Recent Questions - Mathematics Stack Exchange

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• A finite abelian group contains a subgroup isomorphic to the direct product of a group of order $p$ with itself
• Order in gaussian integer moduli
• If R is a regular expression, then is R*R = R?
• local ring of fractions
• Number of possible sets $A_3, \dots A_6$ such that $\{1,2\} \subseteq A_3 \subseteq \cdots \subseteq A_6 \subseteq \{1,2,3,4\}$
• Identifying graph based upon given function
• How to refer to image of homset under a functor.
• Can I replace or simplify the unitary matrix to get the same results of multiplication
• $f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 )$ $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has two distinct real roots. Find $f(x)$.
• Empirical Fisher Information but with unknown true parameters and distribution?
• Finding inverse of a bounded operator
• How to obtain a relation between the following functions to obtain the desired integral
• Why does $\sum_{k=0}^{\infty}\frac{a^k}{k!}e^{-(a+b)} = e^{-b}$?
• The total angle must be $ 90 ^{\circ} $ degrees however as we sum up the each angle , the equation be dishold
• Is the quotient a field?
• A question on completeness condition from Ahlfors' book
• Stolen Rubys Puzzle, How many rubys can you guarantee to win?
• Prove that $g'''(0)$ does not exist
• For help an inequality that for large $n$, whether $\frac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$ for some $\beta>0$
• Propositional Logic Question Related to Understanding Implication and Tautologies
• Sum with Digamma function expressed as hypergeometric function?
• Explicit formula for heat equation in $\mathbb{R}$
• Definition of sine and cosine [duplicate]
• Split $\{1,2,...,3n\}$ into triples with $x+y=4z$
• A problem about analytic geometry
• A Relation between discriminant of individual fields vs the discriminant of their Composite
• Source of the Banach Lemma
• $\int x^{dx}-1$
• Proving if $A$ is an $n\times n$ positive semi-definite matrix, A is Hermitian with non-negative eigenvalues.
• The joint distribution of two linear combinations of independent standard normal variables

A finite abelian group contains a subgroup isomorphic to the direct product of a group of order $p$ with itself Posted: 10 Jul 2021 09:23 PM PDT In Serge Lang's Algebra he leaves a direct proof of the following statement to the reader: Let $G$ be a finite abelian group. If $G$ is not cyclic, then there exists a prime $p$ and a subgroup of $G$ isomorphic to $C$$\times$$C$, where $C$ is cyclic of order $p$. I have come up with this so far: Consider such a group $G$ and let $r$,$s$ be non-identity elements such that neither can be expressed as a power of the other(such elements exist because $G$ is not cyclic.) We then have two cases: either the periods of $r$ and $s$ are coprime or they aren't(and thus both are divisible by some prime $p$.) Suppose the periods are $m$ and $k$ respectively. If their periods are coprime, then let $p$ divide $m$$k$. The group $\langle$$r$$\rangle$$\times$$\langle$$s$$\rangle$ is then cyclic(since $m$ and $k$ are coprime) of order $m$$k$. Given a generator $x$, the cyclic group $\langle$$x$^$m$$k$/$p$$\rangle$ is of order p and we have our group $C$. Now we must prove that $s$^$i$ does not equal $r$^$k$ for any nontrivial $i$,$k$. If this were true, then $r$^$k$ would generate a subgroup of $\langle$$s$$\rangle$. The order of this subgroup then divides both $\langle$$s$$\rangle$ and $\langle$$r$$\rangle$ contradicting $m$ and $k$ being coprime. Hence we have an isomorphism between the subgroup $\langle$$s$,$r$$\rangle$ and $\langle$$x$^$m$$k$/$p$$\rangle$. I can not seem to find a similar result for the case where $m$ and $k$ are not coprime(that is, I cannot figure out why ($r$^$m$/$p$)^$i$=($s$^$k$/$p$)^$j$ cannot happen for nontrivial $i$,$j$.) So, overall, is what I have in the first case correct and how do I prove the second?

Order in gaussian integer moduli Posted: 10 Jul 2021 09:20 PM PDT Say I want to find the order of an element in $\mathbb Z[i]$ modulo $a+bi.$ If this element is $x+yi,$ I could write $(a+bi)(c+di)+1=(x+yi)^n.$ I noticed that if $y=0,$ we could do this pretty easily by considering the order of $x$ modulo $N(a+bi)=a^2+b^2,$ but for nonreal or nonimaginary elements this seems harder and maybe needs more computation. How can I calculate the order $n$? Is it entirely algebraic?

If R is a regular expression, then is R*R = R? Posted: 10 Jul 2021 09:13 PM PDT I found that that ϵ+ RR = R but my intuition is that RR is also R since $$R^* = R^0 u R^2 u R^3 u ... $$ to infinity So, R*R = R ?

local ring of fractions Posted: 10 Jul 2021 09:03 PM PDT Let $P$ be a prime ideal of a commutative ring $R$ that is an integral domain. Show that if $T = R\backslash P,$ then $T^{-1}R$ is a local ring (i.e. $T^{-1}R$ has only one maximal ideal). I know how to show that $T^{-1}P$ is an ideal of $T^{-1}R$, but I'm not sure how to show that it's maximal. Also, how do I show that it is the only maximal ideal? I think I need to use the fact that $R$ is an integral domain and $P$ is prime, so if $ab \in P,$ then $a\in P$ or $b\in P$ for $a,b\in R.$ I also know that $R\backslash P$ is multiplicatively closed.

Number of possible sets $A_3, \dots A_6$ such that $\{1,2\} \subseteq A_3 \subseteq \cdots \subseteq A_6 \subseteq \{1,2,3,4\}$ Posted: 10 Jul 2021 08:58 PM PDT The original problem I encountered is as follows: Given $S = \{1,2,3,4\}$, how many possible pairs (?) of nonempty subsets $A_1, A_2, \dots, A_6$ of $S$, such that $A_1 \cap A_2 \neq \emptyset$ $A_1 \cup A_2 \subseteq A_3$ $A_3 \subseteq \cdots \subseteq A_6$ are there? I'm interested in the case where $|A_1|= |A_2|=1$. How many possible pairs (?) of $A_3, \dots, A_6$ are there? My approach was letting $A_1 = \{1\}$ and $A_2 = \{2\}$, and therefore $A_1 \cup A_2 = \{1,2\}$. Then, I draw all possible combinations of $A_3, \dots, A_6$ as follows: The first row is the possible choices of $A_3$, the second is $A_4$, and so on. Counting the leaves of the tree gives us $25$. I wonder if there's a way for me to find that without drawing all the possible choices. The solution for the original problem has this formula, $$\binom{2}{0} + 4 \left(\binom{2}{1} + \binom{2}{2}\right) + 6 \binom{2}{1} \binom{1}{1} = 25$$ Where does that come from? To recap, my questions are: Is there a way to find all the possible choices without drawing the whole possibilities? Where does that formula come from? How can I derive it myself? How does the formula match with the tree diagram I made? (Which parts correspond with which parts?)

Identifying graph based upon given function Posted: 10 Jul 2021 08:54 PM PDT Hello I am stuck on this problem because I am uncertain what this graph is even supposed to look like. The question reads as follows: One of the curves below is 𝑦=𝑓(𝑥), and the other is 𝑦=𝜅(𝑥). Determine which is which and explain your reasoning. enter image description here Any assistance would be greatly appreciated!

How to refer to image of homset under a functor. Posted: 10 Jul 2021 08:57 PM PDT If I have a functor $F: C \rightarrow Set$, and morphisms between $c \in Ob(C)$ and $c' \in Ob(C)$ (call them $a_i : c \rightarrow c'$), how can I talk about the set of functions $\{F(a_i)| a_i \in Hom(c, c')\}$? Does it have a name? It is an object in $Set$, but it doesn't have to be isomorphic to $Hom(c, c')$. Using a mix of set-theoretic and category theoretic terms, I would describe it as $Hom(c, c') / \sim$ where two functions $F(a_i)$ and $F(a_j)$ are equivalent if their images are the same. But I'd like to describe this set using a category theoretic construction. Maybe using a pushout? Maybe using a nat transform from $Hom(c, \_)$?

Can I replace or simplify the unitary matrix to get the same results of multiplication Posted: 10 Jul 2021 08:47 PM PDT I have two unitary matrices, $F$ and $D$ where $D$ is real and $F$ is DFT matrix (Discrete Fourier Transform) matrix, where $FD = V$, and $V$ is unitary matrix also. My question, is it possible to replace the matrix $D$ by another real unitary matrix, or simplify it by forcing the majority of its elements to be zeros, under condition to have $F*D_2 = V$. Where $D_2$ is the new simplified matrix of the matrix $D$.

$f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 )$ $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has two distinct real roots. Find $f(x)$. Posted: 10 Jul 2021 09:10 PM PDT Can we get a function in the following form and properties ? $f(x) = (x^2 -1)^2(a_0 x^3+a_1 x^2 + a_2 x + a_3 )$ $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has two distinct real roots. Can anyone help me to find a function in that way ? By using calculus , we can say for sure $f'(x)$ will have at least three distinct real roots and $f''(x)$ has at least two distinct real roots. But I am finding a function in that form such that $f'(x)$ has exactly $3$ distinct real roots and $f''(x)$ has exactly $2$ distinct real roots.

Empirical Fisher Information but with unknown true parameters and distribution? Posted: 10 Jul 2021 08:44 PM PDT I am not sure if I ask it correctly. I am working on using Fisher Information to examine the information in a model (say neural networks for simplicity). What I know is that the definition of Fisher Information is $I(\theta^*)=Var[\frac{\partial}{\partial \theta}\log p(Y|\theta,X)|_{\theta=\theta^*}]$ Conceptually I know that Fisher Information is the Variance of the derivative of log-likelihood. But what is the $\theta$ and the distribution? If Fisher Information is to evaluate the variance at the TRUE $\theta^*$, how come we know the true parameters? and also in the classification case, how can we know the also the true distribution? What I have is A neural network model with some trainable parameters it is a classification task, last layer is softmax, loss function is cross-entropy If we know the true parameters then we don't even have to train the model. So, I don't understand how can people compute the fisher information matrix in the research. Thanks!

Finding inverse of a bounded operator Posted: 10 Jul 2021 09:12 PM PDT Let $T$ be a bounded operator on a normed linear space such that $T^2=T$. Find the inverse of $\lambda I-T$ for any complex number $\lambda \neq 0,1$. Now by taking the inspiration from Taylor series expansion of $\frac{1}{1-x}$ I compute the inverse of $ \lambda I-T$ which is $\frac{1}{\lambda}\left(I+T\sum_{n=1}^{\infty}\frac{1}{\lambda^n}\right)$. Now this operator exists if $|\lambda|>1$. But what happens for $|\lambda|<1$ ?

How to obtain a relation between the following functions to obtain the desired integral Posted: 10 Jul 2021 08:39 PM PDT Let $f(x)=\int_0^xg(t)dt,$ where g is a non-zero even function. If $f (x+5)=g(x)$, then $\int_0^xf(t)dt$ is equal to: i)$\int_5^{x+5}g(t)dt$ ii)$\int_{x+5}^5g(t)dt$ Follwing is my approach: since g(x) is even $\implies$ g(x)=g(-x) From the given question $f(x)=\int_0^xg(t)dt$-------(1) ; on replacing 'x' with '-x' we get $$f(-x)=\int_0^{-x}g(t)dt$$ Let $p=-t \implies dp=-dt$. Substituting this we get $$f(-x)=\int_0^{x}g(-p)(-dp)$$ $$\implies f(-x)=-\int_0^{x}g(p)dp\enspace ----(2)$$ As the integral of a function does not depend on the variable when the limits are same, on adding eqation (1) and (2) we get $$f(x)+f(-x)=0\enspace i.e.\enspace an \enspace odd \enspace function$$ It is given in the question that $f(x+5)=g(x)$, and on replacing $'x'$ with $'x+5'$ we get $f(x+10)=g(x+5)-----(3)$ Cosider option i) $$\int_5^{x+5}g(t)dt$$ On shifting the limits we would get $$\int_{5-5}^{x+5-5}g(t+5)dt=\int_{0}^{x}g(t+5)dt$$ $$\implies \int_{0}^{x}f(t+10)dt$$ Here I am stuck. I am unable to further simplify it to get the desired option. Here I used option i) for simplification. The correct answer is option ii)

Why does $\sum_{k=0}^{\infty}\frac{a^k}{k!}e^{-(a+b)} = e^{-b}$? Posted: 10 Jul 2021 08:46 PM PDT I am following along with an example problem, and there seems to be a significant jump from $$ \sum_{k=0}^{\infty}\frac{a^k}{k!}e^{-(a+b)} $$ to $$ e^{-b} $$ I verified using Wolfram Alpha that this result is correct, but I'm having trouble understanding how I might discover this reduction without the help of a computer.

The total angle must be $ 90 ^{\circ} $ degrees however as we sum up the each angle , the equation be dishold Posted: 10 Jul 2021 09:18 PM PDT I've drawn the above diagram as same as the diagram of the book. Currently I can't get how the below 2 angles are obtained. $$ \theta_{1}:= \theta_{} + 45 ^{\circ} =\text{angle between left M and H} $$ $$ \theta_{2}:= \theta_{} - 45 ^{\circ} =\text{angle between right M and H} $$ $$ \theta_{1} + \theta_{2} = 90 ^{\circ} $$ must be held but actually $$ \theta_{1} + \theta_{2} =2 \theta_{} \neq 90 ^{\circ} $$ I've may made some mistake. I think we may can assume $~\theta \ll1~$

Is the quotient a field? Posted: 10 Jul 2021 08:42 PM PDT I'm studying for my qualifying exam and I came across the following problem: Is $\mathbb{F}_{2011^2}[x]/(x^4-6x-12)$ a field? I'm guessing the answer is yes but I don't know how do I prove that $x^4-6x-12$ is an irreducible polynomial over $\mathbb{F}_{2011^2}$.

A question on completeness condition from Ahlfors' book Posted: 10 Jul 2021 09:03 PM PDT I'm sorry I can't think of any better title for this question. Anyways my question is : In Lars Ahlfors-Complex Analysis , the author said that Real Field satisfies completeness condition . He said , for an increasing and bounded sequence : α1<α2<α3....<αn... and where an assumed real number B is bigger than αn for all n , there will exist a number A=limit of αn when n tends to infinity with a special property : given any e > 0 there will exist a natural number n0 such that A - e < αn < A for all n >no. I don't understand this . To be specific , my questions are - Why did he leave the sequence without terminating ? Couldn't he write the equation like this :α1<α2<α3....<αn...<B ? Or is B an arbitrary number from the real field which is not part of the sequence ? And why do I need n to be near infinity ? I could take a quite small sequence where n is countable ..right? Then shouldn't I write : A - e < αn ≤ A ? Thanks for your time. Edit: Off topic question : x2+1=0 has no roots in Real Field because of Order Relation as far as I understood . But in Complex Field there exists two roots for the equation . So does the complex field have Order Relation?

Stolen Rubys Puzzle, How many rubys can you guarantee to win? Posted: 10 Jul 2021 08:51 PM PDT I just saw this youtube video https://www.youtube.com/watch?v=2QJ2L2ip32w Introducing a puzzle with the following rules. There are 30 Rubys, which have to be distributed between three boxes. Every box has to contain atleast 2 rubys. One box has to contain 6 more rubys then one of the remaining two. Are the rubys hidden, I have to guess the number of rubys inside each box. If I guess to high, I get none. If my guess is $\leq$ the actually number, I get my guess. I thought about it for a while, and I got stuck. I also came up with guessing 8 for every box pretty quickly, as I know that at least one box has to contain at least 8 rubys. But I disagree with the presented solution. They claim you can always guarantee that you get 16 rubys, which is not true. You can guarantee to get 8 rubys, but not more. I could always play differently and for example distribute like this: 2 2 26. So when the guess is 8 8 8 I would just lose 8 rubys. So as the puzzle is presented, as a game where I know that you know that I know ... I could not come up with a sensable solution, because I could always play differently, and distribute arbitrarly, and try my luck, instead of losing 16 rubys for sure. So in real life this game should play out very differently. Would you really dare to guess 26 26 26? Why would you distribute like 8 8 14 when you know this is going to lose you 16 rubys for sure? For me it seems like a more sensable way to play, is to really just play random numbers, where each number is different (obeying the rules of the game of course), so guessing three times the same number doesnt work as well. So let me ask a different question: When you really play arbitrarly, what is the expected value of rubys you would lose? Would this expected value still be 16? So we would go through all possible combinations of playing, and we would also guess every possible combination from 2 to 26 (guessing the same number for every box). Also I am interested in what you think of the presented solution. Thanks in advance.

Prove that $g'''(0)$ does not exist Posted: 10 Jul 2021 09:05 PM PDT The original function goes as it follows: $g(x) = \begin{cases} x^3 & \text{if $x \leq 0$} \\ 0 & \text{if $x \gt 0$} \end{cases}$ How do I prove that $g'''(0)$ does not exist? Through continuity tests or through the definition of derivative?

For help an inequality that for large $n$, whether $\frac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$ for some $\beta>0$ Posted: 10 Jul 2021 09:23 PM PDT Can anyone help me to prove that for large $n$ whether $$\dfrac{n\choose \big[\frac{n}{2\log_2n}\big]-1}{(\sqrt{2})^n}\geq(1+\beta)^n$$ for some $\beta>0$ where and $\big[x\big]$ denotes the integer part of $x$, for example, letting $x=5.3222$, then $\big[x\big]=5$. Which is equivalent to ask whether $ n\choose \big[\frac{n}{2\log_2n}\big]-1 $$>(\sqrt{2}+\beta)^n$ for some $\beta>0$. Thanks very much for your help!!

Propositional Logic Question Related to Understanding Implication and Tautologies Posted: 10 Jul 2021 09:20 PM PDT I should preface this with that I have never studied logic before. When answering my question, please assume that I know nothing about formal logic. Just now, I was reading a different question and one of answers gave the statement: $(A⇒B)⇔(¬B⇒¬A)$ (Colloquially known as: the statements "Paris is in France" and "Not being in France means not being in Paris" mutually imply each other.) Initially, I got confused and understood $(A⇒B)⇒(¬B⇒¬A)$, but not $(¬B⇒¬A)⇒(A⇒B)$. Then I realized I was mistakenly thinking of the "$⇒$" in $(¬B⇒¬A)$ as an "$∧$", in which case $(¬B∧¬A)⇏(A⇒B)$. However, this simple issue raised another more fundamental question in my mind: What is propositional logic assuming in such a statement for $(¬Q⇒¬P)⇒(P⇒Q)$ to always be True? Can't there be a case, where $¬P$ is a tautology, in other words, $(¬Q⇒¬P)∧(Q⇒¬P)$, and thus in this case, $(¬Q⇒¬P)⇏(P⇒Q)$? This would be assuming that, in a case where $⊨P$ ("$P$ is a tautology"), that $Q⇒P$ is True. In this case, isn't the above a contradiction? What I'm I confusing here? Conversely, if I assume that in a case where, $⊨P$, that $Q⇒P$ is False, this would seem to suggest that "$⇒$" would denote more than simply the truth value of a proposition. Since, $Q⇒P$ would be False even though $P$ is always True when $Q$ is True, which I now understand would be blatantly wrong. Unrelated to the question, let's say we're in a different logic system where in a case where, $⊨P$, $Q⇒P$ is False. Which is to say, that even when $P$ is Unconditionally True, $Q$ does not "logically imply" $P$. In this logic system, we would be able to make a statement such as: $(⊨P)⇒(Ȣ⇏P)$ Where $Ȣ⇏P$ is a special expression denoting that there are no True "logical statements" which can be made where any expression "logically implies" $P$.

Sum with Digamma function expressed as hypergeometric function? Posted: 10 Jul 2021 09:09 PM PDT The function $$f(x)=\sum_ {k = 0}^{\infty} \frac {(-1)^ k x^{2 k}} {2^k k! (2 k + 2)! }\left(\psi (k + 2)+ \frac{1}{2}\psi\left(k + \frac{3}{2}\right)\right)\tag{1}$$ with $x>0, x\in\mathbb{R}$ is defined by an infinite sum that contains two Digamma functions $\psi$. It was so far not possible to convert $f(x)$ to a known function. If in $f(x)$ the Digamma functions would be replaced by simpler functions (e.g. $k^2$) then the whole expression could be expressed by hypergeometric functions. Is it possible to express this sum as an hypergeometric function or by any other known function? What I tried The Digamma functions can be expressed by finite sums $$\psi(k+2)=-\gamma+\sum_{i=1}^{k+1}\frac{1}{i}\tag{2}$$ $$\frac{1}{2}\psi\left(k + \frac{3}{2}\right)=-\frac{\gamma}{2}-\textrm{ln}(2)+\sum_{i=1}^{k+1}\frac{1}{2i-1}\tag{3}$$ with $\gamma\approx0.577$ (Euler-Mascheroni constant). If eqs.(2,3) are inserted in eq.(1) one gets $$f(x)=-\left(\frac{3}{2}\gamma+\textrm{ln}(2)\right)\sum_ {k = 0}^{\infty} \frac {(-1)^ k x^{2 k}} {2^k k! (2 k + 2)! }+ \sum_{k = 0}^{\infty} \frac {(-1)^ k x^{2 k}} {2^k k! (2 k + 2)! }\sum_{i=1}^{k+1}\left(\frac{1}{i}+ \frac{1}{2i-1}\right)\tag{4}$$ The left sum in eq.(4) can be expressed as generalized hypergeometric function $_0F_2$ $$f(x)=-\frac{1}{2}\left(\frac{3}{2}\gamma+\textrm{ln(2)}\right) {_0}F_2\left(;\frac{3}{2},\frac{1}{2};-\frac{x^2}{8}\right)+ \sum_{k = 0}^{\infty} \frac {(-1)^ k x^{2 k}} {2^k k! (2 k + 2)! }\sum_{i=1}^{k+1}\left(\frac{1}{i}+ \frac{1}{2i-1}\right)\tag{5}$$ The problem is that Eq.(5) now contains a nested sum that does not simplify eq.(1). A nested sum results also if the Digamma functions are replaced by asymptotic series. Is it possible to transform the right nested sum in eq.(5) to a similar form as the left part in eq.(5). The first terms of the nested sum in eq.(5) are $$\sum_{k = 0}^{\infty} \frac {(-1)^ k x^{2 k}} {2^k k! (2 k + 2)! }\sum_{i=1}^{k+1}\left(\frac{1}{i}+ \frac{1}{2i-1}\right)=1 - \frac{17x^2}{288} + \frac{101x^4}{172800} -\frac{ 1579x^6}{812851200} +\frac{5129x^8}{1755758592000}-\frac{59989 x^{10}}{25493614755840000}+\ldots$$ The question would be answered if an alternative rule for the coefficients of the previous power series would be found.

Explicit formula for heat equation in $\mathbb{R}$ Posted: 10 Jul 2021 09:01 PM PDT I'm trying to find an explicit formula for heat equation in the interval $(0,L)$, with $L>0$. I have the next boundary-initial value problem. \begin{equation} \begin{cases} u_t-u_{xx}=0, &\text{in } (0,L)\times (0,\infty), \\ u(0,t)=u(L,0)=0 & \\ u(x,0)=f \end{cases} \end{equation} with $f$ a smooth function such that $f(0)=f(L)=0$. I have the next suggestion: Extend a function $f$ to $\mathbb{R}$ such that $f(x)$ and $f(L-x)$ are odd. My question is: how do I extend this function? After that, I think that I just have to apply the fundamental solution for Heat equation (Evans PDE book). Thanks a lot and sorry for my bad english.

Definition of sine and cosine [duplicate] Posted: 10 Jul 2021 08:44 PM PDT I've seen Sine and Cosine defined as the unique solution to: $$\begin{align} \frac d{dx} \sin(x) &= \cos(x)\\ \frac d{dx} \cos(x) &= -\sin(x) \end{align}$$ with $\sin(0) = 0$ and $\cos(0) = 1$. Is there really only one solution to these functions? How can these functions be defined more formally?

Split $\{1,2,...,3n\}$ into triples with $x+y=4z$ Posted: 10 Jul 2021 08:55 PM PDT A similar question appeared last week. For which $n\in\Bbb N$ can we divide $\{1,2,3,...,3n\}$ into $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$? In this problem, I want to split the integers from $1$ to $3n$ into triples $\{x,y,z\}$ with $x+y=4z$ instead of $3z$. I have a set of solutions, but am looking for more ideas. $\sum(x+y+z)=\sum5z$ so $n=5k$ or $5k+3$. There are already hundreds of solutions for $n=13$, and I imagine that number will only increase with $n$, but the trick is to find one for any particular $n$. Any solution for $n=N$ can be extended to one for each of $n=19N-7,19N+8,19N+13$. That gives solutions for $3^m$ different $n$ below $Z=19^m$, which is more than $\sqrt[3]Z$ different $n$ below $Z$. But most solutions do not contain a smaller one within them. To split the numbers from $3N+1$ to $57N-21$ into triples $$(3N+1+k,33N-9+3k,9N-2+k)$$ for $k =0..6N-4$ and $$(51N-18+k,33N-10+3k,21N-7+k)\\ (27N-9+k,33N-11+3k,15N-5+k)$$ for $k =0..6N-3$. A way to split $\{3N+1,...,3(19N+8)\}$ into triples is $$(3N+1+k,33N+15+3k,9N+4+k)\\ (51N+22+k,33N+14+3k,21N+9+k)$$ for $k=0..6N+2$, and $$(27N+12+k,33N+16+3k,15N+7+k)$$ for $k=0..6N+1$ To split $\{3N+1 ...,3(19N+13)\}$ into triples $$(3N+1+k,33N+23+3k,9N+6+k)$$ for $k=0..6N+4$ and $$(37N+19+k,33N+25+3k,15N+11+k)\\ (51N+36+k,33N+24+3k,21N+15+k)$$ for $k=0..6N+3$. At Split $\{1,...,3n\}$ into triples with $x+y=5z$ - no solutions?, Thomas Andrews has shown there are no solutions for $x+y=5z$.

A problem about analytic geometry Posted: 10 Jul 2021 09:10 PM PDT In the $xy$-plane, the point $(p,r)$ lies on the line with equation $y = x+b$, where $b$ is a constant. The point with coordinates $(2p,5r)$ lies on the line with equation $y = 2x+b$. If $p\neq 0$, what is the value of $r$ and $p$? I have tried using coordinate system and it is not working.

A Relation between discriminant of individual fields vs the discriminant of their Composite Posted: 10 Jul 2021 08:41 PM PDT Let $L_{1}, \ldots, L_{m}$ be field extension of a number field $K$ and $L$ be the composite field of $L_{1}, \ldots, L_{m}$. Let $d$ be the relative discriminant of $L$ (over $K$) and $d_{1}, \ldots, d_{m}$ be the respective relative discriminants of $L_{1}, \ldots, L_{m}$. Then : $$d \mid \prod_{i=1}^{m} d_{i}^{[L : L_{i}]}.$$ One place where I have seen index of the form $[L : L_{i}]$ come up is when we related discriminants over a tower of fields. For example, when $K \subseteq F \subseteq L$ is a tower of number fields then we have $$d_{L/K} = N_{F/K}(d_{L/F}) \cdot d_{F/K}^{[L : F]}.$$ It appears that if I replace $F = L_{i}$ above, then for each respective towers $K \subseteq L_{i} \subseteq L$ we obtain $$d = d_{L/K} = N_{L_{i}/K}(d_{L/L_{i}}) \cdot d_{L_{i}/K}^{[L : L_{i}]} = N_{L_{i}/K}(d_{L/L_{i}}) \cdot d_{i}^{[L : L_{i}]}.$$ From this, it seems that only way $d \mid \prod_{i=1}^{m} d_{i}^{[L : L_{i}]}$ is if none of the $d_{i}$ can be absorbed by any other $N_{L_{j}/K}(d_{L/L_{j}})$, which is some sort of coprime property? Does one need to impose some extra conditions to obtain the result? Thank you.

Source of the Banach Lemma Posted: 10 Jul 2021 09:04 PM PDT I used the Banach Lemma in a paper I'm writing, but I'm not able to find a good source for this. Is there anyone who knows where to find it? The Banach Lemma: Let B be an n x n matrix . If in some induced matrix norm ∥B∥<1, then I+B is invertible and ∥(I+B)∥−1≤1(1−∥B∥).

$\int x^{dx}-1$ Posted: 10 Jul 2021 09:19 PM PDT If you go to Flammable Maths's YouTube channel and scroll through some of his videos you see him solving the following integral: $$\int x^{dx}-1$$ he explains that this is a Product integral. My questions are the following: 1 - What is the geometric meaning of a product integral? 2 - does it make sense to have: $$\int f(x,dx)$$ and if $f(x,dx) = g(x)dx$ then it's just a regular integrals and if $f(x,dx) = g(x)^{dx}$ it's just a product integral? I'll leave the link to the video here.

Proving if $A$ is an $n\times n$ positive semi-definite matrix, A is Hermitian with non-negative eigenvalues. Posted: 10 Jul 2021 09:11 PM PDT I have a test on Monday and the professor hinted that this question might be relevant to the exam, unfortunately, I'm at a loss. As the title states, I would like to prove that if $A$ is an $n\times n$ positive semi-definite matrix then $A$ is Hermitian with non-negative eigenvalues. I would love to be able to work through this and I hope someone can lend a hand. Thank you very much.

The joint distribution of two linear combinations of independent standard normal variables Posted: 10 Jul 2021 09:05 PM PDT If $W$ and $V$ are standard independent normal variables find the joint distribution of $3W+2V$ and $2W-3V$. Since $W$ and $V$ are standard independent normal variables then would it just be the probability density function $f(3w+2v, 2w-3v)=\phi(3w+2v)\phi(2w-3v)=\dfrac{1}{\sqrt{2\pi}}e^{-.5(3w+2v)^2-.5(2w-3v)^2}$ where the mean is $0$ and standard deviation is $1$?

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