Integral operator

Assume that $ A: L^1([0,1]) ]\to L^1[0,1]$ is an integral operator with a real Kernel G(x,y). Then its adjoint operator $ A^*:L^\infty([0,1]) ]\to L^\infty [0,1]$ is an integral operator with a real Kernel G(y,x).

My question is. Whether the norms of $ A$ and $ A^*$ coincide?

Formula for exponential integral over a cone

While reading ‘Computing the Volume, Counting Integral points, and Exponential Sums’ by A. Barvinok (1993), I came across the following:

“Moreover, let $ K$ be the conic hull of linearly independent vectors $ u_{1}, …, u_{n}$ so $ K = co(u_{1}, …, u_{n})$ . Denote by $ K^{*} = \{x\in \mathbb{R}^{n}: <x, y> \le 0 \text{ for all } y\in K\}$ the polar of K. Then for all $ c \in \text{Int} K^{*}$ we have

\begin{equation} \int_{K}exp(<c, x>)dx = |u_{1} \land … \land u_{n}|\prod_{i=1}^{n}<-c_{i}, u_{i}>^{-1} \end{equation}

To obtain the last formula we have to apply a suitable linear transformation to the previous integral. “

I have tried proving this but I can’t find relevant links to help me. Also, I’m unsure what $ |u_{1} \land … \land u_{n}|$ is supposed to mean. Would greatly appreciate if someone could point me to a relevant resource or provide proof. Thanks!

Density of integral values of a rational function

Let $ \mathbf{x} = (x_1, \cdots, x_n)$ , and consider a rational function $ F : \mathbb{R}^n \rightarrow \mathbb{R}$ be given by

$ $ \displaystyle F(\mathbf{x}) = \sum_{i = 1}^m \frac{Q_i(x_1, \cdots, x_{n-1})}{R_i(x_1, \cdots, x_{n-1})} x_n^i,$ $

where $ Q_i, R_i$ are non-constant polynomials with integer coefficients. Moreover we may assume that $ R_i > 0$ for all $ \mathbf{x} \in \mathbb{R}^n$ , so $ F$ is well-defined everywhere.

I am trying to understand the counting function

$ $ N_F(X) = \# \{\mathbf{x} \in \mathbb{Z}^n : \lVert \mathbf{x} \rVert_\infty \leq X, F(\mathbf{x}) \in \mathbb{Z} \}.$ $

In particular, given $ x_1, \cdots, x_{n-1}$ one can always find $ x_n \in \mathbb{Z}$ such that $ F(\mathbf{x}) \in \mathbb{Z}$ , but the smallest possible choice of such $ x_n$ could very well be extremely large. Thus it is perhaps best to consider all of the $ x_i$ ‘s as varying at once.

The above observation also gives a somewhat straightforward upper bound for $ N_F(X)$ . In particular, having chosen $ x_1, \cdots, x_{n-1}$ the resulting function $ f(x) = F(x_1, \cdots, x_{n-1}, x)$ is then a rational polynomial in a single variable, and we can clear its denominator to obtain $ g(x)$ say, and then the question is equivalent asking for the density of $ x \in [-X,X]$ such that $ g(x)$ satisfies a certain congruence. However the modulus, equal to $ \text{LCM}_{1 \leq i \leq m} R_i(x_1, \cdots, x_{n-1})$ is typically much larger than $ X$ , since the $ R_i$ ‘s are assumed to be positive definite and in particular not linear. Therefore usually there is at most one root of $ g(x)$ in $ [-X,X]$ . It thus follows that $ N_F(X) = O\left(X^{n-1}\right)$ . I am looking for a bound that beats this, and perhaps close to what one might expect to be the exact asymptotic order.

Double integral differentiation

Could somebody tell me how do I get the right side from the left side – which rule of differentiation of the double integral produces this?

\begin{align*}\frac{\mathrm{d}}{\mathrm{d}x} \int_{v=x}^\infty &\int_{u=0}^x f(v, u)\cdot g (v)\cdot g(u)\, \mathrm{d} u\, \mathrm{d} v =\ &=\int_0^\infty g(v)\;\mathrm{d}v\cdot \frac{\mathrm{d}}{\mathrm{d}x} \int_0^x f (v, u)\; g(u)\; \mathrm{d}u + \int_x^\infty \;g(u)\;\mathrm{d}u\cdot \frac{\mathrm{d}}{\mathrm{d}x} \int_0^x f (v, u)\; g(v)\; \mathrm{d}v\end{align*}

My ignorance is probably quite profound, and this is something really simple, but I cannot see. It’s been quite a while since I last dabbled in double integrals.

When I type

d/dx(integral from v=x to infinity integral from u=0 to x f(u,v)g(u)g(v) du dv)

into WolframAlpha, I get

\begin{align*}\frac{\mathrm{d}}{\mathrm{d}x} \Bigg(\int_{v=x}^\infty \int_{u=0}^x &f(v, u)\, g (v)\, g(u)\, \mathrm{d} u\, \mathrm{d} v\Bigg) =\ &=\int_x^\infty g(v) g(x) f(x,v) \mathrm{d}v – \int_0^x g(u) g(x) f(u,x) \mathrm{d}u,\end{align*}

which is correct, but I first need to understand how to arrive at the first formula, in order to apply the Leibnitz rule.

If someone knows how to align these equations here beautifully, do tell.

Distribution of a linear pure-birth process’s integral

I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:

$ $ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k \bigg] $ $ where $ (Y_t)_t$ is a Yule process of intensity $ \lambda$ and $ k$ an arbitrary positive integer.

Are there any results regarding the $ Z_t$ ‘s distribution?

Homology of linear groups over integral domains and their field of fractions

Let $ A$ be a noetherian integral domain of finite Krull dimension with the field of fractions of $ F$ . Consider the natural injections $ i_n:GL_n(A)\hookrightarrow GL_{n+1}(A)$ , $ j_n:GL_n(F)\hookrightarrow GL_{n+1}(F)$ , $ \varphi_n:GL_n(A)\hookrightarrow GL_n(F)$ and similarly $ \varphi_{n+1}$ . Let $ x\in H_k(BGL_{n+1}(A))$ and assume that $ \varphi_{*(n+1)}(x)$ lies in the image of $ j_{*n}$ i.e. for some $ y\in H_k(BGL_n(F))$ , $ j_{*n}(y)=\varphi_{*(n+1)}(x)$ . Does that imply that $ y$ is in the image of $ \varphi_{*n}$ ?