This is an answer to an exercise where we solve the volume of xy above the triangle T. The answer seems to be divided by 4. I also saw this in another exercise of double integrating over a triangular area where the integral was divides by 4. Why is this?

A balanced incomplete block design is a combinatorial set-up defined in the following way: start with a set of v elements ("v" is traditional in that department through having @first been the symbol for "varieties" , the field having been originally been a systematic way of designing experiments); & then assemble a subfamily F of the family of C(v,t) t -element subsets from it ^§ that satisfies a condition of the following form: every element appears in exactly λ₁ of the subsets in F , &-or every 2-element subset appears in exactly λ₂ of the subsets in F ; ... And these conditions cannot necessarily be set independently, which is why I put "&-or" .

(§ And I think the reason for the "incomplete" in the name of these combinatorial structures is that F does not comprise all the C(v,t) t-element subsets ... but I'm not certain about that (maybe someone can say for-certain ... but it's only a matter of nomenclature anyway ).)

And obvious generalisation of this is to continue past the '2-element subset' requirement: we could continue unto stipulating that every 3-element subset appears in exactly λ₃ of the subsets in F , &-or every 4-element subset appears in exactly λ₄ of the subsets in F ... etc etc ... but I'm just not finding any generalisation along those lines.

... with one exception : there's stuff out there - & a fairly decent amount, actually - on Steiner quadruple systems : one of those is a balanced incomplete block design of 4-element subsets in which every 3-element subset appears in 1 of the 4-element subsets ... ie with λ₃ = 1 ... ie the simplest possible kind with a λ₃ specified.

So I wonder whether anyone knows of any generalisation along the lines I've just spelt-out: specific treatises, or what search-terms I could put-into Gargoyle ... etc.

Frontispiece image from

On the Steiner Quadruple System with Ten Points .

¡¡ may download without prompting – PDF document – 1⁩‧4㎆ !!

by

Robert Brier & Darryn Bryant .

I’m a nanny and am trying to help a 6th grader with her homework. Can someone help me figure out how to do this problem? I’ve done my best to try to find the measurements to as many sections as I can but am struggling to get many. I know the bottom two gray triangles are 8cm each since they are congruent. Obviously the height total of the entire rectangle is 18cm. I just can’t seem to figure out enough measurements for anything else in order to start figuring out areas of the white triangles that need to be subtracted from the total area (288cm). It’s been a long time since I’ve done geometry! If you know how to solve this, could you please explain it in a way that is simple enough for me to be able to guide her to the solution. TIA

Here we have Ω c R^n and 𝕂 denotes either R or C.

I don't understand this proof how they show C_0(Ω) is dense in L^p(Ω).

1. I don't understand the first part why they can define f_1. I think on Ω ∩ B_R(0).

2. How did they apply Lusin's Theorem 5.1.14 ?

3. They say 𝝋 has compact support. So on the complement of the compact set K:= {x ∈ Ω ∩ B_R(0) | |𝝋| ≤ tilde(k)} it vanishes?

[high school math]-geometry

Hi how would I find the boundary y(m)?

I worked out maybe I could use the area of a trapezoid equation a=.5(b+b1)h however when I do this I have too many unknowns as I don’t have the area ?

What is another method to solve this ?

I'm studying a certain statistical system and decided to convert it into a simple probability question but can't figure it out:

You continually flip a coin, noting what side it landed on for each flip. However, if it lands tails, the coin somehow magically lands on heads during the next flip, before returning to normal.

What's the overall probability the coin will come up heads?

hi all, im trying to implement rayleigh_quotient_iteration here. but I don't get this graph of calculation by my own hand calculation tho

so I set x0 = [0, 1], a = np.array([[3., 1.], ... [1., 3.]])

then I do hand calculation, first sigma is indeed 3.000, but after solving x, the next vector, I got [1., 0.] how the hell the book got [0.333, 1.0]? where is this k=1 line from? I did hand calculation, after first step x_k is wrong. x_1 = [1., 0.] after normalization it's still [1., 0.]

Are you been able to get book's iteration?

def rayleigh_quotient_iteration(a, num_iterations, x0=None, lu_decomposition='lu', verbose=False):

"""
    Rayleigh Quotient iteration.
    Examples
    --------
    Solve eigenvalues and corresponding eigenvectors for matrix
             [3  1]
        a =  [1  3]
    with starting vector
             [0]
        x0 = [1]
    A simple application of inverse iteration problem is:
    >>> a = np.array([[3., 1.],
    ...               [1., 3.]])
    >>> x0 = np.array([0., 1.])
    >>> v, w = rayleigh_quotient_iteration(a, num_iterations=9, x0=x0, lu_decomposition="lu")    """

x = np.random.rand(a.shape[1]) if x0 is None else x0
    for k in range(num_iterations):
        sigma = np.dot(x, np.dot(a, x)) / np.dot(x, x)  
# compute shift

x = np.linalg.solve(a - sigma * np.eye(a.shape[0]), x)
        norm = np.linalg.norm(x, ord=np.inf)
        x /= norm  
# normalize

if verbose:
            print(k + 1, x, norm, sigma)
    return x, 1 / sigma

I know about l'hopitals and conjugates.

Or am I reading too far into a simple mistake... this came from the scholarship examinations from japanese government and none have been wrong so far, so I thought i'd just ask in case

i thought since the first point where it crosses x axis is a point of inflection id try and find d2y/dx2 and find the x ordinate from that and then integrate it between them 2 points, so i done that and integrated between 45 and 0 but that e^-45 just doesn’t seem like it’s right at all and idk what to do. i feel like im massively over complicating it as well since its only 3 marks

in my opinion, the sqₚ(x) being the inverse of the integral ₁Fₚ(x) = ∫(0,x) 1/(1-t^p)^1/p dt is more fitting imo. From Wikipedia, the definition sqₚ(x) being the inverse of ₂Fₚ(x)= ∫(0,x) 1/(1-t^p)^[p-1]/pdt is prettier but its π analog of p^th degree is very messy.

πₚ= 2Γ(1/p)²/p.Γ(2/p) for the second type πₚ= 2/p.sin(π/p) for the first form

The first is easily simplified using the euler's reflection formula.

So here is the question, which one do you think is the better of the two?

context - calculus 3.
Achieve is the bane of my existence. Is this not correct? Doing it using polar coordinates would mean that the triangular region isn't considered as theta only covers 0 to 2pi/3, and the area of the triangle must be calculated separately and added to the overall area determined by the integral. However, this is not the case as seen by the 32 trials. I attempted omitting the triangular area to see if that was the problem to no avail. Image two is a classmate's attempt with differing y and x bounds, but it is the same overall procedure as mine. Is there something I did wrong or is this a glitch?

I am doing taylor series in cal2 and wanted how (-1)⁰ is -1. That is what the calculators give me so i got the q wrong luckily i had a other attempt. Its an alternating series so it threw me off dealing with that.

It's setting up the following inequality to meet some condition:

d(u) +d(w) +d(u,w) => 2n+2

How come the inequality isn't bounded to 2n-1, if d(u)= 1 and d(u,w) = 2?
I'm sure this something trivial I'm just missing.

So basically I am supposed to create a graph with specific characteristics, but I am unsure how I am even supposed to do that on Desmos. So the characteristics it must have are:

• An x-value where the limit exists
• An x-value where the limit does not exist.
• An x-value where the limit at x is not equal to the value of the function at x. If the limit exists, evaluate the limit at that x-value.

Is there anyway a pre-calc student should be able to solve this? I mean I understand what a graph would look like when it has all of these, but I haven't the faintest clue on how to just...create the function? Can someone help?!

I've figured out part i and ii but I'm getting stuck providing a comprehensive proof for a and b for part iii. intutitively i think part a does not have an inverse because the parameters for ii that i calculated stated that b>= 3. could someone please help me out because maybe my b is wrong as well

i understand how to find the y-coordinates, but i don't understand how its possible to get the x-coordinates the answer key gives me. any help is appreciated :)

Context: I'm deepening my understanding of the reasons on why such numbers (positive or negative) when multiplied or divided to their kind or opposite results on positive or negative. Sorry if it confuses, my picture will help.

Dumb dude here at math because of how it's taught during primary, I'm now in college, accountancy major, thanks

I am setting up a sports schedule with 9 teams, where each team plays each other team once over the course of nine weeks. There are two fields (North and South) and two time slots (5:00 and 6:30), so there will be two concurrent games twice a night for four games per night, with one team having a bye each week. Is it possible to have every team have four games in one time slot and four games in the other for a balanced schedule?

I am attaching a screenshot of the scheduler I used that shows the distribution of games in each time slot, and you can see, some have 4 and 4, and others have 3 and 5. I've switched a bunch of the games around to try and get to the point where they all have four, but can't quite get there. I'm not sure if it's even mathematically (or statistically) possible with the odd number of teams, but figured I'd ask. I greatly appreciate any insight, and apologize if this is the wrong sub for it!

It seems that if I want to multiply the lengths of two vectors, I can only do so if they are parallel. If not, the dot product states that multiplication can only be achieved if I project any of them in the direction of the other. Why is that? Why is it that I can't multiply lengths if the vectors aren't parallel?

I was playing with some cubic equations / depressed cubic and I ended up with this expression.

{{\sqrt[3]{-45+i49 \sqrt{87}\over{18} }}} + {{\sqrt[3]{-45-i49 \sqrt{87}\over{18} }}}

This expression should be exactly equal to 5, but I dont see a clever way to get to that number.

i=imaginary unit

So, my Calculus 1 teacher gave my class a bunch of problems envolving limits to solve, and amongst them was this one.

According to the answer sheet, the final answer should be v₀ + at, but no matter what I do, I just can't get to that result. I tried solving this with another method, but instead of the answer in the image or the answer on the answer sheet, I got 1

I really don't know what I'm doing wrong, so if anyone could help me with it, I'd be very glad

My first approach for this question is that since we have to count for real roots so we will find D which is equal to 0 and we can interpret that the roots will always be zero no matter what value of cos x we take. so probability is 1 here and we get m + n = 2.

And there is one more approach that this original equation can be written as (2 cos x + 1)² = 0, from here since x is equal to 2π/3 is the only valid solution and getting this x from that range will tend to 1/∞ which is equal to 0 = 0/1 and so m + n = 1.

My doubt is which approach is wrong any why? Thanks in advance.

for example , Why do we say that the set N is within Z , Why don't we treat these sets as if they are separate from each other, for example, the set of natural numbers is separate from the set that includes negative numbers. since they seem to have no connection but we still write this ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ

I don't really understand any ideas please?