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H: $\{\langle M,q,x\rangle$| $M$ is a Turing machine and $q$ is a state of M and running of $M$ on $w$ visits $q\} \notin R$?
I'm trying to find where does the language $\{\langle M,q,x\rangle$| $M$ is a Turingmachine and $q$ is a state of M and running of $M$ on $w$ passes on $q\}$ belong? whether it's $R,RE$ or non... |
H: How to perform simple linear interpolation on a data set
With the following data set, what is the best way to interpolate the data for each time.
Time X Y
0 10 15
...
...
24 28 17
...
...
49 9 14
AI: You can use Newton's divided differences interpolation polynomial which is easy to use... |
H: Do all angles occur in Hilbert spaces?
Let $X$ be a Hilbert space with scalar product $(\cdot,\cdot)$. Then for two vectors $v,w$ of norm $1$, we can interpret $(v,w)$ as an angle, so that
$(v,w)=\cos(\varphi)$ for a unique angle $\varphi\in[0,\pi)$.
My question is the following: Let $\varphi'\in[0,\pi)$ and $v'\in... |
H: Dealing a deck of cards
The problem is:
How many are ways to deal a deck of 52 cards to 4 players, and every player has at least one card?
The answer with inclusion-exclusion principle is: $$4^{52}-4\cdot 3^{52}+6\cdot 2^{52}-4 $$
But I'm wondering, why isn't it equal to the number of solutions: $$a+b+c+d=52, \ a... |
H: Expected number of jumps in regular jump HMC
Consider a homogeneous Markov Chain $X$ on a countable state space, ie a jump process.
It is said to be regular (does not explode) if there are only a finite number of jumps in every finite interval.
$$X(t) < \infty\quad \forall t > 0$$
What can you say the expected numb... |
H: Finding the convergence interval of $\sum_{n=0}^\infty\frac{n!x^n}{n^n}$.
I want to find the convergence interval of the infinite series $\sum\limits_{n=0}^\infty \dfrac{n!x^n}{n^n}$.
I will use the ratio test: if I call $u_n = \dfrac{n!x^n}{n^n}$, the ratio test says that, if the following is true for some values ... |
H: Are there any synonyms of "pair of pants" in topology?
I used to know a term for pair of pants, but perhaps there is none. It looks like this also.
AI: You could call it a "pretzel", as shown on http://www.maths.ed.ac.uk/~aar/surgery/zeeman.pdf. I've also heard it called a "trophy". |
H: Subspaces of Hilbert Spaces of finite dimension
Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ and trying to show its limit $f \in S$. I keep getting stuck and s... |
H: What is the connection between strong norms and norms coming from scalar products (in pre-hilbert spaces)?
In the best-approximation problem of seperation theorems in convex analysis, there is the notion of a "strong norm", in the sense that
If $\| x^1 + x^2 \| = \| x^1 \| + \|x^2 \| $, $x_1 , x_2 \neq 0$ $\impl... |
H: Drawing elliptic curve
Consider an elliptic complex curve in $\mathbb{C}^2$ given by equation $w^2 = (z-a)(z-b)(z-c)$ where $a,b,c$ are complex mutually distinct constants. It is a $2$-dimensional surface in $4$-dimensional space (If we talk about real dimensions). How to construct a topologically equivalent real s... |
H: How many 4 worded sentences can a list of 5 words make if two of them must be in that sentence?
Suppose we have:
I am new at this - (5 words)
how many 4 worded sentences can we make with this if "new" and "this" must appear in the sentence.
I think its :
.# of sentences we can make with any word in it - # of sente... |
H: Proving determinant product rule combinatorially
One of definitions of the determinant is:
$\det ({\mathbf C})
=\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}})$
I want to prove from this that
$\det \left({\mathbf {AB}}\right) = \det({\mathbf A})\det({\mathbf B})$
W... |
H: counterexample: degree of representation $\leq$ index of normal subgroup
if I have a finite group $G$ with an abelian normal subgroup $N$ and an irreducible representation $\pi$ of $G$ over $K$. Then I know, that $deg(\pi) \leq [G:N]$, if $K$ has positive characteristic and is a splitting field for $G$. My profess... |
H: Name for a set which has an order?
As we all know, a set is a collection of elements which have no particular order and no multiplicity.
So what do you call a construct which does store its elements in a specific order? What is the correct mathematical term for that?
(I looked at "ordered set", but that apparently ... |
H: Range of bounded operator is of first category
Let $T$ be a bounded operator from a Banach Space $X$ to a normed space $Y$ such that $T$ is not onto, but $R(T)\subset Y$ is dense. Prove that $R(T)$ is of first category and not no-where dense.
Since $\mathring{\overline{R(T)}}=\mathring{Y}=Y\not=\emptyset$ the ra... |
H: Retracts are Submanifolds
Looking over some old qualifying exams, we found this:
Let $A\subseteq M$ be a connected subset of a manifold $M$. If there exists a smooth retraction $r:M\longrightarrow A$, then $A$ is a submanifold.
Our thought to prove this statement was that since $r$ is smooth and the identity on $A... |
H: How to tell $i$ from $-i$?
Suppose now we are trying to explain to students who do not know complex numbers, how do we distinguish $i$ and $-i$ to them? They will object that they both squared to $-1$ and thus they are indistinguishable. Is there a way of explaining this in an elementary way without go into introdu... |
H: An interesting pattern in solutions to differential equations
OK, watch this:
Suppose I have a weight on the end of a spring. Assuming the spring obeys Hooke's law, as the weight is displaced from its rest position, the spring exerts a restoring force in the opposite direction who's magnitude is equal to the displa... |
H: Is a total function also a partial function?
Is there a consensus on whether a total function, i.e., a function defined for each element of the domain, is also a partial function?
AI: When someone says "partial function", the usual interpretation is that the function may or may not be defined on the entire domain. ... |
H: Is there a Hamiltonian path for the graph of English counties?
The mainland counties of England form a graph with counties as vertices and edges as touching borders. Is there a Hamiltonian path one can take? This is not homework, I just have an idea for a holiday around England where I visit every county only once!... |
H: Why we need to know how to solve a quadratic?
Five years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic:
Why do I need how to solve a quadratic? I am not going to use it for
my future job!
This question is, largely not mat... |
H: Does the Cartesian product get smaller if I use fewer sets?
My introduction into Axiom of Choice has been kind of confusing (Zorn's lemma) for the start, so it took me some time to realize it's nothing but to say The non-empty product of non-empty sets is non-empty. I still find it quite puzzling that this doesn't ... |
H: Prove $\tan(A+B+Y)=\frac{\tan A+\tan B+\tan Y-\tan A\tan B\tan Y}{1-\tan A \tan B-\tan B\tan Y-\tan Y\tan A}$
I have to prove this most difficult trigonometric identity.
$$\tan(A+B+Y)=\frac{\tan A+\tan B+\tan Y-\tan A\tan B\tan Y}{1-\tan A \tan B-\tan B\tan Y-\tan Y\tan A}.$$
I know
$$\tan(A+B)=\frac{\tan A+\tan B}... |
H: Quadratic field such that a certain finite set of primes split
Given a finite set $S$ of primes, is it possible to find an imaginary quadratic field $K$ such that all primes in $S$ are split completely in $K$?
AI: Sure. Let me just assume WLOG that $S$ contains $2$. Let $D$ be squarefree. If $D \equiv 1 \bmod 4$, t... |
H: How can I determine a formula for an exponential ratio?
I am not very experienced in mathematical notation, so please excuse some terminology misuse or formatting shortcomings.
I have a project in which a value needs to increase from a set minimum to a set maximum in a set number of seconds. It is easy to calculate... |
H: I need help simplifying and reorganizing this algebraic equation
I've developed the following algebraic equation which could probably be simplified further. Also, I need it reorganized to solve for x and Y (in terms of A, x, and Y. Not looking for a numerical answer to x or Y). Any takers?
$$
A=0.0193(\frac{x+0.... |
H: Is the ten's digit even in any power of 20n+c, where c is an odd digit?
Prove or disprove that in any power of $20n+c$, where $c$ is an odd digit (i.e., $1,3,5,7,9$), the ten's digit is even.
This is probably a generalization of this.
I tried in the following way.
I observe that $(20m+c)(20n+d)$ is $20(20mn+m+n)+... |
H: Multiplicative Semigroup of a Ring
Let $I=\{0, 1, \ldots \}$ be the multiplicative semigroup of non-negative integers. It is possible to find a ring $R$ such that the multiplicative semigroup of $R$ is isomorphic (as a semigroup) to $I$?
AI: Suppose that there is such an isomorphism, $\phi:R\to\mathbb N.$ In parti... |
H: Set of all functions from a finite set to a finite set
We consider the set $\mathbb{J}$ of all functions $f_i: \{1,2,...,n \} \to \{1,2,...,n \}$, where $n \in \mathbb{N}, i \in \{1,2,...,n^n \}$.
We define two functions: $e_1(k)=k$; $e_2(k)=n-k+1$; here $k \in \{1,2,...,n \}$.
Let $f^{(m)}=f \circ f \circ ... \ci... |
H: Variation of sum of measure
A variation of arbitrary complex measure $\nu$ on the measurable set $E$ is called the number $\|\nu\|(E)=\sup \sum_{n=1}^\infty |\nu (E_n)|$, where supremum is taken over all sequences $(E_n)$ such that $E_n$ are measurable, pairwise disjoint and their union is $E$.
Let $\mu$ and $\lam... |
H: What would be possible if we could perform repeated "linear mapping" on recurrences?
We can label the terms in a sequence as $a_n$ for the $n$th term. Then the generating function for the sequence can be defined as
$$ A(x) = \sum_{n=0}^\infty{a_n x^n} $$
If we have a closed form for the resulting sum (meaning no s... |
H: Number of paths in regular graphs, where starting and ending nodes are same
Possible Duplicate:
returning paths on cubic graphs
$\hskip2.7in$
In the above undirected, unweighted graph we start with node $Q$. In a single step we can jump to any adjacent node from current node. For given $n$ steps, we've to find o... |
H: Indefinite Integral of $\sqrt{\sin x}$
$$\int \sqrt{\sin x} ~dx.$$
Does there exist a simple antiderivative of $\sqrt{\sin x}$? How do I integrate it?
AI: Since $\sqrt{\sin(x)} = \sqrt{1 - 2 \sin^2\left(\frac{\pi}{4} -\frac{x}{2}\right)}$, this matches with the elliptic integral of the second kind:
$$\begin{align*}... |
H: If a player is 50% as good as I am at a game, how many games will it be before she finally wins one game?
This is a real life problem. I play Foosball with my colleague who hasn't beaten me so far. I have won 18 in a row. She is about 50% as good as I am (the average margin of victory is 10-5 for me). Mathematicall... |
H: Computer Algebra Systems which implement Cylindrical Algebraic Decomposition
My understanding is that Mathematica's Reduce function is based on Cylindrical Algebraic Decomposition (CAD).
The only other system I've seen which implements CAD is QEPCAD. QEPCAD isn't a general CAS like Mathematica or Maple.
Do any main... |
H: Are there nonconstant Hoelder type function?
It is known that nonconstant Hoelder functions $f:R \rightarrow R$ with power $>1$ non exist.
Are there functions $f: R\rightarrow R$ satisfying
$$ |f(x)-f(y)| \leq C|x-y|^p +D$$ for all $x,y \in R$, where $C,D>0$, $p>1$?
AI: Any Lipschitz continuous (Hölder continuou... |
H: Distance between point and line
Using the formula from this page: http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html how can you find the individual x,y, and z distances? I'm trying to figure it out but can't wrap my head around it.
AI: You can find the individual distances using the given formula. T... |
H: Question about a corollary about Finite Fields
Definition: A field extension $E$ of $F$ is of degree $n$ (and is called a finite field extension) if $E$ is an $n$-dimensional vector space over $F$.
Theorem: Let $E$ be a degree $n$ finite extension of a field $F$. If $F$ has $q$ elements, then $E$ has $q^n$ eleme... |
H: Equation of the locus of centre of the ellipse?
An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?
AI: Given the parametric equation of a rotated ellipse
$$
x(t)=x_0+a\cos\theta\cos{t}-b\sin\theta\sin{t}\\
y(t)=y_0+b\cos\theta\sin{t}+a\sin\thet... |
H: Sufficient Estimators and Generalized Likelihood Ratios
If you can make the assumption that a sufficient statistic exists for some parameter - let's call it $\theta$.
How would you show that the critical region of a likelihood ratio test will depend on the sufficient statistic?
AI: If $T(\mathbf{X})$ is a sufficien... |
H: Minimum value of $x+y$ when $xy=36$
How would I calculate minimum value of $x+y$ when $xy=36$ and x and y are unequal positive integer numbers. I don't even know the answer. Any help would be appreciated.
Edit:
Sorry It was the minimum value.
AI: Now that you have specified $x$ and $y$ to be integers, this problem... |
H: Derivative of $x^x$ at $x=1$ from first principles
Find the derivative of $x^x$ at $x=1$ by definition (i.e. using the limit of the incremental ratio).
The only trick I know is $x^x = e^{x \ln x}$ but it doesn't work.
AI: Using the definition:
$$
\begin{align}
f'(1)&=\lim_{x\rightarrow1}\frac{x^x-1}{x-1}\\
&=\lim... |
H: Problem Involving Finite Fields
I've arrived at a Theorem in text that I'm confused about:
Note: My question below is about the statement of this theorem, not about a proof for it. (The proof is supplied in the text)
Theorem: Let $E$ be a field of $p^{n}$ elements contained in an algebraic closure $\tilde{\mathbb... |
H: showing almost equal function are actually equal
I am trying to show that if $f$ and $g$ are continuous functions on $[a, b]$ and if $f=g$ a.e. on $[a, b]$, then, in fact, $f=g$ on $[a, b]$. Also would a similar assertion be true if $[a, b]$ was replaced by a general measurable set $E$ ?
Some thoughts towards the p... |
H: Sums of sums of sums of...of numbers
If we introduce the following notation
$$S_r^q=\overbrace{\sum_{a_{r-1}=1}^q\sum_{a_{r-2}=1}^{a_{r-1}}\cdots\sum_{a_1=1}^{a_2}\sum^{a_1}}^{\mbox{a total of $r$ sums}}1$$
for example, $S^q_1=q$, $S^q_2=q(q+1)/2$ and so on, then one can show that
$$S^p_{q-1}=S^q_{p-1},$$
where $p... |
H: About differentiation under the product integral sign
It is known that
$$\dfrac{d}{dx}\int_{a(x)}^{b(x)}f(x,t)~dt=\dfrac{db(x)}{dx}f(x,b(x))-\dfrac{da(x)}{dx}f(x,a(x))+\int_{a(x)}^{b(x)}\dfrac{\partial}{\partial x}f(x,t)~dt.$$
How about
$$\dfrac{d}{dx}\prod_{a(x)}^{b(x)}f(x,t)^{dt}?$$
AI: Because we are in a commut... |
H: Bounding the $l_1$ norm of a vector
Let $x$ be real vector with $\|x\|_1=x_1+\ldots +x_{2n}$.
How to bound from above $(x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})$ by $l_2$ norm of the vector $x$.
Of course, using $\|x\|\leq\sqrt {2n}\|x\|_2$ I can bound
$$
(x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})\leq\|x\|^2_1\leq 2n\|x... |
H: Limit point and interior point
Is any interior point also a limit point?
Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample?
(Since an interior point $p$ of a set $E$ has a neighborhood $N$ with radius $r$ su... |
H: The product of all roots of $x^2 - 4x + 6 = 3 - |x - 1|$
Prepping for the GMAT, I came across the following question:
What is the product of all solutions of:
$$x^2 - 4x + 6 = 3 - |x - 1|?$$
First, I set up two equations, ie:
$$x^2 - 4x + 6 = 3 - (x - 1),$$
and
$$x^2 - 4x + 6 = 3 - (-1) \times (x - 1).$$
These f... |
H: Intersection of two (specific) convex functions
Given are the following two functions:
$$g(z) = \left(z-2\right)\left(2+z\left(z-2\right)\right)$$
and
$$h(z) = 2\left(z-1\right)^{2}\ln\left(z-1\right),$$
where $z>2$.
I would like to show that these functions intersect only once (for $z>2$, clearly they also inte... |
H: A closed form for $T_N = 1 + \sum\limits_{k=0}^{N-2}{(N-1-k)T_k}$?
I've narrowed down a problem I am working on to the following recurrence:
$$\begin{align*}
T_0 &= T_1 = 1\\
T_N &= 1 + \sum_{k=0}^{N-2}{(N-1-k)T_k}
\end{align*}$$
I'm stuck on how to close it up, or at least make it linear or $O(n\log n)$. Any clue... |
H: On surjective functions
I'm having trouble with this question involving surjective functions.
If $h(x)=2x+1$ and $h$ maps from the integers to the integers, does there exist a function $g$ that maps from the integers to the integers so that $(g \circ h)(x)$ is onto?
If this is true, can someone provide an example... |
H: Application of Lagrange Multiplier?
Let $M$ and $m$ denote resp. the max and min values of the func. $f(x,y,z) = xyz $ over the region defined by the interior and boundary of the sphere $x^2 + y^2 + z^2 = 3$. What is the value of $M + m$?
I tried using the method of lagrange multiplier to find $M$ and $m$, but I a... |
H: The boundedness of the integral $ \int_0^N \sin(P(x))/{x}\; dx $
Let $\{a_i\}$ be real numbers and $P(x)=a_nx^n+a_{n-1}x^{n-1}+ \cdots+a_1x$. Is there a constant $C$ which is independent of $a_i,N,n$, such that
$$\left| {\int_0^N {\frac{{\sin (P(x))}}{x}dx} } \right| \le C?$$
(edit by LK) The OP pointed out in a c... |
H: If $N$ is a normal subgroup of $G$ with $N$ and $G/N$ solvable, prove that $G$ is solvable
I'm trying to prove that if $N$ is a normal subgroup of $G$, with $N$ and $G/N$ solvable, then $G$ is solvable.
Proving that $G/N$ is abelian would of course suffice, but I'm not sure if that's a necessary condition or not. ... |
H: Circular Sector to Circle Intersection
Is there a formula for determining if a sector intersects a circle (as well as determining if the circle/sector are inside each other)?
Sector definition: A center point $P(x,y)$, a starting angle in radians, an ending angle in radians, and a radius $r$ distance from the cente... |
H: Prove whether a relation is an equivalence relation
Define a relation $R$ on $\mathbb{Z}$ by $R = \{(a,b)|a≤b+2\}$.
(a) Prove or disprove: $R$ is reflexive.
(b) Prove or disprove: $R$ is symmetric.
(c) Prove or disprove: $R$ is transitive.
For (a), I know that $R$ is reflexive because if you substitute $\alpha$ in... |
H: Which value is greater ? Sum of same four numbers or 36
I came across the following question
The average of four numbers is 36. Which is greater Sum of same four numbers or $140$
Now the answer states (a- The sum of same four numbers). How did they determine the minimum and the maximum value of the numbers from... |
H: Finding a formula for $f^{(n)} (x)$ if $f(x) = \ln(x-1)$
While working through Stewart's Calculus Late Transcendental 7th edition, I came across this problem:
Find a formula for $f^{(n)} (x)$ if $f(x) = \ln(x-1)$.
Obviously, I calculated the first few derivatives to see if I could spot a pattern:
$$f^{1}(x) = \fr... |
H: commutativity of torsion functor
For a ring $R$ and finitely generated $R$ modules $U,W$,
$${\rm Tor}_i(U,W)={\rm Tor}_i(W,U)$$ for all $i$.
I saw proof in Hatcher's book, but I can understand that proof.
may be I see another proof?
AI: The proof depends on the definitions of $\newcommand{\Tor}{\textrm{Tor}}$ $\Tor... |
H: Determining sparse frequency distribution via discrete Fourier transform
Consider the function $$f(t) = 2 \sin(t)+\sin(2t)+25 \sin(400t)$$
(for example).
In this case, how many samples of this function would I have to take, and at what sampling frequency, to determine the three frequencies it is composed of? And, h... |
H: How does this expand out?
I'm finding myself getting back into math related stuff for the first time in a while. So please be patient with me.
How does $\frac{(n-i)(n-i+1)}{2}$ expand out to:
$\frac{n^2 - (2i - 1)n - i + i^2}{2}$
If you could show me step by step, I would really appreciate it.
Also, I apologise, bu... |
H: Iterated prisoners dilemma with discount rate and infinite game averages
Suppose we have two players who are perfectly rational (with their perfect rationality common knowledge) playing a game. On round one both players play in a prisoners dilemma type game. With payoffs (1,1) for mutual cooperation, (.01,.01) for ... |
H: The abstract definition of commutative monoids
In trying to begin to understand the idea of a $k$-tuply monoidal $n$-category, I'm already a bit stuck on the idea (Baez, nLab) that a commutative monoid can be defined as a monoid object in the category Mon of monoids. So what I have in a monoid object in Mon is a no... |
H: Non-closed subspace of a Banach space
Let $V$ be a Banach space. Can you give me an example of a subspace $W\subset V$ (sub-vectorspace) that is not closed?
Can't find an example of that yet.
Thanks!
AI: A simple example, which gets at the difference between orthonormal and Hamel bases in infinite dimensions, is to... |
H: Difference between "space" and "mathematical structure"?
I am trying to understand the difference between a "space" and a "mathematical structure".
I have found the following definition for mathematical structure:
A mathematical structure is a set (or sometimes several sets) with various associated mathematical ob... |
H: Help on solving DE
How to solve the following DE
$$ {dx \over x^2 - y^2 - z^2} = {dy \over 2xy} = {dz \over 2xz}$$
Equating the last part, I got $y = c_1 z$ and then I'm stuck. substituting the value of $y$ and equating first and last gives
$$ {dx \over dz} = {x^2 - z^2(c_1^2 + 1) \over 2xz}$$
How to solve it?
AI:... |
H: $f(z)=\int_1^\infty e^{-x}x^z\,dx$ is complex analytic
Note: I'm refereshing my complex analysis skills in order to learn some analytic number theory. Here's one (basic) claim I'd like to prove and my attempt. My questions are:
Is my partial attempt correct?
Are there better (or shorter) ways to prove it? I may b... |
H: Question on trace-weighted sums for irrep of finite group
For a finite group $G$, is the following true, where $\rho$ is a finite-dimensional complex unitary irreducible representation?
$$\sum _{g \in G} \mathrm{Tr} (\rho(g)) \rho(g) = \frac{|G|}{n} \mathrm{id}_{\mathbb{C} ^n}$$
I would like a proof or a counterexa... |
H: Calculating start/end points of a line segment given by a set of points and normal direction
I have a set of $3$D points representing a line segment. Points are not equidistant distributed on the line segment. Points are also unordered. I also have the center point and the normal to the line segment
All points in t... |
H: Is there a way to determine how many digits a power of 2 will contain?
Is there a direct way to determine how many digits a power of 2 will contain without actually performing the multiplication?
An estimation would help as well if there is no absolute solution.
EDIT:
In both decimal and binary bases.
AI: If you so... |
H: How to proof $n^2+n \in \Theta(n^2)$?
It stands to reason that $n^2+n \in \Theta(n^2)$.
But how can I formally proof it?
I tried next way:
Generalized to $$f(n)+o(f(n)) \in \Theta(f(n))$$
Separated to
$$\tag{1} f(n)+o(f(n)) \in O(f(n))$$
$$\tag{2} f(n)+o(f(n)) \in \Omega(f(n))$$
Definitions of asymptotics says:
$... |
H: Eigenvalues of tridiagonal symmetric matrix with diagonal entries $2$ and subdiagonal entries $1$
Let $A$ be a square matrix with all diagonal entries equal to $2$, all entries directly above or below the main diagonal equal to $1$, and all other entries equal to $0$. Show that every eigenvalue of $A$ is a real nu... |
H: hints on solving DE
How to solve this DE?
$$ {dx \over x} = {dy \over y} = {dz \over z - a \sqrt{x^2+y^2+z^2}}$$
From the first part, I get $y = c_1x$. How to find the other solution?
The answer according to answer sheet is $ z + \sqrt{x^2 + y^2 + z^2} = c_2$. Thank you for help.
AI: $$ {dx \over x} = {dy \over y} ... |
H: Why is the Wedderburn formula in this case wrong?
in this question counterexample: degree of representation $\leq$ index of normal subgroup there was the answer (in the second comment under the answer), that the dihedral group $D_5$ hat exactly 3 irreducible representations over $\mathbb{F}_3$ (or $\mathbb{F}_{13}$... |
H: A pseudo Fejér-Jackson inequality problem
$x\in (0,\pi)$
,Prove that:
\begin{align}
\sum_{k=1}^{n}\frac{\sin{kx}}{k}>x\left(1-\frac{x}{\pi}\right)^3
\end{align}
the inequality holds for all integer $n$
I tried Fourier, or Dirichlet kernel, but they don't work.Thanks for your attention!
AI: This left hand side is si... |
H: Hilbert spaces other than $L^2$
From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ be a finite measure space that is also a compact topological group and let $p>2$.
No... |
H: Is this a function?
Is the set $\theta=\{\big((x,y),(3y,2x,x+y)\big):x,y ∈ \mathbb{R}\}$ a function? If so, what is its domain, codomain, and range?
This is probably a dumb question. I understand what a function is, but the three elements in the ordered pair got me confused.
AI: Yes it is, presumably one from $\mat... |
H: Show that $\operatorname{rank}(A) = \operatorname{rank}(B)$
Let $A$ and $B$ be $n\times n$ real matrices such that $A^2=A$ and $B^2=B$. Suppose that $I-(A+B)$ is invertible .
Show that $\operatorname{Rank}(A)=\operatorname{Rank}(B)$.
I proceed in this way: Note that $A(I-(A+B))=A-A^2-AB=A-A-AB=-AB$ and simil... |
H: Find the determinant of $I+A$
Let $A$ be a $2\times2$ matrix with real entries such that $A^2=0$.Find the determinant of $I+A$ where $I$ denotes the identity matrix. I proceed in this way :Note that $(I+A)A=A+A^2 \Longrightarrow (I+A)A=A$ (Since $A^2=0$).
Now taking determinant both side we get
$|(I+A)A| = |A| \Lo... |
H: probability involving matching of discrete shapes on a square grid
Figure F exists on a regular square grid.
T transforms F by any combination of horizontal or vertical reflection as well as rotation by 90 or 180 degrees.
A larger background grid of X by Y contains noise, where each square has a 50% chance of being... |
H: Dual of the linear map
I know the meaning of dual of linear map in inner product spaces, also it is defined in Banach space[ Rudin functional analysis].
What is the definition of Dual of linear map if vector space are Frechet space/ or more generally locally convex sapce which is not normable.
AI: Definition of dua... |
H: Nature of algebraic structure
I am given $G = \{x + y \sqrt7 \mid x^2 - 7y^2 = 1; x,y \in \mathbb Q\}$ and the task is to determine the nature of $(G, \cdot)$, where $\cdot$ is multiplication. I'm having trouble finding the inverse element (I have found the neutral and proven the associative rule.
AI: For $a+b\sqrt... |
H: Prove the limit for a series of products.
Let $\beta > 0$, $\lambda > 1$. Show the identity
$$\sum_{n=0}^\infty\prod_{k=0}^{n} \frac{k+\beta}{\lambda + k + \beta} = \frac{\beta}{\lambda - 1}$$
I have checked the statement numerically.
The special case $\beta = 1$, $\lambda = 2$ looks like this
$$\sum_{n=1}^\infty\p... |
H: Why is it that a linear transformation can only preserve or reduce a vector space's dimension?
I am able to prove the above-asked using the fact that a linear map preserves linear dependence. I also vaguely suspect a connection from group theory (homomorphisms?). But I'm having trouble getting an intuition for that... |
H: Number Line Question
I am stomped on the following question
Which is greater judging from the number line if $\rm JL = KM.$
a) $\rm JK$
b) $\rm LM$
(Answer : Both are the same)
I would like to know how they concluded both are same ?
AI: Hint:
We have $\rm JL = JK + KL$ and $\rm KM = KL + LM.$ (Exercise: why does... |
H: What does "$f$ is a function on $S$" mean?
If somebody says "$f$ is a function on $S$", what do they mean?
Does it mean that $S$ is the domain of the function, the codomain, or both?
AI: When dealing with functions, one usually specifies "$f$ is a function from $X$ to $Y$" or "a function on $X$ into $Y$", or $f$ is... |
H: Are the two statements concening number theory correct?
Statement 1: any integer no less than four can be factorized as a linear combination of two and three.
Statement 2: any integer no less than six can be factorized as a linear combination of three, four and five.
I tried for many numbers, it seems the above two... |
H: Generalizing Bernoulli's inequality
We are already familiar with Bernoulli's inequality: $(1+px)\le(1+x)^p$ for $x\ge-1, p\ge1$.
Can this be generalized to say something useful about $(1+p_1x_1+\cdots+p_nx_n)$? For instance, one would hope that something like this could hold:
$$(1+p_1x_1+\cdots+p_nx_n) \le (1+x_1+\... |
H: Show $\lim_{h \to 0} \frac{f(h)+f(-h)}{h^2}=f''(0)$
Let $f$ be a function such that $f(0)=0$ and $f$ has derivatives of all order .Show that $$\lim_{h \to 0} \frac{f(h)+f(-h)}{h^2}=f''(0)$$ where $f''(0)$ is the second derivative of $f$ at $0$. I proceed in this way: Note that $$f''(0)=\lim_{h \to 0} \frac{f'(h)-f... |
H: Is there an algorithm similar to Gram Schmidt?
Everybody knows the Gram-Schmidt algorithm when it comes to basic linear algebra, to take a set of vectors $x_i \in \mathbb{R}^n$ and transform them into a set of vectors that spans the same space, is linear combination of $x_i$ and all vectors in the new set are ortho... |
H: Extending a subset of $\mathbb{Z}^n$ to a basis
Let $v_1,\dots,v_k\in \mathbb{Z}^n$. Is there a nice criterion for the existence of $v_{k+1},\ldots,v_n \in\mathbb{Z}^n$ such that $v_1,\ldots,v_n$ form a basis of $\mathbb{Z}^n$?
For $k=1$, if $v_1 = (a_1,\ldots,a_n)$, it is not hard to see that it is iff $\gcd(... |
H: To show $ax^{2}+bx+c$ is neither injective or surjective.
I want to show that $\displaystyle ax^{2}+bx+c$ is nither injective nor surjective where $a,b,c\in \mathbb{R}$ and $a\neq 0$.
Can i use the following result?
If $f$ is a continuous real valued function on interval $I$ and if $f'(x)>0$ for all $x$ in $I$ exc... |
H: Find distance traveled by tips of hands of clocks?
The short and the long hands of a wall clock are $8$ cm and $12$ cm respectively. Find the sum of the distance traveled by their tips in $3$ days. Give your answer in terms of $\pi$.
My solution:
Short hand:
Distance traveled in $12$ hours $= 2πr = 16π$ cm
$\R... |
H: Smooth curve orthogonal to all hyperbolae $xy = a$ at points of intersection.
Suppose a smooth, connected curve $C$ in $R^2$ is orthogonal to all hyperbolae $xy = a$ whenever they coincide. I'd like to find the point(s) of intersection of $C$ with the hyperbola $xy = 16$ given that $C$ contains the point (1,1).
I f... |
H: What does the continuum hypothesis imply?
Are there any fundamental/interesting results that are a consequence of assuming the continuum hypothesis as an additional axiom?
I'm sorry if this question was already asked. I'm also sorry if there is no rigour at all in the way I asked it.
Thanks!
AI: There are many card... |
H: Proof that a periodic function is bounded and uniformly continuous.
I need to show that if $f:\mathbb{R}\to \mathbb{R}$ is continuous and $\forall x \in \mathbb R, f(x+1)=f(x)$, then:
$f$ is bounded,
$f$ is uniformly continuous,
there exists $c\in \mathbb{R}$ such that $f(c)=f(c+\pi)$.
AI: Let $$g : \begin{array}{... |
H: Why is ergodicity of transformations only defined for measure-preserving transformations?
In ergodic theory, why does the defintion of an ergodic transformation $T$, why do I have to claim that it is measure-preserving?
E.g.
$T$ is ergodic if $\mathbb{P}(A) \in \{0,1\}$ for all $A$ with $T^{-1} (A) = A$
Couldn't ... |
H: Is this a valid proof for recurrence time?
The following is a well known result of Markov chain:
Given a Markov chain $(X_t)_{t \ge 0}$, if $T_{ii}$ denote the time of
the first return to state $i$ when starting at state $i$, then we have
$$ E[T_{ii}] = \frac 1 {\pi_i}, $$
where $\pi$ is the stationary distribut... |
H: Ergodic theory in mathematics and physics
How is the theory of ergodic measure-preserving transformations related to ergodicity in the physical sense (which I understood as, very very roughly speaking, that a physical system is called ergodic if "averaging" over "states" of the physical system equals the "average" ... |
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