Recent Questions - Mathematics Stack Exchange |
- Given that $A$ is a subspace of the lower limit topology on $\mathbb{R}$, is $A$ normal.
- Proof of Jacobi fraction expansion in Triple Product Proof
- Trying to understand Haagerup tensor product
- Set of non-zero numbers with fixed sum and other constrain
- Well defined function given by an infinite series
- How many blowups do we need to make a pencil base point free?
- Prove or disprove: Let R be an Euclidean domain, then $I= \{a\in R\mid \delta(a)>\delta(1_R)\}$ is an ideal in R.
- $f=\text{ker}(g)$ for a short exact sequence in an abelian category.
- Does this sequence always end?
- How to begin solving a question like this? value of $\frac{d}{dx}[f^{-1} (x)]$ when $x=2 \pi$, given that f(x) = 2x-sin x and $f^{-1}(2\pi)=\pi$
- Is there a strengthening of Carmichael's theorem?
- How to prove by strong induction with no given formula?
- "Barn Door" trig problem
- Simple but interesting problem about the binomial coefficient from Olympiad
- First order ODE?
- Can the constraint $|x| + |y| \geq 5$ be written as a combination of linear constraints?
- Homotopy type of space of continuous vs algebraic functions on complex varieties.
- The worst function in Squarified Treemaps
- What is the Dual curve to a straight line defined projectively as x+y+z=0?
- Cardinal of the set $A=\Big\{x=\frac pq\in\mathbb{Q} :|\sqrt 2-x|<\frac1{q^3}\Big\}$
- I don´t understand the difference between initial and terminal objects
- What is wrong with my two fair dice probability question reasoning?
- Alternative definition of the sign of a permutation and its equivalence
- Is it possible to transform a convex optimisation with the max-affine cost into either LP, QP, or conic programming?
- Find an example of an constructible model.
- Existence of Complex Frames on a Complex Vector Bundle
- How can a bipartite graph be Eulerian?
- Do the columns of an invertible $n \times n$ matrix form a basis for $\mathbb R^n$? [duplicate]
- How to find cosh(arcsinh(f(x)))?
| Given that $A$ is a subspace of the lower limit topology on $\mathbb{R}$, is $A$ normal. Posted: 29 Aug 2021 07:45 PM PDT We know that the subspace of normal space may not be normal. But for the normal space $\mathbb{R}_l$, I think we could show that the subspace is always normal. So let $A$ be a subspace of $\mathbb{R}_l$. Let $C$ and $D$ be disjoint closed sets in $A$. For each $c \in C$, we have basis element $[c, x_c) \cap A$ that is disjoint from $D$, since $D$ is closed. Similarly, we can do this for every point in $B$. Now $\bigcup_{c \in C} [c, x_c) \cap A$ and $\bigcup_{d \in D} [d, x_d) \cap A$ are open neighborhood contains $C$ and $D$ respectively. and we can show that they are actually disjoint by a contradiction. Is this proof correct?  |
| Proof of Jacobi fraction expansion in Triple Product Proof Posted: 29 Aug 2021 07:43 PM PDT In the proof of Jacobi's triple product identity by Jacobi, he considers the infinite product; $\frac{1}{(1-qz)(1-q^2z)...}$ and expands it into 1 + $\frac{B_1z}{(1-qz)}$ + $\frac{B_2z^2}{(1-qz)(1-q^2z)}$ + .. How did Jacobi know such a product could be expanded in such a way?  |
| Trying to understand Haagerup tensor product Posted: 29 Aug 2021 07:43 PM PDT I'm self reading Haagerup tensor product of operator spaces from Effros Ruan's book on operator spaces. I am unable to digest it properly.
Thank you so much in advance.  |
| Set of non-zero numbers with fixed sum and other constrain Posted: 29 Aug 2021 07:38 PM PDT everyone. Suppose I have a set of non-zero numbers $\{x_1,\dots,x_N\}$ and they have fixed sum $\sum_i x_i = A_1$. Also, the quantity $\sum_i i x_i = A_2$ is also fixed. Suppose I know $N$, $A_1$ and $A_2$, is there anyway to estimate the number of possibilities? Is any field of math related to this problem? Thanks.  |
| Well defined function given by an infinite series Posted: 29 Aug 2021 07:26 PM PDT Main Reference: O. Christensen, K.L. Christensen, ``Approximation Theory: from Taylor Polynomials to Wavelets'', Birkhauser, Boston, 2nd printing, 2005. Statement of Exercise $2.11 (i)$ on p. 49: ``Prove that $$f(x) = \sum_{n=1}^{\infty} \frac{\cos(2^n x)}{3^n}, \,\, x \in \mathbb{R} $$ is well defined.'' My attempt to prove the above: In general for a function $g: A \rightarrow B$ to be well defined, the following conditions must be satisfied:
In particular, the third condition says that if an element of $A$ has an image in $B$ then this image is unique. $f(x)$ as given above satisfies this condition if the series is absolutely convergent. We have that ``if $${\sum_{n=1}^{\infty} a_n}$$ is absolutely convergent with sum $X$, then every rearrangement is absolutely convergent and has sum $X$.'' (Ref: Dangello and Seyfried, ``Introductory Real Analysis'', p. 158). Suppose the infinite series for $f(x)$ is only conditionally convergent then there could be a rearrangement that gives a different sum than the original arrangement. That is for some $x \in \mathbb{R}$ we could have two different images. Using the Root test we have: $$\lim_{n \to \infty} \sqrt[n]{\left| \frac{\cos(2^n x)}{3^n} \right| } \leq \lim_{n \to \infty} \sqrt[n]{\left| \frac{1}{3^n} \right| } = \lim_{n \to \infty} \frac{1}{3} = \frac{1}{3} < 1$$ So the infinite series is absolutely convergent. Therefore $f(x)$ is well defined. I would like to know if the above proof is correct.  |
| How many blowups do we need to make a pencil base point free? Posted: 29 Aug 2021 07:21 PM PDT Let $\Bbb{P}^2$ be the projective plane over $\Bbb{C}$. Take a pencil of curves of degree $d$ on $\Bbb{P}^2$ given by a dominant rational map $\phi:\Bbb{P}^2\dashrightarrow \Bbb{P}^1$. The pencil has $d^2$ base points, possibly infinitely near. If $p:S\to\Bbb{P}^2$ is the blow up at all $d^2$ base points, then $\phi\circ p:S\to\Bbb{P}^1$ is a base point free pencil on $S$. My question is: do we always have to blow up $d^2$ times in order to get a base point free pencil? I agree this is necessary when the $d^2$ base points are not infinitely near. But if one of the base points has many infinitely near it, then this is not clear to me.  |
| Posted: 29 Aug 2021 07:35 PM PDT Prove or disprove:
I showed that $I$ has the superclosure property because of the property of euclidean domain $\delta(ab) \geq \delta(a)$ so I thought that $I$ should be an ideal However according to the solution manual, the answer is actually false and they claim it is false for even $Z$. However I don't see how that could be possible unless the product of two elements in I is a unit, but the only units are 1 and -1 and they are not in $I$ and all the other elements don't have inverses.  |
| $f=\text{ker}(g)$ for a short exact sequence in an abelian category. Posted: 29 Aug 2021 07:10 PM PDT I am a rank amateur when it comes to category theory, and have gotten myself stuck on what should be a simple exercise. A short exact sequence in an (abelian), category is a sequence of morphisms: $$ 0\rightarrow A\xrightarrow{f}B\xrightarrow{g}C\rightarrow0 $$ such that $\text{ker}(f)=0$$, \text{im}(f)=\text{ker}(g)$, and $\text{im}(g)=0$. I want to show that for such a sequence, $f=\text{ker}(g)$. The obvious way to proceed is to show that $f$ satisfies the universal property of the kernel, and then appeal to uniqueness. To do so, I must first show that $g\circ f=0$, and then show that given a morphism $e:A'\to B$ with $g\circ e=0$, there exists a unique morphism $e':A'\to A$ such that $e=f\circ e'$. I have been able to use the epi-mono factorisation that abelian-ness gives us to show that $g\circ f=0$, but I have been unable to define the required map. Any hints or help would be much appreciated.  |
| Does this sequence always end? Posted: 29 Aug 2021 07:44 PM PDT Let $g$ be the greatest factor of a positive integer($a$) where $g \neq a$ and $l$ be the smallest factor such that $gl = a$. Let $i$ be the $i$th term of a sequence where the starting term is $\frac{g_1!}{(g_1l_1)^n}$ and $n$ is the greatest positive integer such that $(g_il_i)^n|g_i!$. More specifically the sequence $t_i = \frac{g_{i}! }{t_{i - 1}}$ Note that every consecutive term is coprime and that we consider that a sequence ends when $t_i$ is prime. Ignoring values that are the product of a sqaure, My question is that given any value of $g$ and $l$ will this sequence of terms edventually end and if so, provide a proof.  |
| Posted: 29 Aug 2021 07:22 PM PDT How do I begin solving a question like this: What is value of $\frac{d}{dx}[f^{-1} (x)]$ when $x=2 \pi$, given that f(x) = 2x-sin x and $f^{-1}(2\pi)=\pi$  |
| Is there a strengthening of Carmichael's theorem? Posted: 29 Aug 2021 07:06 PM PDT I'm looking for generalizations of Carmichael's theorem. Specifically, I'm seeking a theorem of the form
Is there a theorem of this kind?  |
| How to prove by strong induction with no given formula? Posted: 29 Aug 2021 07:28 PM PDT I got this homework on mathematical induction, but normally the question would give a formula at the end for a sequential type question, but this one does not.The question is: Abby starts with a pile of 2n books and makes a sequence of "moves". In each move, she chooses one pile of books and splits it into two non-empty piles of books, both of which have an even number of books in them. She counts how many such moves she makes, and does not stop until she cannot make another move. Using strong induction, prove a formula for the number of moves that Abby makes (in terms of n)  |
| Posted: 29 Aug 2021 07:13 PM PDT I am mainly a computer science guy, currently trying to create a 3d model of a barn door. This led me into an interesting little problem that my trigonometry skills are apparently too rusty to solve.
In the given illustration, A and B are known as well as W, The hypotenuse(C) and its angle are obvious even to me, but this is not the same thing as the angle of the diagonal board. What is the best way to calculate that? Thanks for your patience.  |
| Simple but interesting problem about the binomial coefficient from Olympiad Posted: 29 Aug 2021 06:59 PM PDT
This problem is from the final round of an old South Korean Mathematical Olympiad (1997 KMO).  |
| Posted: 29 Aug 2021 07:09 PM PDT I am confused as to what exactly this question is asking for. Consider the equation for y = y(s): sy' − y = 0.
I found the general solution for the second question to be y = C*s but I am unsure how to find an ODE that satisfies x using the definiton given. Any help would be appreciated!  |
| Can the constraint $|x| + |y| \geq 5$ be written as a combination of linear constraints? Posted: 29 Aug 2021 06:59 PM PDT Can the constraint $|x| + |y| \geq 5$ be written as a combination of linear constraints? My attempt: We can never draw a linear inequality that includes both sides ($x\geq 5 $ and $x\leq -5$, or similarly, $y\geq 5 $ and $y\leq -5$) but excludes the origin $(0,0)$. Suppose we have any system o inequalities $Ax\leq b$ that includes both sides, so it includes $(5,0)$ and $(-5,0)$ and then $$ A \begin{bmatrix} 5 \\ 0 \end{bmatrix} \leq b \ \text{and} \ A \begin{bmatrix} -5 \\ 0 \end{bmatrix} \leq b \ $$ $$ A \begin{bmatrix} 5 \\ 0 \end{bmatrix}+A \begin{bmatrix} -5 \\ 0 \end{bmatrix} \leq 2b \ $$ $$ A\begin{bmatrix} 0 \\ 0 \end{bmatrix} \leq 2b \ $$ $$ A\begin{bmatrix} 0 \\ 0 \end{bmatrix} \leq b \ $$ Which implies that $(0,0)$ satisfies the system of inequalities as well. In the other words, this region cannot be approximated by a linear program. Is it correct? Thank you!  |
| Homotopy type of space of continuous vs algebraic functions on complex varieties. Posted: 29 Aug 2021 07:13 PM PDT Given a finite type scheme over $\mathbb{C}$ like $\text{Spec}(A)$. What is the homotopy type of space of continuous fucntions on the complex points of $\text{Spec}(A)$? (With compact open topology). What is the homotopy type of subspace of regular functions i.e. $A$? and how does that compare to the continuous ones. P.S. By Stone-Weierstrass the regular functions are dense in continuous functions with the compact-open topology, but I am not sure how their homotopy types compare.  |
| The worst function in Squarified Treemaps Posted: 29 Aug 2021 07:05 PM PDT backgroundhere is a related post How to calculate optimal sizes of rectangles for this type of array visualization?, that post introduces a algorithm Squarified Treemaps, its paper is here questionMy question is about the sub function the
according to my understanding, for getting the highest aspect ratio of a list of rectangles, I will firstly calculate the aspect ratio for each of the list of rectangles then find the highest one I need someone to give me a hand to understand why the  |
| What is the Dual curve to a straight line defined projectively as x+y+z=0? Posted: 29 Aug 2021 07:10 PM PDT I have recently been learning about algebraic geometry and came across this question in "Algebraic Geometry A Problem Solving Approach" by Thomas Garrity et al. His problem 1.12.18 asks about the dual curve to a projective line given by $L=\{(x:y:z)\in \mathbb{P}^2:ax+by+cz\}$ and why it might be strange. When I try and answer the question on my own following the recipe on https://en.wikipedia.org/wiki/Dual_curve I get $X=Y=Z=\lambda$, which doesn't make sense to me. I also looked at the Plücker formula, which I think is saying that a degree 1 curve should have a degree zero dual. Does anyone have any thoughts? Thanks!  |
| Cardinal of the set $A=\Big\{x=\frac pq\in\mathbb{Q} :|\sqrt 2-x|<\frac1{q^3}\Big\}$ Posted: 29 Aug 2021 07:44 PM PDT Let $A=\Big\{x=\frac pq\in\mathbb{Q} :|\sqrt 2-x|<\frac1{q^3}\Big\}$ We want to prove that $A$ is finite and find its cardinal? we can prove that it's finite by using directly Roth's theorem which is a generalized theorem of Lionville's theorem. But, I am stuck on finding the number of its elements. Any help, and thanks in advance.  |
| I don´t understand the difference between initial and terminal objects Posted: 29 Aug 2021 07:44 PM PDT I have trouble understanding the difference between initial and terminal objects in category theory. I get the definitions, but when seeing examples, I am confused. Example 1: The empty set is the unique initial object in the category of sets. Every one-element set is a terminal object in this category.
Example 2: Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category.
I don´t know why I am stuck with this. Maybe I have a wrong idea about how morphisms behave? Thank you for your help.  |
| What is wrong with my two fair dice probability question reasoning? Posted: 29 Aug 2021 07:44 PM PDT There is a game where you are asked to roll two fair six-sided dice. If the sum of the values equals 7, then win £21. However, must pay £5 to play each time both dice are rolled. Do you play this game? One way to think about this is that getting a 7 comes with 1/6 chance, and to make money we need to get 7 at a rate of 1/4, so the answer is not to play. Another way to think about it is: what is my chance of throwing a 7 at least once in every 4 throws? In which case I would calculate a probability of not throwing a 7 4 throws in a row (5/6)^4, and then subtract this from 1 to get a probability of throwing at least one 7. Which is 1 - (5/6)^4 = 0.52. By this logic I would play the game. Both of these answers cannot be correct. Could someone explain to me which one is incorrect and why? Thanks!  |
| Alternative definition of the sign of a permutation and its equivalence Posted: 29 Aug 2021 07:48 PM PDT I have seen that one can define the sign $\varepsilon (\sigma)$ or $\mathrm{sgn}(\sigma)$ of a permutation $\sigma \in S_n$ with the formula $$\varepsilon(\sigma):=\prod_{1\leq i<j\leq n}\frac{\sigma(j)-\sigma(i)}{j-i}. \tag 1$$ The main definition is probably the following: $$\varepsilon(\sigma)=\begin{cases} +1, & \text{if $\sigma$ can be written as a product of even number of transpositions}, \\ -1, & \text{if $\sigma$ can be written as a product of odd number of transpositions}. \end{cases} \tag 2$$ I've also seen that the sign can be defined via the Vandermonde polynomial. That is, $$\varepsilon(\sigma)=\frac{P(X_{\sigma(1)},\dots,X_{\sigma(n)})}{P(X_1,\dots,X_n)}=\prod_{1\leq i<j\leq n}\frac{X_{\sigma(j)}-X_{\sigma(i)}}{X_j-X_i}.\tag 3$$ My question is, how these three definitions are equivalent. Any reference is also welcomed. Thanks. Update: There is another one definition: $$\varepsilon(\sigma):=(-1)^{N(\sigma)},\tag 4$$ where $N(\sigma)$ is the number of inversions in $σ$.  |
| Posted: 29 Aug 2021 07:35 PM PDT QuestionI wonder if it's possible to transform a convex optimisation problem with the max-affine cost into either LP (linear programming), QP (quadratic programming), or (CP) conic programming. That is, the given problem can be expressed as $$ \min_{x} \quad \max{c_{i}^{\intercal}x}, $$ where $c_{i}$'s are prescribed constant vectors ($i=1, 2, \ldots, N$). My approachI guess that it can be transformed into an LP as $$ \min_{t, x} t \\ \text{subject to} \\ c_{i}^{\intercal} x\leq t \quad (i=1, 2, \ldots, N) $$ by introducing a slack variable $t$ but I'm not sure it's correct.  |
| Find an example of an constructible model. Posted: 29 Aug 2021 07:14 PM PDT I have found the rigorous definition of constructible model like this: Let $L$ be a first-order language and $M$ an $L$-structure. If $A\subseteq M$, $M$ is constructible over $A$ if $(M-A)$ can be written as $(c_i\colon i<\lambda)$, where $\lambda$ is an ordinal, such that, for any $j<\lambda$, the type $tp_M(c_j/A\cup \{b_i\mid i<\lambda\})$ is isolated. I would find an example of constructible structure based on this.  |
| Existence of Complex Frames on a Complex Vector Bundle Posted: 29 Aug 2021 07:37 PM PDT $E \rightarrow M $ be a complex vector bundle (of real rank $2r$) with almost complex structure $J:E\rightarrow E \space\space\space(J^2 =-1)$ on it. $U\subset M$ be a trivial neighbourhood. Does there exist a complex frame for $E \rightarrow M$ over U? By a complex frame, I mean a set of $r$ (real) linearly independent $E_U$-sections $\{X_1, ... , X_r\}$, such that $\{X_1, ... , X_r, JX_1, ... , JX_r\}$ forms a complete $E_U$-frame. (By "linearly independent", I mean they are linearly independent in the fibre above every point $x\in U$). It seems like such frames do exist as I recall seeing such frames in definition of complex connection and curvature matrices on complex vector bundles, which then lead to construction of Chern forms and classes. I need such a frame to prove that existence of a $J$ map for $E \rightarrow M $ implies a reduction of structure group of E to $GL(r,C)$. I am stuck with this, even though I can prove the converse by explicitly constructing a complex frame using the reduction. Edit: I am using the following definition for 'complex vector bundle': A real even ranked vector bundle with a J-map. Sorry for the confusion.  |
| How can a bipartite graph be Eulerian? Posted: 29 Aug 2021 07:00 PM PDT From the way I understand it: (1) a trail is Eulerian if it contains every edge exactly once. (2) a graph has a closed Eulerian trail iff it is connected and every vertex has even degree (3) a complete bipartite graph has two sets of vertices in which the vertices in each set never form an edge with each other, only with the vertices of the other set. So by definition a bipartite graph has some edges that are not used (i.e. the edges between vertices of the same set). That would then mean that there are unused edges and so the graph cannot be Eulerian. What am I missing here?  |
| Do the columns of an invertible $n \times n$ matrix form a basis for $\mathbb R^n$? [duplicate] Posted: 29 Aug 2021 07:28 PM PDT The columns of an invertible $n \times n$ matrix form a basis for $\mathbb R^n$. I follow the definition from the text book, then I guess because the matrix is invertible, each vector in the matrix is linearly independent, thus the basis of column space is span in $\mathbb R^n$. However, I am still confused. Could someone tell me why or if I've missed something? Thanks.  |
| How to find cosh(arcsinh(f(x)))? Posted: 29 Aug 2021 07:27 PM PDT With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric functions, I can draw a right triangle to find a formula for this that doesn't involve any trigonmetric functions. How do I find a similar result for hyperbolic functions? For instance, when working a problem recently, I ended up with $\cosh(\operatorname{arcsinh}(3x))$. WolframAlpha told me that it was $\sqrt{1+9x^2}$, but how do I figure that out? What picture can I draw? I'm not sure of the geometry here. I'm pretty sure that hyperbolic functions are related to hyperbolas the way that trig functions are related to circles, but I don't figure out the trig(arctrig) expressions by looking at circles -- I draw a triangle. Is there something similar I can do with hyperbolic functions?  |
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