Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every x′>x and y′>y, we have f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′). Suppose f and g are supermodular, non-negative and increasing in both arguments. Is the function h(x,y)=f(x,y)∗g(x,y) supermodular? Answer The answer is no, not in general. I am going to assume f and g are supermodular, real polynomials. I’ll discuss what needs … Read more

Can a commutator of a special type be conjugate to its inverse?

Let H=H1∗H2 be a free product of non-trivial groups H1 and H2. We call an element h∈H hyperbolic if h∉Hgidef={g−1fg | f∈Hi} for i=1,2 and g∈H. Let h1,h2∈H be hyperbolic. It is not very difficult to prove that if h1 and h2 are hyperbolic and [h1,h2]def=h−11h−12h1h2≠1, then [h1,h2] is hyperbolic. Is it possible that the element [h1,h2] … Read more

The complex trigonometric function degenerates to the positive integer

For any integer $N \geq 2$, we have the identity: $$\frac{\ \prod _{n=1}^{N-1}\ \left(2+2\sum _{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)\ }{\prod _{n=1}^{N-1}\ \left(1+2\sum _{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)}=N$$ So how to prove it? Any help and suggestion will be appreciated, thank you! Answer Following Johann Cigler’s suggestion, set $q=e^{\frac{i\pi}{N}}$. We will need … Read more

Intuitive explanation of regularized products

I’ve come across some regularized product during study of zeta regularization . We can prove various results like : ∞!=∏∞k=1k=√2π I also know the proof using ζλ and all the standard stuff . Also , we know , ∞#=∏∞k=1pk=4π2 Where n# is primorial (product of first n primes) This is the question asked by C.Soulé, … Read more

Function with zeros plus/minus the primes

While playing with Cohen’s pari script prodeulerrat found a function. For s∈C define f(s)=∏p prime(1−s2p2) The product converges everywhere, no poles and the zeros are ±p. At integers one can tell if f(n)=0 via primality testing. Cohen’s script computes f(s) in O(|s|) and it iterates over primes. Q1 Is there an alternative way to compute f(s)? … Read more

Does the category of Lawvere theories have products?

I know Law has a tensor product, is closed with respect to that tensor product, and it has coproducts. Does it have products? My best guess at the cartesian product of Lawvere theories is the “intersection” of the theories: say $Th_1$ has a sort $X,$ function symbols $f_i\colon X^{n_i} \to X$ and a set of … Read more

Why is sup\sup f_- (n) \inf f_+ (m) = \frac{5}{4} ?

This question is an old question from mathstackexchange. Let f_- (n) = \Pi_{i=0}^n ( \sin(i) – \frac{5}{4}) And let f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) It appears that \sup f_- (n) \inf f_+ (m) = \frac{5}{4} Why is that so ? Notice \int_0^{2 \pi} \ln(\sin(x) + \frac{5}{4}) dx = Re \int_0^{2 \pi} \ln … Read more

Is it possible to express the functional square root of the sine as an infinite product?

Cross-post from MSE. It is known that the sine can be expressed as an infinite product: sin(x)=x∞∏n=1(1−x2n2π2). We can define that functional square root of a function g(⋅) to be the function f(⋅) that satisfies f(f(x))=g(x). The square root of the sine function with respect to function composition has been discussed previously on MO on … Read more

Condition to ensure that the product of closed maps be closed

If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map $$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \ldots, x_n) \mapsto (f_1(x_1), \ldots, f_n(x_n))$$ doesn’t need to be closed. However the question is: are there some nice … Read more