calculus – Interpreting line integral definition

Definition: Let F be a vector field with continuous components defined along a smooth curve C parametrized by r(t), a $leq t leq b$. Then the line integral of F along C is $$int_C F cdot T ds= int_C (F cdot frac{dr}{ds}) ds=int_C F cdot dr$$

I am not sure how they are interrelated.I know T is a tangent vector $dr/ds=v/|v|$, but how is $int_C (F cdot frac{dr}{ds}) ds=int_C F cdot dr$? How to intuitively interpret it?

coq – What can we have in exchange if we drop subtyping from definition of Calculus of Inductive Constructions?

If we remove subtyping ( from CIC we will lose some expressive power. But is that power necessary for a programming language?

If we demand programmer to write e.g. Type(1) and Type(2) to mean respectively type and kind, that won’t be anything novel in software engineering.

Can we get something in exchange? For example relaxed requirement on strict positivity of inductive types (without creating inconsistency in the language)?

We should get at least simpler term equivalence check where problem of η-reduction ( doesn’t exist.

Is there some research on this topic? How does it compare to e.g. inductive types from NuPrl (

evaluation – PlusMinus definition unexpectedly affects plotting

It appears that PlusMinus is the only wrapper that is not Protected:

Plot(x, {x, 0, 1});  (* preload *)

Select(Charting`ParserDump`$pAllWrappers, FreeQ(Protected)@*Attributes)

We can remove it from that list to correct this bug, but perhaps induce others:

Plot(x, {x, 0, 1});(* preload; do not remove! *)

With({ov := OwnValues @ Charting`ParserDump`$pAllWrappers},
  ov = DeleteCases(ov, HoldPattern(PlusMinus), {3});

algorithm analysis – Prove little o with just the definition

I have been searching for a while now but couldn’t find anything about this exact pair of functions with the little $mathcal{o}$ notation.

Given the functions $f(n) = 2^{n}$ and $g(n) = n!$ I am supposed to prove, or disprove, the following statement: $f(n) in mathcal{o}(g(n))$.

I am fairly sure that it’s true but now I need an idea of how to show this. We have just started out with this whole concept and this is the second exercise, the first one being a relatively easy big $mathcal{O}$ task. But this exercise is just beyond me right now. The only definition I am allowed to use (meaning: NO LIMITS) is $mathcal{o}(g(n)) = {f(n)|forall C > 0 exists n_{0} forall ngeq n_{0}:f(n) < C * g(n)}$. This means other than with big $mathcal{O}$, where it suffices to show that there’s at least one pair $C$ and a $n_{0}$ so that $f(n) leq C * g(n)$ $forall n geq n_{0}$, I now have to prove that for every $C > 0$, there is such a $n_{0}$ so that the condition stated in the set is true.

I first have been thinking about the functions, and I would have an answer for $mathcal{O}$, because you can prove with induction that $2^{n} < n!, forall ngeq 4$. Meaning my C would be 1 here. However, I have no idea how to prove it for every C and would be grateful for any guidance!
(It would already help to know how to start. Probably like, let $C$ be greater 0, and then I have to show that for any Value of this $C$, there is… because…
My biggest struggle is to find meaningful estimations to get a chain of inequalities.)

nonstandard analysis – Hyperreal Numbers (Sequence Definition)

I am trying to understand the definition of a limit (for a sequence) regarding hyperreal numbers converging to $L$.

The definition (see link here) states a real sequence of numbers converges to $L$ if every infinite hypernatural $H$, $x_H$ is infinitely close to $L$.

Does $L$ have to be a real number or can it be an element in the set of all hyperreal numbers?

Also, I am confused as to what an infinite hypernatural $H$ (see here) is defined to be.

For instance, what does ${}^*lfloor 4.4omega+5.9 rfloor$ equal? For instance, does ${}^*lfloor 4.4omega+5.9 rfloor=4omega+5$ hold true?

object-oriented – what is meant by the definition of polymorphism?

Based on many of the tutorials I've read, the following is the definition of polymorphism:

Polymorphism is the ability of an object to take many forms.

Now let's say we have one Animal Parent class and a Dog and a Cat Children's classes.

Does the above polymorphism definition mean that a Animal Variable can have many forms in the sense that one Animal Variable can be one Animal or it can be one Dog or it can be one Cator does it mean something else?

Resolution – Objective definition of the parameters required for facial recognition, recognition and identification in digital images

Are there peer review studies that observe the effects of pixel density on a person's ability to identify, recognize, and recognize human subjects (especially the face)?

I searched various sources, but only found results that deal with the image resolution and effectiveness of facial recognition software.

Functional analysis – definition of the Lyapunov exponents for compact operators

There is the following known result from Goldsheid and Margulis (see Theorem 1.3) on the existence of Lyapunov exponents:

To let $ H $ be a $ mathbb R $-Hilbert room, $ A_n in mathfrak L (H) $ be compact and $ B_n: = A_n cdots A_1 $ to the $ n in mathbb N $. To let $ | B_n |: = sqrt {B_n ^ ast B_n} $ and $ sigma_k (B_n) $ denote the $ k $the largest singular value of $ B_n $ to the $ k, n in mathbb N $. If $$ limsup_ {n to infty} frac { ln left | A_n right | _ { mathfrak L (H)}} n le0 day1 $$ and $$ frac1n sum_ {i = 1} ^ k ln sigma_i (B_n) xrightarrow {n to infty} gamma ^ {(k)} ; ; ; text {for all} k in mathbb N tag2, $$ then

  1. $$ | B_n | ^ { frac1n} xrightarrow {n to infty} B $$ for some compact non-negative and self-adjuncts $ B in mathfrak L (H) $.
  2. $$ frac { ln sigma_k (B_n)} n xrightarrow {n to infty} Lambda_k: = left. begin {case} gamma ^ {(k)} – gamma ^ {(k -1)} & text {, if} gamma ^ {(i)}> – infty \ – infty & text {, otherwise} end {case} right } tag2 $$ for all $ k in mathbb N $.

Question 1: I have seen this result in many lecture books, but I wondered why it is given in this way. First of all not $ (2) $ clearly equivalent to $$ frac { sigma_k (B_n)} n xrightarrow {n to infty} lambda_i in (- infty, infty) tag3 $$ for some $ lambda_i $ for all $ k in mathbb N $ which in turn is synonymous with $$ sigma_k (B_n) ^ { frac1n} xrightarrow {n to infty} lambda_i ge0 tag4 $$ for some $ mu_i ge0 $ for all $ k in mathbb N $? $ (4) $ seems a lot more intuitive than $ (3) $, not there $ lambda_i $, but $ mu_i = e ^ { lambda_i} $ are exactly the Lyapunov exponents of the limit operator $ B $. I miss something The definition of $ Lambda_i $ (which is the same $ lambda_i $) strikes me as strange.

Question 2: What is the interpretation of $ B $? I usually consider a discrete dynamic system $ x_n = B_nx_0 $. What does $ B $ (or $ Bx $) tells us about the asymptotic behavior / the development of the pathways?

Resolution – Objective definition of parameters for facial recognition, recognition and identification in digital images

Axis Communications has issued guidelines on the operational requirements of a camera to identify, recognize and recognize human subjects ( Density).
You have cited the Swedish national forensic laboratory and the IEC international standard IEC 62676-4 to explain their recommendations, but I cannot find these specific texts.

Are there peer review studies that observe the effects of pixel density on a person's ability to identify, recognize, and recognize human subjects (especially the face)?

I searched various sources, but only found results that deal with the image resolution and effectiveness of facial recognition software.

Measure theory – find the best definition that fits my intuition of the average?

This is not the same as "Find a strict definition for a Riemman-like sum that is easier to calculate?
"Here I assume that my Riemman-like sum is clear enough to understand. If not, try to answer this question.

Consider $ f: A to (0.1) $ Where $ A subseteq (a, b) $ and $ lambda $ is the Lebesgue measure. I want to create a simple, user-friendly average that fits my intuition. Before I get to the definition, here's why an intuitive average is so important to me $ f $.


After reading the entire previous post, remember to read "non-border points", "first order border points", "second order", "third order", etc. The highest order border point should be "infinitely more weight" for to have $ f $Average as lower order boundary points. For several highest-order limit points with the same order, there are the "denser" lower-order limit points one of the highest order pointsthe more "weight" this point of the highest order should have an average of $ f $ compared to other highest order points.

While I found this intuitive, though $ lambda (A) = 0 $I think when $ lambda (A) $ is greater than zero, the average should be

$$ frac {1} { lambda (A)} int P (x) dx $$

Where $ lambda $ is the Lebesgue measure.

In addition, if $ A $ is countless and has a dimension of zero, I want to "ignore" countable subsets of $ A $ since countable subsets are "infinitely smaller" than non-countable subsets. Take the general peice-wise function, for example:

$ f (x) = f_i (x) $when $ x in A_i $ so that $ f_i: A_i bis (a, b) $ and $ A_1, …, A_m $ are not overlapping subsets of $ A $

We "ignore" the countable $ A_i $ (Give them a zero value) and take the average of $ f $ in terms of countless $ A_i $.

The only time I expect an average to have no "intuitive existence" is when $ A $ is countable and more than one $ A_i $ is countable and dense in $ (a, b) $. In this case, it is impossible to find a clear and intuitive average.

To define the average of $ f $ there is no such thing in this case; we need an upper and a lower sum (Riemman-like sums) that do not converge; Before I do this, however, I have to define the following.

First definitions

Consider $ S subseteq A $

$$ M (S) = begin {cases}
frac { lambda (S)} { lambda (A)} & lambda (A)> 0 \
0 & S text {is countable and} A text {is countable, but} lambda (A) = 0 \
1 & text {else}
end {cases} $$

This prevents the average from being zero if $ lambda (A) = 0 $ and allows us to ignore countable subsets of $ A $ when $ A $ is countless and $ lambda (A) = 0 $

Ownership of $ M (S) $

$ M ( Emptyset) = text {undefined} $

(1) $ M (A) = 1 $

(2) When $ lambda (A)> 0 $,

If $ {A_i } _ {i = 1} ^ { infty} $ are disjoint and $ bigcup_ {i = 1} ^ { infty} A_i = A $ then $ M left ( bigcup_ {i = 1} ^ { infty} A_i right) = sum_ {i = 1} ^ { infty} M (A_i) = M (A_1) + … = 1 $

(3) When $ lambda (A) = 0 $,

We split up $ A $ into a union of countable subsets $ A_c $ and a union of countless subsets $ A_u $ from $ A $. If $ M (A_c) = 0 $, then $ M (A_u) = 1 $, because $ M (A_c) + M (A_u) = M (A) = 1 $. If $ M (A_c) = 1 $, then $ M (A_u) = 0 $ for the same reason. (I think the additivity is true).

Upper and lower total

Now we can create upper and lower sums.

Given $ S subseteq (0.1) $, and let $ P $ to be a partition of $ (0.1) $ (Note: a partition is a finite set of disjoint subintervals $ X $) you can define $ P & # 39; (S) = {X in P: X cap S neq Emptyset } $. And you can define $ n & # 39; = | P & # 39; (S) | $ (the cardinality of a finite set). Note each disjoint subinterval $ X $ has the same length.

For future intuition, consider the following:

$$ tilde {L} _ {f, P} = frac {1} {n ^ { prime}} sum_ {X in P ^ { prime} (S)} ( inf_ {t in X} f (t)) $$

Define the limit as $ | P | to $ 0with refinements from $ P $ how so:
$$ lim _ { | P |} ( tilde {L} _ {f, P}) = tilde {L} _f $$

This is the "lower average" of $ f $ on $ (0.1) $ (in relation to the partition $ P $).

Likewise the "upper Kinda average" of $ f $ on $ (0.1) $ in terms of partition $ P $ is:

$$ tilde {U} _f = lim _ { | P |} ( tilde {U} _ {f, P}) $$


$$ tilde {U} _ {f, P} = frac {1} {n ^ { prime}} sum_ {X in P ^ { prime} (S)} ( sup_ {t in X} f (t)) $$

and the limit lasts $ | P | to $ 0.

We want these lower and upper average limits to converge to the same value.

Note that this @ WillieWong's expanded comment and chat are still not strict and successful.

Full definition

However, the last section was only for intuition. Now we make the real definitions.

We define the full "lower average" as:

$$ L_ {f, P} = frac {M (A)} {n ^ { prime}} sum_ {X in P ^ { prime} (A)} ( inf_ {t in X} f (t)) $$

and the full "upper average" as:

$$ U_ {f, P} = frac {M (A)} {n ^ { prime}} sum_ {X in P ^ { prime} (A)} ( sup_ {t in X} ) f (t)) $$

If these lower and upper average limits converge to the same value (id est: are the same), we get "my definition of the average" from $ f $ for each $ A $. If they don't converge, the average is undefined. Note that I define "upper" and "lower" averages to indicate when an average cannot exist.

Example with general piecewise function

Consider a general piecewise function, $ f (x) = f_i (x) $when $ x in A_i $ so that $ f_i: A_i bis (a, b) $ and $ A_1, …, A_m $ are not overlapping subsets of $ A $.

When $ lambda (A)> 0 $, the lower average of $ f $ is

$$ L_ {f, P} = frac {M (A_1)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_1)} ( inf_ {t in X} f_1 (t)) + … + frac {M (A_m)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_m)} ( inf_ {t in X .} f_m (t)) $$

and the top average of $ f $ is

$$ U_ {f, P} = frac {M (A_1)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_1)} ( sup_ {t in X} f_1 (t)) + … + frac {M (A_m)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_m)} ( sup_ {t in X .} f_m (t)) $$

When the upper and lower average limits converge, we have a defined average. If not, the average is undefined. That's why I create upper and lower sums. I want cases where we can't have an average.

Finally, if $ lambda (A) = 0 $countable $ A_i $ are combined in $ A_c $ and countless $ A_i $ are combined in $ A_u $, then with property $ (3) $, the lower average of $ f $ is

$$ L_ {f, P} = frac {M (A_c)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_c)} ( inf_ {t in X} f (t)) + frac {M (A_u)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_u)} ( inf_ {t in X} f (t ))) $$

and the top average of $ f $ is

$$ U_ {f, P} = frac {M (A_c)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_c)} ( sup_ {t in X} f (t)) + frac {M (A_u)} {n ^ { prime}} sum_ {X in P ^ { prime} (A_u)} ( sup_ {t in X} f (t ))) $$


Are they well defined?

Could we find a definition that is easier to calculate and gives accurate values?

My first guess is that we can change the definition of the generalized Riemann integral. This could give the intuitive result $ 0 $for example in $ P (x) = x $ and $ A = left { frac {1} {2 ^ x} + frac {1} {2 ^ y} + frac {1} {2 ^ z}: x, y, z in mathbb { Z} right } $.

My second guess is that we can use translation-invariant metrics along the line from Density with Folner
. Here are some papers that can help

  • Taras Banakh, Extreme densities and dimensions on groups and $ G $-spaces
    and their combinatorial applications
    , arXiv: 1312.5078

  • Taras Banakh, The Solecki sub-measurements and densities on groups, arXiv: 1211.0717

  • Taras Banakh, Igor Protasov, Sergiy Slobodianiuk, Density, partial measures and division of groups, arXiv: 1303.4612

EDIT: I heard that Choquet is integral
different possibility.

According to a comment by Reddit:

You may want to look in papers about OWA operators. If my memory is good, find one
Generalization in "Mm-OWA: a generalization of OWA operators" using the Choquet integral.

I'm not sure if it helps, but good luck.

Edit 2:

According to this post, we can use the Hausdorff measure, but it's not clear how it works
when $ A $ is countable infinite.

Edit 3: Has @WillyWong already solved this problem ?. Here is his comment.

Probably my last comment about it: I think you can probably reach
what you want by looking at the problem differently.

First, we construct a series of limited functions $ g_ sigma $ how
follows: Start with yours $ A $. Consider the crowd $ A_ sigma = cup_ {x in
A} (x – sigma, x + sigma) $
. This is a union of open intervals and
therefore is an open set. As long as $ A $ is not empty this set is
not empty and therefore has a positive Lebesgue dimension.

Just consider $ sigma <1/2 $. To let $ chi_ sigma $ be the indicator
The function of $ A_ sigma $. Define $ g _ { sigma} (x) = frac {1} {| A_ sigma |}
int _ {- 1/2} ^ x chi_ sigma (y) ~ dy $

Here $ | A_ sigma | $ is the Lebesgue measure of $ A_ sigma $.

Notice that $ g_ sigma $ is normalized so that it takes value in between
$ 0 $ and $ 1 $. (It's limited.)

And $ g_ sigma $ is continuous. The question is whether there is and
What is the limit $ lim _ { sigma to 0} g _ { sigma} $.

When $ | A | > $ 0, then the family $ g_ sigma $ is equal to continuous, and it
is not too difficult to see $ g $ is formed as $ frac {1} {| A |}
int _ {- 1/2} ^ x chi (y) ~ dy $
and here $ chi (y) $ is the indicator
The function of $ A $.

The main question is what happens when $ | A | = $ 0. The guess is
the if $ A $ has zero dimension, but has a non-trivial perfect kernel,
then the limit $ g $ is a continuous function (like the Cantor
Function). And when $ A $ is scattered, the limitation $ g $ is a step
Function. In any case, the integral sought should be
the Stieltjes integral with weight function $ g $.