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In one of my sketchbooks, I found the Diophantine equation $$x^2(x+1)^2 = 2y(y+1)$$ and a solution I apparently came up with at the time. The problem is, I can’t figure out where I found this equation, or what it might be related to. I’m guessing it’s a relatively simple reduction (via rational substitutions) from some other problem or equation (or set of equations). Can anyone help me “reverse-engineer” it, or at least suggest problems it might be closely related to?

I have a question about ensemble learning methods. When should ensemble learning be used and when is better performance than a single model guaranteed? More specifically: Are there theoretical guarantees or conditions under which ensemble methods are provably better than individual base models? What are the practical indicators that suggest ensemble learning might improve performance? In which scenarios might ensemble methods fail to improve (or even worsen) results compared to a well-tuned single model? I'm particularly interested in both the theoretical foundations and practical heuristics for deciding when to employ ensemble methods. Any references to relevant articles or theoretical findings would be helpful, as I have not been able to find any good sources in my research so far.

I am studying the characterization of derivatives in real analysis. I already know that if a function $f$ is a derivative of some function $F$, it must satisfy two conditions: 1.It must have the Darboux Property (Intermediate Value Property). 2.It must be a Baire Class 1 function. However, I know these two conditions combined are not sufficient. I came across this paper: [DARBOUX FUNCTIONS OF BAIRE CLASS ONE AND DERIVATIVES] by [Neugebauer] (1962).enter link description here I am having trouble understanding the logical flow (proof strategy) presented in this paper. The conditions derived seem quite complex. My Questions: 1.Could someone explain the intuition behind the necessary and sufficient condition provided in this paper? 2.Is there a more modern or "simpler" form of the necessary and sufficient condition for a function to be a derivative?Thank you!

I have trouble understanding the difference in the following problem. I have the function F(x) which is not differentiable at x=a $$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for $x<a$}\\ G(x) & \text{ for $x\geq a$}\end{array}\right.$$ and i want the derivative of F(x), F'(x) to be continuous in x=a. I can use a cubic smoothstep function $S_3(x)$ such that $$F_{smooth}(x)=S_3(\frac{x-a}{\epsilon})G(x)$$ which by definition is $$S_3(a)=0, \qquad S_3(a+\epsilon)=1\\ S_3'(a)=0, \qquad S_3'(a+\epsilon)=0$$ such that the derivative of the function is $$F'_{smooth}(x)=S'_{3}G(x)+S_{3}G'(x)$$ So for $F'_{smooth}(a)=0$ and $F'_{smooth}(a+\epsilon)=G'(a+\epsilon)$ which makes the derivative continuous. Another definition i found is to use cubic hermite splines such that $$F(x):=\left\lbrace\begin{array}{cc} 0 & \text{ for x<a}\\ P_{3}(x) & \text{ for $a\leq x\leq a+\epsilon$}\\ G(x) & \text{ for $x\geq a+\epsilon$}\end{array}\right.$$ where the cubic polynom $P_{3}(x)$ is defined with $$P_{3}(a)=0 \quad P_{3}(a+\epsilon)=G(a+\epsilon)\\ P'_{3}(a)=0 \quad P_{3}'(a+\epsilon)=G'(a+\epsilon)$$ I plotted both smoothing functions and they look differently.\ So what is the actual difference between these two approaches? Or are they equally? From their definitions, they lead to the exact same values for $x<a$ and $x>a+\epsilon$ but within $[a,a+\epsilon]$ the functions are $$ \begin{array}{cl}F1_{smooth}(x)=S_{3}G(x) & F1'_{smooth}(x)=S'_{3}G(x)+S_{3}G'(x)\\ F2_{smooth}(x)=P_{3}(x) & F2'_{smooth}(x)=P'_{3}(x)\end{array}$$ If they aren't the same, what are the advantages and disadvantages?

Let A = {1, 2, 3, …, n}. Let B be the set of all bijections from A to A (i.e., the symmetric group S_n). Let C be a non-empty subset of B such that for every permutation tau in B\C(B setminus C), there exist permutations f and g in C (which may be the same) such that tau=f •g. Determine the minimum possible cardinality of C.