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Proposition: For a set $X$ and its power set $P(X)$, any function $f\colon P(X)\to X$ has at least two sets $A\neq B\subseteq X$ such that $f(A)=f(B)$.
I can see how this would be true if $X$ is a finite set, since $|P(X)|\gt |X|$, so by the pigeonhole principle, at least two of the elements in $P(X)$ would have to map... |
Must a polynomial function $f \in \mathbb{R}[x_1, \ldots, x_n]$ that's lower bounded by some $\lambda \in \mathbb{R}$ have a global minimum over $\mathbb{R}^n$?
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Let $\kappa$ a cardinal of cofinality $\omega$; let $C \subseteq \kappa$ be a unbounded countable subset. Why is then $C$ closed (and thus a c.u.b.)? This means that if $\delta < \kappa$ is a limit ordinal, and $C \cap \delta$ is unbounded in $\delta$, then $\delta \in C$.
I doubt that this is true; however, it's a sho... |
Let $\Pi_1 : 7x-5y-2z=0, \Pi_2 : 5x-4y-z=0$ and $\mathbb{L}$ the line that passes through points $P=(-2,3,-3)$ and $Q=(-1,2,-1)$. Find all planes $\Pi$ that satisfy:
$\Pi \cap \Pi_1 \cap \Pi_2 = \emptyset$
$d(R,\Pi)=\sqrt{14} \forall R \in \mathbb{L}$
So far, I've found out that $\Pi_1\cap\Pi_2= { X : \lambda (1,1,1)... |
Question If $p > 3$ is a prime and $$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{p} = \frac{a}{b}$$ then prove that $p^{4} \mid (ap-b)$.
There is an exercise in Herstein which states, if $p > 3$ is prime and if $$1 + \frac{1}{2} + \cdots + \frac{1}{p-1} = \frac{a}{b}$$ then $p^{2} \mid a$. (Page 116, Problem 2).... |
Let $G$ be a finite group, and let $p^{\alpha} \mid |G|$, where $p$ is a prime. Now does this imply $p \mid |Aut(G)|$?
Clearly if $|G| \leq 2$, then the Automorphism group is the trivial group, so one can see that this need not be true for $\alpha =1$. I am curious to know for higher powers of $\alpha$.
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If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.
I do not understand anything more than the following.
Elementary row operations.
Linear dependence.
Row reduced forms and their relations with the original matrix.
If the entries of the matrix are not from ... |
as a function of a real variable, apparently.
Part of the freedom in choosing a proof is that you get to choose what definition of $e^{ix}$ to start from -- do you use a differential equation? a power series? a definition in terms of trig functions?
Another bit of freedom is that you get to choose what definition of $\... |
Singular value decomposition (SVD) and principal component analysis (PCA) are two eigenvalue methods used to reduce a high-dimensional data set into fewer dimensions while retaining important information. Online articles say that these methods are 'related' but never specify the exact relation.
What is the intuitive re... |
Let $\kappa$ ba weakly inaccessible cardinal. Why are there $\kappa$ regular cardinals $\lambda < \kappa$? I've tried a recursive construction, but I don't know what to do in the limit step. Supremum does not work, since then we loose regularity.
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Let us suppose to have a set of $N$ elements and to choose $m$ elements out of the set. I know I can do it in $ N \choose m $ ways; but I want to know how many of these combinations contain $k$ predefined elements (say $1,2,...,k$).
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For a periodic (and not so periodic) function, it is always possible to use Fourier series to find out the frequencies contained in the function.
But what about function that cannot be expressed in mathematical terms? For example, this graph (accelerogram):
Is there anyway to apply a sort of Fourier series, in order t... |
This queston can be regarded as a variant of this one:
Does a section that vanishes at every point vanish?
Let $X,Y$ be varieties, $M$ be an $\mathcal{O}_X$-Module and
$$\pi:X\times Y\rightarrow X$$
be the projection. Let $s$ be a local section of $M$, which vanishes along each "horizontal line":
$$\forall y: i_y^* s=... |
Find all polynomials $p(x)$ such that for all $x$, we have $$(x-16)p(2x)=16(x-1)p(x)$$
I tried working out with replacing $x$ by $\frac{x}{2},\frac{x}{4},\cdots$, to have $p(2x) \to p(0)$ but then factor terms seems to create a problem.
Link: http://web.mit.edu/rwbarton/Public/func-eq.pdf
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In Exercise 5 (f) of Angus Taylor's Advanced calculus (p. 659) one is asked to find the value of the following integral if convergent:
$$I:=\underset{R}{\iiint}\dfrac{x^2 y^2 z^2}{r^{17/2}}\mathrm dV$$
where $R$ is the unit sphere $x^2+y^2+z^2\leq 1$ and $r^2=x^2+y^2+z^2$.
Observing that $\dfrac{x^2 y^2 z^2}{r^{17/2}}\... |
I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare stuff and I noticed this assertation in my notes:
Given $a \cos\theta \pm b \sin\theta = c$ then:
$a \sin\theta \pm b\c... |
Let me first start out by saying as you might have guessed, this is a homework problem. Therefore I am not looking for an answer to the question. I am looking for help in how to analyze it (My textbook is horrid).
I have a grid of 9 squares surrounded by a circle.
I can use three colors to color each region with no reg... |
I have a set of RGB colour values detected by a camera $C_{i_{RGB}}$ which are to be described by the following:
$C_{i_{RGB}} = X F_{i_{rgb}}$
where $F_{i_{rgb}}$ is the component incident light at the detector, where the components $rgb$ are fixed at narrow bands and $X$, an $(n\times 3)$ matrix, is a description of t... |
Back in the day, before approximation methods like splines became vogue in my line of work, one way of computing the area under an empirically drawn curve was to painstakingly sketch it on a piece of graphing paper (usually with the assistance of a French curve) along with the axes, painstakingly cut along the curve, w... |
On my homework today, we had to find all the normal subgroups of $D_{n}$, the dihedral group of order 2n. I solved the problem by looking at how the conjugacy classes change based on whether n is even or odd and then constructed the normal subgroups as unions of the conjugacy classes.
I have 2 questions:
(i) Is there a... |
Say I have a triangle with vertices $(0,0), (2,4), (4,0)$ that I want to rotate along the origin. Rotation by multiples of $90^{\circ}$ is simple. However, I want to rotate by something a bit more complicated, such as $54^{\circ}$. How do I figure out where the vertices would be then?
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I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the proof?
Lemma: Let $Z\in\text{GL}(n,\mathbb{R})$ be a general $n\times n$ r... |
How does one prove that for a prime $p \geq 5$ the sum : $$\sum\limits_{0 < k < \frac{2p}{3}} { p \choose k}$$ is divisible by $p^{2}$.
Since each term of $\displaystyle \sum\limits_{0 < k < \frac{2p}{3}} { p \choose k}$ is divisible by $p$, only thing remains is to prove that the sum $$\sum\limits_{ 0 < k < \frac{2p}... |
A $1-1$ function is called injective. What is an $n-1$ function called ?
I'm thinking about homomorphisms. So perhaps homojective ?
Onto is surjective. $1-1$ and onto is bijective.
What about n-1 and onto ? Projective ? Polyjective ?
I think $n-m$ and onto should be hyperjective as in hypergroups.
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What are some interesting applications of the concept of homomorphism?
Example: If there is a homorphism from a ring $R$ to a ring $r$ then a solution to a polynomial equation in $R$ gives rise to a solution in $r$. e.g. if $f:R \rightarrow r$ and $X^2+Y^2=0$ then $f(X^2+Y^2)=f(0), f(X^2)+f(Y^2)=0, f(X)^2+f(Y)^2=0, x^2... |
I was droodling a bit and a given moment I drew the following construction:
It appears that the three blue intersections are collinear (red line), no matter how I draw the construction lines. If this is always true, I assume that this a know fact [otherwise I have my first theorem! :-) erm conjecture, since I can't ... |
First there was arithmetic with numerical calculations (i.e., one unknown on one side of an equation). Then algebra with manipulations of variables (many unknowns anywhere in an equation). Then systems are studied that differ from ordinary arithmetic but share some of the same properties (equations where the unknowns r... |
Using trignometric identities ( double angle forumlas) one can see that $\cos{2x} = 2 \: \cos^{2}{x} - 1$ can be expressed as a polynomial of $\cos{x}$, where $p(\cos{x})=2 \: \cos^{2}{x}-1$. Then its natural to ask the same question for the sine function.
Can we express $\sin{2x}$ as a polynomial of $\sin{x}$?
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I've got a path that is made up of multiple points - i.e. 0,0 0,50 50,50 70,20
If I just draw this line on the screen it looks quite harsh as it puts a sharp angle at the joining of each point.
Hence I was wondering what the bezier curves algorithm/method would look like which I could call that automatically change ... |
Can anyone give me an example of a vector space $V$ such that there is no norm which is complete on $V$?
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Define $R(0)=0=\emptyset, R(n+1)=P(R(n))$ and $R(\omega) = \cup_{n < \omega} R(n)$. Thus $R(\omega)$ is the set of all sets, which are build out of finitely many braces and $0$.
Consider the following relation $E$ on $\omega$: If and only if the $n$th number in the binary representation of $m$ is $1$, then $n E m$. Now... |
This is such a basic question I'm sure, but I've been trying to find a robust solution to it for a while.
Lets say I have the numbers 0-100 in series.
As the number increases, I want another number say 1000, to decrease. so that when you got to 100, it was at 0
0 = 1000
50 = 500
100 = 0
What is this kind of series call... |
Ok so I'm a programmer, I'm not a mathematician, I've got a minor in math but I didn't even do particularly well at it so please bear with some possibly really stupid thoughts. Just please try to explain to me why what I'm asking is stupid so that I don't keep making a fool of myself.
I recently found myself thinking ... |
I hope this question is suitable for the site.
I recently had to work with Mathematica, and the experience was, to put it kindly, unpleasing. I do not have much experience with similar programs, but I remember not liking much Matlab or Maple either. The result is that I am a mathematician who likes programming, but I n... |
I was trying to obtain this result:
integrate e^|x| dx
The result is
$$ \int e^{|x|} dx = \frac12 e^{-x} \left( (e^x-1)^2 \operatorname{sgn}(x) -2e^x +e^{2x} - 1 \right) \color{gray}{\,+ \text{constant}} $$
which is fine but I would like to see the 2 part solution for $x>0$ and $x<0$, I know you can see that because ... |
I have been thinking a little bit about the binary decimal expansion of reciprocal prime numbers; and I have a few questions.
I found this neat table (Wayback Machine) which lists the binary expansion of many fractions, and I was trying to find some patterns.
Here are my questions, for brevity I say a natural number ha... |
Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on.
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I happened to receive this from my friend.
Let $a,b \in \mathbb{N}$, such that $a^{n}+n \: \bigl|\: b^{n}+n$ for all $n \in \mathbb{N}$. Prove that $a=b$. How do we proceed?
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I was going to ask this on Stack Overflow, but finally decided this was more math than programming. I may still turn out to be wrong about that, but...
Given a number represented in binary, it's fairly easy to derive a decimal representation using integer division and remainders to extract digits in reverse order. You ... |
Assume you have a triangle in the plane defined by its three vertices at $(x_0,y_0)$, $(x_1,y_1)$, and $(x_2,y_2)$. Is there a general expression for the moments of the triangle, where the moments are defined as averages by the following expression:
$$\langle x^n y^m \rangle = \frac{1}{A}\int_\triangle x^n y^m \, dA $$... |
I'm working through a discrete maths text book and was stumped as to how to prove the closed form solution of the Hofstadter G-Sequence
$a(0) = 0$ and $a(n) = n - a(a(n-1)), n \geq 1$
The closed form is $a(n) = \lfloor(n+1)\mu\rfloor$ where $\mu$ is the inverse of the golden ratio.
The text book says to prove the f... |
Is a prime number in the decimal system still a prime when converted to a different base?
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We know that a group $G$ cannot be written as the set theoretic union of two of its proper subgroups. Also $G$ can be written as the union of 3 of its proper subgroups if and only if $G$ has a homomorphic image, a non-cyclic group of order 4.
In this paper http://www.jstor.org/stable/2695649 by M.Bhargava, it is shown ... |
Taken from : here
A mall has sweep stakes where they give you a coupon for every X dollars you spend (amount is not important). You fill the coupon with your contact details and put it in the box. Every two weeks, they draw 10 coupons, give their owners prizes, and discard the rest of the coupons.
Now, thanks to some v... |
One of my math professors gave me the following challenge. It isn't graded, it's just for fun.
Consider the function:
\begin{equation*}
f_n(x)=x+3^3x^3+5^3x^5+...+(2n-1)^3x^{2n-1},~x \in (0, 1).
\end{equation*}
I want to find which of the following functions $f_n$ is getting close to as $n$ gets larger:
$ \displaystyle... |
I'm doing some old exams and found following question:
Set $S={{1,2,3}}$ is given. Provide an
example of binary operation in set S,
binary relation in set S and a
function $f:S\rightarrow R$.
So, I'm thinking about $+_4$ as operation, but wouldn't it be a relation too? I can take ordered pair from $S$ and for e... |
Consider the Banach space $\ell^1(\mathbb N)$.
The sequence $(e_n)_{n\in\mathbb N}$ feels like a kind of basis because every element $a\in\ell^1(\mathbb N)$ can be written as an absolutely convergent infinite linear combination $\sum_{n\in\mathbb N}a(n)e_n$ in a unique way.
(Here $e_n$ denotes the vector whose $n$th en... |
Let $X = \mathcal{C}([0,1],\mathbb{R})$ be the ring of all continuous real-valued functions $f:[0,1] \to \mathbb{R}$. For $x \in [0,1]$, let $$M_{x} = \{ f \in X \ | \ f(x)=0\}.$$ One can show by using compactness of $[0,1]$ that every maximal ideal is of this form.
Extending the question to entire functions: Let $\mat... |
I try to solve for the specific function
$f(x) = \frac{2-2a}{x-1} \int_0^{x-1} f(y) dy + af(x-1)$
It looks similar to the function used to find the Renyi's parking constant because it came out from a simple generalization of that problem.
The skill I have gained in my differential class can't even solve
$f(x) = f'(x-1)... |
Here is a list of some applications of the exponential function.
1) The exponential mapping in Lie theory.
I put this first because my intuition tells me that this must be the most fundamental, or deep, way of thinking about the exponential function. I have often been misled by my intuition however, and the main rea... |
Let $p_i$ denote the $i^{th}$ prime number.
Find the smallest positive integer $k$ such that the product $n = p_1 \cdot p_2 \cdots p_k$ satisfies $\sigma(n) > 3n$.
Is there any positive integer $m < n$ satisfying $\sigma(m) > 3m$?
Taken From (Rosen), at the end of the chapter under challenge problems
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I have to prove that
\begin{equation*}
||A-B||=||A+B||\Leftrightarrow AB=0
\end{equation*}
and I was wondering if this approach is correct, or if there's a better/more elegant way to prove this.
Given n-dimensional vectors A and B, we can write $||A-B||=||A+B||$ as:
\begin{equation*}
\sqrt{\sum\limits_{j=1}^{n}(a_j... |
I've recently started making a small fractal app in Javascript using the famous Mandelbrot bulb $(z = z^2 + c)$. I've been trying to find the best method of coloring the points on the complex plane, and I've come across some very interesting ideas:
http://linas.org/art-gallery/escape/escape.html
http://en.wikibooks.org... |
I am working through Theorem 6 in Chapter 8 of Hoffman and Kunze. (The section is Linear Functionals and Adjoints.)
The statement is Let $V$ be a finite-dimensional inner product space, and $f$ a linear functional on $V$. Then there exists a unique vector $\beta$ in $V$ such that $f(\alpha) = (\alpha | \beta)$ for... |
I'm trying to learn predicate logic.
And I'm looking for some good resources on it:
I've seen that I learn better when I can program
So I was wondering if there was a 'predicate logic' programning language?
Maybe an interactive tutorial?
Maybe lots of examples, something like: Predicate Logic by Example?
Or at least so... |
This question may smell like my older question A probability game, but this time my intentions are different.
The above problem came from my friend. Or so I thought, it really came from me incorrectly remembering his problem. The problem he actually gave me was:
Suppose you start with 1 dollar and flip a weighted coin.... |
I'm currently interested in the following result:
Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" on $X$. Namely, the second category consists of quasi-coherent sheaves $\mathcal{F}$ on $X$ with an isomorphism $p_{1}^*(... |
The only place I've encountered well-definition is with proving an operation defined on an equivalence class is independent of the choice of representative.
On my homework, it asks us to show that the ring of formal Laurent series is well-defined, and I don't understand what exactly I need to show. However, I don't und... |
Question: Prove that the Ring of Complex Entire functions is neither Artinian nor noetherian.
Proof: Clearly $R$ is not Artinian because it is a commutative integral domain which is not a field, and $R$ is not noetherian because it is not a factorisation domain.
Is there a proof of this theorem using the Ascending / ... |
There is a theorem that states that if $f$ is analytic in a domain $D$, and the closed disc {$ z:|z-\alpha|\leq r$} contained in $D$, and $C$ denotes the disc's boundary followed in the positive direction, then for every $z$ in the disc we can write:
$$f(z)=\frac{1}{2\pi i}\int\frac{f(\zeta)}{\zeta-z}d\zeta$$
My quest... |
See this SO thread: Calculating which item is next in a percentage distribution.
To summarize,
you calculate
max ((rand1 * 10), (rand2 * 20), (rand3 * 30), (rand4 * 40))
for four random numbers in a range (say 1-100).
If the max is
rand1 * 10, you choose 'A'
rand2 * 20, you choose 'B'
rand3 * 30, you choose 'C'
rand4... |
I've been struggling with figuring out how to add powers of $i$.
For example, the result of $i^3 + i^4 + i^5$ is $1$. But how do I get the result of $i^3 + i^4 + ... + i^{50}$? Writing it all down would be pretty mundane...
It has to do something with division by 4, since the "power cycle" of $i$ repeats every fourth p... |
Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every interval of the form $[a,b]$, where $a,b \in \mathbb{R}$ contain a rational of this kind. If yes, how can we prove it?
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In the book Elliptic Curves - McKean & Moll we are given the outline for a proof of Eulers addition theorem:
The (projective) quartic $\mathbf y^2 = (1-\mathbf x^2)(1-k^2 \mathbf x^2)$ has four intersections with the quadratic $\mathbf y = 1 + a \mathbf x + b \mathbf x^2$. One is located at $(\mathbf x, \mathbf y) = (... |
$$\ln2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - 2(\frac{1}{2} + \frac{1}{4} + \cdots)$$
$$= (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) - (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots) = 0$$
thanks.
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Hello i am having some problems working out how to attack this assignment, and after spending hours on it, have i resolved to ask you guys here for help.
I have been given the following differential equation
y^{4}-16y=u^{1}+u
y′′′′−16y=u′+u
And i have been asked to determine the complete real solution. I just complete... |
Given 3 pairs of coordinates, $x_1, y_1, x_2, y_2, x_3, y_3$, I need a function $y(x)$ that will return the $y$ coordinate of any $x$ coordinate between $x_1$ and $x_3$ (it can be assumed that $x_1 < x_2 < x_3$).
$y(x_1)$ must equal $y_1$, $y(x_2)$ must equal $y_2$, and $y(x_3)$ must equal $y_3$.
However, it must never... |
Say you have a set of points and joining lines as show below:
In this diagram of triangles, T-(or Y-like) junctions are at D, F, and E.
At vertex F for example, there is a triangle FDE, but no line FJ. If there was a line FJ, then there would be no T junction at F. Two triangles would be formed instead out of triangl... |
Follow up to
Say I have triangles as shown:
Its easy to see that $ABC$ is adjacent to $ACJ: ABC$ and $ACJ$ share $2$ vertices ($A$ and $C$).
However $FEC$ and $ACJ$ are also adjacent, but only one vertex is shared.
How can I find if two triangles are adjacent when they differ in size?
|
I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it).
|
I'm trying to show that all graphs with $5$ vertices, each of degree $2,$ are isomorphic to each other. Is there a more clever way than simply listing them all out?
|
Given sample data $x_1, \ldots, x_n$ generated from a probability distribution $f(x|\theta)$ ($\theta$ being an unknown parameter), a statistic $T(x_1, \ldots, x_n)$ of the sample data is called sufficient if $f(x|\theta, t) = f(x|t)$.
However, I'm always kinda confused by this definition, since I think of a sufficient... |
What's a good way of solving the Diffie–Hellman problem when those exchanging the message have chosen a low primitive root $g$ (e.g. $g=3$)?
Of course you could brute force it but I'm interested in knowing whether there is a formula for solving it when you know $g^a \pmod{p}$ and $g^b \pmod{p}$ as well as $p$ and $g$ o... |
Bounty update: this can be solved by change of basis, but I'm intrigued by David's solution relying on Fourier Transform of Dirac Delta function, so the bounty is for whoever finds a way to fix his solution to give the right result.
Suppose I have a non-negative real-valued function over $d$-dimensional real vectors as... |
What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? Are all such theories decidable? If not, by what methods other than Gödelization can undecidability be established?
EDIT: Both answers have indicated Robinson Arithmetic as an intermediate theory, but I don't unde... |
I have been working on the following lemma for a few days. Any hints would be appreciated.
Lemma
Let $G$ be a finite group, $H \le G$ and $P \in Syl_{p}(H)$. If $N_{G}(P) \subseteq H$ then $P$ is a Sylow $p$-subgroup of $G$.
Partial Proof (Contrapostive).
First assume that $P$ is not a Sylow $p$-subgroup of $G$. Theref... |
In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this?
I'm assuming this has something to do with the following observation: $\cap_{n=1}^{\infty} (-\frac{1}{n},\frac{1}{n}) = \{0\}$ and the... |
Given $C=XF$, and only data for $C$, solving for $X$ and $F$ has been described as not tractable (which I'm content to believe, but others aren't). If that's the case, where does the following argument fall apart?
Suppose $C^i=XF^i$ is written:
$\begin{bmatrix}
C_1^i \\\
\vdots \\\
C_m^i
\end{bmatrix} = \begin{... |
In the comments to the question: If $(a^{n}+n ) \mid (b^{n}+n)$ for all $n$, then $ a=b$, there was a claim that $5^n+n$ is never prime (for integer $n>0$).
It does not look obvious to prove, nor have I found a counterexample.
Is this really true?
Update: $5^{7954} + 7954$ has been found to be prime by a computer: http... |
The following is the start of basic corollary in my logic text:
For any set $\Gamma$ of sentences, $\Gamma\subseteq\text{Th}(\text{Mod}(\Gamma))$.
What happens when $\text{Mod}(\Gamma)$ is empty? Suppose $\Gamma={p_1,\neg p_1}$, so that there is no truth value $V$ which models $\Gamma$, or otherwise $V(p_1)=\text{T}=V... |
According to the definitions of algebra over a ring and algebra over a field, are Elementary Algebra and Boolean Algebra examples of algebra over a ring or algebra over a field? How can one show that? What are the ring or field that the algebras are with respect to?
Thanks and regards!
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for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear optimization, convex optimization and quadratic programming, dynamic programming, multicriterial optimizations (did I miss ... |
I have a set containing numbers $1$ to $49: \{1,2,3, \cdots, 49\}$.
Now, I want to divide the set into $7$ subsets such that each subset should contain $7$ elements and sum of the elements of each subset should be $175$.
Is it possible to prove that such subsets exist?
|
How can we show that the number of pairs $(a,b)$ (where the pairs $(a, b)$ and $(b, a)$ are considered same) with $\operatorname{lcm}(a, b) = n$ is equal to
$$\displaystyle\frac{(2e_1+1)(2e_2+1)...(2e_k+1)+1}{2},$$
Where
$n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}$, $p_i$s are prime for all $1\leq i\leq k$.
I managed to gues... |
(This question is inspired by the StackOverflow question Converting prime numbers by @Manas, which turns out to be the problem PPATH of SPOJ.)
Let's construct a graph $G$ of all primes $p$ having $L$ digits in base-$n$. Two primes $p$ and $q$ form an edge in $G$ iff they only differ in $1$ digit (e.g. $1\mathbf{0}09$ a... |
This question isn't about the actual substance of math so much as it is about style. I'm not going to school, so my feedback on these sorts of problems is limited. How would you improve this proof's readability? And if this isn't the sort of question that's suitable here, where would be a more appropriate venue? Th... |
If F and G are functors between two arbitrary categories C and D, does a natural transformation η from F to G always exists? What is the condition for its existence?
Thanks and regards!
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Let $f$ be continuous on $\mathbb{R}$. Then how to find all continuous functions satisfying $f(f(x))=f(x)+x$
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I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
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I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. This is s... |
I'm trying to construct a simple proof of the Propositional Interpolation Theorem. For the following, let $At(\phi)$ be the set of sentence symbols that occur in a sentence $\phi$. Suppose that $\psi$ is a tautological consequence of $\phi$, but neither $\neg\phi$ nor $\psi$ is a tautology, and so $\psi$ is not a tauto... |
Let $m,n\ge 0$ be two integers. Prove that
$$\sum_{k=m}^n (-1)^{k-m} \binom{k}{m} \binom{n}{k} = \delta_{mn}$$
where $\delta_{mn}$ stands for the Kronecker's delta (defined by $\delta_{mn} = \begin{cases} 1, & \text{if } m=n; \\ 0, & \text{if } m\neq n \end{cases}$).
Note: I put the tag "linear algebra" because i think... |
consider the following game: 10 dice are tossed and those showing 3 are more are retained. [those showing 2 or less are discarded.] the remaining dice are tossed again and those showing 4 or more are retained. finally, the remaining dice are tossed once more and those showing 5 or more are retained.
if $Z$ denotes the... |
How does one find the value of the sum : $$\sum\limits_{n=1}^{p-1} [\sqrt{np} \ ]$$ where $p$ is a prime such that $p \equiv 1 (\text{mod} \ 4)$.
If i remember correctly, i got this sometime back, when i was an undergrad.
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In an introductory algebra course, one proves the Structure theorem for finitely generated modules over a principal ideal domain.
http://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
In every book that I have looked in, the uniqueness part of the proof is a mess.
C... |
Given complete fan $\Delta$ defining a projective toric variety (so that $\Delta$ is the normal fan of some polytope). How do one go on to find a defining ideal of the toric variety in projective space?
Or, going the other way, given binomial equations defining a projective toric variety, is there some way to recover ... |
I am reading in Jech's Set Theory the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in the book.
I went back to the chapter dealing with weakly-compact cardinals and began retracing the definition.
Eventually, it came ... |
Usually, applying the conversion formulae
$r^2=x^2+y^2$
$\cos\;\theta=\frac{x}{r}$
$\sin\;\theta=\frac{y}{r}$
to transform an equation in polar coordinates to an implicit Cartesian equation is quite straightforward for an equation of the form
$r=f(\cos(n\theta),\sin(n\theta))$
with $n$ an integer, thanks to multiple an... |
Let $X$ be a complex manifold of dimension $n$. Thus, it's a real manifold of dimension $2n$.
Now cohomology is a topological concept so it should not depend upon the structure given on a topological space.
We know that $k^{th}$ Singular cohomology of $X$ is $0$ for $k \gt 2n$. We can also define a sheaf cohomology on ... |
One of my Fellows asked me whether total induction is applicable to real numbers, too ( or at least all real numbers ≥ 0) . We only used that for natural numbers so far.
Of course you have to change some things in the inductive step, when you want to use it on real numbers.
I guess that using induction on real numbers... |
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