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Consider a $2 \times 4$ matrix $A = \begin{bmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ \end{bmatrix}$. Its all minors of order 2, such as $A_{13} = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \\ \end{vmatrix}$ representing the minor formed by selecting the first and third columns, satis...

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The functional equation identity, (assuming also $\,f(-x)=-f(x)\,$ for all $\,x$), $$f(a)f(b)f(a\!-\!b) + f(b)f(c)f(b\!-\!c) + f(c)f(a)f(c\!-\!a) + f(a\!-\!b)f(b\!-\!c)f(c\!-\!a) = 0 \tag{1}$$ for all $\,a,b,c\,$ has solutions $f(x)=k_1\sin(k_2\,x)$ and $f(x)=k_1\tan(k_2\,x)\,$ with $\,k_1,k_... 7 hours later… 12:45 PM 6 I was reading about Many-sorted logic and I kept seeing a lot of authors claiming that "When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first-order logic". I get that this is done by introducing, for every sort$a, a unary predic... 32 Here is something I've found on the internet \begin{aligned} f-\int f&=1\\ \left(1-\int\right)f&=1\\ f&=\left(\frac1{1-\int}\right)1\\ &=\left(1+\int+\int\int+\dots\right)1\\ &=1+\int1+\int\int1+\dots\\ &= 1+x+\frac{x^2}2+\dots\\ &= e^x \end{aligned} At first I interpreted this as a joke, but... 3 Appell's two-variable functionsF_1, F_2, F_3$and$F_4\$ are known to have numerous uses in applied mathematics, notably mathematical Physics. I am looking for generalized Laplace transforms (if they exist) of these functions relative to one of the two variables (I found at least two references ...