@BalarkaSen So, consider the set of smooth curves where the endpoints and the tangents there are fixed. Is that set simply connected? Can I do a nontrivial loop?
I feel like the Thurston wiggles from that Inside Out video should do it but I'm not sure
"The general element is now [set of equations including $z=z_B+z_S$] where $z_B$ is sometimes referred to as the body and $z_S$ as the soul of the supernumber $z$."
"Dual numbers were introduced in 1873 by William Clifford, and were used at the beginning of the twentieth century by the German mathematician Eduard Study, who used them to represent the dual angle which measures the relative position of two skew lines in space. Study defined a dual angle as $\theta + d\epsilon$, where $\theta$ is the angle between the directions of two lines in three-dimensional space and $d$ is a distance between them."
i hate that notation btw. $d$ as a variable is gross
namely, take one loop to be a current-carrying wire. that generates a magnetic field per the Biot-Savart law. if you compute the circulation of said field along the other loop, then Ampere's law tells you it's the linking number (up to irrelevant physical constants)
it's going to be back pain for me. my daughter somehow broke her leg in day care today. her leg has been immobilized but she will need to be carried around until we can see an orthopedist. probably next week at the earliest.
indoors. it sounded like she was moving around very rapidly. per my daughter, she was 'spinning around' in a room with a lot of low-lying kid-sized furniture. her foot probably got caught on something, and she crashed into furniture on the way down.
she's at the age where she can move extremely fast, but not old enough to control her movement very well or appreciate risk.
yeah, something like that. completions of tensored spaces/normed algebras is an interesting subject. grothendieck was an early contributor to the field although i understand he is better known for other things.
and without dissing it, his work in functional analysis turned out to be less impactful than his work in other areas. he was wise to switch.
she was picking at her splint. we put her in a tight pair of pajamas to hopefully limit this behavior. i will scream if we cannot get a cast tomorrow.
semiclassical an interesting and fairly introductory treatment of aspects of some of the theory a monograph by raymond ryan, titled something like 'tensor products of banach spaces,' from springer.
no. see if you can prove that f^{-1} (which exists by your hypotheses) is continuous and use the fact that the continuous image of a compact set is compact.
or get there the same way using, i dunno, sequences or something.
Write a short program, that would generate the longest possible error message, in a standard C++ compiler (gcc, cl.exe, icc, or clang).
The score of each entry is the number of characters in the longest error message the compiler emitted. Types included in your source code and quoted by the comp...
oh! the author does add $\epsilon(h,k) \to 0$ as $\sqrt{h^2 + k^2} \to 0$, which i'm guessing then has to be interpreted as: the error goes to zero as the change in x,y goes to 0
@leslietownes i do find it amusing that the business of trace class vs Hilbert-Schmidt class (and thus whether the Hilbert-Schmidt class is a categorical tensor product) hinges on "$\sum_k 1/k^2$ converges but $\sum_k 1/k$ diverges"
actually, come to think of it. is there another 'obvious' example of a sequence $\{a_k\}$ such that $\sum_k a_k^2$ converges but not $\sum_k a_k$? I'm sure one can do some variations on the harmonic series but i can't think of another kind of example
(again, failure of imagination for analysis pathologies)
@Semiclassical you could do $\sum_{n=1}^\infty\frac1{n^{3/4}}$ which diverges and $\sum_{n=1}^\infty\frac1{n^{3/2}}$ which converges, or did you mean something other than $p$-series?
Hi I want to use { bracket. As of now I am using it like this: \{. But its size is small. I am not able to use \left{, the way we can use for \left( to increase the size of (. What is the code for bigger {? Thanks.
@Semiclassical even with \big, I am getting small size of the bracket. Maybe after I post it, the formatting may look better. If not I'll post the link here. I may take more than an hour to finish. Thanks for your help.
@Semiclassical the answer can be found by using this formula median$=l + \frac {\frac{s}{2}-c}{f}×i$ where[here we are taking specific values of 3rd entry cz 6/2=3, (even) no of entries = 6] $l=500$ lower limit of 3rd entry $f=$frequency of 3rd term i.e. 23 $c=$cumultative frequency i.e. 12+10=32 $s=$total frequency 100 $i=$ limit width 750-500= 250. So median in this way = 500+ [{100/2-32}/23]250 or 500 + 0.782608696×250 or 695.652174
it's a bit delicate. suppose you insert a factor of $x^n$ into that series. then for $-1<x<1$ the series definitely converges per the root test, and indeed sums (via the binomial theorem) to $(1-x)^{-1/2}$
so $x=1$ lies right on the radius of convergence.
at the very least, the ratio test is not going to do a bit of good here
should have "ratio test" above, not "root test" (though that's also not helpful)
When two planes intersect, the intersection is a line. The cross product of the normal vectors of both planes will lie on that line of intersection. I understand that there is several ways in which 3 planes can intersect. When 3 planes intersect at a point, will the direction vector of the line (when two planes intersect) be parallel to all 3 planes?
if $a_n=\frac{(2n)!}{4^nn!^2}$, then $a_0=1$ and $$ \begin{align} a_n &=\frac{2n(2n-1)}{4n^2}a_{n-1}\\ &=\frac{2n-1}{2n}a_{n-1} \end{align} $$ Therefore, $$ a_n=\underbrace{\frac{2n-1}{2n}\frac{2n-3}{2n-2}\frac{2n-5}{2n-4}\cdots\frac34}_{\substack{\text{we bound the squares}\\\text{of these terms from below below}}}\frac12 $$ cross-multiplying shows that $\frac{2n-1}{2n}\ge\frac{2n-2}{2n-1}$; therefore, $\left(\frac{2n-1}{2n}\right)^2\ge\frac{2n-2}{2n-1}\frac{2n-1}{2n}=\frac{n-1}{n}$. Therefore,
@barista This shows that $\frac{(2n)!}{4^nn!^2}\ge\frac1{\sqrt{4n}}$
Hey guys! I've almost solved this interesting question on second-order nonlinear ODE, but I can't seem to get the last part. My question of Math SE details the question and my current solution; I would really appreciate if anyone here could take look and give me any advice on how to continue
The following question is from Courant's Differential and Integral Calculus, Vol. 1:
A particle of unit mass moves along the x-axis and is acted upon by a
force $f(x) = -\sin x$.
(a) Determine the motion of the point if at time $t=0$ it is at the
point $x=0$ and has velocity $v_0=2$. Show that a...
@DavidChoi this is equivalent to the ODE for a mathematical pendulum. the integrals actually can be carried out...well, at least if you count "expressed as elliptic integrals/functions"
that said, the "simple pendulum" version gives the intuition immediately: If you hit a simple pendulum hard enough (thereby giving it an initial speed) it'll pass over the top and swing back down
and since no energy is lost, it'll just keep repeating that motion forever
if it's not hit hard enough, by contrast, it'll just swing back and forth
Excuse me, but do I need some sort of common algebra for working with boolean algebra and other algebras like elementary algebra? I'm trying to figure out why wolfram here can't recognize something like wolframalpha.com/input/…
that said, while WolframAlpha can't do it, Mathematica can. the `Solve` function doesn't owrk here, but `FindInstance` does: `FindInstance[a||b==a+b,{a,b},Integers]` yields `{{a->0,b->0}}`
hmm. i must be forgetting how to format code in chat
The relevance of solving something like $a + b = a \lor b$ as specified in my query for both $a$ and $b$ solution sets gives a nice pattern matching function on the basis that a number $n = m(a + b)$ to help me find a substring within a string.
I'm sure you can see how this helps for finding substrings. Evaluating the solution set as a boolean function tells us whether or not a substring exists in the string at all. Evaluating the solution set as a set of numbers and computing a function that maps all a to all b can give us an occurrence of that substring within the string assuming it is a true substring.
So correction: not a function that maps from a to b. Rather, a function $f : a\to n$ for $n = a + b$ and $n$ satisfying $a + b = a \lor b$. That is the full set of constraints.
Yes, wolfram can be retarded sometimes. Severely retarded. Sometimes it parses latex correctly; other times, "Wolfram|Alpha doesn't understand your query".
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?
It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natura...
Also, I determined long ago that zero is a "neutral" number so-called (because there is no such thing as neutral in reality) with its own sign in my eyes because it lacks the properties of positive and negative numbers, and it cannot be both negative and positive at the same time.
(that said, i do think there are contexts where 0 is natural and contexts where it isn't. for power series, you start with the x^0 term so including 0 makes sense. but 0 is quite unnatural when it comes to multiplication.)
(if all you're doing is addition, 0 is natural as the identity element. but if you're also doing multiplication, it's not.)
I always took natural to be a definitive property accepted by all mathematicians as the way to describe positive integers because that is the accepted name of the set of naturals, and the numbers in that set.
@Semiclassical It is impossible that anyone with a disordered foundation and understanding of reality can come up with consistent conventions that are coherent, and worse, is still capable of making internally consistent systems that are erroneous as a result.
Yes, well that in summary means that it is best to have a single, common, objective standard because without that standard, no conversion can take place between the subjective models.
setting aside irritating questions of whether {0,1,2,..} and {1,2,3,...} exist in a Platonic sense or not, the name 'natural' is not an objective property
I partly come at this from a physics mindset. In special relativity, there's no preferred reference frame: each one is just as good as another. You could insist that one of them is the "true" reference frame, with all others being defined relative to it, but there would be no empirical content to that
i.e., I'd get exactly the same predictions based on whether I said that my reference frame is the "correct" one, or if I said that yours was
there's an Einstein quote i'm struggling to remember
@Semiclassical There always exists object and subject. Without the objective, the subjective is meaningless. Einstein's (special) relativity is nothing more than the realization that object and subject are always present in reality, but the philosophical concepts of object and subject are universal, generally speaking.
The objective perspective or "correct" frame of reference is one that would transcend both the coil and the magnet themselves such that the coil and magnet become subjects.
@Semiclassical That is incorrect. Whether or not an object is stationary depends on whether or not the object is in fact stationary, not whether the object appears to move according to some observer. Hence why I said the objective perspective is one that transcends both the coil and the magnet as such an observer understands what is stationary and what is moving with absolute certainty regardless of other observers.
@Semiclassical Precisely because we do not transcend the confines of reality being that we are both rational spirit and material body, and reality being purely material with the instruments we use being material. A spiritual entity can easily observe these things, however. Thus, in the Catholic model of reality, even things such as Godel's incompleteness are not contradicted because there is always an outside observer or outside means of observation.
@Semiclassical Well that's because science has abandoned reason. It has become a cesspool of pride and ignorance. Harsh, for sure, but that is reality.
And then they claim that we who believe are pseudoscientists. How ironic.
