Calculus – Area of ​​the sector bounded by line and curve

I have a line segment $ overline {AB} $with endpoints $ (x_1, y_1) $ and $ (x_2, y_2) $, Draw any function $ f (x) $Through these 2 points, the function must have an average speed $[x_1,x_2]$ from $ frac {y_2-y_1} {x_2-x_1} $, However, the area is bounded by the line and $ f (x) $ can be very different, and what I'm trying to find is the relationship between $ f & # 39; (x) $ and this area. I am not sure how to formally express this next part, but as the average speed of both the line and the function over $[x_1,x_2]$ is the same, the average of the sum of the values ​​of $ f & # 39; (x) $ at all infinite points $ f (x) $ about the domain $[x_1,x_2]$ is the average of the sum of the slope / value of the derivative of the line at all infinite points of the line in the domain $[x_1,x_2]$, The actual values ​​of $ f & # 39; (x) $ can vary considerably. Visually, it seems to be the larger y-value of $ f & # 39; (x) $ over $[x_1,x_2]$The bigger the area. For example, with the line segment with endpoints (0,0) $ and (2.4) $and the function $ f (x) = x ^ frac {3} {2} * sqrt {2} $the area is limited by $ y = 2x $ and $ f (x) $ is $ frac {16-2 sqrt {2}} {5} $and the largest y-value of $ f & # 39; (x) $ over $[1,2]$ is 3. If $ f (x) = frac {x ^ 3} {2} $is then the area 4 and the largest y-value of $ f & # 39; (x) $ over $[0,2]$ is 6.

Am I correct in saying the larger the largest y-value of is $ f & # 39; (x) $ over $[x_1,x_2]$The bigger the area? And if so, there is a relationship where a line segment is over $[x_1,x_2]$, can give the area bounded by the line segment and $ f (x) $, only the largest y-value of $ f & # 39; (x) $ over $[x_1,x_2]$?

Number theory – Parametric elliptic curve over Q with a rank of at least 1

Which tools are available without explicit numbers to prove the rank of an elliptic curve of at least 1?

Look at the curves to get a concrete example of the discussion:
$$ y ^ 2 + y = x ^ 3 – 18 (k + 1) $$
with the restriction on $ k $ The $ p = 72k + $ 71 is a prime number.

I searched many such curves for small ones $ k $ (The first 1000 that meet this restriction) and rank (or analytical rank if Sage has difficulty computing the base explicitly) is always odd.

How can we prove (or refute) that this applies to everyone? $ p $?

The above was chosen so that the discriminant has a known factorization:
$$ Delta = -2 ^ 4 3 ^ 3 p ^ 2 $$
If it is not a Weierstrass form, it is easier to approach with elemental techniques. Note, however, that the curve is birationally equivalent to:
$$ y ^ 2 = 4x ^ 3 – p $$

nt.number theory – $ p $ – Primary torsion of an elliptic curve in the $ mathbb {Z} _p $ cyclotomic extension of a $ p $ adic field

To let $ K $ be a number field and $ v $ be a fixed prime above $ p $, To let $ k = K_v $, We have the cyclotome $ mathbb {Z} _p $ extension $ K_ infty / K $ and if $ w $ is a prime number above $ v $ in the $ K_ infty $ we write $ k_ infty = K _ { infty, w} $,
To let $ E $ be an elliptic curve that is over defined $ k $ and suppose it has a good ordinary reduction over $ k_ infty $,

Is there a nice explicit description for the $ p $– Primary pivots $ E (k_ infty) _ {p ^ infty} $?

algebraic geometry – elliptic curve for group law with divisor class group

To let $ E $ be an elliptic curve and $ k $ a field That's well known $ E (k) $ has an (additive) group structure and in fact there are many sources that describe what is going on geometrically.

My concern is to find a derivation of this group law in the language of divisors – in particular using properties of the Divisor class group,

In fact the divisors $ Div (E) $ are formal amounts $ sum_ {P in E (k)} n_P (P) $ With $ n_P in mathbb {Z} $ and the main dividers $ div (f) = sum_P ord_P (f) (P) $ form a subgroup of $ Div (E) $; label it $ PrDiv (E) $,

The divisor class group is the quotient $ Cl (E) = Div (E) / PrDiv (E) $,

Of course, we can canonically define a map $ E (k) to Cl (E), p to (P) – (O) $ from where $ O $ is the special point (= neutral element).

How to show that this card determines the group law $ E (k) $, Especially why is $ (O) $ a major diver?

What is this curve?

We consider the distribution of the roots of some Catalan polynomials.
And we get the following curve that the roots are approaching.
What is this curve?

Distribution of the roots of the Catalan polynomial

Algorithms – Practical optimization of the transmission line curve

I am trying to find a mathematical solution to a problem resulting from the curve generation and practical design of radio frequency (RF) circuits. Essentially, I would like to create a bow that connects the pads of two components while minimizing the spacing and sharpness of the bow bend as both contribute to a greater loss in the electrical / RF signal. The idea is that the user clicks on the pads of each component and automatically creates a curve between them. Note that the components can be positioned arbitrarily (it does not necessarily have to be a quarter circle).

The problem

When designing an RF circuit, connecting a copper trace between two fixed components usually has the following characteristics:

  1. The longer the track, the greater the signal loss. This is how the path of the shortest distance is important. Ideally, two components are placed on the same axis so that a straight line can be connected to both pads.

  2. If a straight line can not connect the pads to two components, a curved line must be used. At lower frequencies, a 45 ° connection may suffice; at higher frequencies, a 45 ° connection will cause too much reflection. A useful analogy is how water flows through a hose: it is much less tight if the hose has a larger curvature. Thus, an arc is desired and the resulting arc must bend slightlyor have the largest possible radius of curvature.

  3. To further extend the second attribute, the endpoints of the arc should affect the vector of a given pad. By default, the pad vectors are calculated from the midpoint of the component to the midpoint of each pad (left image). However, the user can turn this vector in a different direction (right image).

Demonstration of pad vectors

Mathematical research

My intuition tells me that cubic Bézier curves, in particular the algorithm of De Casteljau, would work well with this problem, both in terms of implementation and computation time. The resulting arc could consist of several line segments connected together:

Connected line segments

Note that this is an interesting problem of line segment length versus resolution (which I need to think more about). It would be more intuitive to calculate the error compared to a continuous arc. If you increase two arcs, this indicates that this error would fluctuate.

Two segments

The questions)

It seems that this is becoming a classic optimization problem. I'm curious about the following:

  1. Is the Bézier curve the right approach to this type of problem?
  2. If so, should I minimize the arc distance (calculated by summing the discrete line segments) and the relative slope of each line segment? How do I calculate the latter?
  3. How would you approach the mentioned error calculation?

Many Thanks!

Plotting – Tracing the existing curve with star symbol

b1 = {1.8743,1.8784,1.88248,1.89049,1.89828,1.90587,1.91327,1.96335,2.03035,2.12536,2.23701,2.30098,2.34255,2.37175,2.3934,2.42334,2.44307,2.48725,2.51208,2.5173,2.52799,2.53164,2.53348,2.53533 2,53625,2,53745,2,53793,2,53894,2,53909,2,53543};
ks1 = {0.01, 1, 2, 4, 6, 8, 10, 25, 50, 100, 200, 300, 400, 500, 600, 800, 1000, 2000, 4000, 5000, 10000, 15000, 20000, 30,000, 40,000, 70,000, 100,000, 1 * 10 ^ 6,1. * 10 ^ 8,1. * 10 ^ 12};
data1 = Transpose[{ks1, b1}];
s1 = ListLogLinearPlot[data1, Joined -> True, 
  PlotStyle -> {Black, Thickness[0.01]}, AxesStyle -> Black,
PlotRange -> All]

How do you put a star symbol on the same drawing that follows the curve? I know we can do this with the plot marker, but I want the track to be tracked equally from start to finish. With my data points, I do not get the tracking right. How do you achieve this?

lightroom – How do I handle the tone curve (linear / medium / high contrast) and other development settings?

I use Lightroom 6.x and usually start with a linear curve. The sliders in the "Basic" area are adjusted accordingly.

I'm not a professional, I never adjust the tone curve point by point.

However, there are other predefined settings for the tone curve: medium or high contrast.

How should I choose a starting point? I think the tone curve, even if placed visually lower, must be selected before the remaining sliders are retouched.

Medium contrast, followed by an increase of the contrast with the base slider or a medium contrast, followed by a manual decrease?

What kind of photos are better retouched with each standard tone curve?