Recent Questions - Mathematics Stack Exchange |
- To find gradient of cost function using Wirtinger calculus
- Give all the nodes on a tree a positive number $v_i$ such that no adjacent nodes share the same number, what is the minimum $\sum v_i$?
- How to match a table to a contour diagram.
- Let f : R -> R be a twice continuously differentiable function and f(0) = f'(0) = 0. If |f''(x)| is ≤ 1 then prove f(x) is ≤ 1/2 for x in [−1, 1].
- Given that n and m are integers, if $n^2 + 1 = 2m$, prove that m is the sum of the squares of 2 non-negative integers.
- Geodesics on oblique helicoid using isometry
- About random variables and expected value
- Does $a \equiv b \equiv 5 \pmod {12}$ imply $A \equiv 3 \pmod 4$, under the given conditions?
- If I "randomly" come up with curve, how do I obtain it's parametrization?
- If A is the 4 by 4 matrix of ones, find the eigenvalues and the determinant of A−I
- Is the series $\sum_{i=k}^\infty\frac{\sin\left(\frac{x}{i}\right)}{i}$ bounded?
- If $X^{\star} \times Y^{\star}$ is a KC space, then $X\times Y$ is a $k$-space
- Fourier Series of f(x) = 1 (exponential version)
- If I have a complex valued odd/even holomorphic function, can I say anything about parity of its real/imaginary part as $R^2$ functions?
- On an example of an idempotent which is not a projection
- False proofs that there are finitely many primes
- Combination of Addition and Multiplication of 1 and 0 done 13^12 times to never reach above 3^3^3^3^3
- showing a ring is isomorphic to a localization of $R$ at $S$
- Vector-space, Linear algebra, Span of a vector space
- Proving an inequality given $\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}\le 1$
- What can be said about the coefficients of linear transformations which transforms the real axis into the imaginary axis?
- Convex subsets of $\mathbb{R^2}$
- Show a matrix with the "even" rows of the binomial coefficients is invertible
- 2 standard 52 card decks, how many sequences with exactely 12 adjacent pairs of identical cards?
- Sum of terms in a reduced residue system
- Is $\lim_{n \to \infty} n((1-\frac{1}{n})^n - \frac{1}{e}) = 0$?
- Minimizing the expectation of the loss function
- What exactly is a basis in linear algebra?
- Reduced Residue Systems: $r_1+r_2+...+r_{\phi(m)} \equiv0 \pmod m$
- Bounded and finite - examples of distinction
| To find gradient of cost function using Wirtinger calculus Posted: 11 Jul 2021 09:37 PM PDT I want to get gradient of the following cost function. $f(g,g^*) = \|(f-|Ag|)\|^2_2$ where $\|.\|_{2}$ is the $L_2$ norm, $g$ is $n\times1$ vector and $g^*$ is its complex conjugate $f$ is real positive $m\times1$ vector and $A\in L(R^n,R^m)$ (Linear operator that maps a $n\times1$ vector to $m\times1$ vector) and adjoint of $A =A^{\dagger}$ defined such that $\langle g_{01},Ag_{i2}\rangle=\langle A^\dagger g_{01}, g_{i2}\rangle$. any help regarding this will be appreciated.  |
| Posted: 11 Jul 2021 09:46 PM PDT
I came across this problem on a programming platform. I had solved it but I had a few doubts concerning it.
If this is a well-known problem (especially on a tree), please provide me with some reference links cuz I can't find any. Thanks in advance. Example of $\max(v_i)$ being $3$:
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| How to match a table to a contour diagram. Posted: 11 Jul 2021 09:15 PM PDT picture of the tables and the contour diagrams I need to know how to match contour diagrams to tables. I'm really at a loss for how to do this. This wasn't discussed in my class, and there's nothing about it in my textbook. I feel like I'm missing something. I know two of these are right (I tried guessing the right answers) but I don't know which are right. If someone could explain the steps to solving this, I would greatly appreciate it.  |
| Posted: 11 Jul 2021 09:05 PM PDT Let f : R → R be a twice continuously differentiable function and suppose f(0) = f'(0) = 0. If |f''(x)| ≤ 1 for all x in R, then prove that |f(x)| ≤ 1/2 for all x in [−1, 1].  |
| Posted: 11 Jul 2021 09:44 PM PDT I have absolutely no idea how to approach the question posed in the title, and would like a hint towards the answer. I tried multiple approaches, such as multiplying both sides by 2, or subtracting 2 from both sides to make the LHS a difference of squares, or adding $2n$ to both sides to make the LHS a perfect square. I had no idea how to progress by doing any of these methods, though.  |
| Geodesics on oblique helicoid using isometry Posted: 11 Jul 2021 09:02 PM PDT I am trying to find the non trivial geodesics on an oblique helicoid(non-minimal) surface apart from the rulings. Instead of using the usual geodesic equations and also because I was interested in seeing the correspondence between geodesics through isometry in this case, I was looking at using the hyperboloid of one sheet(of revolution) which is isometric to the oblique helicoid. By the hyperboloid I mean the surface given by the equation $$\dfrac{x^2+y^2}{a^2}-\dfrac{z^2}{b^2} = 1$$ The parametrisation of the oblique helicoid considered is this: \begin{equation} M(u,v) = \bigg\lbrace\dfrac{ub}{\Delta}\cos{\frac{v}{b}}, \dfrac{ub}{\Delta}\sin{\frac{v}{b}}, \dfrac{au}{\Delta}+v\bigg\rbrace \end{equation} where $\Delta = \sqrt{a^2+b^2}$. The central circle on the hyperboloid is a geodesic which corresponds to the straight axis of the oblique helicoid. But as the hyperboloid is also a surface of revolution, the meridians of which are hyperbolas and are also geodesics. I am trying to see what the images of these geodesics are on the oblique helicoid, but haven't had any luck. Any hints or suggestions are welcome. Thanks in advance.  |
| About random variables and expected value Posted: 11 Jul 2021 09:38 PM PDT Let $X$ and $Y$ be two independent random variables defined on $(\Omega, \mathcal G, P)$. If $M \in \mathbb B(\mathbb R)$ prove that $$ \int_{\{Y \in M\}} X dP = E(X) P_Y(M), $$ where $P_Y$ is the probability measure induced by $Y$ on $(\mathbb R, \mathbb B(\mathbb R))$. So far I tried to work with the RHS: $E(X)P_Y(M) = (\int_{\Omega}X dP) P_Y(M) = \int_{\Omega}P_Y(M) X dP = \int_{\Omega}P(\{ Y \in M\}) X dP $, but don't know what else to do. Also, I don't see how the fact that $X$ and $Y$ are independent can help.  |
| Does $a \equiv b \equiv 5 \pmod {12}$ imply $A \equiv 3 \pmod 4$, under the given conditions? Posted: 11 Jul 2021 09:24 PM PDT Let $a$ be prime, and let $b$ be a positive integer. Suppose that I know that $a \equiv b \equiv 1 \pmod 4$ holds in general. Additionally, assume that I know that the following biconditionals hold: $$(A \equiv 1 \pmod 4) \iff (a \equiv b \pmod 8)$$ $$(A \equiv 3 \pmod 4) \iff (a \equiv b + 4 \pmod 8)$$ If I know that $a \equiv b \equiv 5 \pmod {12}$, then what can I conclude about $A$? MY ATTEMPT Since $a \equiv b \equiv 1 \pmod 4$ holds in general, then $4 \mid (a - b)$. Suppose to the contrary that $A \equiv 1 \pmod 4$. This means that $a \equiv b \pmod 8$, which is equivalent to $8 \mid (a - b)$. But we know that $a \equiv b \equiv 5 \pmod {12}$. In particular, $4 \mid 12 \mid (a - b)$. This contradicts (?) $8 \mid (a - b)$. Hence, we conclude that $$a \equiv b \equiv 5 \pmod {12} \implies A \equiv 3 \pmod 4.$$ QUESTION
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| If I "randomly" come up with curve, how do I obtain it's parametrization? Posted: 11 Jul 2021 09:11 PM PDT Sorry If this question is too cumbersome, I am still a bit lost at the basics of differential geometry: Given the curve $(\cos t,\sin 3t)$, how to obtain the parametrization of this curve? I've been thinking about using the relation $\tilde\gamma(t)=\gamma(\phi(t))$ and comparing it with the arc length parametrization. I'm a bit confused.  |
| If A is the 4 by 4 matrix of ones, find the eigenvalues and the determinant of A−I Posted: 11 Jul 2021 08:45 PM PDT So I want to find the eigen values and eigen vectors of a matrix with all 1's \begin{bmatrix}1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1\end{bmatrix} Only 1 independent would be left, \begin{bmatrix}1&1&1&1\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix} Now, Let's assume λ= 1. A-λI would give me, \begin{bmatrix}0&1&1&1\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix} With eigen values as, -1,-1,-1 and 3. Ultimately, λ1=3 and λ2=-1. But this is something I have assumed, how can I get eigen values and its vectors by a method? Or what steps should I take ahead? Thanks in advance  |
| Is the series $\sum_{i=k}^\infty\frac{\sin\left(\frac{x}{i}\right)}{i}$ bounded? Posted: 11 Jul 2021 09:05 PM PDT Comparison with $\frac{1}{i^2}$ shows that the series $$\sum_{i=k}^\infty\frac{\sin\left(\frac{x}{i}\right)}{i}$$ is convergent. However, the function of $x$ thus obtained appears to be bounded, with the bound approaching zero in $k$. I have no idea how to prove this, every method I know gives me no better a bound than the obvious $\frac{\pi^2x}{6}$. My suspicion is that prior to reaching $\left|\frac{x}{i}\right|<1$, the values of $\frac{x}{i}$ modulo $2\pi$ have an asymptotic distribution which helps similar to other identities with alternating signs bounded by $\frac{1}{i}$.  |
| If $X^{\star} \times Y^{\star}$ is a KC space, then $X\times Y$ is a $k$-space Posted: 11 Jul 2021 09:35 PM PDT The following theorem is found in the article "ON KC AND k-SPACES, A. García Maynez. 15 No. 1 (1975) 33-50" Theorem 3.5: Let $X, Y$ be topological spaces. If $X^{\star} \times Y^{\star}$ is KC then $X\times Y$ is a $k$-space Definitions:
The article refers that the theorem 3.5 is a consequence of theorem 3.4 Theorem 3.4: Let $X,Y$ be non-compact spaces and let $Y^{\star}=Y\cup \{ \infty \}$ be the one-point compactification of $Y$. Assume $Y$ is a $KC$ space. Then a set $C\subset X\times Y$ is compactly closed in $X\times Y$ if and only if $C\cup (X\times \{ \infty \})$ is compactly closed in $X\times Y^{\star}$. I fail to understand why theorem 3.5 is a consequence of theorem 3.4. Could someone explain me with more details?  |
| Fourier Series of f(x) = 1 (exponential version) Posted: 11 Jul 2021 09:12 PM PDT I'm trying to find just the Fourier series of f(x) = 1 from 0 $\leq$ x $\leq$ $\pi$ using the exponential definition of the coefficients (f(x) = 0 from $-\pi \leq x < 0$). I'm running into a divide by 0 problem. So I know that $c_n$ = $\frac{1}{2*\pi}$ $ \int_{-\pi}^{\pi} f(x) e^{(-inx)} \,dx $. f(x) = 1 and integral of $e^{-inx}$ is $\frac{-e^{-i*\pi*n}}{in}$. Integrating, I get $\frac{1}{2\pi} (\frac{-e^{-i*\pi*n}}{in} + \frac{1}{in})$ = $\frac{1}{2\pi} (\frac{-(-1)^{-n}+1)}{in})$. So this seems like it should be the formula for the $c_n$? But I don't think it is. Because we have to take the sum $\sum_{n=-\infty}^{\infty} c_n e^{inx}$. But $c_0$ = $\infty$. Also my understanding is that for Fourier Series, $\sum_{n=-\infty}^{\infty} c_n e^{inx}$ = f(x) for all x. But Wolfram Alpha shows that's not the case here. So what am I doing wrong?  |
| Posted: 11 Jul 2021 08:31 PM PDT Let $f(z)=f(x_1,x_2)=f(x_1+i x_2)=u(x_1,x_2)+iv(x_1,x_2)$ be a holomorphic function, then by Cauchy-Riemann: $\frac{\partial u}{\partial x_1}(x_1, x_2)=\frac{\partial v}{\partial x_2}(x_1, x_2), \quad \frac{\partial u}{\partial x_2}(x_1, x_2)=-\frac{\partial v}{\partial x_1}(x_1, x_2).$ If I'm not wrong it means that if $u$ is odd in $x_i$, $v$ must be even in $x_i$, right? My question is: suppose $$f(-z)=-f(z),$$ then it must be true that $u(-x_1,-x_2)=-u(x_1,x_2),v(-x_1,-x_2)=-v(x_1,x_2)$. But, can I say anything about $u(-x_1,x_2),v(-x_1,x_2),u(x_1,-x_2),v(x_1,-x_2)$? EDIT: From Ted Shifrin comment I realize that it should have to be a relation with the power series. Let's say: $$f(z)=\sum_{i=-\infty}^\infty c_i z^i.$$ as $$u=(f+\bar{f})/2=\sum_{i=-\infty}^\infty \frac{c_i z^i + \bar{c}_i \bar{z}^i}{2}$$ then if $c_i$ are reals $u(x_1,-x_2)=u(x_1,x_2)$. Is it a sufficient condition to say $c_i$ are reals? What should be the condition for $u(-x_1,x_2)=-u(x_1,x_2)$?  |
| On an example of an idempotent which is not a projection Posted: 11 Jul 2021 09:32 PM PDT I am working through a solution of the following exercise in Conway's functional analysis:
I am very confused with this question. I will write my try can you please help me why I can't get the solution? My questions : 1- $Ε_θ$ is an operator s.t. it transforms all points of $R^2$ to x-axis. So intuitively $Ε^2_θ=Ε_θ$. But how one write an explicit formula for $Ε_θ$. Isn't it $Ε_θ (x,y)=(x,0)$ and if so it doesn't depend on θ then? 2- $Ker (Ε_θ)$ is the set of point s.t. $Ε_θ (x,y)=(0,0)$ then it must be the y-axis, then why it is {(x,xtan0): $x \in R$}? 3- how $||Ε_θ||=(sin θ)^{-1}$? Calculating $||Ε_θ||=sup Ε_θ$ depends on two things to know before that : is sup taken on the circle in $R^2$? what is explicit formula for $Ε_θ$ so that to know on what sup is taken? 4- this exercise is designed to indicate an example of an idempotent which is not a projection? ker E is the y-axis and ran E is the x-axis so how $ker E = (ran E)^{\perp}$ does not hold when it holds?  |
| False proofs that there are finitely many primes Posted: 11 Jul 2021 08:50 PM PDT There are numerous proofs of the infinitude of primes. I'm interested in FALSE proofs that there are finitely many primes. I looked online, but the only proof I found is this: https://jeremykun.com/2011/07/05/there-are-finitely-many-primes/ (Essentially, it falsely claims that there is a bijection between the square-free naturals and the power set of the set of primes). Does anyone know other nice fake proofs?  |
| Posted: 11 Jul 2021 09:40 PM PDT I have been doing this question for some time. If the question is asking us to find the combinations of adding and multiplying 1 and 0 why can't it be 1+1+1+1... but it can be 1+1=2 or just 1? Below is the question:
Any tips for how to get to the answer or what this question is asking us would be welcome! Thanks.  |
| showing a ring is isomorphic to a localization of $R$ at $S$ Posted: 11 Jul 2021 08:41 PM PDT
I know that localizations are unique in the sense that if $R$ is a commutative ring, $S$ is a multiplicatively closed subset of $R$, and $Q$ is a commutative ring so that there exists an injective homomorphism $\phi : R\to Q$ so that
then $Q$ is isomorphic to the localization of $R$ at $S.$ So I think one way to prove part $3$ is to show these two conditions are satisfied for $S(c).$ Alternatively, is there a more "direct" approach to this problem? I was thinking of defining the map $\phi : \mathbb{K}[x]_{(x-c)}\to S(c), \phi(\frac{a}b) = \frac{a}b$. However, it seems that showing that this map is an isomorphism would also show that $S(c) = \mathbb{K}[x]_{(x-c)}.$ I know $S(c)$ is a subring of $\mathbb{K}(x)$ and hence is commutative. Let $R = \mathbb{K}[x], S = \mathbb{K}[x]\backslash (x-c).$ Define $\phi : R\to S(c)$ as follows. $\phi(x) = \frac{x}1.$ Observe that this map is clearly injective. Since for $f=g/h\in S(c), h(c)\neq 0, h\not\in (x-c)\Rightarrow h\in S.$ So $f = \phi(g)\phi(h)^{-1}.$ Then define $\tilde{\psi} : S(c)\to T$ so that $\tilde{\psi}(\frac{a}b) = \psi(a)\psi(b)^{-1}.$ Then one can verify that $\tilde{\psi}$ has the required properties.  |
| Vector-space, Linear algebra, Span of a vector space Posted: 11 Jul 2021 09:44 PM PDT Question: Will a set of all linear combinations of the basis of a vector space give the span of that vector space? This is what I have understood from the meaning of the span of a vector space:
Question: Am I correct in understanding what a span of a vector space is?  |
| Proving an inequality given $\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}\le 1$ Posted: 11 Jul 2021 09:30 PM PDT Given that $a,b,c > 0$ are real numbers such that $$\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}\le 1,$$ prove that $$\frac{1}{b+c+1}+\frac{1}{c+a+1}+\frac{1}{a+b+1}\ge 1.$$ I first rewrote $$\frac{1}{a+b+1} = 1 - \frac{a+b}{a+b+1},$$ so the second inequality can be rewritten as $$\frac{b+c}{b+c+1} + \frac{c+a}{c+a+1} + \frac{a+b}{a+b+1} \le 2.$$ Cauchy-Schwarz gives us $$\sum \frac{a+b}{a+b+1} \geq \frac{(\sum \sqrt{a+b})^2}{\sum a+ b+ 1}.$$ That can be rewritten as $$\frac{2(a+b+c) + 2\sum \sqrt{(a+b)(a+c)}}{2(a+b+c) + 3},$$ which is greater than or equal to $$\frac{2(a+b+c) + 2 \sum(a + \sqrt{bc})}{2(a+b+c) + 3} = \frac{4(a+b+c) + 2 \sum \sqrt{bc}}{2(a+b+c) + 3} \geq 2,$$ which is the opposite of what I want. Additionally, I'm unsure of how to proceed from here.  |
| Posted: 11 Jul 2021 09:12 PM PDT Question: What can be said about the coefficients of linear transformations which transforms the real axis into the imaginary axis? Thoughts: If I wanted to transorm the real axis into itself, then I can show that I can do this using the cross ratio $(w,w_1,w_2,w_3)=(z,1,0,-1)$, and it turns out the we get the coefficients are real. So, if I wanted to transform the real axis into the imaginary axis, could I set up a cross ratio as $(z,1,0,-1)=(w,i,0,-i)$? But I don't have anything that tells me, for instance, that $1$ is necessarily going to $i$. So I am sort of stuck. I would like to do this using the cross ratio, but maybe that is the wrong approach? Any help is greatly appreciated! Thank you.  |
| Convex subsets of $\mathbb{R^2}$ Posted: 11 Jul 2021 08:56 PM PDT I have some troubles with proving that $A$ and $B$ are convex. $A = \left\{(x,y)\in\mathbb{R}^2\,:\, y > \frac{1}{|x|}, x<0\right\} \quad \mbox{and}\quad B = \left\{(x,y)\in\mathbb{R}^2\,:\, y > \frac{1}{x}, x>0\right\}$. The definition of convex is: A set A is convex iff for $a,b\in A$ and $\alpha\in[1,0]$, then $\alpha a+(1−\alpha)b\in A$. I have some troubles with proving that $A$ and $B$ are convex. My attempt for $A$ (I think that $B$ should be the same idea) Let $a=(a_1,a_2),b=(b_1,b_2)\in A$, and $\alpha \in (0,1)$, its clear that $$\alpha a_1+(1-\alpha)b_1<0.$$ On the other hand, we know that $$b_2> \frac{1}{|b_1|} \quad \mbox{and} \quad a_2> \frac{1}{|a_1|} $$ How can I arrange the inequalities? Any hint?  |
| Show a matrix with the "even" rows of the binomial coefficients is invertible Posted: 11 Jul 2021 09:43 PM PDT I have a $n \times n$ matrix $\mathbf{M}$ defined as, \begin{equation}(\mathbf{M})_{ij} = \begin{cases} {2i \choose j} \quad 1 \le j \le 2i, 1 \le i \le n \\ 0 \quad \mathrm{otherwise} \end{cases} \end{equation} I want to prove that this matrix is invertible i.e. the determinant is non-zero. I tried various various by induction, and using some tricks in this question on Pascal matrices, but I was not able to reach a resolution. What would be a good way to approach this?  |
| 2 standard 52 card decks, how many sequences with exactely 12 adjacent pairs of identical cards? Posted: 11 Jul 2021 09:45 PM PDT I am doing recreational math as a pastime. I like to do math exams and such and also to come up with math-puzzles on my own. Recently i was doing the following MIT math exam: It was super fun! :-) Anyway, here is problem 7:
That problem got me thinking about the problem I described below. Take 2 standard 52 card decks. Then you can take those cards and form sequences. There are a total of $104! / 2^{52}$ possible different sequences. From all of those only $52!$ exist that have $52$ pairs of identical cards placed right next to one another. For example ( Ad,Ad,Ah,Ah,As,As,...) How many sequences are there with exactly 12 pairs? Can anyone help me out?  |
| Sum of terms in a reduced residue system Posted: 11 Jul 2021 09:23 PM PDT Problem: Prove that if $r_1, r_2, ... r_{\phi(m)}$ is a reduced residue system modulo m, and m is odd, then $r_1 + r_2 + ... + r_{\phi(m)} \equiv 0\mod m$. I know that $r_1 + r_{\phi(m)} \equiv 0$ because $r_{\phi(m)}=m-1$. My intuition tells me that $r_2 + r_{\phi(m)-1}$ is also equivalent to 0 so on and so forth. I can also see that, since m is odd, the reduced residue system has an even number of elements. So each $r_i$ has a partner, namely $r_{\phi(m)-i+1}$ How do I show this rigorously?  |
| Is $\lim_{n \to \infty} n((1-\frac{1}{n})^n - \frac{1}{e}) = 0$? Posted: 11 Jul 2021 09:28 PM PDT Intuitively I think this statement is true, but I am unable to proof it. Can someone help me? If possible, I would like bound $\vert(1-\frac{1}{n})^n - \frac{1}{\mathrm{e}}\vert$ (or even $\vert{(1+\frac{1}{n})^n - \mathrm{e}}\vert$) because I think this is a good think to know in general.  |
| Minimizing the expectation of the loss function Posted: 11 Jul 2021 09:03 PM PDT So i was reading Elements of Statistical Learning and found this in the Statistical Decision Theory part.I Did not understand it. The expected (squared) prediction error . By conditioning on $X$, we can write EPE as $$EPE(f) = E_x E_{y|x} ([Y − f(X)]^2|X) \qquad (2.11)$$ and we see that it suffices to minimize EPE pointwise: $$f(x) = \operatorname{arg\,minc} E_{y|x} ([Y − c]^2|X = x)$$
Can someone explain me what exactly happened here with proper mathematical formulae and some intuition as well. Is it to assume that conditioning over $x$ implies assuming $x$ to be constant in some sense. And if possible please try to explain using a density and the definition of expectation.  |
| What exactly is a basis in linear algebra? Posted: 11 Jul 2021 09:34 PM PDT I have a brief understanding of bases. But I don't know if it is right or not. So, I just need someone to correct me if it's not. When we look for the basis of the image of a matrix, we simply remove all the redundant vectors from the matrix, and keep the linearly independent column vectors. When we look for the basis of the kernel of a matrix, we remove all the redundant column vectors from the kernel, and keep the linearly independent column vectors. Therefore, a basis is just a combination of all the linearly independent vectors. By the way, is basis just the plural form of base? Let me know if I am right.  |
| Reduced Residue Systems: $r_1+r_2+...+r_{\phi(m)} \equiv0 \pmod m$ Posted: 11 Jul 2021 09:23 PM PDT Prove that if $r_1,r_2,...,r_{\phi(m)}$ is a reduced residue system modulo $m$, and $m$ is odd, then $r_1+r_2+...+r_{\phi(m)} \equiv0 \pmod m$.  |
| Bounded and finite - examples of distinction Posted: 11 Jul 2021 09:19 PM PDT What is the distinction between a bounded function, and a finite function? Is there any example of two functions that satisfies only one of them?
Thank you.  |
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