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Find the maximum number of possible real roots of the equation $a x^4+ b x^3+ x^2+ x+1=0$ is equal to where $a \ne 0$. | The question goes on like this : "Let $a$ and $b$ be real numbers such that $a \ne 0$ . Then the maximum number of possible real roots of the equation $ax^4+bx^3+x^2+x+1=0$ is equal to " My attempt: First I differentiate wrt $x$ , but due to variable: $a$ and $b$ , I can't make any direct conclusion Also by writing $-a... | Let $\alpha,\beta,\gamma,\delta$ be real roots of $ax^4+bx^3+x^2+x+1=0$ . Then $\displaystyle \frac{1}{\alpha},\frac{1}{\beta},\frac{1}{\gamma},\frac{1}{\delta}$ be roots of $\displaystyle x^4+x^3+x^2+bx+a=0.$ Then $\displaystyle \sum \bigg(\frac{1}{\alpha}\bigg)^2=\sum \frac{1}{\alpha^2}+2\sum \frac{1}{\alpha}\cdot \f... | |calculus|algebra-precalculus|polynomials| | 0 |
Is $\textit{affine space}$ the same as $\textit{quotient space}$? | From the answer for this question , I understand that affine subspace is the same as affine subset , however (despite the somewhat misleading question's title), it doesn't say that affine space is the same as quotient space . Also, I found the following definition of affine space on Wikipedia : An affine space is a set... | The concepts of "affine space" and "quotient space" are by no means identical. In the linked question it was observed that the elements of a quotient space $V/V_0$ , where $V$ is a vector space and $V_0 \subset V$ is a linear subspace, are the subsets of $V$ having the form $x + V_0$ with $x \in V$ . Thus $$V/V_0 = \{ ... | |vector-spaces|affine-varieties| | 0 |
Is $\textit{affine space}$ the same as $\textit{quotient space}$? | From the answer for this question , I understand that affine subspace is the same as affine subset , however (despite the somewhat misleading question's title), it doesn't say that affine space is the same as quotient space . Also, I found the following definition of affine space on Wikipedia : An affine space is a set... | The vocabulary "transitive and free action" refers to group action on a set. You know vectors and points and relation between three points $A,B, C$ : $$\vec{AB}+\vec{BC}=\vec{AC}$$ Describing an affine space as in wikipedia definition you refer in OP is an equivalent way of describing a point space where two points $A$... | |vector-spaces|affine-varieties| | 1 |
Solve $\sqrt{\dfrac{a}{x}}-\sqrt{\dfrac{x}{a}}=\dfrac{a^2-1}{a}$ | Solve $\sqrt{\dfrac{a}{x}}-\sqrt{\dfrac{x}{a}}=\dfrac{a^2-1}{a}$ Let $u^2=\dfrac{a}{x}$ : $\Rightarrow \sqrt{u^2}-\sqrt{\dfrac{1}{u^2}}=\dfrac{a^2-1}{a} \tag{1}$ $\Rightarrow u-\dfrac{1}{u}=\dfrac{a^2-1}{a} \tag{2}$ $\Rightarrow a(u^2-1)=u(a^2-1) \tag{3}$ $\Rightarrow au^2-a^2u=a-u \tag{4}$ $\Rightarrow au(u-a)=a-u \ta... | Now: $$ \sqrt{\frac{a}{x}} - \sqrt{\frac{x}{a}} = \frac{a^2 - 1}{a} $$ multiply by $a$ $$ a\sqrt{\frac{a}{x}} - a\sqrt{\frac{x}{a}} = a^2 - 1 $$ Set $t=a/x$ $$ a\sqrt{t} - a\sqrt{\frac{1}{t}} = a^2 - 1 $$ solve for $t$ by squaring $$ a^2t-2a^2+a^2/t=a^4-2a^2+1 $$ $$ a^2t^2+a^2-a^4t-t=0 $$ $$ a^2t(t-a^2)-(t-a^2)=0 $$ an... | |algebra-precalculus| | 0 |
Problems with integration $\int_{-\infty}^{\infty}\frac{xe^{a - x}}{(1+e^{a-x})^2}dx$ | I have this integral: $\int_{-\infty}^{\infty}\frac{xe^{a - x}}{(1+e^{a-x})^2}dx$ I've tried to find its: $\int_{-\infty}^{\infty}\frac{xe^{a - x}}{(1+e^{a-x})^2}dx = | a-x = t| = -\int_{-\infty}^{\infty}\frac{(a-t)e^{t}}{(1+e^{t})^2}dx$ , then I used integration by parts: Let $u = a-t$ so $du = -dt$ and let $dv = \fra... | Let $t=a-x$ . Then $\mathrm{d}t=-\mathrm{d}x$ . This will give you a minus sign. But you also need to apply your substitution to the boundaries of the integral. This gives you another minus sign. So your substituted integral becomes $$ I = \int \limits_{-\infty}^\infty (a+t) \frac{e^t}{(1+e^t)^2} \mathrm{d}t = a\underb... | |integration|definite-integrals| | 1 |
Evaluate $\int{\frac{1}{x^3}}$ using u-sub. | I tried solving $\int{\frac{1}{x^3}}$ using u-sub instead of power rule and I got $-\frac{1}{2}x^{-2}$ instead of $\frac{x^4}{4}$ . Its very possible I've made a very simple mistake or their is something fundamentally wrong with my idea, if someone could show how/if this problem can be done with u-sub it would be great... | Your answer is fine maybe there is a typo in your textbook. Also an easier u-sub would be letting $u=\frac{1}{x} $ then $du=-\frac{1}{x^2} dx $ and then notice $\frac{1} {x^3} = \frac{1}{x^2} \cdot \frac{1}{x}$ So the integral becomes $\int (-u) du = -\frac{u^2}{2}+C $ and after subbing $u $ back you get $-\frac{1}{2} ... | |calculus|indefinite-integrals|substitution| | 0 |
Determining algebraically a point of intersection. | A student I was tutoring posed the question: "I know how to solve $$e^{-x} = \ln x$$ graphically, however how do you solve this algebraically?" I have been fiddling around with it for a while and I feel like I'm missing something. I have tried various methods involving series expansions and de Moivres theorem but I fee... | I think that it could be better to write the equation as $$e^x\,\log(x)=1$$ By inspection, the solution is $\in (1,2)$ . Expand the lhs as $$f=e^x\,\log(x)=e \sum_{n=1}^\infty a_n\,(x-1)^n$$ where the first $a_n$ (which are defined by recursion) are $$\left\{1,\frac{1}{2},\frac{1}{3},0,\frac{3}{40},-\frac{7}{144}, \fra... | |algebra-precalculus|numerical-methods| | 0 |
List of geometric theorems linked by two squares | I'm trying to create a classification for geometric theorems that relate to two squares As a type of organization and classification And the curiosity to explore I have collected some theorems of this type that I will put in an answer/answers. I hope you can help me expand my list. It's important to note that I'm not l... | Here are somewhat-natural generalizations of a couple of the two-squares-joined-at-a-vertex results from OP's answer . Potema's Theorem Let squares $\square A'B'C'D'$ and $\square A''B''C''D''$ be as shown, with $X$ the midpoint of $X'X''$ . If segment $A'A''$ remains fixed, and segment $D'D''$ keeps a constant length ... | |geometry|euclidean-geometry|big-list| | 0 |
How to pull back the differential form $\omega = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}$ to $S^2$ | Consider the stereographic projection chart on $S^2$ which doesn't include the north pole $$(X,Y)=\varphi(x,y,z)=\left(\frac{x}{1-z}, \frac{y}{1-z}\right).$$ I want to pull back the 1-form $\omega = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}$ to $S^2$ from $\mathbb{R}^2$ to $S^2$ but I am not sure about a step in the calculation.... | Imho the simplest but still a bit tedious approach is to use the inverse stereographic projection : \begin{align} \pmatrix{ x\\y\\z}=\frac{1}{1+X^2+Y^2}\pmatrix{2X\\2Y\\-1+X^2+Y^2} \end{align} where $x,y,z$ are the coordinates in $\mathbb R^3$ and $X,Y$ the coordinates on $S^2\,.$ By ordinary calculus, \begin{align} dx... | |calculus|geometry|analysis|manifolds|differential-topology| | 0 |
If $u^2 \ge -\dfrac{8}{3}$, then $u \ge -\sqrt{\dfrac{8}{3}}$. | If $u^2 \ge -\dfrac{8}{3}$ , then $u \ge -\sqrt{\dfrac{8}{3}}$ . Is this the correct convention? I was confused because initially I thought the negative sign would go inside the square root, but then that would lead to imaginary numbers. Thanks. | NOTE that $\forall a \in \mathbb{R} , a^2 \geq 0$ $ so your first inequality is always true . $u^2 \geq 0 > -\frac{8}{3 } \implies u \in \mathbb{R} $ If you're still not convinced we can move everything to one side and solve the quadratic inequality $u^2+ \frac{8}{3} \geq0$ which has a negative determinant $D=b^2-4 \cd... | |algebra-precalculus| | 0 |
Form of Hypergeometric Differential Equation - possible mistake? | I'm performing a close critical study of David Nelson's Penguin Dictionary of Mathematics (4th ed., 2008). Under the entry hypergeometric differential equation , it suggests the form: $$x (1 - x) \dfrac {\mathrm d^2 \phi} {\mathrm d x^2} + [c - (a + b - 1) x] \dfrac {\mathrm d \phi} {\mathrm d x} - a b \phi = 0$$ Howev... | Any homogenous linear equation of second order with singular points $(0,1,\infty)$ $$x(1-x)f'' + (a + b x) f'[x] +c f[x]=0$$ is a hypergeometric differential equation, the regular solution at (0,1) being the hypergemetric series $$\, _2F_1\left(-\frac{1}{2} \sqrt{b^2+2 b+4 c+1}-\frac{b}{2}-\frac{1}{2},\frac{1}{2} \sqrt... | |ordinary-differential-equations|hypergeometric-function| | 0 |
Complete Riemann Manifold | I was trying to understand this article about the existence of complete Riemannian metrics by Nomizu Ozeki, see The Existence of Complete Riemannian Metrics, Proceedings of the American Mathematical Society, Vol. 12, No. 6, pp. 889-891, 1961. I understand the general idea of building this function that increases indefi... | The key observation is that subsets of relatively compact are relatively compact: if $X \subseteq Y$ and $Y$ is relatively compact, then $\overline{X} \subseteq \overline{Y}$ is a closed subset of a compact set, hence is compact. So, if $B(x, d(x, y)+a)$ is relatively compact, then $B(y, a)$ is as well. Suppose for a c... | |metric-spaces|riemannian-geometry| | 0 |
Name for elements from domain used in mapping | As I understand, in a function, the subset of the codomain actually mapped to is called the range. What about the domain? Is there a name for the subset of the domain actually used in a mapping? | I'm not sure if I fully understand your question but here's an answer to what I think you're confused about. Let's use examples so that it's easier to see it "in action" Let $f(x)=\sqrt{x} $ ; the Domain is $[0,+ \infty)$ but notice that $ [0,+ \infty) \subset \mathbb{R}=(- \infty , + \infty)$ the symbol " $\subset$ " ... | |functions|terminology| | 0 |
Notation clarification on Allen hatcher section 3.1 | I am reading Hatcher(alg.top). in the chapter 3, section 3.1, the universal coefficient theorem, there it is being argued that, In the original chain complex the homology groups are $\mathbb{Z}$ 's in dimensions 0 and 3, together with a $\mathbb{Z}_2$ in dimension 1. The homology groups of the dual cochain complex, whi... | Have a look at the top sequence. Kernel of the last $0$ map $C_1\to C_0$ is of course whole $\mathbb{Z}$ . While the image of $x\mapsto 2x$ map is $2\mathbb{Z}$ . The corresponding homology is the quotient of kernel by image, and gives us $\mathbb{Z}/2\mathbb{Z}$ , also commonly referred to as $\mathbb{Z}_2$ . The vert... | |algebraic-topology|notation| | 0 |
Determining whether a housing allocation is in the Core | I have recently been thinking about the housing allocation problem where we have a set of players and a set of houses where players have strict preferences over the houses. I am aware of the Top Trading Cycle algorithm which can be used to assign houses to players in a Pareto Optimal, Strategy Proof, and Individual Rat... | There are two ways to interpret the question. First, is an allocation in the core for the initial assignment of houses? Second, is the allocation in the core if every agent owns their assigned house. For the second question, just run the top trading cycle algorithm starting from the allocation (that seems to be your an... | |game-theory|economics|matching-theory| | 0 |
Determinant of $n \times n$ matrix of a sort of skew symmetric matrix plus some diagonal | Given, a matrix: $$\begin{pmatrix} a & b & \ldots & b & b \\ -b & a & \ldots & b & b \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ -b & -b & \ldots & a & b \\ -b & -b & \ldots & -b & a \end{pmatrix}.$$ I need to find a determinant. So initially what I did, was I added the first column to other ones: $$\begin{pmatrix... | Let $U$ be the upper triangular matrix of ones. The matrix in question can be expressed as $$ A_n=aI_n+b(U-U^T)=UB_nU^T $$ where $$ B_n=aU^{-1}(U^{-1})^T+b\left[(U^{-1})^T-U^{-1}\right]. $$ Clearly, we have $\det A_n=\det B_n$ because $\det U=1$ . It is straightforward to verify that $$ U^{-1}=\pmatrix{1&-1\\ &\ddots&\... | |linear-algebra|abstract-algebra|matrices|determinant|problem-solving| | 1 |
The selection of the direction of the auxiliary curve when applying green's theorem to line integral with a singular point seems to change the answer | Problem Compute $$ \oint_L\frac{xdy-ydx}{4x^2+y^2} $$ where $L$ is a circle centered at $(1, 0)$ with a radius of $R > 1$ , and the direction of $L$ is counterclock-wise. Solution To by pass the singular point at $(0, 0)$ which is inside the circle $L$ surrounds, adding an auxiliary curve: $$ C:4x^2+y^2 = \delta^2 $$ T... | The "pitfall" here is that Green's theorem only works with positively oriented boundaries/curves, which means that you need the correct orientation for $C$ , which should be the opposite of that of $L$ (see for example this answer for the reasoning behind that: https://math.stackexchange.com/a/141869/1104384 ), because... | |calculus|multivariable-calculus|line-integrals| | 0 |
How to compute $\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x}$? | I'm trying to compute the following limit: $$L=\lim_{x\to 0} \frac{e^{ax}-e^{bx}}{x} \tag{1}$$ And I have to use some of the following limits for it: $$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}$$ I tried some substitutions, specially the first limit but I got only to: $$L=\lim_... | $$ L = \lim _{x\to 0}\left(\frac{e^{ax}-e^{bx}}{x}\right) $$ $$ L = \lim _{x\to 0}\left(\frac{e^{ax}\left(1 - e^{(b-a)x}\right)}{x}\right) $$ $$ L = \lim _{x\to 0}\left(e^{ax} \cdot \frac{1 - e^{(b-a)x}}{x}\right) $$ Notice that as $x \to 0, e^{ax}\to 1$ , so we can focus on the second part of the product: $$ L = \lim ... | |limits|exponential-function|limits-without-lhopital| | 0 |
Request: Does notion of natural density $1$ with specified lower bound already exist | Suppose that $A\subseteq\mathbb{N}$ . Then one can look at the function $f(n)=|\{0,...,n\}\setminus A|$ ( $|S|$ denoting the cardinality of $S$ ). I am interested in the case when $f(n)\leq Cn^\alpha$ with $C>0$ and $\alpha . In this case $A$ obviously has natural density $1$ . My question is: Is there already a name f... | I think this is a bad question, because it doesn't really matter if there is a name for sets with this specific property, as the property isn't sufficiently interesting to warrant a name being described to it. I also don't see how the sets you describe are "refined notions of natural density." Looking for names of sets... | |number-theory| | 0 |
for any positive integer $n$, we can find numbers$ 1, 2, \cdots, n$ that the mean of any two of them will not appear between these two numbers | Prove that, for any positive integer $n,$ we must be able to find a permutation of the numbers $1, 2, . . . , n$ such that the mean of any two numbers in the permutation will not appear between these two numbers. I hope that someone can help me. Thanks! | I am assuming you're asking Let $n$ be a possitive integer , prove that the mean let's call it $m$ of $0 , $m \notin \{1,2,...,n\}$ , $\forall a,b \in\{1,2,...,n\}$ If that is what you mean it is clearly wrong Let $n=10$ and we chose $a=1 , b=9 $ the mean of $a,b$ is $\frac{1+9}{2}=5$ and $5 \in \{1,2,3,...,10\} $ If I... | |algebra-precalculus| | 0 |
MLE's for ANOVA Model | Given the ANOVA model $Y_{ij} = \mu_i + \varepsilon_{ij}, \varepsilon_{ij}\sim N(0, \sigma^2)$ , $i = 1, 2, \ldots , I, \space j = 1,2, \ldots, n_i$ , I am trying to find the MLE's $\hat\mu_1, \hat\mu_2, \ldots , \hat\mu_n, \hat\sigma^2$ . I have that the likelihood function is $L = \prod_{i=1}^{I}\prod_{j=1}^{n_i} \fr... | Denote $\sigma ^ 2 = \theta $ , thus \begin{align} L(\mu, \theta) = \left( \frac{1}{(2 \pi \theta ) ^{0.5}} \right)^ {\sum_{i=1}^I n_i} \exp\{ - (Y_{ij} - \mu_i ) ^ 2 / (2\theta) \}, \end{align} \begin{align} l(\mu, \theta) = 0.5\sum_{i=1}^I n_i \ln \left( 2\pi \theta \right) + \frac{1}{2\theta}\sum_i^I \sum_j^{n_i} (Y... | |statistics|regression| | 0 |
find square root of $x^2+x^3 $ in formal power series $k[[x,y]]$ | I am trying to show that the polynomial $y-x^2-x^3$ is reducible in the formal power series ring $k[[x,y]]$ . I am attempting the question by finding a polynomial in $k[[x,y]]$ which is the square root of $x^2+x^3$ . In order to find the square root I wrote the general polynomial in $k[[x,y]]$ , $$a_{00}+a_{10}x+a_{01}... | By the generalized binomial theorem we have $$ \sqrt{x^2+x^3}=|x| (1+x)^{\frac{1}{2}}=|x| \sum_{i=0}^\infty \binom{\frac{1}{2}}{i}x^i, $$ where $$ \binom{\frac{1}{2}}{i}=\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2) \cdots (\frac{1}{2}-i+1)}{i!} $$ | |abstract-algebra|systems-of-equations|formal-power-series| | 0 |
Existence of an $L^2$ function. | Let $f_n \in L^2([0,1])$ be non-zero for all $n \in \Bbb{N}$ . Prove there exists a function $g \in L^2([0,1])$ such that $$\int_0^1 g(x) f_n(x) dx \neq 0.$$ So first I tried contradiction which implies $g(x)f_n(x)$ need be zero almost everywhere which would imply $g$ to be the zero function. But then again I thought m... | For each $n \in \Bbb{N}$ , put $$U_n:=\{g \in L^2: \int_0^1 gf_n \neq 0\}.$$ Next, define linear functionals $L_n$ on $L^2$ via $$L_n(g):=\int gf_n dx.$$ Then $U_n=L^{-1}(\{0\})^c$ . And by continuity of $L_n$ , $U_n$ is open as it is the preimage of an open set. Moreover, given any scalar $t$ , \begin{align*} L_n(g+tf... | |real-analysis|integration|measure-theory| | 1 |
Finding matrix to transform one vector to another vector | If I have an arbitrary vector $A = (a,b,c,0)$ how can I find a transformation matrix $M$ such that $M \times A = (0,1,0,0)$ ? We can assume $A$ has a magnitude of $1$ if it helps simplify the derivation process. The trivial case $A = (0,1,0,0)$ would cause $M$ to be the identity matrix. If $A = (0,-1,0,0)$ then $M$ wou... | Since $B=(0,1,0,0)^T$ is very simple and $A=(a,b,c,0)^T$ is not we can consider $$M_1=\left[\begin{array}{cccc}0&a&0&0\\0&b&0&0\\0&c&0&0\\0&0&0&0\end{array}\right]$$ which satisfies $ M_1B=A$ which is not satisfactory since we are looking for all $M$ such that $MB=A$ . Furthermore $M_1^{-1}$ does not exist. So we rathe... | |matrices| | 0 |
Must a countable disjoint union of closed balls in $\mathbb{R}^n$ with positive radius be disconnected? | A disjoint union of open balls is of course disconnected. Here it is proved that a locally compact, connected, Hausdorff space is not a countable disjoint union of compact subsets, so a countable disjoint union of closed balls in $\mathbb{R}^n$ can't be connected and locally compact (hence cannot be open or closed conn... | Example of countable family of pairwise disjoint closed sets on a plane whose union is connected Here is a relatively simple example. The other section has an older, more complicated example in $ℝ^3$ . Let $D = \{d_n : n=0,1,\dots\}$ be a dense subset of unit sphere with $d_n \neq d_m$ . For each $n=1,2\dots$ define th... | |real-analysis|general-topology|analysis| | 1 |
Solve $\sqrt{\dfrac{a}{x}}-\sqrt{\dfrac{x}{a}}=\dfrac{a^2-1}{a}$ | Solve $\sqrt{\dfrac{a}{x}}-\sqrt{\dfrac{x}{a}}=\dfrac{a^2-1}{a}$ Let $u^2=\dfrac{a}{x}$ : $\Rightarrow \sqrt{u^2}-\sqrt{\dfrac{1}{u^2}}=\dfrac{a^2-1}{a} \tag{1}$ $\Rightarrow u-\dfrac{1}{u}=\dfrac{a^2-1}{a} \tag{2}$ $\Rightarrow a(u^2-1)=u(a^2-1) \tag{3}$ $\Rightarrow au^2-a^2u=a-u \tag{4}$ $\Rightarrow au(u-a)=a-u \ta... | Question Summary (for Easier Reference) Solve: $$\sqrt{\dfrac{a}{x}}-\sqrt{\dfrac{x}{a}}=\dfrac{a^2-1}{a} =a-\dfrac{1}{a} \text{ for }x\text{ in terms of }a \tag{Eq. 1}$$ Solution Steps Start by observing that the roles of $\sqrt{\frac{a}{x}}$ and $a$ are very similar, and same with $-\frac{1}{a}$ and $-\sqrt{\frac{x}{... | |algebra-precalculus| | 0 |
Joint distribution of two conditional distributions | I am trying to understand how a joint distribution is formed when two regular conditional distributions are involved that are conditional with respect to different random variables. Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, and let there be the three random variables $X:(\Omega, \mathcal{A})\right... | If $Y$ and $Z$ are conditionally independent when given $X$ , then $$\begin{align}\mathsf P_{X,Y\mid Z=z}(x,y\mid z)=\mathsf P_{Y\mid X=x}(y)\,\mathsf P_{X\mid Z=z}(x)\\\mathsf P_{Y\mid Z=z_0}\!(E)=\int_\mathcal X \mathsf P_{Y\mid X=x}(E)\,\mathsf P_{X\mid Z=z_0}\!(\mathrm dx)\end{align}$$ If that was not the case, the... | |probability-distributions|conditional-probability|conditional-expectation| | 0 |
Find the inverse Laplace transform of F(s) = 1/(s+exp(-sτ)), where τ is a positive real parameter. | I'm looking for the inverse Laplace transform of $$F(s) = \frac{1}{s + e^{-s\tau}}$$ where τ is a positive real parameter. I am trying to use general inverse formula of Laplace transformation to solve it. But then, I need to find the singularities of F(s), that is, $ s + e^{-s\tau} = 0$ . Transform the euqation, I can ... | $$\mathcal{L}_s^{-1}\left[\frac{1}{s+\exp (-s \tau )}\right](t)=\sum _{m=0}^{\infty } \frac{(-t+m \tau )^m \theta (t-m \tau )}{\Gamma (1+m)}$$ where: $\theta (t-m \tau )$ is HeavisideTheta function. $$\mathcal{L}_s^{-1}\left[\frac{1}{s+\exp (-s \tau )}\right](t)=\\\mathcal{M}_q^{-1}\left[\mathcal{L}_s^{-1}\left[\mathca... | |complex-analysis|partial-differential-equations|laplace-transform|residue-calculus|inverse-laplace| | 0 |
Show that if a Mobius transformation has 3 fixed points then it is the identity map. | I have that any non trivial Mobius transformation has at most 2 fixed points since f(z)-z=0 has at most 2 roots. But I cannot deduce why it must then be the identity. | Lemma: If $f$ fixes $1,0,\infty$ , it must be the identity map. Proof: Indeed, take: $$f(z)=\frac{az+b}{cz+d}$$ Because it fixes $0$ , then $b=0$ . Because it fixes $\infty$ then, $f(z)=\frac{a}{c+d/z}$ must have $c=0$ and we have $f(z)=az/d$ . But it fixes $1$ , so $f(z)=z$ . $\square$ Theorem: If $f$ fixes any distin... | |complex-analysis| | 0 |
Which theorem should be used to solve this question? | My friend sent me that question and said "another Carnot theorem" is used to solve this question but i couldnt find that theorem. Can you help me? Additional explanation: $$ \widehat{ABD} = 30^{\circ} $$ $$ \widehat{DBC} = 1^{\circ} $$ $$ \widehat{ACD} = 89^{\circ} $$ $$ \widehat{BAD} = \widehat{DAC} $$ $$ x(\widehat{B... | Take $E$ - reflection of $C$ about $AD$ , it belongs to $AB$ and $\widehat{BED}=91^\circ$ , i.e. $O$ , the circumcenter of $\triangle BDE$ belongs to $BC$ and $BO=OD=DE=CD$ , hence $\widehat{BCD}=\widehat{COD}=2^\circ$ . Best regards | |geometry|triangles| | 0 |
Source of the definition of integrating a form along a curve in a manifold | Suppose that $M$ is a smooth manifold. Let $\omega$ be an $n-$ form on $M$ with compact support. Then we define $\int_M\omega$ using partitions of unity. If $M$ is covered by a single chart $h:M\to \mathbb R^n$ , then we define $\int_M\omega:= \int_{\mathbb R^n} (h^{-1})^\ast \omega$ , where $\ast$ denotes pullback. $\... | This notion is known as a "line integral" and you should have already encountered it in the special case that $M=\mathbb{R}^n$ in a lecture on analysis. Lee has an entire chapter on this concept, this exact definition is stated on p.289 (in the Second Edition). For the other question, yes, integrating over $[0,1]$ is a... | |integration|multivariable-calculus|differential-geometry|algebraic-topology|reference-request| | 0 |
How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ | I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice. $$\sum_{n=3}^ \infty \frac{2^n}{2^n n\ln(2)(\ln(n \ln(2)))^2}=\sum_{n=3}^\infty \frac{1}{n \ln(2)(\ln(n \ln(2)))^2} $$ and $$\sum_{n=3}^\... | Not a complete answer, but maybe it helps.. Here, have some derivatives: $$\frac{d}{dx}\log x = \frac{1}{x}$$ $$\frac{d}{dx}\log\log x = \frac{1}{x\log x}$$ $$\frac{d}{dx}\frac{1}{\log\log x} = \frac{-1}{x\log x(\log\log x)^2}$$ Therefore, take $$F(x) = -\frac{1}{\log\log x}$$ $$f(x) = \frac{1}{x\log x(\log\log x)^2}$$... | |real-analysis|calculus|limits|summation|numerical-methods| | 0 |
Show the linear functional relating to the solution of ODE is bounded | This is an exercise in the book by B. Daya Reddy . For $f \in L^2(0, 1)$ , let $u_f$ be the solution of the ODE: $u'' + u' - 2u = f$ , $u(0) = u(1) = 0$ . Define the functional $\ell$ by $$ \langle \ell, f \rangle = \int_0^1 u_f(x) dx \ \forall f \in L^2(0, 1) $$ Show that $\ell$ is a bounded linear functional. I have ... | Multiply your ODE by $u$ throughout, then integrate from $0$ to $1$ : \begin{align} \int_0^1u’’u +\int_0^1u’u-\int_0^12u^2&=\int_0^1fu. \end{align} Integrate by-parts on the first term to get $[u’u]_0^1-\int_0^1|u’|^2=-\int_0^1|u’|^2$ , since $u(0)=u(1)=0$ . For the second term, it can be integrated to give $\left[\fra... | |functional-analysis|ordinary-differential-equations| | 1 |
How to evaluate $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ | I saw this problem : Prove that $\sum\limits_{n=3}^ \infty \frac{1}{n \ln(n)(\ln(\ln(n)))^2}$ converges, this is an easy problem could be proved using Cauchy condensation test twice. $$\sum_{n=3}^ \infty \frac{2^n}{2^n n\ln(2)(\ln(n \ln(2)))^2}=\sum_{n=3}^\infty \frac{1}{n \ln(2)(\ln(n \ln(2)))^2} $$ and $$\sum_{n=3}^\... | In general we can approximate a sum $S=\sum_{i=m}^n f(i)$ by an integral $I=\int_m^n f(x)dx$ , more precisely for $n=\lfloor x\rfloor$ and $f(x)=1/(x\log(x)(\log\log(x))^2)$ we have $f(n)\geq f(x)\geq f(n+1)$ and integrating this inequality we obtain \begin{align} 0&\leq \sum_{n=n_0}^\infty f(n)-\int_{n_0}^\infty f(x)d... | |real-analysis|calculus|limits|summation|numerical-methods| | 0 |
If $R$ is a ring that is finitely generated as an additive group, then R is Noetherian | Recall that a ring $R$ is called right (left) Noetherian if every right (left) ideal $I$ of $R$ is a finitely generated $R$ -module, i.e., there exists $x_1,\ldots,x_m \in I$ such that $I=x_1R+\ldots+x_mR$ (or $I=Rx_1+\ldots+Rx_m$ ). Suppose that $R$ is finitely generated as an additive group, i.e., $R=\mathbb{Z}x_1+\l... | Yes, it is true. The ring $\mathbb{Z} $ is a PID, hence noetherian. Thus an abelian group (i.e. $\mathbb{Z} $ -module) is a noetherian $\mathbb{Z} $ -module iff is finitely generated. Now, given any ring homomorphism $S\to R$ and an $R$ -module $M$ , if $M$ is noetherian as an $S$ -module, so it is as an $R$ -module. T... | |modules| | 0 |
Limit of lacunar power series in $1^-$. | Let $\sigma:\mathbb{N}\longrightarrow\mathbb{N}$ be strictly increasing, and consider the power series $$ S_{\sigma}(x)=\sum_{n=0}^{+\infty}(-1)^nx^{\sigma(n)}. $$ Can any real number in $[0,1]$ be obtained as the limit $\lim\limits_{x\rightarrow 1^-}S_{\sigma}(x)$ for some $\sigma$ ? According to this answer, the limi... | My remark from MO ... $$\lim_{x \to 1^-}\sum_{k=0}^\infty \big(x^{10k}-x^{10k+3}\big) = \frac{3}{10} .$$ Similarly, get any rational in $(0,1)$ . | |real-analysis|limits|power-series|analytic-number-theory|lacunary-series| | 0 |
When is digital sum applicable and when it isn't? | I am a high school student and there is something I want to ask about the application of digital sums. Let's say there is a fraction " 520/7", let 520/7=a , so 520= a × 7 , so if we now calculate the digital sums, it would be like 7= a × 7 , it means the digital sum of a should be 1 and nothing else so it means the rem... | As explained here , modular fractions are well-defined if the denominator is coprime to the modulus, and they obey common fraction laws, e.g. $\!\bmod 9\!:\ 520\equiv 5\!+\!2\equiv 7\,\Rightarrow\, \frac{520}7\equiv \frac{7}7\equiv 1$ . If you knew only the decimal we can still use this method to compute its value modu... | |elementary-number-theory|modular-arithmetic| | 0 |
Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}$ | Let $X$ be a normed linear space over the field $\mathbb C$ with the norm $\|\cdot \|$ . Let $x,y \in X$ . Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\Vert^2$ . Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}.$ My attempt: \begin{align*} \overline{\lan... | You cannot bring the complex conjugate inside the norm as $\|x+iy\|$ is not necessarily equal $\|x-iy\|.$ Instead I would use $$i^k\|x+i^ky\|^2= i^k\|i^{-k}x+y\|^2=\overline{i^{-k}\|y+i^{-k}x\|^2}$$ On summing up the terms you get the conclusion. | |functional-analysis|normed-spaces|hilbert-spaces| | 0 |
Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}$ | Let $X$ be a normed linear space over the field $\mathbb C$ with the norm $\|\cdot \|$ . Let $x,y \in X$ . Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\Vert^2$ . Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}.$ My attempt: \begin{align*} \overline{\lan... | Since the norm is a positive real number by definition/construction, it isn't affected by the complex conjugation. Then, it is only a matter of rearranging the terms. At the end, we have : $$ \begin{align} \overline{\langle y, x \rangle} &= \overline{\frac{1}{4} \sum_{k =0}^{3} i^{k} \|y + i^kx\|^2} \\ &= \frac{1}{4} \... | |functional-analysis|normed-spaces|hilbert-spaces| | 1 |
$n^3 \equiv n^5 \pmod{12} $? | I am proving that $$5n^3 + 7n^5 \equiv 0 \pmod{12}$$ It would suffice to show $$n^3 \equiv n^5 \pmod{12}$$ How would I go about doing that? I suppose I could just go through each $n \equiv r \pmod{12}$ with $r$ from $1$ to $11$ and show that $n^3 \equiv n^5 \pmod{12}$ for each, but that would be tedious. Surely there's... | $$\begin{align}n^5-n^3&=6\binom{n+1}{3}n^2\\&=4\binom{n+1}{2}\binom{n}{2}n\end{align}$$ Since $6,4\mid n^5-n^3$ we also have that $\text{lcm}(4,6)=12\mid n^5-n^3$ | |elementary-number-theory| | 0 |
Values of Lebesgue integrable function from the integral of the product of a known function. | Let's say we have three complex absolutely integrable and square integrable functions $A,B,C\in \mathbb{L}^1(\mathbb{C})\cup \mathbb{L}^2(\mathbb{C})$ such that the following holds: $$A(y) = \int_{-\infty}^\infty{B(x)C(xy)dx}$$ If we can measure or calculate $A$ and $C$ for all values, what information (if any) could w... | There is an algorithm similar to what I was looking for in this physics paper . It uses the de-convolution theorem and polynomial series to "invert" the integral. I use the term "invert" loosely here because information is definitely lost, but for some functions it clearly produces decent approximations. | |functional-analysis|measure-theory|fourier-analysis|information-theory| | 0 |
Is '$10$' a magical number or I am missing something? | It's a hilarious witty joke that points out how every base is ' $10$ ' in its base. Like, \begin{align} 2 &= 10\ \text{(base 2)} \\ 8 &= 10\ \text{(base 8)} \end{align} My question is if whoever invented the decimal system had chosen $9$ numbers or $11$ , or whatever, would this still be applicable? I am confused - Is ... | To avoid confusion, the following somewhat cumbersome notation seems appropriate to me : Let us write $(2:1:7)_{ten}$ instead of $217$ . It means $$(\color{red}2:\color{green}1:\color{blue}7)_{ten}=\color{red}2\times ten^2+\color{green}1\times ten^1+\color{blue}7\times ten^0$$ That's base ten So let's look at base four... | |notation|number-systems| | 0 |
Solve for all $x $ such that $16\sin^{3}x -14\cos^{3}x = \sqrt[3]{\sin x\cos^{8}x + 7\cos^{9}x}$ | The original question : Find all $x$ in $\mathbb R$ such that $16\sin^3(x) -14\cos^3(x) = \sqrt[3]{\sin x\cos^8(x) + 7\cos^9(x)}$ It's a tough question I've found. I've tried using $16\tan^3(x) -14 = \sqrt[3]{\tan x + 7 }$ By inspection, $\tan x=1$ is one of the answers. but According to WA , $\tan x$ is not equal to $... | The function $f\colon \mathbb{R} \to \mathbb{R}$ , $f(a) = (2a)^3-7$ , is bijective and increasing. Since $f(a) - a = (a-1)(8a^2+8a+7)$ , the only solution of the equation $f(a) = a$ is $a=1$ ; $f(a) > a$ if $a>1$ and $f(a) if $a . Then the equations $f(f(a)) = a$ and $f(a) = a$ are equivalent. In particular, $f^{-1} (... | |algebra-precalculus|trigonometry|complex-numbers|radicals| | 0 |
What numbers can be written uniquely as a sum of two squares? | What numbers can be written uniquely as a sum of two squares? I was looking at sequence A125022 , which shows the numbers that can be uniquely written as a sum of two squares. Here are a few things that I noticed from the first numbers. We have $1$ , $2$ , $4$ , $8$ , $16$ , $32$ , $64$ , $128$ . It is then safe to ass... | Your conjecture is correct. You are missing no numbers. Any number not on your list would contain a prime power $q^e$ with $q$ of the form $4k+3$ , $e$ odd, or a prime power $p^e$ with $p$ of the form $4k+1$ with $e\ge2$ , or at least two primes $p_1$ , $p_2$ of the form $4k+1$ . In the first case $n$ cannot be written... | |number-theory|elementary-number-theory|algebraic-number-theory|diophantine-equations| | 1 |
Introduce a parameter to determine the value of $\int_0^1\frac{\log(1+x)}{1+x^2}dx$ | How could I introduce a parameter to determine the value of $\int_0^1\frac{\log(1+x)}{1+x^2}dx$ ? | You can write the integral as $$I(\alpha)=\int_0^1 \frac{\log(1+\alpha x)}{1+x^2}\,dx$$ The target integral is the case $\alpha=1$ , note that for $\alpha=0$ we find $I(0)=0$ which is going to be useful later. Now we want to differentiate $I(\alpha)$ with respect to $\alpha$ , so we write: $$I'(\alpha)=\frac{\partial}{... | |definite-integrals| | 0 |
Given $x_1^3+x_2^3+...+x_9^3=0$. Find the maximum value of $S=x_1+x_2+...+ x_9$. | Given 9 real numbers $x_1, x_2, ... , x_9\in [-1,1]$ such that $x_1^3+x_2^3+...+x_9^3=0$ . Find the maximum value of $S=x_1+x_2+...+ x_9$ . I have tried ordering the numbers from smallest to largest and then dividing the set of integers $\{x_1, x_2, ... , x_9 \}$ into two subsets of only negative numbers and only posit... | Notice that for $-1\leq x \leq 1$ there exists a number $t$ such that $\cos(t)=x$ . We have: $$ 4\cos^3(t)-3\cos(t) = \cos(3t) \geq -1\\ \Rightarrow 4x^3 - 3x \geq -1\\ \Rightarrow x\leq \frac{4x^3+1}{3}$$ Or you can prove by showing that $4x^3-3x+1\geq0$ . Applying this to our sum, we obtain: $$\sum_{i = 1}^9 {x_i} \l... | |inequality|a.m.-g.m.-inequality| | 0 |
Deriving Schwarzian Action in SYK Theory | I am trying to derive the Schwarzian action for the $q=4$ SYK model following "An introduction to the SYK model" by V. Rosenhaus. I understand that we have the solution $$G(\tau_1, \tau_2) = \frac{b {\rm sgn}(\tau_1-\tau_2)}{ J ^{2\Delta}} \frac{f^\prime (\tau_1) ^\Delta f^\prime(\tau_2)^\Delta}{\vert f(\tau_1)-f(\tau_... | Hints: Define $$ \tau~:=~\tau_+~:=~\frac{\tau_1+\tau_2}{2}, $$ $$ 2\delta~:=~\tau_{12}~:=~\tau_1-\tau_2, $$ and the Schwarzian derivative $${\rm Sch}(f(\tau),\tau)~:=~\frac{f^{\prime\prime\prime}(\tau)}{f^{\prime}(\tau)}-\frac{3}{2}\left(\frac{f^{\prime\prime\prime}(\tau)}{f^{\prime}(\tau)}\right)^2.$$ Then $$\begin{al... | |derivatives|taylor-expansion|physics|conformal-geometry|conformal-field-theory| | 0 |
Tractable formulation of a mixed integer program | Given constant matrices $A_1\in\mathbb{R}^{1\times l}$ and $A_2\in\mathbb{R}^{1\times l}$ , and constants $b_i$ , $i=1,\dots,n$ . Consider the following mixed integer program (MIP) with decision variables $c_i\in\{0,1\}$ and $X=[x_1,\dots,x_n]\in\mathbb{R}^{l\times n}$ with $x_i\in\mathbb{R}^{l}$ for $i=1,\dots,n$ . Ob... | You can linearize the problem as follows. Introduce nonnegative decision variables $y_i$ to represent $|c_i A_1 x_i|$ , change the objective to minimizing $\sum_i y_i$ , let $M_i$ be a small constant upper bound on $|A_1 x_i|$ , and impose additional linear big-M constraints \begin{align} A_1 x_i - y_i &\le M_i(1-c_i) ... | |optimization|convex-optimization|mixed-integer-programming|linearization| | 1 |
Solve $y+3=3\sqrt{(y+7)^2}$ | Solve $y+3=3\sqrt{(y+7)^2}$ $\Rightarrow y+3=3(y+7)$ $\Rightarrow y+3=3y+21$ $\Rightarrow 2y=-18$ $y=-9$ But $-9+3\ne3\sqrt{(-9+7)^2} \Rightarrow-6 \ne6$ How is this humanely possible? What's going on here??? | $\sqrt{a^2}$ is not equal to $a$ , it’s equal to $|a|$ . So $$y+3=3\sqrt{(y+7)^2}$$ $$y+3=3|y+7|$$ We see that $y+3\ge0$ , so $y+7>0$ , and $|y+7|=y+7$ . $$y+3=3(y+7)$$ $$2y=-18$$ $$y=-9$$ But $y+3\ge0$ doesn’t hold. Hence, no solutions. | |algebra-precalculus| | 1 |
Solve $y+3=3\sqrt{(y+7)^2}$ | Solve $y+3=3\sqrt{(y+7)^2}$ $\Rightarrow y+3=3(y+7)$ $\Rightarrow y+3=3y+21$ $\Rightarrow 2y=-18$ $y=-9$ But $-9+3\ne3\sqrt{(-9+7)^2} \Rightarrow-6 \ne6$ How is this humanely possible? What's going on here??? | The error that you have is very subtle. If you do not make any simplifications, the right-hand side of the equation will never be negative, that is because $\sqrt{(x+7)^2} = |x+7|$ . So actually you have a different equation that doesn't have solutions (you can check it with simple algebra) | |algebra-precalculus| | 0 |
Natural transformations of Hom-sets “transport” natural transformations from one pair of functors to another? (Reference) | Question 1: Does anyone know a name, or have a reference, for the following lemma? $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\operatorname{O... | $\newcommand{\Hom}{\operatorname{Hom}}$$\newcommand{\F}{\mathscr{F}}$$\newcommand{\G}{\mathscr{G}}$$\newcommand{\op}{\operatorname{op}}$$\newcommand{\C}{\mathscr{C}}$ $\newcommand{\Id}{\operatorname{Id}}$$\newcommand{\Ob}{\operatorname{Ob}}$ Lemma: Given functors $G_1,G_2: \C \to \G$ such that there exists a natural tr... | |reference-request|category-theory|terminology|natural-transformations|yoneda-lemma| | 0 |
Jacobian and vectorization | Given the matrix function $h(Q) = Q^{T}AQ$ . The derivative can be obtained as $$\lim_{\epsilon \to 0} \frac{h(Q - \epsilon H)-h(Q)}{\epsilon} = H^{T} A Q + Q^{T} A H$$ Then, I saw that the Jacobian $J_h(vec(Q)) = ((AQ)^{T} \otimes I)\Pi + I \otimes Q^{T} A$ . I have some issues with obtaining this identity. Notice tha... | Using the more standard $K$ $($ instead of $\Pi)$ to denote the Commutation Matrix , an uppercase $H$ to denote your $h$ matrix, lowercase letters $(h,q)$ to denote the vectorized form of the matrices $(H,Q),\,$ and using differentials instead of limits, the calculation runs as follows $$\eqalign{ \def\p{\partial} \def... | |linear-algebra|matrix-equations|matrix-calculus| | 0 |
Eigendecomposition of the direct sum of two operator on Hilbert spaces | Let the (finite dimensional) Hilbert space $\mathcal{H}$ be the direct sum of $\mathcal{H}_A$ and $\mathcal{H}_B$ . Let $A$ be a linear operator on $\mathcal{H}_A$ and $B$ be a linear operator on $\mathcal{H}_B$ . Let $A = \sum_j \lambda_j^A |\psi_j^A\rangle \langle \psi_j^A|$ and $B = \sum_j \lambda_j^B |\psi_j^B \ran... | The eigendecomposition of $A \oplus B$ is just the sum of the two eigendecompositions, that is, $$A \oplus B = \sum_j \lambda_j^A | \psi_j^A \rangle \langle \psi_j^A | + \sum_k \lambda_k^A | \psi_k^A \rangle \langle \psi_k^A |$$ Where I identified $H_A$ as a subspace of $H = H_A \oplus H_B$ by $H_A \ni h \mapsto h \opl... | |linear-algebra|operator-theory|hilbert-spaces| | 1 |
Finding value of $(a+b)^5$ using $2$ cubic equation in $a$ and $b$ | If $a,b\in\mathbb{R}$ and $\displaystyle \frac{a^3+4a}{3a^2+5}=-1.$ and $\displaystyle \frac{b^3+4b}{3b^2+5}=1$ . Then $(a+b)^5=$ What I try : From the above data, we have $\displaystyle a^3+3a^2+4a+5=0\cdots (1)$ $\displaystyle b^3-3b^2+4b-5=0\cdots (2)$ Adding both , We get $\displaystyle a^3+b^3+3(a^2-b^2)+4(a+b)=0$... | First, we can prove that there is only one real value of $a$ that satisfies $\frac{a^3+4a}{3a^2+5} = -1$ . We can see this from the cubic equation $$a^3 + 3a^2 + 4a + 5 = 0.$$ The derivative of the cubic polynomial is $3a^2 + 6a + 4 = 3(a+1)^2 + 1$ , which is always positive. This means that the cubic polynomial is alw... | |polynomials| | 0 |
Finding value of $(a+b)^5$ using $2$ cubic equation in $a$ and $b$ | If $a,b\in\mathbb{R}$ and $\displaystyle \frac{a^3+4a}{3a^2+5}=-1.$ and $\displaystyle \frac{b^3+4b}{3b^2+5}=1$ . Then $(a+b)^5=$ What I try : From the above data, we have $\displaystyle a^3+3a^2+4a+5=0\cdots (1)$ $\displaystyle b^3-3b^2+4b-5=0\cdots (2)$ Adding both , We get $\displaystyle a^3+b^3+3(a^2-b^2)+4(a+b)=0$... | Observe that $$a^2-ab+b^2+3a-3b+4=\left(\frac{a+b}{2}\right)^2+3\left(1+\frac{a-b}{2}\right)^2+1$$ | |polynomials| | 1 |
Detail in standard measure theory I cannot seem to obtain | There is a standard result in measure/integration theory which I just cannot seem to obtain. If $f \colon X \to \mathbb{C}$ is measurable ( $X$ is any measurable space), there exist simple measurable functions $\phi_k \colon X \to \mathbb{C}$ such that $\phi_k \to f$ pointwise and all $|\phi_k| \le |f|$ . This is fine.... | You need to also use the fact that these approximations are subordinate to the positive and negative parts of $u,v$ . Here’s a more precise version of what I mean: Lemma Let $u:X\to[-\infty,\infty]$ be a given function and $u^+,u^-$ its positive, negative parts. Suppose $0\leq s_1\leq u^+$ and $0\leq s_2\leq u^-$ are g... | |integration|measure-theory|proof-explanation|simple-functions| | 0 |
Prove Localization in CRing is Epimorphism | Give a commutative ring $R$ and a multiplicative subset $S$ of $A$ , we have the normal localization map $\lambda_S: R \rightarrow S^{-1}R$ . How does one prove that this is an epimorphism? So given some $f, g: S^{-1}R \rightarrow C$ such that $f \circ \lambda_S = g \circ \lambda_S$ , I can see that $f$ and $g$ agree o... | How about lets just use universal property of localization. Lets denote $k = f\lambda = g\lambda : A\to C$ , then we can verify that $k$ maps every thing in $S$ to invertible element in $C$ . Basically for any $s\in S$ , $\lambda(s)$ is invertible in $S^{-1}A$ , so there is $t\in S^{-1}A$ , such that $\lambda(s)t = 1$ ... | |abstract-algebra|ring-theory|category-theory| | 0 |
How many nonnegative integers $x_1, x_2, x_3, x_4$ satisfy $2x_1 + x_2 + x_3 + x_4 = n$? | Can anyone give some hints about the following question? How many nonnegative integers $x_1, x_2, x_3, x_4$ satisfy $2x_1 + x_2 + x_3 + x_4 = n$ ? Normally this kind of question uses stars and bars but there are $2x_1$ , which I don’t know how to handle. Help please! Ps :I think may be we can use recurrence relation. | I can’t post my ask on new thread. Problem 1. Let be $a$ is positive integer $(a\le 5)$ . Define $\|1,1,1,a;n\|$ are Number of non-negative integer solutions of equation $\quad x_1+x_2+x_3+ax_4=n$ Proof $\|1,1,1,a;n\|=\left\lfloor\dfrac{(n+2)(n+a+2)(2n+a+1)}{12a}\right\rfloor$ Problem2. Applies result on problem 1, Cou... | |combinatorics|recurrence-relations| | 0 |
Confusion in applying the Implicit function theorem | Consider the following equations $$\begin{cases} 2(x^2+y^2)-z^2=0\\ x+y+z-2=0\end{cases}$$ Prove that the above system of equations defines a unique function $\phi: z\mapsto (x(z),y(z))$ , from a neighborhood of $z=2$ to a neighbor hood $V$ of $(1,-1)$ and $\phi\in C^1$ on $U$ . My idea is to use The implicit function ... | No, you must check that the $2\times 2$ Jacobian matrix $\dfrac{\partial (F,G)}{\partial (x,y)}$ is invertible at the given point $(1,-1,2)$ . With regard to your final question, read the statement of the Implicit Function Theorem very carefully. What specific question do you have? | |real-analysis|calculus|multivariable-calculus|implicit-function-theorem| | 0 |
Does $A/B \cong C/D$ and $B \cong D$ imply $A \cong C$? | Say that for some group $A$ who has a normal subgroup $B$ , and for some group $C$ who has a normal subgroup $D$ , we know that $A/B$ is isomorphic to $C/D$ and that $B$ is isomorphic to $D$ . Is $A$ necessarily isomorphic to $C$ ? EDIT: What if there is a homomorphism $\sigma: A \to C$ ? | This is one everyone knows: $n\neq m\implies \Bbb Z/n\Bbb Z\not \cong \Bbb Z/m\Bbb Z, $ but $n\Bbb Z\cong m\Bbb Z.$ But you still see the mistake all the time. | |group-theory|normal-subgroups|group-isomorphism|quotient-group| | 0 |
Technique for generating Lie point symmetries | Consider I believe that there is something wrong with this text. In particular, how is $$\Delta=0 \quad \Longrightarrow \quad V(\Delta)=0$$ completely non trivial by linearity of operators? Moreover, I do not understand how this is used to find lie point symmetries. More context For context, the text also mentions so I... | The implication $\Delta = 0 \Longrightarrow V(\Delta) = 0$ is not trivially true, despite the linearity of $V$ , because the object $\Delta$ has to be understood as a formal expression representing the differential equation for $u$ itself (when set to zero). As an element of the tangent space, the vector field $V$ is d... | |ordinary-differential-equations|partial-differential-equations|lie-algebras|integrable-systems| | 0 |
Proving my IVP for a Piecewise Decay Function (Diff Eq) | Setup So... I kinda handled most of my proof but I need help with some of the stuff I just kinda went with until it worked out. The problem relates to medicine and its decay in the body. We are given that the medicine will release over a period of $b$ hours and another dose is given at time $T$ . Known Values Decay Con... | Introducing some formalism, calling the unit step as $\unicode{x1D7D9}(t)$ we have that the medicine delivery is made as $$ g(t) = \frac 1b \sum_{k=0}^n \left(\unicode{x1D7D9}(t-kT)-\unicode{x1D7D9}(t-kT-b)\right) $$ so the differential relationship reads $$ y'(t) = g(t) - \gamma y(t) $$ with Laplace transform $$ (s+\g... | |ordinary-differential-equations|solution-verification|exponential-function| | 0 |
Proving Euler product related to Riemann zeta function | Let $\omega(n)$ denote the number of prime factors of a positive integer $n$ . Prove that \begin{equation}\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}\end{equation} My attempt: note that $2^{\omega(n)}$ is multiplicative, since $\omega(n)$ is clearly additive. Therefore the Dirichlet series... | As commenters have pointed out, "the number of prime factors of $n$ " can be ambiguous: does it mean the number of distinct prime factors, or the number of prime factors counted with multiplicity. It turns out that it's currently standard in analytic number theory to use $\omega(n)$ to denote the number of distinct pri... | |real-analysis|number-theory|elementary-number-theory|analytic-number-theory| | 1 |
Prove that $a * b = a + b - ab$ defines a group operation on $\Bbb R \setminus \{1\}$ | So, basically I'm taking an intro into proofs class, and we're given homework to prove something in abstract algebra. Being one that hasn't yet taken an abstract algebra course I really don't know if what I'm doing is correct here. Prove: The set $\mathbb{R} \backslash \left\{ 1 \right\}$ is a group under the operation... | First and foremost you have to prove that " $*$ " is well defined, namely that: $$a,b\ne1\Longrightarrow a+b-ab\ne1$$ or, equivalently, that: $$a+b-ab=1\Longrightarrow (a=1)\vee(b=1)$$ And this is indeed the case, as: $$a+b-ab=1\iff a(1-b)=1-b$$ which splits into the two cases: $b=1\wedge a\in\mathbb R$ , or (exclusive... | |abstract-algebra|group-theory|solution-verification|proof-writing| | 0 |
Give an interpretation where a predicate logic formula is true | Give an interpretation where $$∃x(\neg P(x) ∨ Q(x)) \to (∃xP(x) ∧ ∀x\neg Q(x))$$ is false. How does someone even begin with questions like this? I have interpreted it in my head and I kind of get it in a sense. But seems like the only thing I know is that since it is an implication, the only way it will be false is if ... | The main strategy for finding a counter-model for a sentence in FOL is similar to the strategy you use to factor equations in algebra. You’re looking for a way to show a formula is falsifiable similarly to how you look for numbers and roots of polynomials that produce a polynomial expression. It takes practice to learn... | |logic|propositional-calculus|computer-science|first-order-logic| | 1 |
Exercise 6, Section 47 of Munkres’ Elements of Algebraic Topology | I've been reading through the section on cohomology of Munkres' elements of algebraic topology book and I'm having some problems trying to solve one exercise (Exercise 6 of section 47). The exercise asks to compute the cohomology of the 5-fold dunce cap $X$ (which is defined as a pentagon with all its sides identified)... | Use the most obvious triangulation. I have left edges unlabelled for legibility; there are two vertices, $v$ and the central $w$ , five faces $\sigma_\bullet$ and five unmarked edges which run $v\to w$ , where $e_1$ is thought to be the base (the " $d_2$ " face) of $\sigma_1$ , etc. and a sixth unmarked edge $e_0$ whic... | |algebraic-topology|homology-cohomology| | 0 |
Why is the difference of consecutive primes from Fibonacci sequence divisible by $4$? | The primes represented in the Fibonacci sequence are written in the form $6n + 1$ and $6n -1$ , respectively. $$5=6\times1-1$$ $$13=6\times2+1$$ $$89=6\times15-1$$ $$233=6\times39-1$$ $$1597=6\times266+1$$ $$28657=6\times4776+1$$ $$514229=6\times85705-1$$ $$433494437=6\times 72249073-1$$ $$2971215073=6\times495202512+1... | To expand on the comments: All Fibonacci primes $>3$ are of the form $4k+1$ . To prove this, suppose it were otherwise. That is, suppose we had an index $n$ for which $F_n\equiv 3 \pmod 4$ . We wish to show that $F_n$ is composite. But, The sequence $\{F_n\}$ is periodic $\pmod 4$ with cycle of length $6$ . $\{0,1,1,2,... | |sequences-and-series|prime-numbers|fibonacci-numbers|difference-sets| | 0 |
Prove that sum of integrals $= n$ for argument $n \in \mathbb{N}_{>1}$ | ORIGINAL QUESTION (UPDATED): I have a function $f:\mathbb{R} \rightarrow \mathbb{R}$ containing an integral that involves the floor function: $$f(x):= - \lfloor x \rfloor \int_1^x \lfloor t \rfloor x \left( -\pi t x \left( \frac{\cos \left(\pi (1-t) x \right)}{\pi (1-t) x} - \frac{\sin \left(\pi (1-t) x \right)}{\pi^2 ... | Let $I$ be the required integral. First, I assume that $\operatorname{sinc}$ mentioned in $I$ is $\sin$ . Second, I think your approach is overcomplicated, because $I=\lim_{\epsilon\to 0+}\int_{1+\epsilon}^x \dots$ , and the integrals $\int_{1+\epsilon}^x\dots $ are proper. Third, in the final sum all summands are cont... | |real-analysis|integration|summation|indefinite-integrals|ceiling-and-floor-functions| | 1 |
Using Euclidean geometry, how to find $x$? | This question comes from a friend exam that I'm helping to review. I've been trying hard but can't find the answer. Using Euclidean geometry, how to find the angle $x$ ? I've been able to work out all the angles based on $x$ and $180^{\circ} $ , but then I got stuck. Here's my calculation: I named the center point as $... | As already shown, $BD=AB$ , also since $\widehat{DAC}=\widehat{ACB}=x$ we infer $BC\parallel AD$ , hence $\widehat{DBC}=\widehat{ADB}=3x$ . Take $E$ reflection of $D$ about $BC$ , it belongs to $AB$ and $BE=BD$ , thus, since $AB=BD=BE$ , $\widehat{ADE}=90^\circ$ , thus $\widehat{CDE}=\widehat{CED}=90^\circ-2x$ , thence... | |geometry|euclidean-geometry|triangles| | 0 |
Are "infinitesimal rotations" commutative? If so, which mathematical fact allows it? | I was reading Moysés Nussenzveig's "Basic Physics Course 1" when I came across this excerpt in chapter 11, about rotations and angular momentum, in section 11.2, vector representation of rotations: We could then think about associating a vector “θ” to a rotation through the angle θ, the direction of this vector being g... | Rotations in $\Bbb{R}^3$ are element of the Lie group $SO(3)$ . Ultimately, it is possible to write a rotation $R_\theta$ as $R_\theta = e^{\theta L}$ , where $L$ is the generator of that rotation of angle $\theta$ . As a side note, $L$ is an element of the Lie algebra associated to $SO(3)$ and can be interpreted physi... | |lie-groups|lie-algebras| | 0 |
The function $\log^+x=\max\{1, \log x\}$. | I was reading Marcinkiewicz-Zygmund (MZ) law of large numbers for random fields and came across necessary and sufficient condition $E(|X|\log^+|X|) for MZ-SSLN to hold true. I have a question about this function $\log^+|X|$ . Why don’t they just need condition without this max, that is, $E(|X|\log|X|) ? | @Shyam, you were right after all! Its unnecessary to state that for $\log^+$ instead of $\log$ , as it can be shown easily that $$ \operatorname{E}[|X|\log^+|X|] The reason is that the function $x\mapsto x\log x$ is bounded in $(0,1)$ and it can be continuously extended to $[0,1)$ . Let $Y:=|X|$ then note that $$ \oper... | |probability|probability-limit-theorems|law-of-large-numbers| | 1 |
$x, y \in \mathbb{N} \setminus \{0\}.$ Find the smallest value of $P = |36^x - 5^y|.$ | $x, y \in \mathbb{N} \setminus \{0\}.$ Find the smallest value of $P = |36^x - 5^y|.$ Here's my attempt using calculus: Fix $x. P'(y) = 0 \iff y = y_{0} = x\log_5 36 \approx 2.23x.$ Also, $P'(y) > 0$ for $y > y_{0}$ and $P'(y) for $y Since $y \in \mathbb{N},$ $P$ reaches its smallest value when $y = 2x$ or $y = 3x$ - t... | Let's look at $|36^x-5^y|\pmod{180}$ . I chose $180$ because $36\cdot 5=180$ , so it seemed like an interesting modulus to try. Notice that $36^2\equiv 36\pmod{180}$ , so by induction, we can show $36^x\equiv 36\pmod{180}$ for all $x\geq 1$ . Ergo, the expression is equivalent to one of the following: $$ (36-5^y)\pmod{... | |elementary-number-theory| | 0 |
Proving Density for Function Approximation with Hidden Layer Perceptron | I'm working on a problem related to function approximation within the $L^2\left(I_n\right)$ space of square-integrable functions: Problem Statement: Given a lemma without proof: $\textit{Lemma}$ : Let $g \in L^2\left(I_n\right)$ such that $\int_{\mathcal{H}} g(x) d x=0$ , for any half-space $\mathcal{H}:=\left\{x: w^T ... | Here is a fully detailed proof based on your given Lemma, which I will restate below (Lemma) : Let $g \in L^2\left(I_n\right)$ such that $\int_{\mathcal{H}} g(x) d x=0$ , for any half-space $\mathcal{H}:=\left\{x: w^T x+\theta>\right.$ $0\} \cap I_n$ . Then $g=0$ almost everywhere. You didn't define it, but assuming th... | |functional-analysis|measure-theory|hilbert-spaces|approximation-theory|neural-networks| | 1 |
explanation required for the logic of a proof step regarding set membership, conjunction, and implication | This question is asking for an explanation of a step in the following segment of someone else's proof of a textbook exercise regarding set membership, conjunction and implication. Consider the following: $$ (x \in A \land y \in B) \implies (x \in C\land y \in D) $$ Let me check I understand the meaning. It says that if... | I think I understand where your confusion lies. In short, you are not paying attention to the fact that $\,x\,$ and $\,y\,$ are arbitrary. I expound below. To get your answer, first consider a different scenario than yours. Imagine that $\,x=y\,$ . In this scenario, your statement would be $$ \forall x,\qquad x\in A\ \... | |elementary-set-theory|logic| | 0 |
$x, y \in \mathbb{N} \setminus \{0\}.$ Find the smallest value of $P = |36^x - 5^y|.$ | $x, y \in \mathbb{N} \setminus \{0\}.$ Find the smallest value of $P = |36^x - 5^y|.$ Here's my attempt using calculus: Fix $x. P'(y) = 0 \iff y = y_{0} = x\log_5 36 \approx 2.23x.$ Also, $P'(y) > 0$ for $y > y_{0}$ and $P'(y) for $y Since $y \in \mathbb{N},$ $P$ reaches its smallest value when $y = 2x$ or $y = 3x$ - t... | You have $$|36^x - 5^y|=|6^{2x}-5^y|=|(5+1)^{2x}-5^y|=\left|5^{2x}-5^y+\sum^{k=2x}_{k=1}\binom{2x}{k}5^{2x-k}\right|$$ It is clear that $(x,y)=(1,2)$ gives the minimum $11$ . | |elementary-number-theory| | 0 |
A game on a rectangular board | Setup Let there be a board looking like a rectangular table. A piece is placed at any square of the board. Two players play a game. They move the piece in turns. The piece can only be moved to an adjacent square (no diagonal moves). The piece can’t be moved to a square that it has already visited (the starting square c... | Yes, the first player wins on an odd x odd board if the knight starts on gray square (where the board is colored like a checkerboard such that the corners are blue). To describe the first player's winning strategy, tile almost all of the board with dominoes, where only one corner is uncovered. The first move will be to... | |combinatorial-game-theory| | 1 |
Are "infinitesimal rotations" commutative? If so, which mathematical fact allows it? | I was reading Moysés Nussenzveig's "Basic Physics Course 1" when I came across this excerpt in chapter 11, about rotations and angular momentum, in section 11.2, vector representation of rotations: We could then think about associating a vector “θ” to a rotation through the angle θ, the direction of this vector being g... | This answer is a little loose in terms of what an infinitesimal means, but I think it provides some intuition. Consider any set of matrices that are really close to identity. In other words, each matrix can be written as $I+\delta_A A$ . If you multiply two of these together, you get: $$ \begin{align} &(I+\delta_A A)(I... | |lie-groups|lie-algebras| | 0 |
$x, y \in \mathbb{N} \setminus \{0\}.$ Find the smallest value of $P = |36^x - 5^y|.$ | $x, y \in \mathbb{N} \setminus \{0\}.$ Find the smallest value of $P = |36^x - 5^y|.$ Here's my attempt using calculus: Fix $x. P'(y) = 0 \iff y = y_{0} = x\log_5 36 \approx 2.23x.$ Also, $P'(y) > 0$ for $y > y_{0}$ and $P'(y) for $y Since $y \in \mathbb{N},$ $P$ reaches its smallest value when $y = 2x$ or $y = 3x$ - t... | Inspired by Nobie Mushtak's solution... We have $$P\equiv\pm 1\pmod2$$ $$P\equiv\pm 1\pmod3$$ $$P\equiv\pm 1\pmod5$$ The smallest such $P$ is $11$ , which occurs when $x=1$ , $y=2.$ $P=1$ is not possible due to Mihailescu's theorem. | |elementary-number-theory| | 0 |
Definition of Schubert Variety | Let $V$ be a full flag, $\lambda$ a partition. Consider $$\sigma_\lambda(V) = \{ \Lambda \in G(k,n): \Lambda \cap V_{n-k+i-\lambda_i} \geq i \}.$$ If you have another full flag $V'$ , are $\sigma_\lambda(V)$ and $\sigma_\lambda(V')$ isomorphic to each other? It seems that in intersection theory, they only care about th... | Yes, $GL_n$ acts on the Grassmannian by linear change of coordinates, which changes the choice of auxiliary flag. For intersection theory, it also matters that $GL_n$ is rationally connected. That is, the two subvarieties are not only isomorphic but rationally equivalent (essentially the algebraic version of being homo... | |algebraic-geometry|intersection-theory|schubert-calculus| | 0 |
Is every extreme point in a compact convex set contained in a defining supporting hyperplane? | Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$ . Let $k \in K$ be an extreme point. Question 1: Does there exist a supporting hyperplane of $X$ containing $k$ ? I think the answer is “yes” via some Hahn-Banach argument, although I’m a little confused about this at the moment. But what I re... | The answer to question 2 is: "no" even in $X = \mathbb R$ . Indeed, $$ [0,\infty) = \bigcap_{n \in \mathbb N} [-1/n, \infty) $$ and the extreme point $0$ is not a boundary point of any $[-1/n, \infty)$ . After the edit, we need at least two dimensions, I guess: Take $X = \mathbb R^2$ and $$ X = \bigcap_{q \in \mathbb Q... | |functional-analysis|convex-analysis|compactness|convex-geometry|locally-convex-spaces| | 1 |
Why is the difference of consecutive primes from Fibonacci sequence divisible by $4$? | The primes represented in the Fibonacci sequence are written in the form $6n + 1$ and $6n -1$ , respectively. $$5=6\times1-1$$ $$13=6\times2+1$$ $$89=6\times15-1$$ $$233=6\times39-1$$ $$1597=6\times266+1$$ $$28657=6\times4776+1$$ $$514229=6\times85705-1$$ $$433494437=6\times 72249073-1$$ $$2971215073=6\times495202512+1... | There are plenty of relations with Fibonnaci numbers , for example $$\forall n\geq 1,F_{2n}=F_{n+1}F_n+F_nF_{n-1}=F_n(F_{n+1}+F_{n-1})$$ which explain @lulu's answer. From his response, we know that $F_{2n}$ is not prime; The remainders when you do Euclidean division by 4 are $$0,1,1,2,3,1,\color{red}{0,1,1,2,3,1,},0,1... | |sequences-and-series|prime-numbers|fibonacci-numbers|difference-sets| | 0 |
Non-Abelian groups exact sequences, right split and left-split are different? | I am learning about exact sequences that split, in the context of modules. In this context, as I understand it, sequences that split on the left are the same as sequences that split on the right. But in non Abelian groups, is there an easy example of an exact sequence $1 \rightarrow G \rightarrow H \rightarrow K \right... | In the full category of groups, $\bf Grp,$ left and right split aren't equivalent. If it's left split then it's right split, and then a direct product. There's a little example on Wikipedia of one that is right but not left split, with $S_3$ and $A_3.$ But, any semi-direct product that is not a direct product should wo... | |abstract-algebra|exact-sequence| | 0 |
How to find multiple solutions for 3 variable, 2 degree Diophantine equation? | I have the equation $x^2+xy+y^2=z^2$ to solve it in natural numbers. The discriminant of it $D=4z^2-3y^2$ and must be perfect square. I wrote Python program to get solutions for $1 by enumeration. def Solution(): A=[] nMaximum=10**2 for x in range(1,nMaximum): dTemp1a=3*x**2 for z in range(x+1, nMaximum): dDiscriminant... | Can't fit this into a comment so I'll make it an answer. Firstly, your question isn't clear. Three variable, second degree diophantine equation doesn't explain how to get other solutions when we know the first solution $(3,5,7)$ If you are expecting to generate ALL triples from $(3,5,7)$ then I don't think this is poss... | |diophantine-equations| | 0 |
Confusion in applying the Implicit function theorem | Consider the following equations $$\begin{cases} 2(x^2+y^2)-z^2=0\\ x+y+z-2=0\end{cases}$$ Prove that the above system of equations defines a unique function $\phi: z\mapsto (x(z),y(z))$ , from a neighborhood of $z=2$ to a neighbor hood $V$ of $(1,-1)$ and $\phi\in C^1$ on $U$ . My idea is to use The implicit function ... | In this particular case you can solve the system explicitly. Namely $$ z^2+4xy=2(x+y)^2=2(2-z)^2$$ Thus $$xy={1\over 4}z^2-2z+2,\ x+y=2-z$$ By the Vieta formulas $x,y$ are solutions of the quadratic equation $$ u^2-(2-z)u+\left ({1\over 4}z^2-2z+2\right )=0$$ The discriminant is equal $4z -4.$ Therefore $$x,y ={1\over ... | |real-analysis|calculus|multivariable-calculus|implicit-function-theorem| | 0 |
Uniqueness and continuous dependence on the data of Heat equation. | Let two smooth $v_1$ and $v_2$ both satisfy the system $$\partial_t{v}-\Delta v=f \quad \text{in} \quad U \times (0,\infty), $$ $$v = g \quad \text{on} \quad \partial U \times (0,\infty),$$ for some fixed given smooth $f: \bar{U}\times (0,\infty) \rightarrow \mathbb{R}$ and $g: \partial U \times (0,\infty).$ $U$ is ope... | To fix the gap $(\star)$ in my previous answer. In my other answer, I claimed that $\Vert u(t,\cdot)\Vert_2\to 0$ as $t\to\infty$ implied that $u(t,x)\to 0$ for almost all $x\in U$ . Note that the restriction to "almost all" is necessary, because $L^p$ convergence does not imply everywhere pointwise convergence, not ev... | |analysis|partial-differential-equations|heat-equation|gronwall-type-inequality| | 0 |
Let $u$ and $w$ be complex numbers such that $|u|=5, |w|=3,$ and $|u+w|=6$. Calculate $|u+2w|$ with proof. | This came up on my homework and I don't understand how to calculate $|u+2w|$ . How do I get from $|u+w|$ to $|u+2w|$ ? I'm guessing that I have to square $|u+w|$ and then add $|u|$ and $|w|$ in a way that which would have a square root of $|u+2w|$ but I don't know how to get there. | We are given that $|u+w|=6$ . (eq. 1) On squaring eq. 1, we get $|u|^2+|w|^2+u\bar{w}+\bar{u}w=36$ $\implies u\bar{w}+\bar{u}w=36-25-9=2$ (eqn. 2) Now, $|u+2w|^2=|u|^2+4|w|^2+2(u\bar{w}+\bar{u}w)=25+36+4=65$ Hence, $|u+2w|=\sqrt{65}$ . | |algebra-precalculus|complex-numbers| | 1 |
Finding matrix to transform one vector to another vector | If I have an arbitrary vector $A = (a,b,c,0)$ how can I find a transformation matrix $M$ such that $M \times A = (0,1,0,0)$ ? We can assume $A$ has a magnitude of $1$ if it helps simplify the derivation process. The trivial case $A = (0,1,0,0)$ would cause $M$ to be the identity matrix. If $A = (0,-1,0,0)$ then $M$ wou... | Since you are mentioning Rodrigues' rotation formula, you may be interested in this alternative method: Represent A and B as quaternions: $$Q_A=a*1+b*i+c*j\\ Q_B=1*i.$$ Now compute the transformation quaternion $Q_R$ = $Q_B/Q_A$ by using Hamilton's product, getting $Q_R*Q_A=Q_B$ . Next you take (one of the) real 4*4 ma... | |matrices| | 0 |
Derive $\sin x$ expansion without using calculus | We know that $$\sin x = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \cdots = \sum_{n\ge 0} \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ But how could we derive this without calculus? There were some approach using $e^{ix} = \cos x + i \sin x$ , pls notice I also would like to avoid such definition as to prove $e^{ix} = \cos x + i \sin... | "Without calculus" is a pretty big ask, given that the modern definition of sine is the power series representation you gave, which is the result of analysis. It arises from solving the differential equation $f(x)=-f''(x), f(0)=0, f'(0)=1$ . I think the best anyone will be able to do is to show you how to recover the p... | |sequences-and-series|algebra-precalculus| | 0 |
why is this associative? | I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already The fourth exercise asks me to determine whether the following operation is compliant with the associative principle: $$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$ The answer says that it is, because: $$(αγ − βδ)ε − (αδ + βγ)ϕ,(αγ − β... | It's best to look at each component, one at a time. I will also use English letters. The operation is defined as $$(a,b) \cdot (c,d) = (ac - bd, ad + bc).$$ Then $$\bigl((a,b) \cdot (c,d)\bigr) \cdot (e,f) = (ac - bd, ad + bc) \cdot (e,f).$$ The first component is $$(ac-bd)e - (ad+bc)f = ace-bde-adf-bcf. \tag{1}$$ The ... | |associativity| | 1 |
Why does this multiplication trick work | Why does this multiplication trick works-- I sort of discovered it on my own. 985 x 974: (1000 x 1000) - (15 x 1000 + 26 x 1000) + (-15)(-26) 997 x 989: (1000 x 1000) - (3 x 1000 + 11 x 1000) + (-3)(-11) 1003 x 976: (1000 x 1000) - (-3 x 1000 + 24 x 1000) + (3)(-24) 1005 x 1007: (1000 x 1000) - (-5 x 1000 - 7 x 1000) +... | You are using the distributive property, sometimes called FOIL ("first outers inners last"): $$(a+b)\cdot (c+d) = a\cdot c + b\cdot c + a\cdot d + b\cdot d.$$ If $b$ and/or $d$ happen to be negative, then you get your earlier results. You can understand why this works by going back to first principles with multiplicati... | |algebra-precalculus| | 1 |
Radius of Convergence of Laurent Series Confusion | Determine the largest number $R$ such that the Laurent series of $$f(z)= \dfrac{2sin(z)}{z^2-4} + \dfrac{cos(z)}{z-3i}$$ about $z=-2$ converges for $0 ? I know the maclaurin series for sine and cosine which are valid for all complex numbers. For $\frac{1}{z^2-4} = -\frac{0.25}{z+2} + \frac{0.25}{z-2} = \frac{-0.25}{z+2... | The other poles are at $2,3i.$ So compute the distances from $-2.$ We get $4,\sqrt {13},$ the smaller one being $\sqrt {13}.$ Since the poles of a meromorphic function are isolated, we get $\sqrt {13}.$ Or an annulus $0\lt\mid z+2\mid \lt\sqrt {13}.$ | |sequences-and-series|complex-analysis|analysis|taylor-expansion| | 1 |
But what is with the other cyclic groups? Doesn't one also have to consider them? | I'm currently reading a textbook about abstract algebra. There is a proof that every subgroup of a cyclic group is cyclic. This proof is using the fact as every proof I have found on the Internet that all cyclic groups have the form $ \langle a\rangle=\{a^n\}$ , where $n \in \mathbb{Z}$ . But I don't think that this is... | It's just notation. For each group $(G,\ast)$ (for any binary operation $\ast$ that defines a group on the set $G$ ), we may write the set $G$ under concatenation (which is the fancy term for putting symbols next to each other and it does not always denote multiplication ), via the inclusion map $\iota(g)=g$ because $$... | |abstract-algebra|group-theory|notation|proof-explanation|cyclic-groups| | 0 |
If $x,y\in\mathbb{N},\varepsilon>0$ then are there infinitely many positive integer pairs $(n,m)$ s.t. $\vert\frac{x^n}{y^m}- 1\vert < \varepsilon?$ | Proposition: If $x,y\in\mathbb{N}_{\geq2}$ then for any $\varepsilon>0,$ there are infinitely many pairs of positive integers $(n,m)$ such that $$\frac{\left\lvert y^m-x^n \right\rvert}{y^m} i.e. $\displaystyle\large{\frac{x^n}{y^m}} \to 1\ $ as these pairs $(n,m) \to (\infty,\infty).$ I think this is true, and I want ... | Since $$ \left|\,e^x-1\,\right|\le\frac{|x|}{1-|x|}\tag1 $$ if $|n\log(x)-m\log(y)|\le\frac\epsilon{1+\epsilon}$ , then $$ \begin{align} \left|\,\frac{x^n}{y^m}-1\,\right| &=\left|\,e^{n\log(x)-m\log(y)}-1\,\right|\tag{2a}\\ &\le\frac{|n\log(x)-m\log(y)|}{1-|n\log(x)-m\log(y)|}\tag{2b}\\[3pt] &\le\epsilon\tag{2c} \end{... | |number-theory|diophantine-approximation| | 0 |
find condition on $a(n)$ such that the $\lim_{n\rightarrow\infty}\frac{a(n)\cdot (n-a(n))}{\log\binom{n}{a(n)}}=+\infty$ | I met the following problem in my research but I don't know how to deal with it: What is the condition on $a(n)$ such that $$\lim_{n\rightarrow\infty}\frac{a(n)\cdot (n-a(n))}{\log\binom{n}{a(n)}}=+\infty$$ note that $a$ can be (no neccessary) function on $n$ thus $\binom{n}{a}$ is $\binom{n}{a(n)}$ Example. If $a(n)=n... | Let $s=1/n$ and $r = a/n = a \cdot s $ We have ( see ) the bound $$\frac{1}{a \cdot(n-a) }\log \binom{n}{a} \le \frac{n}{a \cdot (n-a) } H(r)= s \frac{H(r)}{r \cdot (1-r)} \tag 1$$ where $H(r)= -r \log r -(1-r)\log (1-r)$ is the binary entropy function. The bound is quite tight (we might add the corresponding lower bou... | |asymptotics|binomial-coefficients| | 0 |
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