Recent Questions - Mathematics Stack Exchange |
- Infinite nested radicals from Putnam exam 1953
- does A intersection B =A' intersection B mean that b is empty
- Reference of a famous theorem of Topology is required
- Help with finding reason behind weighting factor
- beginner multivariable calculus
- Let n be natural numbers. Show that if √ n is rational, then n = a^2 for an integer a
- Solving a system of polynomials over the third roots of unity
- Congruence of 3 Equilateral Triangles
- implementation of a b-spline curve n points
- Bernoulli distribution Estimation
- Given $n$ points in the domain(graph) of a function, what is the simplest function constructible?
- Quartic equations having no real roots
- The set of all rational numbers in $[0,1]$ is countable [closed]
- Hyperbolic identities proof
- Proof of existence of a lift in a fiber bundle
- Prove $\forall m,n,p\in \mathbb{Z}:m=p\implies m+n=p+n$
- There is always and $x$ such that $|T'(x)| > 1$
- Definition of bounds
- time-dependent inflection points
- Let A and B be sets. Show that A + B - C = (A - C) + (B - C)
- How to write 1 tuple as sets?
- The domain of a function for a series of functions
- Disk glued to a Torus. Classifying the resulting spaces - How/Where to begin?
- Show that $f:\mathbb R^n\to \mathbb R$ defined as $f(x)=\inf_{y\in K}||x-y||$ continuous on $\mathbb R^n$
- On the intuitive explanation for the discriminant property of conics
- Complex number related problem
- Self-adjoint extension of closed symmetric operator
- Second derivatives, Hamilton and tangent bundle of tangent bundle TTM
- How to solve this differential equation involving a polynomial function?
- Negative & Positive Shear Factor
| Infinite nested radicals from Putnam exam 1953 Posted: 28 Aug 2021 08:58 PM PDT Prove that the following sequence is convergent and find the limit: $$\sqrt{7}, \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7+\sqrt{7}}}, ... $$ with $x_{n+2}=\sqrt{7-\sqrt{7+x_{n}}}$. Notice that $$2=\sqrt{7-3}=\sqrt{7-\sqrt{7+2}}=\sqrt{7-\sqrt{7+\sqrt{7-\sqrt{7+2}}}} $$ $$=\sqrt{7-\sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-\sqrt{7+...}}}}}}$$ Is this proof valid? Since this is a contest problem and the trick is well known (but not at that time i guess)  | ||||||||||||||||||||||||||||||||
| does A intersection B =A' intersection B mean that b is empty Posted: 28 Aug 2021 08:58 PM PDT if $$A \cap B =A' \cap B$$ then can it be said that B is an empty set  | ||||||||||||||||||||||||||||||||
| Reference of a famous theorem of Topology is required Posted: 28 Aug 2021 08:57 PM PDT The following is attributed to Hausdorff by a Professor of mine. I need to confirm the reference, in order incorporate in a book I am writing: "A metric space is compact if and only if its complete and totally bounded."  | ||||||||||||||||||||||||||||||||
| Help with finding reason behind weighting factor Posted: 28 Aug 2021 08:54 PM PDT I am trying find a solution to figure out a formula for how a specific weighting factor was decided. I am trying to determine a rarity ranking amongst the species based on traits that it has. My sample set is a total of 9971 species:
In a tool that I used, it gave the following weighted scale for each of the trait category. I was able to find the weighting scale by cross referencing my score that was calculated with the following : Rarity Score = 1 / (Number of Species with that Trait / Total Number of Species)
To further explain the data with an example, Species #1: Trait1 = Strong. There are 148 Species with the value of Strong in Trait1. Thus, with my rarity scoring, it gives a ~67.37. The tool gave a score of 46.92 (~0.696 multiplier from my score) Species #1: Trait 7 = Green and this trait is extremely rare with only 17 species with this trait. and thus my score is 586.53 . The tool gave a score of 1470.59 (~2.5 multiplier) And again, I am hoping to find out why those specific multipliers were chose for the weighting factor. I tried looking through normalization or using Standard Deviation for something but couldnt get to those number. Thanks for the help in advance!  | ||||||||||||||||||||||||||||||||
| beginner multivariable calculus Posted: 28 Aug 2021 08:50 PM PDT . Let f(x, y) = min(|x|, |y|), where min(a, b) is the minimum of a and b. (a) Sketch three level sets of f(x, y). (b) Calculate the partial derivatives fx(0, 0) and fy(0, 0). (b) Show that f is not differentiable at (0, 0)  | ||||||||||||||||||||||||||||||||
| Let n be natural numbers. Show that if √ n is rational, then n = a^2 for an integer a Posted: 28 Aug 2021 08:47 PM PDT Let n be natural numbers. Show that if √ n is rational, then n = a^2 for an integer a  | ||||||||||||||||||||||||||||||||
| Solving a system of polynomials over the third roots of unity Posted: 28 Aug 2021 08:35 PM PDT So, given this function: $$L(A) = \sum_{IJKLM} \alpha^J_L \beta^K_M \gamma^{ILM} A_I A_J A_K$$ We desire to find the matrices $\alpha$ and $\beta$, where all elements of $\alpha$, $\beta$ and $\gamma$ are $\in \{0, 1, w, w^2\}$ and $\gamma$ is already fully known. Where $w$ is the principle root of unity and hence has the properties that $w^3 = 1$ and $1 + w + w^2 = 0$. I,J,K,L are all over the same set (with 10 elements), and M is a set with 6 elements (19 if I made a false assumption). The 10 element set is: {(0,0), (1,1), (2,2), (3,3), (0,1), (0,2), (0,3), (2,3), (1,3), (1,2)} The 6 element set is: {(0,0,1,1), (0,0,2,2), (0,0,3,3), (1,1,2,2), (1,1,3,3), (2,2,3,3)} $\gamma$ has the property that if an index is repeated more than 2 times it is zero. Otherwise it is $w$ raised to power $(c\mod 3)$, where $c$ is the number of swaps required to sort the indices in ascending order. Essentially a brother to the standard Levi-Civita symbol. Next, we define this polynomial: $$J_3(a,b,c) = (a + b + c)(a + wb + w^2c)(a + w^2b + wc) = a^3 + b^3 + c^3 - 3abc$$ If we set: $$F = [V_0, V_1, V_2, V_3, X_1, X_2, X_3, Y_1, Y_2, Y_3]$$ Then the current working assumption is that we require: $$L(F) = J_3(V_0, V_1, X_1) + J_3(V_0, V_2, X_2) + J_3(V_0, V_3, X_3) + J_3(V_2, V_3, Y_1) + J_3(V_1, V_3, Y_2) + J_3(V_1, V_2, Y_3)$$ There may be more non-zero terms in $L(F)$, I'm not 100% certain, but it is unlikely. If there are more terms, they must also use $J_3()$. Now, the challenge is to find the $\alpha$ and $\beta$ matrices. I think I have correctly narrowed it down from 160 to a total of 60 free parameters $\in \{1, w, w^2\}$ by finding the zero terms. And I have working code in SageMath/Jupyter/sympy that implements $L(A)$, and normalizes the roots of unity terms, including applying $1 + w + w^2 = 0$. Using sympy, and isolating the coefficients of terms such as $V_0^3$ or $V_1^3$ or $V_0 V_1 X_1$ or $V_0 X_1^2$ or $V_0^2 V_1$ then gives me a large collection of equations of form: $$a_{01} b_{05} + a_{02} b_{04} w + a_{03} b_{03} w^{2} = 3$$ $$a_{10} b_{15} w + a_{12} b_{12} w^{2} + a_{13} b_{11} = 3$$ $$a_{04} b_{15} w + a_{14} b_{05} w + a_{40} b_{05} w + a_{41} b_{15} + w^{2} + w = - 3$$ $$a_{04} + a_{41} w^{2} + b_{05} w = 0$$ $$a_{01} b_{15} + a_{02} b_{02} w^{2} + a_{02} b_{14} w + a_{03} b_{01} + a_{03} b_{13} w^{2} + a_{12} b_{04} w + a_{13} b_{03} w^{2} + b_{05} w^{2} + b_{05} = 0$$ At which point I'm stumped. At least our equations have the right number of terms though, so a solution is at least plausible. Eg, to get $-3$, we need $6$ terms (which makes use of $w + w^2 = -1$). To get $0$ we need a multiple of $3$ terms (which makes use of $1 + w + w^2 = 0$). For reference, here are my current $\alpha$ and $\beta$ matrices:  | ||||||||||||||||||||||||||||||||
| Congruence of 3 Equilateral Triangles Posted: 28 Aug 2021 08:43 PM PDT I've been having some trouble with this question from my geometry booklet, any help would be appreciated Let ABC be an acute-angled triangle. To the outside of triangle ABC attach equilateral triangles ABD, BCE and CAF, prove that line segments AE, BF and CD all have equal length Thank you in advance! Edit: I have tried to learn the Menelau's Theorem in order to attempt this question but did not see any relevance, I have learnt that they all intersect at the same point but am unable to describe that in mathematical terms  | ||||||||||||||||||||||||||||||||
| implementation of a b-spline curve n points Posted: 28 Aug 2021 08:12 PM PDT I am trying to draw a b-spline, the code below does not work correctly. control point px [0] p and [0] appears to be at 0.0. but it should be x = 22 y = 33, that is, the curve should start from 22.33. someone who understands spline curves can tell me it's wrong. I am using ide processing to draw the curve. processing is a java ide, you can download it to test the code below.  | ||||||||||||||||||||||||||||||||
| Bernoulli distribution Estimation Posted: 28 Aug 2021 08:44 PM PDT Let X1, ..., Xn ∼ Bernoulli(p=0.5). Let Yn = max{X1, ..., Xn}.
For a Bernoulli distribution with p as 0.5, wont Yn be 0.5? Also won't E[Yn] be just Sigma(x*p(x)) which is 0.5? I have no clue on how to plot a graph for the same or the distribution. Any leads would be helpful.  | ||||||||||||||||||||||||||||||||
| Given $n$ points in the domain(graph) of a function, what is the simplest function constructible? Posted: 28 Aug 2021 08:06 PM PDT "simplest" as in utilizing the fewest operations (primary) linking the fewest other digits (secondary) preferably of fewer occurrence(s) of the variable(s) as possible, and secondarily (as a tie-breaker, if applicable) closest in form to identity function $x↦x$ ≡ $f(x)=x^1$ (linear) preferred to $x↦x^0$ (constant) preferred to (some heirachy.. perhaps other positive integer power preferred to rational power preferred to exponential,logarithmic preferred to irrational power, ..but only in the case of tie-breaking complexity#). "constructible" as in analytic (yielding an exact solution); and perhaps as a tie-breaker, more "elementary operators" would be preferred in the target function (e.g. addition preferred over trig-function preferred over tetration preferred over integration). So a univariate function of form $x↦ax^5-321$ would preferred to one of $x↦ax^3+bx^2+cx+0.7$. They don't need to be represented in base-10, though in any integer base the number of digits will monotonically increase with regularity (never decrease) as the modulus of the value increases. I'm seeking algorithm most for general-purpose mapping, i.e. $\{f_m\}:ℂ→ℂ$ (or even more general field) where $m∈ℕ_1$ and $f_1$:=top-pick (best solution), but also in how to modify it for constraints on the image and codomain, as well as implementing particular preferences (perhaps e.g. favoring a polynomial of degree between 2 and 4, even if yields "complexity#" greater than the minimum by 1 or 2). I'm also curious how to implement it to allow some wiggle-room around the points (as simple as ±$c$ for some chosen value $c$ uniformly distributed around the points, to more descriptive distributions giving preference to the discrete point correlating somehow with giving preference scheme to ordering the constructible functions), though this would shift the problem toward one of regression (curve of best-fit). A related question is deciding an algorithm to determine maximum #$m$ for up to a certain complexity-value. For an example of $n=2$ (since $n≥3$ requires more reverse-engineering irrational values for $m>1$ if requiring exact fits) occupying origin, consider ($p_k$ := $(x_k,y_k)$ for a known point on the graph of $f$): $\{p_n\}=\{(0,0),(3,364.5)\}$ ⇒ $\{f_{m≥2}\} ⊑ \{\{x↦121.5x\},\{x↦\frac{1}{6}x^7\}\}$. Notice that if the algorithm gave strict preference to fewest digits over "number of operations", then the preference ordering of the two mappings would be reversed. In reverse-construction, special constants could count as some number of digits close to 1 (say 1.5), though in allowing $m>1$ likely would require to allow a continuum of values around $p_n$.  | ||||||||||||||||||||||||||||||||
| Quartic equations having no real roots Posted: 28 Aug 2021 08:30 PM PDT Let $f(n,x)=1+\dfrac{x}{2^n}+\dfrac{x^2}{3^n}+\dfrac{x^3}{4^n}+\dfrac{x^4}{5^n}$. The value(s) of the positive integer $n$ such that $f(n, x) = 0$ has no real roots is/are
My attempt: I tried to compute the derivatives of $f(n, x)$ with respect to $x$ and found that the second derivative was strictly increasing for such values of $n$ given in the options. That gives me $f'(n,x)$ has only one real root. I am unable to observe anything about $f(n, x)$. Any constructive hint is appreciated.  | ||||||||||||||||||||||||||||||||
| The set of all rational numbers in $[0,1]$ is countable [closed] Posted: 28 Aug 2021 08:02 PM PDT Let $S$ be the set of all rational numbers in $[0,1]$. How to construct a explicit mapping between $S$ and $\Bbb N$ to show that $S$ is countbale?  | ||||||||||||||||||||||||||||||||
| Posted: 28 Aug 2021 08:37 PM PDT I am trying to prove this hyperbolic identity: $ \sinh ^{2}x= \frac{1}{2}(-1+ \cosh (2x)) $ $ \sinh ^{2}x= \frac{1}{2}(-1+ (\cosh ^{2}x-\sinh ^{2}x)) $ $ \sinh ^{2}x= \frac{1}{2}(-1+(1-\sinh ^{2}x- \sinh ^{2}x)) $ $ \sinh ^{2}x= \frac{1}{2}(-1+1-2 \sinh ^{2}x) $ $ \sinh ^{2}x= -\sinh ^{2}x $ In the last step, what is my mistake?  | ||||||||||||||||||||||||||||||||
| Proof of existence of a lift in a fiber bundle Posted: 28 Aug 2021 08:10 PM PDT I have a doubt on what seems to be a simple step on a proof I've seen, but I just can't seem to get it. Theorem. Suppose $\pi:E\rightarrow B$ is a fiber bundle with fiber $F$. For every path $a:[s_0,s_1]\rightarrow B$ and point $x_0\in E$ such that $\pi(x_0)=a(s_0)$, there exists a lift $\tilde a:[s_0,s_1]\rightarrow E$ such that $\tilde a(s_0)=x_0$. The first part of the proof I was shown goes as follows: Proof. Initially, suppose $a([s_0,s_1])$ is contained in a trivializing neighbourhood $U$ (that is, an open set $U$ such that there exists a homeomorphism $\varphi_U:U\times F\rightarrow\pi^{-1}(U)$ satisfying $\pi\circ\varphi_U={proj}_1$). Then, fixing any point $y_0\in F$, defining $\tilde a$ by $\tilde a(s)=\varphi_U(a(s),y_0)$ gives the desired lift. Now, I can't see why $\tilde a(s_0)=x_0$ as desired. I was thinking of the bundle $B=S^1$, $E=\mathbb R$, $F=\mathbb Z$ and can't see how the choice of $y_0$ would be arbitrary. You'd have to choose $y_0\in\mathbb Z$ corresponding to the "floor" in which $x_0$ is when looking at $\mathbb R$ as a coil, right? I hope my question was clear enough and thank you in advance for your attention and any help provided.  | ||||||||||||||||||||||||||||||||
| Prove $\forall m,n,p\in \mathbb{Z}:m=p\implies m+n=p+n$ Posted: 28 Aug 2021 08:46 PM PDT My question is whether proving $\forall m,n,p\in \mathbb{Z}:m=p\implies m+n=p+n$ is equivalent to prove $\forall n,p\in \mathbb{Z}:p+n=p+n$ and if it is why?  | ||||||||||||||||||||||||||||||||
| There is always and $x$ such that $|T'(x)| > 1$ Posted: 28 Aug 2021 08:18 PM PDT Let $T(x)$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$ such that $T(x) = x \iff x \in \mathbb{Z}$. Prove that there is always a set of points such that $|T'(x)| > 1$ Proof: Lets consider the interval $[n,n+1]$ where $n\in \mathbb{Z}$. By the Mean Value Theorem we know that there must exist a point $x_0 \in (n,n+1)$ such that: $$ T'(x_0) = \frac{T(n+1) - T(n)}{(n+1) - n} = \frac{n+1 - n}{n+1 - n} = 1 $$ Given that $T(x)$ has fixed points only when $x \in \mathbb{Z}$ we know that $T(x)$ is not the identity function and therefore we will have two cases:
For case 1. we can use the MVT again to show that there exist a point $x_1 \in (n,x_0)$ such that $$ T'(x_1) = \frac{T(x_0) - T(n)}{x_0-n} = \frac{T(x_0) - n}{x_0 - n} > \frac{x_0 - n}{x_0 - n} = 1 $$ For case 2. we can also use the MVT to show that there exist a point $x_2 \in (x_0, n+1)$ such that $$ T'(x_2) = \frac{T(n+1) - T(x_0)}{n+1 - x_0} = \frac{(n+1) - T(x_0)}{(n+1) - x_0} > \frac{n+1 - x_0}{n+1 - x_0} = 1 $$ This would show that there always points for which $T'(x) > 1$ and therefore $|T'(x)| > 1$. However, I would like to also show that there necessarily exist points where $T'(x) < -1$.  | ||||||||||||||||||||||||||||||||
| Posted: 28 Aug 2021 08:27 PM PDT What is the definition of bounds? I read somewhere that in the set: $\{3,5,11,20,22\}$, $3$ is a lower bound, and $22$ is an upper bound. In the same explanation it is said that $2$ is also a lower bound, but it does not mention wether $45$ is also an upper bound. Is $45$ an upper bound as well? (Im assuming the author wrote it for numbers $\in\mathbb{N}$. Does this last bit matter? In the same explanation there is an example of an human, about how tall that human is. It's noted that it can't be lesser than $0$, so $0$ is a lower bound. Suppose that this human can be negative in length, i.e. it's length is some integer defined as $x\in\mathbb{Z}$. If we have a set of human lengths, though the number of humans can vary: $\{-6,-5,-4,-3,-2,-1,1,2,3,4,5,6,7\}$ Can the lower bound be $-7$ feet tall? Would it mean that the lengths also vary?  | ||||||||||||||||||||||||||||||||
| time-dependent inflection points Posted: 28 Aug 2021 08:09 PM PDT For a function $u:\mathbb{R}\times [0,\infty)\rightarrow\mathbb{R}$, let $\bar{x}$ be an inflection point of $u_0(x):=u(x,0)$ to the right of its maximum. Further, let the inflection point evolve with time $t\mapsto\bar{x}(t)$ such that $\bar{x}(0)=\bar{x}$. Surely we then have $(u_0)_{xx}(\bar{x}(0))=u_{xx}(\bar{x}(0),0)=0$, but why is it true that $u_{xx}(\bar{x}(t),t)=0$ for all $t\geq 0$? I'm assuming this follows since $(\bar{x}(t),t)$ is also an inflection point of $u$, but I can't see why this is true. For reference, I'm trying to understand the proof of Lemma 6 from this paper: https://arxiv.org/pdf/1707.09000.pdf. It is in this proof that I encountered the claim above. I'm pretty confused about the entire idea of a time-dependent inflection point, and in particular, about going from an inflection point of a function of a single variable, i.e. an inflection point of $u_0(x)$, to an inflection point of the two-variable function $u$. I would really appreciate a clear, pedagogical answer if possible.. Thanks in advance.  | ||||||||||||||||||||||||||||||||
| Let A and B be sets. Show that A + B - C = (A - C) + (B - C) Posted: 28 Aug 2021 08:55 PM PDT can someone help me with this problem please. thank you. Let A and B be sets. Show that A + B - C = (A - C) + (B - C).  | ||||||||||||||||||||||||||||||||
| Posted: 28 Aug 2021 08:54 PM PDT A 2-tuple is $(a, b) = \{\{a\}, \{a, b\}\}$  | ||||||||||||||||||||||||||||||||
| The domain of a function for a series of functions Posted: 28 Aug 2021 08:08 PM PDT I'm here again trying to solve a problem about series of functions. I have to study the pointwise and uniform convergence of the following series:
I tried to study the pointwise convergence first, but there's a problem: I need to find the domain of the functions, so : $1+\frac{\vert\sin(x)\vert^n}{1+x^n}\gt0$ and $1+x^n\neq0$ The problem is that I can't solve the first inequality at all and the second one clearly depends on n. I'm not even sure if I should solve them in order to fin the pointwise convergence . Help please.  | ||||||||||||||||||||||||||||||||
| Disk glued to a Torus. Classifying the resulting spaces - How/Where to begin? Posted: 28 Aug 2021 08:09 PM PDT My algebraic topology professor gave me an exercise and I feel very lost. The exercise is the next:
Obs: The gluing is only at the disk's border. On a first try, I was thinking of gluing the border of the disc to a single side of the torus (>), then to two sides (maybe continuous sides (>,>>) or inverting the orientation of the first gluing(>,<)), etc.
After a week, I gave up and asked my prof for hints. He said that could be useful to think in $\mathbb{R}^2/\mathbb{Z}^2$ and he also said that at some point I would be working on an algebra problem instead of a topology problem. To do the classification of the spaces, he also mentioned that Van Kampen would be necessary. Any ideas/suggestions about this?  | ||||||||||||||||||||||||||||||||
| Posted: 28 Aug 2021 08:29 PM PDT A question from my analysis course reads
Now, I can very well understand that the described function has to be continuous because $f(x)=0$ when $x\in K$ and as $x$ slowly moves out of $K$, the distance increases continuously. But, of course, that's far away from a proof, and we need to use the property
to arrive at the proof. But, I'm new to functions whose domain is not $\mathbb R$. So, I'm a little confused about how to construct the proofs. Please help me to do it. I was asked in the comments whether I'm allowed to use the $\epsilon-\delta$ property to prove continuity. Well, it wasn't explicitly mentioned in the exercise that I need to use the preimage property, but this question came just a couple of pages after this property was introduced. That's why I assumed they expect me to use that technique. Although I will also appreciate a detailed $\epsilon-\delta$ proof, it would be better if I can have a proof using the preimage method.  | ||||||||||||||||||||||||||||||||
| On the intuitive explanation for the discriminant property of conics Posted: 28 Aug 2021 08:22 PM PDT In this old post detailing the intuition for the classification of conics based on the sign of the discriminant, the author has considered the coordinates $x$ and $y$ of the point on a conic to become very large as you zoom out on it, and used that idea in the initial simplification of the general conic equation. This is apparent when you consider, say, a hyperbola with the coordinate axes as its axes. However, this simplification seems to run into a problem when you consider a hyperbolas like $xy=c^2$ or even just a parabola of the form $y^2=4ax$. For the hyperbola case, when you zoom out, one coordinate dwindles into zero close to either axis while the other gets large as stated. For the parabola case, $y$ becomes negligible compared to $x$. Thus, the original treatment approximating the conic equation seems to fail here. I managed to work things out for these specific cases; for $xy=c^2$, the coefficients $A$ and $B$ as in the original post would be zero anyway, and the sole real solution for $\frac{y}{x}$ results (zero, representing the $x$-axis). Similar justification can be provided for the parabola case I brought up. But that could still be just coincidence; I can't seem to find a general justification of this intuitive proof for conics which pose similar problems as the above examples. For example, hyperbolas obtained by slightly tilting $xy=c^2$ pose the same problem; one variable does not become larger, but remains small compared to the other variable. Can someone help prove how the original author's initial treatment of the general equation, with the coordinates growing large, still holds for such cases?  | ||||||||||||||||||||||||||||||||
| Complex number related problem Posted: 28 Aug 2021 08:03 PM PDT Let $z_1,z_2,z_3$ be complex numbers such that $|z_1|=|z_2|=|z_3|=|z_1+z_2+z_3|=2$ and $|z_1–z_2| =|z_1–z_3|$,$(z_2 \ne z_3)$, then the value of $|z_1+z_2||z_1+z_3|$ is_______ My solution is as follow ${z_1} = 2{e^{i{\theta _1}}};{z_2} = 2{e^{i{\theta _2}}};{z_3} = 2{e^{i{\theta _3}}}$ & $Z = {z_1} + {z_2} + {z_3} = 2\left( {{e^{i{\theta _1}}} + {e^{i{\theta _2}}} + {e^{i{\theta _3}}}} \right)$ $\left| {{z_1} - {z_2}} \right| = \left| {{z_1} - {z_3}} \right| \Rightarrow \left| {{e^{i{\theta _1}}} - {e^{i{\theta _2}}}} \right| = \left| {{e^{i{\theta _1}}} - {e^{i{\theta _3}}}} \right|$ Let ${\theta _1} = 0$ $\left| {{z_1} - {z_2}} \right| = \left| {{z_1} - {z_3}} \right| \Rightarrow \left| {1 - \left( {\cos {\theta _2} + i\sin {\theta _2}} \right)} \right| = \left| {1 - \left( {\cos {\theta _3} + i\sin {\theta _3}} \right)} \right|$ $ \Rightarrow \left| {1 - \cos {\theta _2} - i\sin {\theta _2}} \right| = \left| {1 - \cos {\theta _3} - i\sin {\theta _3}} \right| \Rightarrow \left| {2{{\sin }^2}\frac{{{\theta _2}}}{2} - 2i\sin \frac{{{\theta _2}}}{2}\cos \frac{{{\theta _2}}}{2}} \right| = \left| {2{{\sin }^2}\frac{{{\theta _3}}}{2} - 2i\sin \frac{{{\theta _3}}}{2}\cos \frac{{{\theta _3}}}{2}} \right|$ $\Rightarrow \left| { - 2{i^2}{{\sin }^2}\frac{{{\theta _2}}}{2} - 2i\sin \frac{{{\theta _2}}}{2}\cos \frac{{{\theta _2}}}{2}} \right| = \left| { - 2{i^2}{{\sin }^2}\frac{{{\theta _3}}}{2} - 2i\sin \frac{{{\theta _3}}}{2}\cos \frac{{{\theta _3}}}{2}} \right| \Rightarrow \left| { - 2i\sin \frac{{{\theta _2}}}{2}\left( {\cos \frac{{{\theta _2}}}{2} + i\sin \frac{{{\theta _2}}}{2}} \right)} \right| = \left| { - 2i\sin \frac{{{\theta _3}}}{2}\left( {\cos \frac{{{\theta _3}}}{2} + i\sin \frac{{{\theta _3}}}{2}} \right)} \right|$ $ \Rightarrow \left| { - 2i\sin \frac{{{\theta _2}}}{2}\left( {{e^{i\frac{{{\theta _2}}}{2}}}} \right)} \right| = \left| { - 2i\sin \frac{{{\theta _3}}}{2}\left( {{e^{i\frac{{{\theta _3}}}{2}}}} \right)} \right|$ $ \Rightarrow \left| {2\sin \frac{{{\theta _2}}}{2}\left( {{e^{ - i\frac{\pi }{2}}}} \right)\left( {{e^{i\frac{{{\theta _2}}}{2}}}} \right)} \right| = \left| {2\sin \frac{{{\theta _3}}}{2}\left( {{e^{ - i\frac{\pi }{2}}}} \right)\left( {{e^{i\frac{{{\theta _3}}}{2}}}} \right)} \right| \Rightarrow \left| {2\sin \frac{{{\theta _2}}}{2}\left( {{e^{i\left( {\frac{{{\theta _2}}}{2} - \frac{\pi }{2}} \right)}}} \right)} \right| = \left| {2\sin \frac{{{\theta _3}}}{2}\left( {{e^{i\left( {\frac{{{\theta _3}}}{2} - \frac{\pi }{2}} \right)}}} \right)} \right|$ $\Rightarrow \left| {2\sin \frac{{{\theta _2}}}{2}} \right|\left| {\left( {{e^{i\left( {\frac{{{\theta _2}}}{2} - \frac{\pi }{2}} \right)}}} \right)} \right| = \left| {2\sin \frac{{{\theta _3}}}{2}} \right|\left| {\left( {{e^{i\left( {\frac{{{\theta _3}}}{2} - \frac{\pi }{2}} \right)}}} \right)} \right| \Rightarrow \left| {2\sin \frac{{{\theta _2}}}{2}} \right| = \left| {2\sin \frac{{{\theta _3}}}{2}} \right|$ ${\theta _2} \ne {\theta _3}$ How do I proceed further?  | ||||||||||||||||||||||||||||||||
| Self-adjoint extension of closed symmetric operator Posted: 28 Aug 2021 08:06 PM PDT
This question is an exercise in our class "Functional analysis II", but I have no idea how to answer it. Is there any hint or solution? Thank you!  | ||||||||||||||||||||||||||||||||
| Second derivatives, Hamilton and tangent bundle of tangent bundle TTM Posted: 28 Aug 2021 08:42 PM PDT I'm learning the Hamilton formalism of classical mechanics, where a second order differential equation is formalized as two first order differential equations on the cotangent bundle of the configuration manifold. I find the concept of tangent spaces and the notion of the derivative $f_*: TM \to TN$ as a function between tangent spaces very elegant, natural and intuitive. I still struggle, though, with an intuitive understanding of tangent spaces of tangent spaces. Let the $n$ dimensional configuration space $M$ be a smooth manifold, $\pmb{q} \in M$, then $TM$ is the tangent bundle and $\pmb{v} \in TM$ a tangent vector. Even without local coordinates, every tangent vector can canonically be split into a point $q$ and a vector $\dot q \in T_qM$. Therefore $\pmb v = (q, \dot q)$. The intuitive notion of a tangent vector is the notion of a change of position or a velocity (thus the notation) starting at a point. Now lets look at the tangent space of the tangent space $TTM$. Let $\pmb a \in TTM$ be tangent vector to $TM$. The intuitive notion of $\pmb a$ is a change of velocity or acceleration. Just as we could do for $TM$, we can split $\pmb a$ into a "point" $(q, \dot q)$ in $TM$ and a vector in $T_{(q, \dot q)}TM$ given by $(\dot{q}, \ddot q)$, with $\dot{q}$ denoting a change of the fiber and $\ddot{q}$ denoting a change of the vector within the same fiber. Combining with the previous, $\pmb a \in TTM$ consists of $(q, \dot{q}, \dot{q}, \ddot{q})$. What you might disregard as a double occupancy in notation, is a real problem for my understanding. It seems like the information about the position change is duplicated, not even necessarily consistently.
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| How to solve this differential equation involving a polynomial function? Posted: 28 Aug 2021 08:41 PM PDT I am struggling with this question please help.. Suppose $f(x)$ is a polynomial function as well as continuous in $\mathbb{R} \to \mathbb{R}$. Given that $f(2x)=f'(x) f''(x)$, then find $f(3)$.  | ||||||||||||||||||||||||||||||||
| Negative & Positive Shear Factor Posted: 28 Aug 2021 08:07 PM PDT My question relates to constructional geometry & matrices aren't to be involved in the solution because stated Math level is up to O Levels... The figure below shows shear with y=3 as invariant line & shear-factor of 3 http://i.stack.imgur.com/ysr4y.png My question is if you are provided the original polygon & asked to do shear with y=3 as invariant & shear-factor 3 or -3 how would I know whether to slide segment AD to right or left? Same confusion thus occurs with segment BC? Moreover shear-factor is defined via (object-image dist)/(object-invariant dist) thus if the object polygon ABCD & its image say the blue one is given & you are asked to completely define the transformation I can give the invariant line & the factor's magnitude but I can't tell the sign (+ or -) of the shear-factor because distances are always positive? Plz help this teacher  |
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