed
I want to find an infinite sequence that conerges to e so that the kth term of the sequence is less than 10^-k away from e. Obviously, I've considered the Taylor series, but asymptotic bounds on the truncation error don't tell you what the actual upper bound for a spe... |
H: logical statement: proving $\mathrm{len}(\psi)\leq4\cdot\mathrm{lenz}(\psi)+1$
Given a logical statement $\psi$ I want to prove $\mathrm{len}(\psi)\leq4\cdot\mathrm{lenz}(\psi)+1$
with $\mathrm{lenz}:=\textrm{number of all logical connectives}$ and $\mathrm{len:=\textrm{number of all signs}}$ of the logical stateme... |
H: What does $i^i $ equal and why?
I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator and it returned 0.2078795764 (!) How is this possible and why is the result a real n... |
H: Derivative of polynomial division in Maple
This is proably a beginner's question about Maple. I'm trying to use Maple to differentiate:
$$\frac{(z^2-1)^2}{(az-1)(z-a)}$$
Where $a$ is a constant.
On the first line, is there a way to tell Maple not to expand the denominator?
How can you tell Maple that $z$ is var... |
H: Is the cofinite topology on an uncountable set first countable?
Let $X$ be any uncountable set with the cofinite topology. Is this space first countable?
I don't think so because it seems that there must be an uncountable number of neighborhoods for each $ x \in X$. But I am not sure if this is true.
AI: You are co... |
H: Prove that for all integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
the homework question I'm having trouble with is this one.
Write a detailed structured proof to prove that for all the integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
I have the proof format and structure down... |
H: About the Collatz conjecture
I worked on the Collatz conjecture extensively for fun and practise about a year ago (I'm a CS student, not mathematician). Today, I was browsing the Project Euler webpage, which has a question related to the conjecture (longest Collatz sequence). This reminded me of my earlier work, so... |
H: Basis of the polynomial vector space
I don't understand how to find a basis for a polynomial vector space.
Can someone help me with an example?
AI: The simplest possible basis is the monomial basis: $\{1,x,x^2,x^3,\ldots,x^n\}$.
Recall the definition of a basis. The key property is that some linear combination of b... |
H: Proof that the trace of a matrix is the sum of its eigenvalues
I have looked extensively for a proof on the internet but all of them were too obscure. I would appreciate if someone could lay out a simple proof for this important result. Thank you.
AI: These answers require way too much machinery. By definition, the... |
H: Help with inf sup concept
I am very bad at maths , by definition we have
$$\limsup_{n\to\infty} x_n:=\inf_n(\sup_{k\geq n}x_k)$$
where $(x_n)$ is a sequence.
I was doing this in a finite sequence to understand this better: for example with
$$(2,3,5,4)$$
we start with
$n=1$ then we have $\sup\{ 2,3,5,4\}=5$.
$n=2$... |
H: Why does the maximal irrelevant is out of this correspondence?
I'm solving the Hartshorne's questions and I didn't understand why $S_+$ doesn't occur in this equivalence:
My reasoning
By the previous exercise, if $X$ is an algebraic set in $\mathbb P^n$, we have $Z(I(X))=\overline X= X$.
If $\mathfrak a$ is a rad... |
H: Finding the limit of a sequence; difficult
I have a sequence $(a_n)$ where $a_n=\sqrt[n]{3^n +5^n +7^n}$
I have to find the limit of this sequence. By intuition I should find the limits of two sequences $(a_n)$ & $(b_n)$ such that $(a_n)\le(x_n)\le(b_n)$
Does anyone if this will be the correct way of going for it o... |
H: Is this valid? ( (easy?) Limit Manipulation question)
I'm current studying series for a calc 2 exam.
The problem I'm on is to determine the convergence/divergence of this series:
$$\sum_{n=1}^\infty \frac n{(n+1)^n} $$
The ratio test is appropriate here, so I've set it up:
$$\lim_{n\to\infty} \frac {(n+1)}{(n+2)^{... |
H: In how many ways can four students be chosen from a group of 12 students?
Myself and my Math teacher are at a disagreement in to what the proper method of solving the question In how many ways can four students be chosen from a group of 12 students? is.
The question comes straight out of a Math revision sheet from ... |
H: Proving a triangle is isoceles
In the graphic we have an isosceles triangle, and the problem is
Calculate $\text{m}\angle BCD$
I added the point $E$ at distance $x$ from $C$ because it causes $DE=x$, after playing with geogebra. With this, the question is easily solved. Of course since the triangle is determine... |
H: Continuity type problem involving a limit?
I am trying to solve this problem involving a limit / continuity.
https://www.dropbox.com/s/aglohigapdfalny/Screenshot%202013-10-30%2020.22.40.png
I've set the equations equal to each other and ended up with:
x^2 - 4x - 20 = 0
However, using the quadratic equation and gett... |
H: Derivative of $\sin(-x) = -\cos(x)$?
Doesn't derivative of $\sin(-x)$ equal:
$f'(x) = \cos(-x)(-1)$
$f'(x)= -\cos(-x)$
how do you get $-\cos x$?
AI: $\cos(x)$ is an even function which means $\cos(-x)=\cos(x)$. Using this we have $-\cos(-x)=-\cos(x)$ |
H: Question on the formal completion of a ring $R$ w.r.t. an ideal $J$
I am trying to understand the completion $\hat R$ of a commutative unitary ring $R$ w.r.t. an ideal $J$. Please let me first recall, what I think is true (since if there is a misunderstanding already, I would be very glad for pointing it out).
The ... |
H: probability of exactly, atleast, and expected number
The people discovered that 54% of refugees who ask for asylum in the New York immigration court win asylum, but only 12% are granted asylum in the Florida immigration court.
Assume that you randomly select 20 refugees who are asking for asylum in the Florida immi... |
H: Ultrametric on a normed space (real or complex)
Given some normed space $E$ (real or complex), why is it impossible that $E$ can't be an ultrametric space?
My professor briefly said something along these lines today and I didn't follow...
Also, I am not sure what the strong triangle inequality would be for a norme... |
H: Solving integral using contour integration: $I = \int_0^{+\infty} \frac{x^a}{x^2 + 1} = \frac{\pi / 2} {\cos \frac{\pi a}{2}}$ provided $-1 < a < 1$.
I am trying to show that $I = \int_0^{+\infty} \frac{x^a}{x^2 + 1} = \frac{\pi / 2} {\cos \frac{\pi a}{2}}$ provided $-1 < a < 1$.
So I consider $I = \int_{C_R}\frac{... |
H: Determine a function that satisfies the differential equation $f''(x) = -4y$?
How would I come to this solution? Is it a matter of working backwards?
$f'(x)= -y^4 + 3$?
AI: Assuming you meant:
$$y''(x) = -4y(x)$$
The solution for these type of equations is known to be a linear combination of $\sin$ and $\cos$ funct... |
H: Partial fraction integration problem in calculus
$$ \int \frac{x^2-3x+7}{(x^2-4x+6)^2} dx = \int
\frac{1}{x^2-4x+6} + \frac{x+1}{(x^2-4x+6)^2} dx
$$ $$=\int \frac{1}{(x-2)^2 +2} +\frac{1}{2}\frac{2x-4}{(x^2-4x+6)^2} +
\frac{3}{(x^2-4x+6)^2} dx $$
Here how can we calculate last term ?
[Supplement] --------
In fol... |
H: Prove that a continuous $f$ in $(0,1)$ can be extended into its one-point compactification if the limit at both end point exist and equal
Let $X = (0,1)$. Consider the one-point compactification of $X$ (which is homeomorphism to $S^{1}$). Prove that a bounded continuous function $f:(0,1) \rightarrow R$ is extenda... |
H: Probability, Statistics - Whats the answer?
Assume that a programmer makes on average two errors in every hundred lines of code
written and that errors occurring in different lines of code are independent. Suppose the
programmer writes a software application consisting of 75 lines of code.
a. What is the probabilit... |
H: Given some ultrametric space $X$, is its completed metric $\hat{X}$ necessarily an ultrametric?
I have a lot of difficulty understanding completed metrics. In fact, I don't think I understand them at all!
How could I start showing this? I'm nearly certain that the first two properties of a metric would hold, but th... |
H: Repeated Summation function
I am writing a solution to a question, and the solution requires a lot of $\sum$ functions, is there a way to notate many $\sum$ functions in a row? for example is there one function that can simplify: $$\sum_{x=0}^n\left(\sum_{y=0}^{n-x} \frac{n!}{x!y!(n-x-y)!}\times a^xb^yc^{n-x-y} \ri... |
H: Undetermined Coefficients: $y''-2y'-3y=-3te^{-t}$
I am trying to solve the following DFQ: $y''-2y'-3y=-3te^{-t}$.
I solved the homogeneous equation to find that the general solution is $y(t)=c_1e^{3t}+c_2e^{-t}$, and therefore a good guess for the particular equation would be $Y_p(t)=Ate^{-t}$b because it is not p... |
H: Throw 10 dice, probability of getting 6 identical numbers?
I got this result when I threw a set of 10 dice (btw. for the first with this set):
What is a probability of getting 6 identical numbers (any) with 10 dice, with no particular order?
AI: If you count only throws that have exactly 6 identical dice, the prob... |
H: What is the proof that the total number of subsets of a set is $2^n$?
What is the proof that given a set of $n$ elements there are $2^n$ possible subsets (including the empty-set and the original set).
AI: Suppose you want to choose a subset. For each element, you have two choices: either you put it in your subset,... |
H: Convergence or divergence
The sum is $$\sum_{n=1}^{\infty} \frac{n+2^n}{n+3^n}$$ Is this convergent or divergent? I tried to use the divergent test but the test fail because $a_n = (n+2^n)/(n+3^n) = 0 $ as $n$ goes to infinity. Could someone point me to the right direction?
Thanks
AI: If you don't want to resort to... |
H: Why are Polynomial Time Problems Considered Tractable, and Larger Times are Not?
I've been reading up on $P=NP$, problem tractability, etc. Here's my question:
Why is it that we consider problems that can be solved in polynomial time - or algorithms/problem-solvers running in polynomial time - nice, tractable, solv... |
H: Inputting complicated equation into Wolfram
I am having a bear of a time getting this equation into Wolfram so I can solve it for E(r) = 1000.
Everything is constant except for little r, which is the cursive r in each term. In case anyone is wondering, this is the expression for the electric field from the center... |
H: Finding slope of a curve by finding the limits of secant slopes
Find the slope of the curve $y=x^2-4x-5$ at the point $P(3,-8)$ by finding the limit of the secant slopes through point $P$.
My try:
I picked another point $Q$ to get the secant $PQ$. Since $P$ is $(3,-8)$, $Q$ is $(3+h, x^2-4(3+h)^2-5)$.
The se... |
H: Proof that $e^{n}-\lfloor e^{n} \rfloor \neq \frac{1}{2} $ for all $n\in\mathbb{N}$
Let $n\in\mathbb{N}$, how can I proof that $e^{n}-\lfloor e^{n} \rfloor$ is never equal to $\frac{1}{2}$?
Thanks
AI: Hint: If we have
$$e^{n} = \frac{1}{2} + \lfloor e^n \rfloor \in \Bbb{Q}$$
then $e$ would be an algebraic number. |
H: With the pigeon hole principle how do you tell which are the pigeons and which are the holes?
For example, I was reading this example from my textbook:
Let S be a set of six positive integers who maximum is at most 14.
Show that the sums of the elements in all the nonempty subsets of S
cannot all be distinct.
... |
H: Help with area of surface of revolution $x=\frac{1}{3}(y^2+2)^{\frac{3}{2}}, 4 \le y \le 5$
The question is:
Find the exact area of the surface obtained by rotating the curve about the x-axis.
$$x=\frac{1}{3}(y^2+2)^{\frac{3}{2}}, 4 \le y \le 5$$
I'm really confused by how the solution is presented. The formulas I... |
H: Indefinite Integration (1) $ \int 2^{\log_{e}(x)}dx$ (2) $\int 2^{mx}\cdot 3^{nx}dx$
calculation of some Indefinite Integration
(1) $\displaystyle \int 2^{\log_{e}(x)}dx$
(2) $\displaystyle \int 2^{mx}\cdot 3^{nx}dx$
$\bf{My\; Try}::$ for (1) $\log_{e}(x)=t\Rightarrow x=e^t$ and $dx = e^tdt$
$\displaystyle \int 2^... |
H: Sum of multinomial coefficients with constraints
The title doesn't reflect the question properly, since I don't know enough
about combinatorics to get it right, here. Feel free to change the title.
From the multinomial theorem, we can deduce, that the sum over all
multinomials is
$
\sum_{k_1+\ldots+k_m=n}\binom{n}... |
H: Global approximation theorem in Sobolev space
${\bf Global\ Approximation\ Theorem}$(251 page inEvans's PDE book) : If $U$ is ${\bf bounded}$ in ${\bf R}^n$, then for $u\in W^{k,p}(U)$,
there exists $u_n \in C^\infty (U)\cap W^{k,p}(U)$ such that
$$u_n \rightarrow u\ in\ W^{k,p}(U)$$
For the proof, we used parti... |
H: In metric spaces, is a function uniformly continuous iff $\delta$ depends on $\varepsilon$?
Most book examples end with an expression for $\delta$ that depends on $\varepsilon$ when proving uniform continuity. What I am wondering is whether a function can be uniformly continuous as long as the distance between any ... |
H: probability of no matching or exactly one matching and generalization
I had the following question in a midterm today:
There are $10$ pairs of shoes. One randomly selects $8$ shoes. What is the probability that :
$\textbf{a)}$ There are no matching pairs of shoes in the selected shoes.
$\textbf{b)}$ There is exactl... |
H: If $G$ is simple with diameter two and maximum degree $|V(G)| - 2$, then $|E(G)| \geq 2|V(G)| - 4$
This is my try:
Because the diameter of $G$ is two and have maximum degree the number of vertex: $|V(G)| - 2$, where $|V(G)|$ is the number of vertex, then the grade for any vertex in $G$ is greater than or equal to t... |
H: Cesaro mean approaching average of left and right limits
Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$, where $\mathbb{R}/2\pi\mathbb{Z}$ means that $f$ is periodic with period $2\pi$. Let $\sigma_N$ denote the Cesaro mean of the Fourier series of $f$. Suppose that $f$ has a left and right limit at $x$. Prove that as ... |
H: Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$.
So, I can intuitively see that this is true. My proof m... |
H: Show that $\sqrt{n^2+1}-n$ converges to 0
I want to use the definition of the limit to show that $\sqrt{n^2+1}-n$ converges to 0.
The definition is as follows: if $\sqrt{n^2+1}-n$ converges to 0, then $\forall \epsilon>0$, there exists an $N>0$ such that $n\ge N \implies \mid\sqrt{n^2+1}-n\mid<\epsilon$.
Now I want... |
H: Fundamental Theorem of Calculus help
I am working on a problem set assigned to us by my professor. One item is to find the derivative of g
$g(x)=\int_a^b \dfrac{u^2-1}{u^2+1}~du$ where a=2x and b=3x.
A hint was given and it confuses me:
$g(x)=\int_a^b f(u)~du= \int_a^0 f(u)~du + \int_0^b f(u)~du$ where a=2x and b=3... |
H: Intersection of a Infinite Collection of Sets - null set or infinity?
Let's say we have a collection of sets $\bigcap_{i=1}^\infty A_i$ where $A_i=[i,\infty]$. In other words:
$$ \bigcap_{i=1}^\infty A_i = [1,\infty] \cap [2,\infty] \cap [3,\infty] \cap ... [\infty,\infty] $$
I was thinking that there is no interse... |
H: How many sequences of n letters chosen from { A,B, ..., Z } are in non-increasing, or non-decreasing order
I am studying for a test and this is one of the practice questions. I really don't understand how to start this? It looks like a derangement question to me but I might be overthinking it
AI: Let me see if I un... |
H: Calculus questions
Could anyone tell me how to solve 9b and 10? I've been thinking for five hours, I really need help.
AI: Well, Notice $f'' = \frac{1}{x} \implies f''' = -\frac{1}{x^2} \implies f'''' = \frac{2}{x^3} \implies f^{5} =-\frac{2 \times 3 }{x^4}$
$$ \therefore f^{(n)} = \frac{(-1)^{n}(n-2)!}{x^{n-1}}$$... |
H: Determining the order of the kernel and image
Let $G$ be a finite group. Let $G'$ be a group and let $\phi : G \to G'$ be a homomorphism. Let $K \leq G$ be the kernel of $\phi$ and $I \leq G'$ be the image of $\phi$.
(a) Find a formula that relates the number of elements in $G$, $K$, and $I$.
(b) Suppose $H \le... |
H: ${\rm erf}$ question: I am stuck
can somebody help me? I am stuck at this problem.
Show that
$\displaystyle{\int_{a}^{b}{\rm e}^{y}\,{\rm d}t
=
\dfrac{1}{2}\sqrt{\pi\,}\,\left[{\rm erf}\left(b\right)
-
{\rm erf}\left(a\right)\right]}$ where $y = -t^{2}$
Thanks
AI: If you can't use integration by parts, then I g... |
H: Let $f(x)=|x|$ for $x$ rational and $f(x)=1$ for $x$ irrational. Show $f$ has limits at $1$ and $-1$. Find them.
Let $f(x)=|x|$ for $x\in\mathbb{Q}$ and $f(x)=1$ for $x\in\mathbb{R}/\mathbb{Q}$, where $\mathbb{Q}$ is the set of rationals. (a) Show $f$ has limits at $1$ and $-1$. Find them. (b) Show that if $c\not=... |
H: Given probability of event
If I'm given the probability that a certain event will occur, how can I find the probability of at least $x$ events occurring given $y$ opportunities?
For example, the probability of a single coin landing heads is 50%. What are the chances of landing at least 2 heads out of 5 coins?
AI: M... |
H: Trapezium drawn in the circle
A circle is drawn inside a trapezium such that it touches all the sides of trapezium. The line joining the midpoints of the non parallel sides divides the trapezium in two parts with the area in the ratio of 3:5. If the length of the non parallel sides are 6 cm and 10 cm, then what is ... |
H: Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge?
Let $a_n = \int\limits_{0}^{n} e^{-x^4} dx$. Does $\{ a_n \}_{n \rightarrow \infty}$ converge?
$\{ a_n \} =\{ \int\limits_{0}^{1} e^{-x^4} dx, \int\limits_{0}^{2} e^{-x^4} dx, ..., \int\limits_{0}^{\infty} e^{-x^4} dx \}$... |
H: H ow to compute the integral of an absolute value
How do we compute the integral of an absolute value?
$\int |x|\,dx$
$\int_0^1 |x|\,dx$
AI: $\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left... |
H: About Diophantine Equation
This is a problem about Diophantine equation.
The problem is the following.
If $ax+by=c$ is solvable and $b\ne0$, then prove that it has a solution $x_0$, $y_0$ with $0 \le x_0 <|b|$
First I thought that $x=x_1-\frac bg k$ , $y=y_1+\frac ag k$, where $g=gcd(a,b)$, $k$ is integer and $x_... |
H: Find $f(4)$ if $x \sin(\pi x) = \int_0^{x^2} f(t)\,\mathrm dt$
The problem I am currently working on is this:
Suppose $x \sin (\pi x) = \int_0^a f(t)~\mathrm dt$ where $a=x^2$. Find $f(4)$.
I have tried this.
$$\sin (\pi x) + \pi x \cos (\pi x)= f'(x) 2x$$
$$f'(x) = \dfrac{\sin \pi x + \pi x \cos \pi x}{2x}$$
$$... |
H: How many ways to arrange $20$ items on $4$ towers
Suppose you have $20$ different rings and $4$ display towers. On each tower the rings are stacked one above another. In how many ways can they be arranged if:
[a]: The order of rings on each tower does not matter:
[b]: The order of rings on each tower matters, and t... |
H: Find $\int_0^1 \mathrm{\frac{x-1}{ln(x)}}\,\mathrm{d}x$
Find $\int_0^1 \mathrm{\frac{x-1}{ln(x)}}\,\mathrm{d}x$
I tryed this:
$\int_0^1 \mathrm{\frac{x-1}{ln(x)}} = \int_0^1 \mathrm{\frac{x}{ln(x)}} - \int_0^1 \mathrm{\frac{1}{ln(x)}}\,\mathrm{d}x$
To $\int_0^1 \mathrm{\frac{1}{ln(x)}}\,\mathrm{d}x$
Let $t=lnx $ th... |
H: Arranging numbers so that $i$ is not immediately followed by $i+1$
How many arrangements of the integers $1,2,\ldots,n$ are such that no number $i$ is ever immediately followed by $i+1$?
AI: This is sequence 255 from the Online Encyclopaedia of Integer Sequences
https://oeis.org/A000255 |
H: $G$ abelian. If $G\cong \sum G_i$ then $mG \cong \sum mG_i$
Let $G$ be an abelian group and $m \in \mathbb{Z}$. If $G\cong \sum_{i \in I} G_i$, then $mG \cong \sum_{i \in I} mG_i$.
$$\sum_{i \in I} G_i = \{ f:I\rightarrow \cup G_i \mid f(i) \in G_i \text{ and } f(i)=e_i \text{ for all but finitely many $i$}\}$$
I... |
H: Cancellation of products in an arbitrary category does not hold; does it hold with this extra condition?
Let $\mathcal{C}$ be a category with products, and let $A,B,C$ be objects of $\mathcal{C}$. Certainly, it need not be true that
$$A\times B\cong A\times C\implies B\cong C.$$
For an easy example, we could have $... |
H: strict separation theorem?
Im learning and we have a theorem that says:
Let $C$ be a non-empty, convex subset of $\mathbb R^d$ and let $p \in \mathbb R^d$ be a point which is not in the closure of $C$.
Then there exists a strict separating hyperplane, which means there exists an $\eta \in \mathbb R^d\backslash \{0... |
H: X is the largest sum of rupees which can never be paid using any number of coin of denominations Rs. 4, Rs. 8, Rs. 13 and Rs. 18
'X' is the largest sum of rupees which can never be paid using any number of coin of denominations Rs. 4, Rs. 8, Rs. 13 and Rs. 18. What is the sum of digits of 'X'?
Answer is 9. But how?... |
H: Find the value of $a>1$ such that the curve $y=a^x$ meets the line $y=x$ once and only once.
Find the value of $a>1$ such that the curve $y=a^x$ meets the line $y=x$ once and only once.
AI: The derivative of $y=a^x=e^{x\ln a}$ is $e^{x\ln a}\ln a= y\ln a$ and shall equal the derivative $1$ of $y=x$ at a point where... |
H: Find $f(\dfrac{\pi}{2})$. Provided two functions
Suppose
$f(x)=\int_0^{g(x)}\dfrac{1}{\sqrt{1+t^2}}~dt$ and $g(x)=\int_0^{\cos x}1+\sin t^2~dt$ Find $f(\dfrac{\pi}{2})$
I evaluated everything and ended up with
$f(\dfrac{\pi}{2})=\dfrac {2}{1+\int_0^{\cos\dfrac{\pi}{2}} 1+\sin t^2~dt}$
As usual I am stuck at ev... |
H: Find the number of positive integers whose digits add up to 42
Find the number of positive integers $$n <9,999,999 $$ for which the sum of the digits in n equals 42.
Can anyone give me any hints on how to solve this?
AI: Let the digits be $d_1,d_2,d_3,d_4,d_5,d_6$, and $d_7$, where we allow leading zeroes so as to... |
H: Basic Issue With the Hom Functor on Commutative Rings
In the category of $A$-modules, one has the following property: if $f:M \rightarrow M''$ is a map of $A$-modules, and the induced map $f^*:Hom(M'',N) \rightarrow Hom(M,N)$ is injective for all $N$, then $f$ itself must be surjective. The proof I know of this fac... |
H: Prove that for positive number, some multiple only has 0 and d as it's digits
Let $ n$ be a positive integer, and let $1<=d<=9$. Show that some multiple
of $n$ has $0$ and $d$ as its only digits.
I don't know how to even start this question. It's under the pigeonhole section of the text book so I am guessing that w... |
H: How prove this $ab|a^8+b^4+1$
show that: there exsit infinite $(a,b)$ such
$$ab|a^8+b^4+1$$
my try: let $a^8+b^4+1=kab,k\in N^{*}$
and I can't work,Thank you
AI: We show that for any solution $(a, b), a, b \in \mathbb{Z}^+$, we can get another solution $(a', b'), a', b' \in \mathbb{Z}^+$ such that $a'+b'>a+b... |
H: Is this "set quotient" known?
Let $A,B$ be subsets of a set $X$. Then there is a largest subset $C \subseteq X$ such that $C \cap A \subseteq B$. Explicitly, we have $C = \{x \in X : x \in A \Rightarrow x \in B\} = (X \setminus A) \cup B$. Does $C$ have a name? I would call $C$ the set quotient and denote it by $(B... |
H: Linear independence of vectors over larger fields
I was just wondering whether anyone knows an answer to the following:
Suppose that ${\mathbb F}$ is a subfield of a field ${\mathbb G}$ and that $v_1,\ldots ,v_k$ are linearly independent vectors in ${\mathbb F}^n$ (over $\mathbb F$). Is it necessarily true that $v_... |
H: Arranging 7 rows of 3 motercycles of 5 different types and 4 different colours
A playground equipment manufacturer makes a biker formation comprised of $21$ wooden
motorcycles that are fixed in place, three abreast, to form $7$ rows of $3$ motorcycles each.
She has $5$ different types of wooden motorcycle that can ... |
H: Characterization of an invertible module
Let $B$ be a commutative ring.
Let $A$ be a subring of $B$.
If $M$ and $N$ are $\mathbb{Z}$-submodules of $B$,
we denote by $MN$ the submodule of $B$ generated by the subset $\{ab\mid a \in M, b\in N\}$.
If $M$ and $N$ are $A$-submodules of $B$, $MN$ is clearly an $A$-submod... |
H: Formal proof for $\lim_{x\to\infty}x\exp(-x) =0$
I intuitively understand that
$$\lim_{x\rightarrow\infty} xe^{-x}=0$$
as the $e^{-\infty}$ approaches zero faster than $x$ approaches infinity. But this requires one to have a knowledge of the property of exponents. Is there any way to prove this formally that is mat... |
H: Help with rearranging equation to get real and imaginary parts..
I know this is so simple but my algebra is totally failing me.. I have the equation 1/1+2i and I want to extract the real and imaginary parts so I have it in the form..
Re+Im could someone just show me the algebra steps for doing this please.. Tha... |
H: Help with exponential integrals
I'm trying to find a nice expression for the following function
\begin{equation}
f_k(x)=\int_0^\infty y^k (x+y) e^{-(x+y)^2} \text{d}y.
\end{equation}
So far I know that
\begin{equation}
f_k(x)=P_k(x)e^{-x^2}+Q_k(x)\text{ erfc}(x),
\end{equation}
where $P_k(x)$ and $Q_k(x)$ are polyn... |
H: tail events and tail sigma-field
I'm working on tail-events.
I have a sequence $(X_{n})_{n}$ of random variables. Let $\tau$ be its $\sigma$-field.
From this, I defined $G_n:=\sigma (X_n,X_{n+1},...)$, so that $\tau = \bigcap_{n\geq 1}{G_n}$.
Now I should say if the event {$X_n \rightarrow \infty$} is in $\tau$, bu... |
H: How to weigh up to 100kg with 5 weights
1) You are a shopkeeper who is selling sugar between 1-100 kg .Now you have to design 5 weights in such a way that any integer weight between 1-100 can be measured in a single attempt ,without using more than 5 weights.You can't repeat weights.
He gave me time like 1 hour to ... |
H: Designing Context-Free Grammars for Sets of Strings
I'm pretty lost, and would appreciate help or solutions to the following two exercises. I don't really know where to begin or even how to correctly denote a context-free grammar. I have to design CFGs for the following two sets:
1) $\{a^ib^jc^k \mid i \neq j \;or... |
H: Binomial distribution false reasoning
While reading the answer of a previous question Binomial Distribution Question (Exactly/At Least $x$ Trials for Success), it got me thinking a little. I know the reasoning must be flawed somewhere, but I can't tell where. So here it goes:
Given
I have a biased coin, 5% chance o... |
H: Constructing an increasing function on a set A that is continuous only at the irrational points in A.
Exercise 6.2
Show that there is a strictly increasing function on $[0,1]$ that is continuous only at the irrational numbers in $[0,1]$ .
Proof Let $C=[0,1] \cap \mathbf{Q}$ and let $\left\{q_{n}\right\}_{n=1}^{\in... |
H: Solving integrals by substitution with exponent $du$
Given the integral $\int_0^4 x^3(x^2 + 1)^{-\frac{1}{2}}dx$, I have tried to choose $u = x^2 + 1, du = 2x\space dx$, thus being left with the integral $\int_1^{17} \frac{1}{2}(du)^ 3\space u^{-\frac{1}{2}}$
Is there a simple way to solve this, or do I need to fin... |
H: Computing $a^b$ as $\lim_{n\to\infty} a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n}$
Let $b \in \mathbb{R}$, then $ \forall n \in \mathbb{N}(\frac{\lfloor b \cdot 10^n\rfloor}{10^n} \in \mathbb{R})$, but does $\frac{\lfloor b \cdot 10^n\rfloor}{10^n}$ have a particular name?
And is the following correct?
Let $a,b \i... |
H: Why is the geometric multiplicity of an eigen value equal to number of jordan blocks corresponding to it?
Geometric multiplicity of an eigen value is $$ \dim \mathrm{null} (A -\lambda I)\tag 1.$$
Suppose $A$ is in jordan normal form and has two Jordan forms with eigen value $\lambda$, one of size $2 \times 2$ and... |
H: Difference between interior and set of accumulation points
I don't understand the difference between the interior of a set, and the set of all its accumulation points.
My understanding of an accumulation point is any point in a set which has an epsilon neighborhood around it, which is contained in the set- not nece... |
H: Improper integrals are "not totally Improper"
Question is to evaluate $$\int _{-\infty}^{\infty} \frac{dx}{(x^2+a^2)^2}\text {for } a>0$$
Idea is to calculate this using complex analysis/residue theory/contour integration.
Approach is consider contour $D_R$ consisting of a semicircle in upper half plane of radius ... |
H: Differential - Aproximation to $ \triangle A$ (area variation)
The central angle of a cirular sector is $80°$. It is desired to reduce it by $1°$. By how much should the radius of the sector be increased so that the area will remain unchanged if the original length of the radius is $20cm$?
Let $A$ be the area of th... |
H: Another proof for Liouville's Theorem
I'm having trouble completing a homework question which will produce an alternative proof for Liouville's Theorem. The question reads
Let $f$ be an entire function. Evaluate, for $|a|<R,|b|<R,\int_{|z|=R}\frac{f(z)dz}{(z-a)(z-b)}$. When $f$ is bounded, let $R\to \infty$ and de... |
H: Does this function satisfy "Intermediate Value Property"?
Although this problem may look easy, but I am very much confused over this.
Consider the function
$$f(x) = \begin{cases}(1-x)/|x|, & x\neq 0\\1, & x=0\end{cases}$$
Does $f$ satisfy the Intermediate Value Property on $[-2,2]$?
Thanks.
AI: $f$ does not satis... |
H: Calculating $\lim_{x \to 0}\frac1x \int_{0}^{x}f(y)dy$ when $f$ is continuous
I was trying the following problem which is :
Let $f \colon \Bbb R \to \Bbb R$ be a continuous function such that $$\lim_{x \to 0}f(x)=a .$$ Then $$\lim_{x \to 0}\frac1x \int_{0}^{x}f(y)dy $$ is which of the following:
$(A)1 \,\,(B)a \,\... |
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