Recent Questions - Mathematics Stack Exchange |
- The radius of the circle which just touches the outer circle
- Find the values of a and b that make the hyperbola congruent to $xy=1$
- Constructing a DFA for a language
- Lower bounding differences of a non linear function
- Question about pinch singularities
- integro- (partial) differential equation
- Understanding the proof of an equivalent condition for a graph to be a tree
- condition of solution existence of system of linear equations where elements are either 0 or 1
- Proof of the discrete-time optional sampling theorem
- Use strong induction to show that postage of 24-cents or more can be achieved by using only 5-cent and 7-cent stamps
- if there exists a non-zero divisor $\beta \in R$ with $\beta \alpha^n \in R$ for all $n \geq 1$, then $\beta$ is integral
- Equivalence relations and Stirling numbers
- When a subset $Y \subset \Bbb{Z}$ is periodic and closed under powering, what can we say about the size of a maximal interval contained in it?
- Enumerating how many ways two sets of symbols can be written in order.
- $L^p$-norm is *monotonic* in $p$.
- Basis of a vector
- Closed geodesics on $O^2$
- Prove that $\sum a_{n}$ converges. (I have some doubts on an expert's solution)
- Conditional expectation based on two bivariate normal RVs?
- Mixing inequality in the paper "The absence of mixing in special flows over rearrangements of segments"
- How to find symmetry of a function y=f(x) if it is not centered on the y-axis?
- Let $S = \{\frac{n^2+\sqrt{5}}{n}, n\in\mathbb{N}\}$ and $f : \mathbb{N} \to S$ be defined by $f(n) = \frac{n^2+\sqrt{5}}{n}$. Show $f$ is injective
- If $R$ is a UFD, then $R[x]$ is Noetherian?
- What are the subfunctors of $\operatorname{Hom}_{\Bbb Q}(q,-)$
- Is there a function which has limit only given $a_1,a_2...a_n....$ infinite points.
- A possible error in Villani's monograph “Hypocoercivity”
- Proof of rational number
- If the row echelon form of a linear equation system has a line with zero entries, then it has more than a solution.
- Unique steady state vector in relation to regular transition matrix
- Is the "$p$-norm" with $0<p<1$ a concave function
| The radius of the circle which just touches the outer circle Posted: 30 Mar 2021 10:38 PM PDT Suppose two concentric circles with radius $a$ and $b\ (>a)$ and origin as their center.
I wanted to put another circle whose center lies in a line $x=a$ (that is red line) in such a way that it just touches the outer circle at a single point. Something like this
As an equation solving a problem we can solve $$(x-a)^2+(y-r)^2=r^2$$ $$x^2+y^2=b^2$$ find the quadratic in $x$ and make the roots equal to each other. But that's too long a calculation, I wanted to know if I can solve it from the geometry of the figure. Or some much simpler method. Mainly I'm interested in $r$. Can any help me with this?  |
| Find the values of a and b that make the hyperbola congruent to $xy=1$ Posted: 30 Mar 2021 10:37 PM PDT I need to find a and b that would make the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ congruent to the rectangular hyperbola $xy=1$. I know that the answer is $a=b=\sqrt{2}$, and I've found some answers that prove it (using polar coordinates), however, I haven't found anything on the process of how to actually find a and b. Furthermore, we haven't gone over polar coordinates yet so I need an answer that doesn't involve that. I've been searching for over an hour both on here and other websites and I can't seem to find what I'm looking for.  |
| Constructing a DFA for a language Posted: 30 Mar 2021 10:29 PM PDT Yesterday I saw there was a discussion of the following problem, which I'm interested in too: Given that L is a regular language, construct a DFA for L-pref, where L-pref is defined as follows: L-pref = {w | at most one prefix of w is not in L} I've read the solution suggested here, How to build a DFA for the given language and prove its correctness? but I don't understand the solution entirely. I understand the case where epsilon belongs to L, but I don't understand the solution in the case in which epsilon doesn't belong to L (i.e the two machine copies thing). I'd be glad if you could explain how to transition between the two copies, preferably referring to an example of an accepting run of a word and a non-accepting run of some other word. Thanks!  |
| Lower bounding differences of a non linear function Posted: 30 Mar 2021 10:24 PM PDT Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be an injective function which admits a first order taylor series expansion of the form $$ f(x) - f(x_0) = \langle D_{x_0} , x- x_0 \rangle + o( \| x - x_0 \|) $$ Under what conditions can we say that there exists some $c > 0 $ for which $$ \left| f(x) - f(x_0) \right| \geq c \left| \langle D_{x_0} , x- x_0 \rangle \right| $$ for all sufficiently small $\| x - x_0 \|$. I think this is some kind of curvature condition but I'm not able to find a reference anywhere.  |
| Question about pinch singularities Posted: 30 Mar 2021 10:19 PM PDT I am reading the book "Analytic S-matrix" by Eden,Landshoff, Olive and Polkinghorne. In Sec.2, they discuss the different kinds of singularities of a function $f(z)$ defined as an integral of an analytic function $g(z,\omega)$ in the complex $\omega-$plane along some closed contour $\mathcal{C}$, i.e. $$ f(z) = \int_\mathcal{C}\, d\omega\, g(z,\omega)\,. $$ It is assumed that we know the singularities of $g(z,\omega)$. They discuss the case of pinch singularities, i.e. those singularities that approach the contour from the opposite sides and eventually coincide. As an example, they consider $$ f(z) = \int_0^1 \frac{d\omega}{(\omega-z)(\omega-a)} $$ Their result is $$ f(z) = \frac{1}{z-a}\log{\left[\frac{a(1-z)}{(1-a)z}\right]}\,,\qquad(a>1) $$ To perform this integral, I was thinking to take the parameterization $\omega(t) = \sqrt{t} e^{i \arccos{\sqrt{t}}}$ where $t\in [0,1]$. This parameterization actually selects the path of a semi-cicle in the upper-half complex $\omega-$plane, running from $\omega=0$ to $\omega=1$. The integral then can be computed as $$ f(z) = \int_0^1 dt \frac{d\omega(t)}{dt}\frac{1}{(\omega(t)-z)(\omega(t)-a)} $$ If $z=a>1$ or $z=a<0$, the singularities of the integrand lie outside my integration path, and the integral is perfectly convergent. Under Eq.(2.1.5) of their book, the authors say that
Can anyone explain to me what they mean by the last sentence? It is not clear to me which kind of contour can give a singularity at $z=a$.  |
| integro- (partial) differential equation Posted: 30 Mar 2021 10:13 PM PDT I'm dealing with a certain kind of integro-differential equation. The equation reads as : \begin{equation} \frac{du(t,x)}{dt} = \int_{\Omega} u(t,y)K(x,y) dy \end{equation} for some nice kernel function $K(x,y)$. I want to do a kind of decay estimate of $\|u(t,\cdot)\|_{L^2(\Omega)}$. can anyone help me? Any kind of reference would be grateful. Thanks.  |
| Understanding the proof of an equivalent condition for a graph to be a tree Posted: 30 Mar 2021 10:18 PM PDT I am studying graph theory from the book of Harary. There is a proof given in the book for the following: If G is a connected graph and $p=q+1$ then G is acyclic, where, $p$= No. of points in the graph and $q$= No. of lines in the graph. Proof given is as follows: Assume that G has a cycle of length $n$. Then there are $n$ points and $n$ lines on the cycle and for each of the $p-n$ points not on the cycle, there is an incident line on a geodesic to a point of the cycle. Each such line is different, so, $q\ge p$, which is a contradiction. I am facing difficulty in understanding the proof since it is too brief for me to understand. Can anyone explain me the line For each of the $p-n$ points not on the cycle, there is an incident line on a geodesic to a point of the cycle. Each such line is different. I have just started with the basic definitions of graph and a few of its properties. It will be helpful if someone explains the above proof with an example pictorially.  |
| condition of solution existence of system of linear equations where elements are either 0 or 1 Posted: 30 Mar 2021 10:06 PM PDT I would like to know a condition of solution existence of linear equations where the elements in the system are either 0 or 1. Formally, let $\vec{x} \in \mathbb{R}^N$, $\vec{1} = (1,1,1,\cdots,1) ∈ \mathbb{R}^N$, and $W$ is a $N \times N$ matrix of which elements are either 0 or 1. Then, what is a condition to guarantee existence of solution of the following system of the equations? $$W\vec{x} = \vec{1}$$ I think there are several trivial answers like $rank[W] = rank[W\ \vec{1}]$. I wish to get a answer which is a specific one for this situation.  |
| Proof of the discrete-time optional sampling theorem Posted: 30 Mar 2021 10:02 PM PDT Let $(X_n)_{n\in\mathbb N_0}$ be a submartingale on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_n)_{n\in\mathbb N_0},\operatorname P)$ and $X=M+A$ denote the Doob decomposition of $X$, i.e. $$M_n:=X_0+\sum_{i=1}^n\left(X_i-\operatorname E\left[X_i\mid\mathcal F_{i-1}\right]\right)\;\;\;\text{for }n\in\mathbb N_0$$ is an $(\mathcal F_n)_{n\in\mathbb N_0}$-martingale and $$A_n:=\sum_{i=1}^n\left(\operatorname E\left[X_i\mid\mathcal F_{i-1}\right]-X_{i-1}\right)\;\;\;\text{for }n\in\mathbb N_0$$ is predictable.
This must be an application of the submartingale property; but I really don't see how we obtain it.  |
| Posted: 30 Mar 2021 10:16 PM PDT I'm not entirely sure where to go from where I'm at. Right now all I have is the base step. Base step: P(24) is true because $7 + 7 + 5 + 5 = 24$. P(25) is true because $5 + 5 + 5 + 5 + 5 = 25$ A possible solution that I could use would be calculating P(26),P(27), and P(28), and then stating that any P(n) where n > 28 can be reached by adding some number of 5-cent stamps to one of P(24) through P(28). This solution doesn't really seem right, as I don't really know how I'd prove the previous statement, and especially how I would prove it using induction. Also, what's the difference between induction and strong induction?  |
| Posted: 30 Mar 2021 10:32 PM PDT Let $R$ be a Noetherian integral domain with $K$ as its quotient field. Then show that $ \alpha \in K$ is integral over $R$ if there exists a non-zero divisor $\beta \in R$ with $\beta \alpha^n \in R$ for all $n \geq 1$.  |
| Equivalence relations and Stirling numbers Posted: 30 Mar 2021 09:56 PM PDT Show that there are $\sum_{k=1}^{n}S_{n, k}$ equivalence relations on an n-element set. The numbers Sn,k are Stirling numbers of the second kind. I am learning discrete mathematics from different books and I have come across the problem above and I couldn't find a way to prove it.  |
| Posted: 30 Mar 2021 10:16 PM PDT Let $$ Y = \bigcup_{n = 1}^N (p_n\Bbb{Z} \pm 1) $$ Then the characteristic function $\chi_Y :\Bbb{Z} \to \{0,1\}$ is periodic with period $p_1 p_2 \cdots p_N = m$. By a basic Chinese remaindering argument. But $Y$ can also be written: $$ Y = \bigcup_{n=1}^{p_N} (n\Bbb{Z} \pm 1) $$ by a a basic sieve argument. Also, note that: $$ (p \Bbb{Z}\pm 1)^k = p\Bbb{Z} \pm 1 $$ since modulo $p$, the set $\{\pm 1\}$ is always a multiplicative subgroup of $\Bbb{Z}/p$. Thus $Y$ is closed under powering in particular. But it's also closed under things such as: $ab = (pz \pm 1)(pw \pm 1) = p^2zw \pm pz \pm pw \pm 1$, whenever both $a,b \in p\Bbb{Z} \pm 1$. So $x \in Y \implies p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in Y$ for any $a_i \in m\Bbb{Z}$. Thus $\text{ev}_y: m \Bbb{Z}[X] \to \Bbb{Z}$ is such that $\operatorname{im}(\text{ev}_y) \subset Y$ for any $y \in Y$. The set of all finite combinations of evaluation maps $R = \{ f(\text{ev}_{y_1}, \dots, \text{ev}_{y_k}) : y_i \in Y, f \in \Bbb{Z}[X_1, X_2, \dots, X_k], k \geq 1\}$ then forms a ring. Let $I = [a,b]$ be a maximal interval of integers such that $I \subset Y$. Question 1. Given the above data, can we come up with a formula for the maximal size of any such $I$?  |
| Enumerating how many ways two sets of symbols can be written in order. Posted: 30 Mar 2021 09:56 PM PDT Let $m,n \in \mathbb{N}$ and suppose that I have two ordered tuples of symbols $A = (a_1,\ldots,a_m)$ and $B = (b_1,\ldots, b_n)$. Using each of the symbols in $A$ and $B$ once and only once, we can form words of length $m+n$ with the condition that the symbols from each of $A$ and $B$ appear in the correct order. For example, $$ a_1 \cdots a_m b_1 \cdots b_n, \qquad a_1 \cdots a_{m-1}b_1 a_m b_2 \cdots b_n, \quad \text{and} \quad b_1\ldots b_n a_1 \cdots a_m $$ are all valid words. Let $\Sigma(m,n)$ denote the total number of words of this allowed form. For example $\Sigma(m,1) = m+1$ since we can slot the single element of $B$ at the start or end of the string of elements $a_1 \cdots a_m$ or anywhere in between two elements. We have $\Sigma(2,2) = 6$ with allowable strings $$ a_1a_2b_1b_2 \quad a_1 b_1 a_2 b_2 \quad b_1a_1a_2b_2 \quad a_1 b_1 b_2 a_2 \quad b_1a_1b_2a_2 \quad b_1b_2 a_1a_2. $$ Is there a "nice" way to compute $\Sigma(m,n)$? In particular, is there a way of expressing $\Sigma(m,n)$ in terms of $\Sigma(k,l)$ for $k < m$ and $l < n$?  |
| $L^p$-norm is *monotonic* in $p$. Posted: 30 Mar 2021 10:24 PM PDT Let $(X,\,\mu)$ be a positive measure space and assume $\mu(X)=1$. Let $f:X\to \mathbb C$ be a measurable function. Let $\|f\|_p$ denote the $L^p$-norm of $f$. If $0<r<s\le \infty$, then $\|f\|_r \le \|f\|_s$. How to prove this? I've tried but no progress. By the way I proved that the map $p \mapsto (\|f\|_p)^p$ is convex. Does it make any help?  |
| Posted: 30 Mar 2021 10:39 PM PDT I am struggling with this question about basis vectors and would love your help. The question is: Let W be the solution space to $Ax=0$ for the matrix $$ A=\begin{bmatrix} 1 & -2 & 0 & 4 & -7 \\ 0 & 0 & 1 & 0 & 1\\ \end{bmatrix} $$ Our task is to see if set $B$ and $C$ are the basis for $W$. $$ B=\begin{bmatrix} 1 & 2 & 1 \\ -1 & 1 & -2 \\ 2 & 0 & 0 \\ 0 & -1 & 0\\ 0 & 0 & 1 \end{bmatrix} $$ $$ C=\begin{bmatrix} 2 & 0 & 0 & 1 & -4 \\ 1 & 2 & -7 & -1 & 0 \\ 0 & 0 & -2 & -1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 2 & 1 & 0\\ \end{bmatrix} $$ My approach: To be a basis the set has to span $W$ and be linearly independent. I found that $C$ is linearly dependent so that set is not a basis. But I am confused with set $B$. This set is linearly independent and since $W$ is $3$-dimensional and $B$ has 5-dimensional vectors, it should also span $W$. However, my teacher says that this is not true because none of the vectors in $B$ are in $W$? What does she mean by this? I thought linearly independence and span was the only necessary condition to have a basis vector.  |
| Posted: 30 Mar 2021 10:27 PM PDT I'm reading a snapshot of what modern mathematics looks like: Closed geodesics on surfaces and Riemannian manifolds. Define$$ O^2:=E^1 \times E^1 $$ where $$ E^1:= \big\{(x,y)\in\Bbb R^{2+}:\log^2(x)+\log^2(y)=1\big\}. $$
I know that there must be at least one closed geodesic, and I think there could be infinitely many closed geodesics due to a theorem: Theorem 1: (Gromoll–Meyer [2], Sullivan–Vigué-Poirrier [6]) Any Riemannian manifold whose rational cohomology ring is generated by at least two elements contains infinitely many distinct closed geodesics. Also the meridian and longitude geodesics on $O^2$ are (under a coordinate change) equivalent to the meridian and longitude geodesics on $T^2$ (torus). Could this be the case for all the geodesics? That is, could all the geodesics on $O^2$ be related by a coordinate change to the geodesics on $T^2?$  |
| Prove that $\sum a_{n}$ converges. (I have some doubts on an expert's solution) Posted: 30 Mar 2021 09:57 PM PDT Let $\left ( a_{n} \right )_{n=1}^{\infty }$ be a sequence of real numbers and $s_{k}=a_{1}+a_{2}+\cdot \cdot \cdot +a_{k}$ be the $k$-th partial sum. Suppose that $\lim_{}a_{n}=0$, and there exists a $m\in \mathbb{N}$ such that the sequence $\left ( s_{mk} \right )_{k=1}^{\infty }=(s_{m},s_{2m},s_{3m}...)$ converges. Prove that $\sum a_{n}$ converges. Here, I have some expert's solution, which I doubt if it's really correct. Proof: $\forall \varepsilon >0$, since $\lim_{}a_{n}=0$, $\exists N_{1}>0$ s.t $\left | a_{n} \right |<\frac{\varepsilon }{4m} $ $\forall n>N_{1}$. Also, since $\left ( s_{mk} \right )_{k=1}^{\infty }$ converge, $\exists N_{2}>0$ s.t $\left |a_{km+1}+\cdot \cdot \cdot +a_{lm}\right |<\frac{\varepsilon }{2}$ $\forall l>k>N_{2}. $ Let $N=\max \{ N_{1},m(N_{2}+2) \}$. Then, $\forall q\geq p>N$,$$|a_{p}+a_{p+1}+\cdot \cdot \cdot+a_{q}|$$ $$\leq|a_{km+1}+\cdot \cdot \cdot+a_{lm}|+2m\cdot\frac{\varepsilon }{4m}$$ $$<\varepsilon$$ where $k$ is the least integer s.t $km+1\geq p$ , and $l$ is the largest integer s.t $lm\leq q$. $$\blacksquare $$ The most tedious part to understand is "how do we know there is some number which is $0$ (mod $m$), and some other number which is $1$ (mod $m$)?" (What if $p=q$?) And, "why do we need to set $N$ as this solution did?" Lastly, "how do we know that there are less than $m$ terms between $p$ and $km+1$, and there are less than $m$ terms between $lm$ and $q$?"  |
| Conditional expectation based on two bivariate normal RVs? Posted: 30 Mar 2021 10:19 PM PDT Suppose $X$ and $Y$ are bivariate normal, both with mean 1 and sd 1. Considering the correlation $\rho$, what should $E[X|X,Y]$ be? I think the answer should be just the RV $X$ as we conditioned $X$ itself. But is it possible that, since $Y$ and $X$ are correlated, it has some effect on the conditional expectation? Thanks!  |
| Posted: 30 Mar 2021 10:21 PM PDT I can't verify an inequality in the paper: https://link.springer.com/content/pdf/10.1007/BF02110361.pdf The inequality is at the end of the second page in the proof of the second theorem and is shown in this image: inequality It is in the proof of showing that special flow over a transformation with an ND approximation is not mixing. The author makes a sequence that contradicts mixing. I tried using density of the ergodic transformation to try and understand the intersection of sets but it wasn't insightful. Thanks for your help.  |
| How to find symmetry of a function y=f(x) if it is not centered on the y-axis? Posted: 30 Mar 2021 10:35 PM PDT I'm thinking about the example $f(x)=(x-1)^2$ which is clearly symmetric about the line $x=1$. The question is really how do you show that it is symmetric about $x=1$ algebraically? I notice that if you plug in $-x+2$ you end up with $y=(-x+1)^2$ which is equivalent to the original function. I'm looking for this answer because I'm curious how one would show a similar property for a function that is more tricky to graph.  |
| Posted: 30 Mar 2021 10:14 PM PDT I am really confused on this one. I am not sure how exactly to proceed. Furthermore, is my understanding of functions and mapping misguided or is this function not one-to-one on $S$? When I graph $f(n)$, it's clear to see that $f(1) = f(2.236) = \frac{1+\sqrt{5}}{1}$, and doesn't $f : \mathbb{N} \to S$ mean that the inputs lie within $\mathbb{N}$ (so $n = 1, 2, 3, ...$) and all the outputs are in the form $\frac{1+\sqrt{5}}{1}, \frac{2+\sqrt{5}}{2}, ...$ for all values of $\mathbb{N}$? Clearly, there is an output shared for two values of $n$, rending this not one-to-one? Here is my work thus far: Suppose $x, y, \in \mathbb{N}$, then $\frac{x^2+\sqrt{5}}{x} = \frac{y^2+\sqrt{5}}{y}$ $ = x + \frac{\sqrt{5}}{x} = \ y + \frac{\sqrt{5}}{y}$ $ = x - y = \frac{\sqrt{5}}{x} - \frac{\sqrt{5}}{y}$ I do not know how else to continue.  |
| If $R$ is a UFD, then $R[x]$ is Noetherian? Posted: 30 Mar 2021 10:24 PM PDT If $R$ is Noetherian, then $R[x]$ is Noetherian. However, if $R$ is a UFD, then $R$ might not be Noetherian. $\DeclareMathOperator{\Frac}{Frac}$ I want to show that if $R$ is a UFD, then $R[x]$ is a UFD. I think (hope) I have passed through the gauntlet of proving the uncomfortably diverse range of slightly nontrivial results that are lumped together as the chameleonic "Gauss's lemma" (and cited succinctly in many textbooks/websites/pdfs as "by Gauss's lemma" whenever convenient, even though it is often totally unclear to me which particular result the author is talking about). These include the following results when $R$ is a UFD. (Say $\sum_i a_ix^i \in R[x]$ is primitive iff $\gcd_R(a_0, a_1, \dots, a_n) = 1$.)
Say $R$ is an integral domain and $0 \neq r \in R - R^\times$. Pass $r$ into the following algorithm:
Artin's textbook says factoring terminates in an integral domain $R$ to mean that this algorithm is guaranteed to terminate. He proves that "factoring terminates" is equivalent to "$R$ does not contain an infinite strictly increasing chain $(a_1) \subsetneq (a_2) \subsetneq (a_3) \subsetneq \dots $ of principal ideals". If either of these equivalent conditions hold, then an integral domain $R$ is a UFD if and only if "irreducible $\implies$ prime". Therefore, it suffices to prove that $R[x]$ is Noetherian if $R$ is a UFD. How do we show this? Artin says "It is easy to see that factoring terminates in $\mathbb{Z}[x]$", but this is unclear to me as well.  |
| What are the subfunctors of $\operatorname{Hom}_{\Bbb Q}(q,-)$ Posted: 30 Mar 2021 10:00 PM PDT Let $\Bbb Q$ be the linearly ordered set of rational numbers, and $q\in\Bbb Q$ . I think the functor $\operatorname{Hom}_{\Bbb Q}(q,-)$ is in bijection with $A=\{t : t\in \Bbb Q, t>q\}$. I want to see if it is possible to show using Dedekind's that the subfunctors of $\operatorname{Hom}_{\Bbb Q}(q,-)$ are in bijection with $\{r: r\in\text{ extended real numbers}, r>q\}?$ (I need this to prove in category $Sets^Q$ theory, the subobject classifier $S$ has $S (q)=\{r: r\in\text{ extended real numbers}, r>q\}$ ). Please help....  |
| Is there a function which has limit only given $a_1,a_2...a_n....$ infinite points. Posted: 30 Mar 2021 10:22 PM PDT Find a $function $ which has limit only in advanced marked points$(a_1,a_2,....a_n)$. Here is my example. $g(x) = \begin{cases} 1, & \text{if $x$ $\in$ $Q$} \\ 0, & \text{if $x$ $\in$ $R/Q$} \end{cases}$ $f(x) = (x-a_1)(x-a_2)*...*(x-a_n)*g(x)$ But the problem I can't solve is that,is there a function which has limit only in advanced given $a_1,a_2...a_n....$ infinite points.  |
| A possible error in Villani's monograph “Hypocoercivity” Posted: 30 Mar 2021 10:29 PM PDT
I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (8.1). The in the Theorem 40 below (page 64, with the choice $N = M^3$) this inner product is changed to (8.4), which is definitely different from (8.4). Also, even though we accept (8.4) rather than (8.1) as the definition of $((\cdot,\cdot))$, I don't see why (8.5) implies the advertised conclusion (which basically says the existence of some constant $\lambda$ such that $((h,Lh)) \geq \lambda\,((h,h))$ (please ignore the real part as I am only looking at the real case). (Background information: here $L := A^*A + B$ and $B$ is antisymmetric). I sometimes feel that celebrated mathematicians are so careless and they didn't pay great attention to details. Thanks for any help!  |
| Posted: 30 Mar 2021 10:25 PM PDT Prove that if $a$ and $b$ are rational numbers satisfying $a^5+b^5=2a^2b^2$, then $1-ab$ is the square of a rational number. I am just a Year 2 student learning Abstract Algebra. This problem is a challenging one that my teacher gives us. However, I have no ideas about how to solve it. I tried to represent rational numbers by $p/q$ ($p$ and $q$ are both integers) but failed. I would appreciate it very much if anyone can help me.  |
| Posted: 30 Mar 2021 10:13 PM PDT
Let \begin{align} \text{I}& \qquad 3x_1&=2\\ \text{II}& \qquad 0x_1&=0 \end{align} a linear equation system in row echelon form and a line with zero entries. If we solve this equation system, the solution $\mathcal{L}$ would be $\mathcal{L}=\{x_1=\frac23\}$. Contradiction. The statement is wrong. This looks way too easy. Is there anything I might forgot about? EDIT:
This statement is wrong (too), right?  |
| Unique steady state vector in relation to regular transition matrix Posted: 30 Mar 2021 10:02 PM PDT
Consider the following theorem: Suppose $T$ is a regular transition matrix such that $x_k = T^kx_0$ (a discrete dynamic system), then there is a unique steady-state vector $v$ such that $Tv = v$. Furthermore, if $x_0$ is any probability vector: $$\lim_{k\to\infty} x_k = \lim_{k\to\infty} T^kx_0 = v$$ Here's the confusion: Suppose $\lambda$ is an eigenvalue for the $x_0$ mentioned above, then $$T^kx_0 = \lambda^kx_0 \Rightarrow \lim_{k\to\infty} T^kx_0 = \lim_{k\to\infty} \lambda^kx_0$$ Suppose we have real entries and real eigenvalues, then we have few cases: 1) $\lim_{k\to\infty} \lambda^k = \pm\infty$ when $\lambda > 1$ or $\lambda < -1$ 2) $\lim_{k\to\infty} \lambda^k = 0$ when $ -1 < \lambda < 1$ 3) $\lim_{k\to\infty} \lambda^k = 1$ when $\lambda = 1$ 4) $\lim_{k\to\infty}$ oscillate between $1$ and $-1$ when $\lambda = -1$ It seems to me that, besides the trivial case of 3), none of these cases can produce $\lim_{k\to\infty} \lambda^kx_0 = v$. Since $v$ is a unique steady-state vector, in the other $3$ cases, $v$ becomes $0$, $\pm\infty$, or oscillate between $x_0$ and $-x_0$. I think I'm missing something but I don't know what it is. Can someone care to explain what I'm missing?  |
| Is the "$p$-norm" with $0<p<1$ a concave function Posted: 30 Mar 2021 10:15 PM PDT We know that for $p>1$ by the triangle inequality $$\|tx + (1-t)y\|_p \leq t\|x\|_p + (1-t)\|y\|_p,$$ when $t\in[0,1]$. Now the "$p$-norm" for $0<p<1$ is not actually a norm but we can still define it to be $$\|x\|_p=\left(\sum_{i=1}^n |x_i|^p\right)^{1/p},$$ we do not have the triangle inequality but can we at least say that this function is concave?  |
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