## Is this the simplest way to visually prove the “scalene trapezoid area formula”

Please refer to image. Is there a simpler way to visually prove the scalene trapezoid area formula?

My colleague has a spreadsheet full of links. She copy / pasted this from a variety of different sources.

For a number of reasons, I want to convert these from just plain copy / pasted links to HYPERLINK formula cells.

Is there anyway to do this?

## Google Sheets: Conditional Formatting – Custom Formula – Text Contains

Okay, so I have dates in column B that include weekdays in the following format: DDD, MMM DD, YYY (For example, Sat, Jan 19, 2019).

I want each row to auto format to a specific background colour based on the weekday in column B.

So, if B1 contains “Sat” I want row 1 to turn blue. If B100 contains “Sat” I want row 100 to turn blue.

I have tried using suggestions from these threads like regexmatch: Conditional formatting based on portion of text

or countif: https://stackoverflow.com/questions/27723102/finding-partial-texts-in-conditional-formatting-custom-formula-is

But neither of these seem to work for me.

I am at a loss. Any help is appreciated.

## A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?

I have the following conjecture involving a possible new formula for the class number of the quadratic field $$\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$$ with $$p$$ an odd prime.

Conjecture. Let $$p$$ be an odd prime and let $$p^*=(-1)^{(p-1)/2}p$$. Then the class number $$h(p^*)$$ of the quadratic field $$\mathbb Q(\sqrt{p^*})$$ coincides with the number
$$\frac{(\frac{-2}p)}{2^{(p-3)/2}p^{(p-5)/4}}\det\left[\cot\pi\frac{jk}p\right]_{1\le j,k\le (p-1)/2},$$ where $$(\frac{\cdot}p)$$ is the Legendre symbol.

This is Conjecture 5.1 in my preprint arXiv:1901.04837. I have checked it for all odd primes $$p<29$$. Note that $$h(p^*)=1$$ for each odd prime $$p<23$$, and $$h(-23)=3$$.

Here I invite some of you to check this conjecture further. My computer cannot check it even for $$p=29$$.

## Derivation of Perceptron weight update formula

I’ve started out studying Machine Learning and am currently reading up about how a single perceptron works. From the wikipedia page, my understanding is as follows: suppose we have an input sample $$\mathbf{x} = [x_1, \ldots, x_n]^T$$, an initial weight vector $$\mathbb{w} = [w_1, \ldots, w_n]^T$$. Let the true output corresponding to $$\mathbf{x}$$ be $$y’$$.

The output given by the perceptron is $$y = f(\sum_{i=0}^n w_ix_i)$$, where $$w_0$$ is the bias and $$x_0=1$$. If $$\eta$$ is the learning rate, the weights are updated according to the following rule: $$\Delta w_i = \eta x_i(y’-y)$$

This is according to wikipedia. But I know the weights are updated on the basis of the gradient descent method, and I found another nice explanation based on the gradient descent method HERE. The derivation there results in the final expression for weight update:

$$\Delta w_i = \eta x_i(y’-y)\frac{df(S)}{dS}$$

where $$S = \sum_{i=0}^{n}w_ix_i$$. Is there a reason why this derivative term is ignored? There was another book that mentioned the same weight update formula as Wikipedia, without the derivative term. I’m pretty sure we can’t just assume $$f(S) = S$$.

## Fórmula compleja con PHP/MySqli

Estoy haciendo unos cálculos en mysqli que no encuentro la forma de hacerlo. Tengo una tabla, que suma los puntos de cada equipo, perfecto, pero necesito hacer un cálculo adicional en la cual necesito ayuda.

Mi problema es con la columna en negrilla marcada como SB. Para calcularla se debe proceder de la siguiente manera:
1.- Cada valor a sumar es el de la columna SUM.
2.- Solo se suma el valor, cuando un equipo ganó, es decir tiene 1 punto.
3.- Sumamos la mitad de puntos, de aquellos con quienes entablamos.
4.- Si un equipo perdió, no se suma nada.

Ver la tabla:

Ejemplo 1: El equipo RDO:
RDO perdió con PRC, suma 0
RDO ganó a BRA, suma 2 de la columna SUM
RDO empató con ARG, suma 1/2 de 4.5 de la columna SUM
RDO ganó a Ven, suma 2 de la columna SUM
RDO ganó a Nic, suma 2 de la columna SUM
SB de RDO = 2+4.5/2+2+2 = 8.25

Todo esto lo hago en excel si problema, pero en php/mysqli, necesito ayuda.

Alguna idea?

## Auto-Sort script which excludes formula if ascending False

I am looking to exclude formulas or blank cells when I am running my auto sort. I used the following… It sorts fine, but I wanted the numbers to be ascending: False. When I change this, it takes account of the formula I have set for my column I am sorting by and puts them all at the top and my actual data points to at the bottom where my formulas stopped… Is there a way to exclude this?

function onEdit(e) {   var ss = SpreadsheetApp.getActiveSpreadsheet();   var sheet = ss.getSheetByName("G2 List")   var range = sheet.getRange("A2:T200");   var columnToSortBy = 14;   // Sorts by the values in column 14 (N)  range.sort({column: 14, ascending: false});  } 

## Google Sheets Formula for Proper Case with Apostrophe

I’m using a Google Docs spreadsheet glued together (not be me) from multiple sources where some of the names are in all caps. In trying to clean web ready content, I have good success with the proper(#ref) function except when it is a name with an apostrophe.

If the cell contains SMITTY'S running it through the proper() function returns Smitty'S and is ugly. Any clever ideas around this?

## Generalize Wu formula to general Bockstein homomorphisms

The classical Wu formula claims that $$Sq^1(x_{d-1})=w_1(TM)\cup x_{d-1}$$ on a $$d$$-manifold $$M$$, where $$x_{d-1}\in H^{d-1}(M,\mathbb{Z}_2)$$.

I wonder whether there is a generalization of the classical Wu formula to general Bockstein homomorphisms. We consider the Bockstein homomorphism $$\beta_{(2,2^n)}:H^*(-,\mathbb{Z}_{2^n})\to H^{*+1}(-,\mathbb{Z}_2)$$ which is associated to the extension $$\mathbb{Z}_2\to\mathbb{Z}_{2^{n+1}}\to\mathbb{Z}_{2^n}$$.

I guess there is a generalized Wu formula: $$\boxed{\beta_{(2,2^n)}(x_{d-1})=\frac{1}{2^{n-1}}\tilde w_1(TM)\cup x_{d-1}}$$ on a $$d$$-manifold $$M$$, where $$x_{d-1}\in H^{d-1}(M,\mathbb{Z}_{2^n})$$.

Here $$\tilde w_1(TM)$$ is the twisted first Stiefel-Whitney class of the tangent bundle $$TM$$ of $$M$$ which is the pullback of $$\tilde w_1$$ under the classifying map $$M\to BO(d)$$. Let $$\mathbb{Z}_{w_1}$$ denote the orientation local system, the twisted first Stiefel-Whitney class $$\tilde w_1\in H^1(BO(d),\mathbb{Z}_{w_1})$$ is the pullback of the nonzero element of $$H^1(BO(1),\mathbb{Z}_{w_1})=\mathbb{Z}_2$$ under the determinant map $$B\det:BO(d)\to BO(1)$$.

The right hand side makes sense since $$2\tilde w_1(TM)=0$$.

Can you help me to prove or disprove the boxed formula above?

Thank you!

## Proof that 2^yx+2^y-1 is an closed formula of f(1,x,y)

I should proof that 2^yx+2^y-1 is an closed formula of g(1,x,y) with Induktion or somthing else. Given is

$$g(n,x,y)=\begin{cases} x+y,\quad if\quad n=0 \ x,\quad \quad \quad if\quad n>0\quad and\quad y=0 \ f(n-1,f(n,x,y-1),f(n,x,y-1)+1,\quad else \end{cases}$$