Does exponential decay have to be related to $ln()$? Can't $r^{-x}$ be considered exponential decay? And in the inverse square case, x is a constant ($x=2$) ie. $r^{-2}$?
@antimony for the first part, yes, and in applications it’s inevitably so. One speaks of exponential decay having some time constant T, such that $y=y_0 e^{-t/T}$. But that can equally well be written as $(e^{1/T})^{-t}=r^{-t}$
For the second, tho, not really. $r^{-x}$ is an exponential function of $x$, but it’s a power-law function of $r$
It being exponential in $x$ is irrelevant if $x$ isn’t changing, eg when $x=2$ to get inverse-square behavior in $r$
@Semiclassical thanks i've revised my view in terms of the definition of 'exponential decay'. however re. power law i think this seems to be a preferred terminology
i don't think dictionaries are very useful in math. the best source of definitions for anything is the materials in front of you, which might not agree with dictionaries or wikipedia or even what someone with a phd in math says is in common use
i think it is the case that exponential to refer to non-natural-exponential power-law type things is going out of fashion, rather than being "incorrect"
there is too much variation in how people use terminology and it isn't practical to list all of them. wikipedia is OK but not always best at signaling when there are multiple definitions
wikipedia is funny: "The exponent is usually shown as a superscript to the right of the base. In that case, b^n is called "b raised to the nth power", "b raised to the power of n",[1] "the nth power of b", "b to the nth power",[3] or most briefly as "b to the nth"."
weirdly none of those are how i would say it. "b to the n"
the page reads like it was written by a chatterbot. fairly common problem with topics applicable to a broad range of contexts. every editor adds or edits one the sentences they are interested in and you get this frankenstein monster of facts
> It being exponential in $x$ is irrelevant if $x$ isn’t changing @Semiclassical do you pls know where i can read more about that ^ so i can learn what the distinction is?
i was wondering where the complex plots were for complex exponents. whenever anything can be graphed in the complex plane, it should be graphed in the complex plane, according to wikipedia. even if those pictures might not tell you anything, or at least not be one of the first 10 things you learn about a concept. but they're stored separately, in he page for "exponential function"
no offense if anyone around here is like the world curator-in-chief of these wikipedia pages
The idea is this: If $T\in L(V,W)$. Let $B=\{u_1,...,u_m\}$ be a basis of $V$ and let $C=\{w_1,...,w_n\}$ the given basis of $W$. Matrix of $T$ with respect to $B$ and $C$ has $Tu_i$'s as columns. Applying elementary column operations on the matrix won't change the column space of $T$ and the co...
Let $N_2 \rtimes G \le N_1 \rtimes G$ be internal semidirect products, where $N_2 \le N_1$. Is there a nice formula for the index $|N_1 \rtimes G : N_2 \rtimes G|$? I think it's equal to $|N_1 : N_2|$; is that right?
I want to check if the following is true : If $n$ is a number of distinct subgroup of order $6$ of some large group $G$, then there are at least $6n-3n+3$ distinct elements in $G$.
@Prithubiswas anyone can edit wikipedia pages (unless the page is locked), that page is probably locked right now since it was recently edited for mischievous means
Thank you very clear. Last question please, should not we have $\mathbb{T}(2^u)=2^{u-1}+\frac 12 \mathbb{T}(2^{u-1})$ instead of $ \mathbb{T}(u)=2^{u-1}+\frac 12 \mathbb{T}(u-1)$ as I did substitutions based on values you wrote for $u=\log{z}$
it's a definition chase more than it is linear algebra. someone should fill it in pretty quickly. it might help to prove at the outset that d(a,b) = -d(b,a) for any a, b; can you prove that?
2) For v in V, and U a subspace of V , the affine subset v+U is said to be parallel to this. My confusion is: are (1) and (2) different? If not, why were they stated in two different points?
@leslietownes I think you're referring to some other question. I have not used T here. Are you referring to the question that I asked the other way about finding a basis of U?
the conditions (1) and (2) to be proved are different, one talks only about T, the other requires the restriction to X' to be bijective (when the definition only gives you the restriction to X being bijective)
i'm thinking out loud about epsilon emperor's question, koro
not whatever you're thinking about
what a misfortune to have X and (1) and (2) having independent meaning in two problems at the same time
koro, in your setting, (1) defines the term "affine subset", and (2) defines what "parallel to" means, so yes, they're different? they involve the same set of notions, but define different things?
he doesn't define 'parallel affine subset', he defines a relation ("parallel to") between an affine subset and a vector subspace
by his definitions, every affine subset is parallel to a subspace, if that's what you're asking, so "being parallel to a subspace" is not a new property of affine subspaces but one that every affine subspace has by definition
i agree that these definitions aren't "doing" very much, but you do sometimes see this. keeping definitions as simple as possible so they are easier to verify
Ok. So here, since x can be any non-negative number. The first two are ruled out. (Note that (100, 10) lies on the parabola but sine and cosine have maximum value 1)
and similarly, what can you say about $C$ ?@Wolgwang
@Wolgwang The language on that is confusing. From the options, I assume they are looking for a parameterization of the whole parabola, not just points that lie on the parabola (it would have been more confusing if "all of the above" were a choice). In which case, $(0,0)$ is only in (A) and (B), whereas $(4,2)$ is only in (C).
$\left(\tan^2(\theta),\tan(\theta)\right)$ would do.
the language in that parabola question is indeed confusing. I think the question asker's intention was: "the parametric coordinates of all points" on the parabola...
Let $I$ be a nonempty directed set, and $H_i \le G_i$ a directed sequence of groups living in some ambient group. Does the following index formula hold: $$|\bigcup_{i \in I} G_i : \bigcup_{i \in I} H_i| = \lim_{i} |G_i : H_i|?$$
For one-one and onto functions can i bypass the usual methods, by saying for any y=f(x) if there exist dy/dx=0 then the function is not onto and one-one? And also I ensure that the point where dy/dx=0 is a local minima/maxima?
for reasons that i will not disclose at the moment, consider function $y=\cos(x)(\sin(x)+\sqrt{\sin(x)^2+a})$ with $a\geq 0$.
If $a=0$, this simplifies to $y=\cos x\sin x$ with max value $y(x=\pi/4)=1/2$
But at a glance, that looks incredibly painful to maximize for arbitrary $a$
And yet, the solution is way nicer than you'd expect: the critical point occurs when $x=\tan^{-1}(1/\sqrt{1+a})$ and yields a max value of $y=\sqrt{1+a}$
i can show that in various ways, including without use of calculus at all. and yet it still just baffles me how the answer can be as simple as it is, starting from such an ugly equation
can you construct a smooth manifold which is a surface built by taking a rotating homotopy that projects onto planar homotopic curves?
plane homotopy has two fixed endpoints
I can only do this for a simple case using rotation matrices to rotate the curve by a certain angle and then projecting down onto the planar curve. then repeating the process
the hard part is if the plane homotopy is say analytic chiral, you have to "deform/twist" the curve s.t. it projects precisely onto the planar curve, probably sacrificing analyticity of the surface
what i meant specifically was: suppose you define and use a new command in an answer. the wysiwyg editor on the main site will properly format that command
if you delete the definition from the post, the use of the command will persist in the display
I wanted to define operatornames span, null, range etc.
@Semiclassical: I was trying to solve one problem on linear maps. I discussed that problem here also the other day. I have tried to present my understanding here in this answer math.stackexchange.com/questions/1924952/…
I request you to please take a look at it. Thanks.
i think he does talk a little bit about change of basis and matrices in a later chapter, but maybe only in the context of operators on a single vector space? memory is hazy about this.
some books go the route of, the following are equivalent for a linear map T: V to V. [list of about 20 things that are equivalent to "T is invertible"]. then another one for T: V to W possibly of different dimensions with stuff about rank, nullity, left invertibility, right invertibility.
koro yeah. note that the other answer solves the problem very quickly and directly. the approach about column operations is a little more 'conceptual,' in that it suggests a whole family of exercises on the same subject and hints at a larger point on the kinds of matrices that can be chosen to represent a given operator.
joe: the mathcal P is often used to denote a fixed probability measure on some space. we don't seem to have a notation for that space. here X is a random variable on that space. X does induce a measure on the real numbers: assign the subset E of the real numbers the measure P(X in E).
so maybe P^X is that measure. this is just a guess.
if my guess is right, it strikes me as weird to use a superscript for that purpose. robjohn is interpreting the superscript in a more standard way.
I have a quick question about logic - how can two things be tautologically equivalent but not equal? The example given in the slides for my CS class were [ not (P and Q) ] and [ (not P) or (not Q) ] - firstly with De Morgan's laws I think these are equal, but even if we're not allowed to use that in this specific field, how can two formulas be equivalent but not equal?
