Recent Questions - Mathematics Stack Exchange |
- n-fold composition of a circle diffeomorphism is identity
- Conditional density and conditional probability given $\sigma$-algebra
- how to writing long Equations with in Latex [closed]
- Tensor Product of 4 matrix
- What is the cardinality of $ \{ (A,B) \in \mathbb{P}(E)^2: A \subseteq B \}$ and $ \{ (A,B) \in \mathbb{P}(E)^2: A \cup B = E \}$
- Properties of homeomorphisms
- Help with Bolzano Weierstrass theorem (Real analysis)
- How can I prove that $(1+\frac{1}{k})^{2k}$ is increasing? [closed]
- Complex derivative vs two-variable derivative: requiring same limit from different directions
- Classifying the critical points of $\arctan(1+y^2-x^2y+x^2)$
- Relation between upper (or lower) riemann sums based on norm.
- How to prove this R is a relation on A, symmetric iff R = R²
- Martingale Convergence under equivalent probability measures
- Substituting $it+\ln t$ with $u$ in challenging integration procedure
- Quiz show contestant
- What is $\mathbb{F}_q((1/T))/\mathbb{F}_q[T]$ as a topological space?
- Let $a,b,c,d,e$ be five numbers satisfying the following conditions...
- In a dense-in-itself Polish space there is a "continuous" Borel probability measure [closed]
- Irreducible polynomials over non-UFD
- Find the distance between a function f(x, y) and a given point P
- Do both real and imaginary roots of a cubic equation need to continuous?
- Does every strip of positive irrational slope contain a perfect square point?
- Good presentation of the Witt numbers
- symbolic definition of bijection?
- What is the intuition behind the definition of a simply strongly normal number?
- Prove that for any integer $n, n^2+4$ is not divisible by $7$.
- If $Y=\tan^{-1} x$ , obtain an equation showing the relationship between $Y_{n+2}$ , $Y_{n+1}$ and $Y_{n}$.
- Can this quick way of showing that $K[X,Y]/(Y-X^2)\cong K[X]$ be turned into a valid argument?
| n-fold composition of a circle diffeomorphism is identity Posted: 03 Feb 2022 04:21 AM PST Suppose that $f(x):\mathbb{T}\to \mathbb{T} $ is a smooth diffeomorphism of the circle, whcih is suffciently close to the rigid rotation $R_{\frac{1}{n}}: x\mapsto x+\frac{1}{n}$. If the $n$-fold composition $f^n=f\circ f\cdots \circ f =id$, does it follows that $f$ is conjugate to $R_{\frac{1}{n}}$? (That is, does there exist a conjugacy $h$ such that $h\circ f\circ h^{-1}=R_{\frac{1}{n}}$ ?) |
| Conditional density and conditional probability given $\sigma$-algebra Posted: 03 Feb 2022 04:10 AM PST I am reading a proof concerning a conditional density $f(t|\mathcal{G_n})$ where $\mathcal{G_n}=\sigma(T_1,T_2,...,T_n)$ is a sub $\sigma$-algebra generated by random variables $T_1,...,T_n$. In the proof, they express this conditional density in terms of the probability of $T_{n+1}$ being in some infinitesimal interval $ds$ around $t$. They write: $$f(t|\mathcal{G})=\frac{\mathbb{P}(T_{n+1}\in[t,t+ds]|\mathcal{G}_n)}{ds}$$ Intuitively, this makes sense to me, but I'm not really sure how to understand this more rigorously. I know that a conditioal probability given a $\sigma$-algebra is the same as a conditional expectation of an indicator function, but how do we make sense of the left hand side? |
| how to writing long Equations with in Latex [closed] Posted: 03 Feb 2022 04:27 AM PST I am new in Latex, I am trying to write the Equation below, but I have some errors, I couldn't correct them. \begin{align} \frac{Y_{\tau(j)}\left[\left(\begin{array}{l} \prod\limits_{1}^{k m+ \left\lfloor\frac{j}{2}\right\rfloor-1} \end{array} c_{2 k_{1}+\tau(j)}\right)\right. +\frac{1}{Y_{\tau(j)}} \sum\limits_{l=0}^{k m+\left\lfloor\frac{j}{2}\right\rfloor-1}-\left(\begin{array}{c} k m+ \\ \left.d_{2 l+\tau(j)} \sum\limits_{k_{2}=l+1}^{\lfloor} \frac{i}{2}\right\rfloor-1 \\ \end{array} \quad c_{2 k_{2}+\tau(j)}\right)}{% {% \bigl[X_{\tau(k+j)}\left[\left(\prod_{k_{1}=0}^{\left.\frac{k+j}{2}\right\rfloor-1} a_{2 k_{1}+\tau(k+j)}\right)\right. +\frac{1}{X_{\tau(k+j)}} \sum_{l=0}^{k m+}}{% \left(\begin{array}{c} \left\lfloor\frac{k+j}{2}\right\rfloor-1 \\ b_{2 l+\tau(k+j)} \prod_{k_{2}=l+1}^{k m+} a_{2 k_{2}}+\tau(k+j) \end{array}\right)}} \notag\\[1ex] \end{align} |
| Posted: 03 Feb 2022 04:08 AM PST How can we show that $$(A \otimes B)(C \otimes D) = AC \otimes BD $$ I have tried to show that in generalised way by taking the size of the each matrix A,B,C,D |
| Posted: 03 Feb 2022 04:07 AM PST
In this problem, we set $|E| = n$, we can notice that $C$ is equal to how many ways we can distribute the $n$ elements of $E$ over $A$ and $B$. For every elements in $E$, we can either put it in $A$ or $B$, so there are $2$ possibilities for every element of $E$. Hence there are $2^n$ ways to distribute the elements. As for the first one, I don't know how to approach it. Can you please help? :) |
| Posted: 03 Feb 2022 04:07 AM PST I'm having an intrudoctory chair of dynamical system's at my faculty. I've read the teacher's notes and shearched the internet, but can't seem to find if the following affirmations are true or false: (1) If a homeomorphism of a compact metric space has a dense orbit, then it has no periodic orbits. (2) There exists a homeomorphism of $\mathbb{R}$ with dense orbit. (3) If $f:\mathbb{S}^1 \longrightarrow \mathbb{S}^1$ is a homeomorphism that inverts orietations, then $f$ has two fix points. Any help would be greatly appreciated. |
| Help with Bolzano Weierstrass theorem (Real analysis) Posted: 03 Feb 2022 04:03 AM PST I have a hard time understanding the Bolzano Weierstrass theorem. Help would be appreciated Question: What does the Bolzano Weierstrass theorem tell you about the following sets: a) e^(− cos n) : n ∈ N} b) tan (π/(2n+1)) : n ∈ N |
| How can I prove that $(1+\frac{1}{k})^{2k}$ is increasing? [closed] Posted: 03 Feb 2022 04:22 AM PST How can I show that the sequence $$a_k=(1+\frac{1}{k})^{2k}$$ is increasing for $$k=2,4,6,...?$$ I have tried proving that $$a_{k+2}>a_k$$ but wasn't able to do it. Thankful for any tips. :) |
| Complex derivative vs two-variable derivative: requiring same limit from different directions Posted: 03 Feb 2022 03:53 AM PST I am trying to understand how the CR equations encode the field properties present in $\mathbb{C}$ that aren't present in $\mathbb{R^2}$. Yes, I know their derivation involves multiplying by the inverse of a complex number, which you can't do for a vector, hence the different definition of a multivariable derivative. My question is about approaching along different paths: the crux of the CR derivation seems to involve forcing the limit of the difference quotient of two complex numbers to be the same along both the real and imaginary directions. My question is how this is different from the derivative of a function from $\mathbb{R^2}$ to $\mathbb{R^2}$: do such function not require limits along different paths to be the same? I get that the complex derivative, as a $2 \times 2$ matrix is a special case of a general $Df$ for $f: \mathbb{R^2} \to \mathbb{R^2}$ that enforces the CR condition amongst the entries of the matrix. I have looked at this question, this question, and a few other questions, and it doesn't seem to answer the heart of my confusion: CR conditions do nothing more than enforce that limits of quotients along two orthogonal axes are the same. Are limits along different (orthogonal) paths allowed to be different in a well defined $Df$? Is this why complex derivatives are a stronger condition than multivariate real derivatives? I also understand that complex functions can be thought of as functions in conservative vector fields, in some analogous sense (though they are not totally isomorphic). |
| Classifying the critical points of $\arctan(1+y^2-x^2y+x^2)$ Posted: 03 Feb 2022 04:18 AM PST In the problem I am asked to classify and find the critical points of the function $$\arctan(1+y^2-x^2y+x^2)$$ I solved the question initially by differentiating the arctan(x,y) with derivatives $$ \frac{\partial f}{\partial x} = \frac {1} {(1+y^2-x^2y+x^2)^2}(-2xy+2x)$$ and $$ \frac{\partial f}{\partial y} = \frac {1} {(1+y^2-x^2y+x^2)^2}(2y-x^2) $$ because it is a composite function but my professor finds the solution only by studying the argument inside the arctangent hence using $$ \frac{\partial f}{\partial x}=(-2xy+2x) $$ and $$ \frac{\partial f}{\partial y}=(2y-x^2) $$ I have not asked him because tutoring in the university is now closed. |
| Relation between upper (or lower) riemann sums based on norm. Posted: 03 Feb 2022 04:21 AM PST I can prove that if $Q$ is a refinement of $P$, ie $P\subseteq Q$ then $L(P,f)≤ L(Q,f)$ and $U(Q,f)≤ U(P,f)$. An interesting thing to note here is that $P \subseteq Q \implies ||Q||<||P||$ but the converse is not true. However, I'm wondering whether the following implication holds: If $||P_1|| < ||P_2||$ then $L(P_2,f)≤L(P_1,f)$ and $U(P_1,f)≤U(P_2,f)$. Intuitively, I think it should hold as $P_1$ is in some sense finer than $P_2$ but I'm not sure how to prove it rigorously because the definition of norm as maximum of the lengths of the subintervals doesn't seem easy to use. Any help? |
| How to prove this R is a relation on A, symmetric iff R = R² Posted: 03 Feb 2022 03:56 AM PST Let R be a relation on A, Show the R i: symmetric if and only if R = R². could anybody tell me the answer? |
| Martingale Convergence under equivalent probability measures Posted: 03 Feb 2022 04:15 AM PST I am trying to spot the mistake in the following simple equation: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a prob. space, $(\mathcal{F}_n)_n \nearrow \mathcal{F}$ and let $\mathbb{Q} \approx \mathbb{P}$, thus $\mathbb{Q}$-a.s. is equivalent to $\mathbb{P}$-a.s.. For a $X \in L^\infty$ (thus we can freely apply the dominated convergence theorem) we have \begin{align*} \mathbb{E}_\mathbb{P}[X] &= \mathbb{E}_\mathbb{P}[\lim_n\mathbb{E}_\mathbb{Q}[X | \mathcal{F}_n]] = \lim_n\mathbb{E}_\mathbb{P} [ \mathbb{E}_\mathbb{Q}[X | \mathcal{F}_n]]\\ &= \lim_n \mathbb{E}_\mathbb{P} [\mathbb{E}_\mathbb{P}[d\mathbb{Q}/ d\mathbb{P} \text{ } \text{ } X | \mathcal{F}_n]] = \lim_n \mathbb{E}_\mathbb{P} [ d\mathbb{Q}/ d\mathbb{P} \text{ } \text{ } X] = \mathbb{E}_\mathbb{Q}[X] \end{align*} I really cannot see the mistake, and I also think that this cannot be true. I would be really grateful if you could help me with this. |
| Substituting $it+\ln t$ with $u$ in challenging integration procedure Posted: 03 Feb 2022 04:26 AM PST Regarding this post I would like to ask the community for hints regarding this problem. Since the steps done on that original post, did not yield the correct result, I have done a different approach. \begin{equation} \int_0^\infty \frac{\sin t}{\sqrt{t}}dt=\frac{1}{2i}\int_0^\infty \frac{e^{it}-e^{-it}}{\sqrt{t}}dt \end{equation} for simplicity we split in two integrals, and solve the first here, which gives the solution to the second too \begin{equation} \frac{1}{2i}\int_0^\infty \frac{e^{it}}{\sqrt{t}}dt=\frac{1}{2i}\int_0^\infty e^{\frac{1}{2}\ln t}e^{it}dt=\frac{1}{2i}\int_0^\infty e^{{\frac{1}{2}\ln t}+it}dt \end{equation} So we want to solve this now, \begin{equation} \frac{1}{2i}\int_0^\infty e^{{\frac{1}{2}\ln t}+it}dt \end{equation} and it is tempting to set $u=\frac{1}{2}\ln t+it$, but then $du=\frac{1}{2t}+i$, so we can't get rid of that $t$ in the substitution. Alternatively, we can write $e^{it}=z$, and form the complex integral \begin{equation} \frac{1}{2i}\int_0^\infty \sqrt{t}z dz \end{equation} But here we need to change the t-variable into a complex variable. So I suggest doing a Möbius transformation on of t, but since it is real, it doesn't apply. Is there any chance here to solve this by some other way, or use this with some modifications? |
| Posted: 03 Feb 2022 03:58 AM PST A contestant on a quiz show is presented with two questions, questions 1 and 2, which he is to attempt to answer in some order he chooses. If he decides to try question i first, then he will be allowed to go on to question , j (if i=1, j=2. If i=2, j=1), only if his answer to question i is correct. If his initial answer is incorrect, he is not allowed to answer the other question. The contestant is to receive Vi, dollars if he answers question i correctly, i = 1,2. For instance, he will receive V1 + V2 dollars if he answers both questions correctly. If the probability that he knows the answer to question i is Pi, = 1,2, which question should he attempt to answer first so as to maximize his expected winnings? Assume that the events Ei = 1,2, that he knows the answer to question i are independent events. I understand this question completely except in the example in the book it shows that for: P1: Probability he knows the answer to question 1 P2: Probability he knows the answer to question 2 V1: prize value of correct answer to question 1 V2: prize value of correct answer to question 2 Ei: event he chooses question i as the first question, for i=1,2. When: E[E1] = V1(P1)(1-P2) = V2(P2)(1-P1) = E[E2] Which represents the expected value of choosing question 1 first being equivalent to the expected value of the person choosing question 2 first The contestant should: Choose question 1 first I do not understand why if they are equal question 1 is the better choice. I am missing something? Thank you for any help. |
| What is $\mathbb{F}_q((1/T))/\mathbb{F}_q[T]$ as a topological space? Posted: 03 Feb 2022 04:12 AM PST We fix a prime power $q$, and we let $\mathbb{F}_q((1/T))$ be the completion of $\mathbb{F}_q[T]$ at infinity. We have the absolute value on $\mathbb{F}_q((1/T))$, $$\bigg{|}\sum_{i=-\infty}^n a_iT^i \bigg{|} = n $$ if $a_n \neq 0$. Now we can define a metric on $\mathbb{T} := \mathbb{F}_q((1/T))/\mathbb{F}_q[T]$ as $$||f|| = \inf_{f' \sim f}|f'| $$ where $\sim$ is the usual relation from the quotient. I am wondering if $\mathbb{T}$ is homeomorphic to a well known topological space? I know that this is quite analogous to $\mathbb{R}/\mathbb{Z}$, which is homeomorphic to a circle. |
| Let $a,b,c,d,e$ be five numbers satisfying the following conditions... Posted: 03 Feb 2022 03:58 AM PST
My Approach: $$(a+b+c+d+e)^3 = \sum_{a,b,c,d,e}{a^3} + 3\sum_{a,b,c,d,e}{a^2b} + 6\sum_{a,b,c,d,e}{abc} $$ Taking $\mod (a+b+c+d+e)$, $$(a+b) ≡ -(c+d+e)$$ $$ab(a+b) ≡ -ab(c+d+e)$$ $$\sum{a^2b} ≡ -ab(c+d+e) -bc(a+d+e) -cd(a+b+e)-... = -\sum_{a,b,c,d,e}{ab(c+d+e)} = -3\sum_{a,b,c,d,e}{abc}$$ Therefore, $\sum{a^2b} = p(a,b,c,d,e) . (a+b+c+d+e) - 3\sum_{a,b,c,d,e}{abc}$ Since, $(a+b+c+d+e) = 0$ $$\sum{a^3} = (3×3 -6)\sum{abc} = 3×33 = \color{red}{99}$$ But the answer key shows: $$\frac{\sum{a^3}}{\color{blue}{502}} = 99$$ Where is my mistake? |
| In a dense-in-itself Polish space there is a "continuous" Borel probability measure [closed] Posted: 03 Feb 2022 04:26 AM PST Let $X$ be a complete separable metric space with no isolated points. Prove that there is a $\mathbf{p} \in \Delta(X)$ where $\Delta(X)$ is the set of all Borel probability measures on $X$, such that additionally $\mathbf{p}\{\omega\}=0$ for every $\omega \in X$. |
| Irreducible polynomials over non-UFD Posted: 03 Feb 2022 04:24 AM PST It is well known that if $R$ is a UFD with field of fractions $K$ and $f \in R[x]\setminus R$, then the following holds: $f$ is irreducible in $R[x] \iff f$ is primitive and irreducible in $K[x]$ So far, I've always seen the term "primitive" and thus the statement with $R$ being a UFD. I'm wondering what happens in the general case of $R$ being an arbitrary integral domain. It is pretty easy to show the following ($f$ being a non-constant polynomial): $f$ is irreducible in $R[x] \implies f$ primitive $f$ is irreducible in $R[x] \impliedby f$ primitive and irreducible in $K[x]$ So the interesting point for me here is the following: $f$ is irreducible in $R[x] \implies f$ irreducible in $K[x]$. My question is: Under which condition is the last-mentioned statement true? When I tried to find a counterexample over a non-UFD which is at least integrally closed, I've found $f=2x^2-2x+3 \in \mathbb Z[\sqrt{-5}][x]$, which should be irreducible over $\mathbb Z[\sqrt{-5}]$, but reducible over the field of fractions $\mathbb Q(\sqrt{-5})$ due to $f=\frac12(2x-1-\sqrt{-5})(2x-1+\sqrt{-5})$. My next question is: Is there a monic polynomial being irreducible over $\mathbb Z[\sqrt{-5}]$, but reducible over $\mathbb Q(\sqrt{-5})$? All I know is that it must necessarily have degree $4$ (otherwise it would have a root in $\mathbb Q(\sqrt{-5})$ and thus in $\mathbb Z[\sqrt{-5}]$ because $\mathbb Z[\sqrt{-5}]$ is integrally closed). |
| Find the distance between a function f(x, y) and a given point P Posted: 03 Feb 2022 04:04 AM PST Uni student here, we have assignments that are miles harder than what we learn in the lectures and today I am stuck on the following task: Given the function $f(x,y) = x^2 - \frac{1}{2}y^2$, find the distance between $P = (2, 0, \frac{1}{2})$ and the graph of the function. I am aware that there is a distance formula which goes something like this: $d = \sqrt{(x_0-x_1)^2+(y_0-y_1)^2}$ but I struggle to see how this could be applied in a scenario where you have a function with more than one variable. |
| Do both real and imaginary roots of a cubic equation need to continuous? Posted: 03 Feb 2022 04:08 AM PST I have a cubic equation: $X^3-UX^2-KX-L=0$ (1) with $X=1-E+U$, $K=4(1-\gamma^2-\lambda^2)$, $L=4\gamma^2U$. I solve Eq. (1) for the variable $E$ numerically for $U=2$ and different sets of parameter $\gamma =$ 0.1, 0.25, and 0.45 and plot real and imaginary parts of $E$ against $\lambda$. The real parts of the solutions for $X$ and $E$ differ by a constant shift for a fixed value of $U$ while the imaginary parts remain identical. See the plots below. The plots plot all the roots or solutions. They are colored with three different colors by looking at the possible continuity of the roots or the solutions (here stressed on the continuity of the imaginary parts).
Here we see for the first case $\gamma = 0.1$, the continuity in the real parts of $E$ (Re $E$) break down while the imaginary parts (Im $E$) remain continuous for all parameter values of $\gamma$. Since the solutions are found for discrete values of $\lambda$, we may think that extreme left and right of the cyan and blue curves of the first plot can be interchanged and hence made Re $E$ continuous all the way. However, that may lead Im $E$ to be discontinuous for other parameter values of $\gamma$ (I can add images if further clarity is needed). How can we interpret or understand this? Or is there something fundamentally wrong? The alternative coloring for the first two plots could be the following (in this sense all the curves appear continuous function of $\lambda$). Real part:
Imaginary part:
Though the above fixes the continuity issue, it appears that all three curves do not smoothly evolve with parameter $\gamma$. Experts' opinions awaited. |
| Does every strip of positive irrational slope contain a perfect square point? Posted: 03 Feb 2022 04:31 AM PST Does every strip between two parallel lines of positive irrational slope contain a point with perfect square coordinates? Equivalently (I think), are there perfect square points arbitrarily close to any positively irrationally sloped line? EDIT: [wrong, retracted]. EDIT 2: Numerical experiments seem to hint at a counterexample for $\alpha = \frac{3-\sqrt{5}}{2} = (\phi -1)^2$. With $p_k/q_k$ being convergents of the continued fraction $[0;\overline{1}]$ for $\phi-1$, it seems that $$\lim\limits_{k \to \infty}{\left|p_k^2-\alpha q_k^2\right|} = 0.55278\dots$$ so square points other than $(0,0)$ are at least that constant away vertically from $y = \alpha x$. Squared perpendicular distance from the points to the line seems to tend to a rational, $4/15$. EDIT 3: The convergents in EDIT 2 are the ratios of consecutive Fibonacci numbers $F_n$. With the help of wxMaxima I managed to calculate: $$\lim_{n\to \infty}\left| {{\left(\phi-1\right)^2\ F_{n+1}^2}}-F_n^2\right| = {{\sqrt{6\,\sqrt{5}-10}}\over{5^{{{3}\over{4}}}}} ≈ 0.55278... > 0$$ which settles the general question negatively, and I'm nearly sure that a similar limit exists whenever the CF for $\sqrt{\alpha}$ is eventually periodic (i.e. $\alpha$ is a square of a quadratic irrational) — or more precisely, a finite set of limits, one for each offset within the period, and one of them smallest. |
| Good presentation of the Witt numbers Posted: 03 Feb 2022 04:25 AM PST Let $A$ be a $\mathbb{F}_p$-algebra of finite presentation. Then we know that $W(A)$ is a $W(\mathbb{F}_p)=\mathbb{Z}_p$-algebra. My question is if there is a "good" presentation of $W(A)$ as $\mathbb{Z}_p$-algebra. If it is helpful for this, I'm perfectly happy to assume that $A$ is smooth or a hypersurface (by which I mean $A\cong \mathbb{F}_p[x_1,\ldots,x_n]/f$). |
| symbolic definition of bijection? Posted: 03 Feb 2022 04:23 AM PST As I understand it, bijection works if each element of the co-domain is mapped to by exactly one element of the domain. Could I write it as $$f(X) = \{\forall y \in Y \exists! x \in X : f(x) = y\}$$ I have just started trying my hand at symbolic math and the two Wikipedia articles that I skimmed over on bijection didn't have a symbolic definition unfortunately. So is this correct or can be made clearer in some way? |
| What is the intuition behind the definition of a simply strongly normal number? Posted: 03 Feb 2022 04:25 AM PST In a thesis written by A. Belshaw (On the normality of numbers), they define a new normality criterion named simply strongly normal. The motivation behind this definition is:
Definition Let $\alpha$ and $m_k(n)$ be defined as above. Then $\alpha$ is simply strongly normal to base $r$ if $\displaystyle \limsup_{n \to \infty} \displaystyle\frac{(m_k(n) - n/r)^2}{\frac{r-1}{r^2}n^{1+\varepsilon}} = 0 \hspace{2cm}$ and $\hspace{2cm} \displaystyle \limsup_{n \to \infty} \displaystyle\frac{(m_k(n) - n/r)^2}{\frac{r-1}{r^2}n^{1-\varepsilon}} = \infty$ for any $\varepsilon > 0$, where the constant $\frac{r-1}{r^2}$ is derived from the variance of the binomial distribution. The definition as stated above, makes me think of a (binomially) normalised random variable. As the random variable $m_n(k)/n$ should converge to $1/r$ for $\alpha$ to be simply normal, the first criterion seems more intuitive to me than the second (as the numerator 'should converge to zero'). Also, intuitively, I'd say that they normalise because they want to look at the rate of convergence (towards the asymptotic frequency) in order to analyse the asymptotic behaviour of different normal numbers. E.g. Champernowne's number in base 2 has an increasing excess of ones in the expansion, but the asymptotical frequency of the digit 1 is still 1/2. This is as far as my intuition brought me. The motivation given in the paper does not give me more understanding of the choice for these criteria. Thus, what is the intuition behind the above definition? / Why do these criteria give a "stronger" form of normality? |
| Prove that for any integer $n, n^2+4$ is not divisible by $7$. Posted: 03 Feb 2022 04:04 AM PST The question tells you to use the Division Theorem, here is my attempt: Every integer can be expressed in the form $7q+r$ where $r$ is one of $0,1,2,3,4,5$ or $6$ and $q$ is an integer $\geq0$. $n=7q+r$ $n^2=(7q+r)^2=49q^2+14rq+r^2$ $n^2=7(7q^2+2rq)+r^2$ $n^2+4=7(7q^2+2rq)+r^2+4$ $7(7q^2+2rq)$ is either divisible by $7$, or it is $0$ (when $q=0$), so it is $r^2+4$ we are concerned with. Assume that $r^2+4$ is divisible by 7. Then $r^2+4=7k$ for some integer $k$. This is the original problem we were faced with, except whereas $n$ could be any integer, $r$ is constrained to be one of $0,1,2,3,4,5$ or $6$. Through trial and error, we see that no valid value of $r$ satisfies $r^2+4=7k$ so we have proved our theorem by contradiction. I'm pretty sure this is either wrong somewhere or at the very least not the proof that the question intended. Any help would be appreciated. |
| Posted: 03 Feb 2022 04:09 AM PST This is what I did : Since, $Y = \tan^{-1} x$, by differentiating we can get, $Y_1 = \dfrac{1}{x^2 + 1},\\ Y_2 =\dfrac{ -2x}{(x^2 + 1)^2},\\ Y_3 = \dfrac{2 (3x^2 -1)}{ (x^2+1)^3}$ and so on... As per the above pattern, I know the formula for $Y_n$ will have $(x^2+1)^n$ in denominator but I'm unable to figure out the numerator. Even If I get the formula for $Y_n$ , How am I supposed to proceed? Please help. Thanks. EDIT : I solved the question by using Leibniz theorem. Thanks to all who tried to help :) |
| Can this quick way of showing that $K[X,Y]/(Y-X^2)\cong K[X]$ be turned into a valid argument? Posted: 03 Feb 2022 04:12 AM PST I've been trying to show that $$ K[X,Y]/(Y-X^2)\cong K[X] $$ where $K$ is a field, $K[X]$ and $K[X,Y]$ are the obvious polynomial rings over the indeterminates $X$ and $Y$ and $(Y-X^2)$ is the ideal generated by the polynomial $Y-X^2$. Though I'm sure there's a fairly easy way to find an explicit isomorphism between the two rings, the following argument jumped out at me: If we substitute in a value for $X$ - $x$, say, then the ideal $(Y-x^2)$ is a maximal ideal of $K[Y]$. So the quotient $K[Y]/(Y-x^2)$ is a field; in fact, the homomorphism $K[Y]\to K:P(Y)\mapsto P(x^2)$ and is clearly surjective, so the quotient is isomorphic to $K$. I'd like to be able to deduce from this that $K[X,Y]/(Y-X^2)\cong F[X]$, but I can't see a nice way to do it. I know that the 'substitution' maps $P(X,Y)\mapsto P(x,Y)$ are homomorphisms, but I can't see a nice way of pulling all these homomorphisms back to the polynomial ring in two variables. Or maybe I'm completely wrong and there is no way to turn this into a valid argument. Can anyone help me? |
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