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Philipp J. di Dio

Publications and source records attributed to Philipp J. di Dio.

At least 19 recordsLinked to original sources
arXiv2026

Time-dependent moments from partial differential equations and the time-dependent set of atoms

We study the time-dependent moments and associated polynomials arising from the partial differential equation $\partial_t f = νΔf + g\cdot\nabla f + h\cdot f$, and consider in detail the dual equation. For the heat equation we find that several non-negative polynomials which are not sums of squares become sums of squares under the heat equation in finite time. We show that every non-negative polynomial in $\mathbb{R}[x,y,z]_{\leq 4}$ becomes a sum of squares in finite time under the heat equation. We solve the problem of moving atoms under the equation $\partial_t f = g\cdot\nabla f + h\cdot f$ with $f_0 = μ_0$ being a finitely atomic measure. The time evolution $μ_t = \sum_{i=1}^k c_i(t)\cdot δ_{x_i(t)}$ of the atom positions $x_i(t)$ are described by the transport term $g\cdot\nabla$ and the time-dependent coefficients $c_i(t)$ have an explicit solution depending on $x_i(t)$, $h$, and $\mathrm{div}\, g$.

math.FA↗
arXiv2026

Making Non-Negative Polynomials into Sums of Squares

We study linear operators $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$, especially for the purpose to move sets $S\subseteq\mathbb{R}[x_1,\dots,x_n]$ into cones $C\subseteq\mathbb{R}[x_1,\dots,x_n]$: $TS\subseteq C$. We develop the theory of (semi-)groups of operators $(e^{tA})_{t\in\mathbb{R}}$ on $\mathbb{R}[x_1,\dots,x_n]$, which requires techniques from regular Fréchet Lie groups. We study the special case of making non-negative polynomials $\mathrm{Pos}(K)_{\leq 2d}$ with $K\subseteq\mathbb{R}^n$ and $\mathrm{int}\, K\neq \emptyset$ into sums of squares: $\tilde{T}\mathrm{Pos}(K)_{\leq 2d}\subseteq \sum\mathbb{R}[x_1,\dots,x_n]_{\leq d}^2$. With $N:=\dim\mathbb{R}[x_1,\dots,x_n]_{\leq 2d} = \binom{n+2d}{n}$, for $\tilde{T}$, a memory of at most $2N+1$ is required. Matrix multiplications $\tilde{T}M$, $M\tilde{T}$, $\tilde{T}^{-1}M$, and $M\tilde{T}^{-1}$ of $\tilde{T}$ with any $M\in\mathbb{R}^{N\times N}$ require at most $4N^2+1$ operations. Transformations $\tilde{T}v$ and $\tilde{T}^{-1}v$ of vectors $v\in\mathbb{R}^N$ require at most $4N+1$ operations. Calculating $\tilde{T}^{-1}$ of $\tilde{T}$ requires only one (!) operation.

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arXiv2025

Characterization of Matrix $K$-Positivity Preserver for $K=\mathbb{R}^n$ and for Compact Sets $K\subseteq\mathbb{R}^n$

For any closed $K\subseteq\mathbb{R}^n$, in [P.\ J.\ di\,Dio, K.\ Schmüdgen: $K$-Positivity Preserver and their Generators, SIAM J.\ Appl.\ Algebra Geom.\ 9 (2025), 794--824] all $K$-positivity preserver have been characterized, i.e., all linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $Tp\geq 0$ on $K$ for all $p\geq 0$ on $K$. An important extension of polynomials $\mathbb{R}[x_1,\dots,x_n]$ with real coefficients are polynomials $\mathbb{R}^{m\times m}[x_1,\dots,x_n]$ with matrix coefficients. Non-negativity on $K$ for matrix polynomials with Hermitian coefficients $\mathrm{Herm}_m$ is then $p(x)\succeq 0$ for all $x\in K$. In the current work, we investigate linear maps $T:\mathrm{Herm}_m[x_1,\dots,x_n]\to\mathrm{Herm}_m[x_1,\dots,x_n]$. We focus on matrix $K$-positivity preserver, i.e., $Tp\succeq 0$ on $K$ for all $p\succeq 0$ on $K$. For $K=\mathbb{R}^n$ and compact sets $K\subseteq\mathrm{R}^n$, we give characterizations of matrix $K$-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets $K\subseteq\mathbb{R}^n$ with $K\neq \mathbb{R}^n$ and $K$ non-compact.

math.FA↗
arXiv2025

Absolutely continuous representing measures of complex sequences

In 1989, A. J. Duran [Proc. Amer. Math. Soc. 107 (1989), 731-741] showed, that for every complex sequence $(s_α)_{α\in\mathbb{N}_0^n}$ there exists a Schwartz function $f\in\mathcal{S}(\mathbb{R}^n,\mathbb{C})$ with $\mathrm{supp}\, f\subseteq [0,\infty)^n$ such that $s_α= \int x^α\cdot f(x)~\mathrm{d}x$ for all $α\in\mathbb{N}_0^n$. It has been claimed to be a generalization of the result by T. Sherman [Rend. Circ. Mat. Palermo 13 (1964), 273-278], that every complex sequences is represented by a complex measure on $[0,\infty)^n$. In the present work we use the convolution of sequences and measures to show, that Duran's result is a trivial consequence of Sherman's result. We use our easy proof to extend the Schwartz function result and to show the flexibility in choosing very specific functions $f$.

math.FA↗
arXiv2024

The Hadamard Product of Moment Sequences, Diagonal Positivity Preservers, and their Generators

In this work we investigate special aspects of positivity preservers and especially diagonal positivity preservers, i.e., linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $Tx^α= t_αx^α$ holds for all $α\in\mathbb{N}_0^n$ with $t_α\in\mathbb{R}$ and $Tp\geq 0$ on $\mathbb{R}^n$ for all $p\in\mathbb{R}[x_1,\dots,x_n]$ with $p\geq 0$ on $\mathbb{R}^n$. We discuss representations of $T$, give characterizations of diagonal positivity preservers, and compare these to previous (partial) results in the literature. On the side we get a full characterization of linear maps preserving moment sequences and a new proof of Schur's product formula. The tool of diagonal positivity preservers simplifies several other existing proofs in the literature. We give a full characterization of generators $A$ of diagonal positivity preservers, i.e., $e^{tA}$ is a diagonal positivity preserver for all $t\geq 0$. We give the connection of these generators to infinitely divisible moment sequences.

math.FA↗
arXiv2024

$K$-Positivity Preservers and their Generators

We study $K$-positivity preservers with given closed $K\subseteq\mathbb{R}^n$, i.e., linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $T\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds, and their generators $A:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$, i.e., $e^{tA}\mathrm{Pos}(K)\subseteq\mathrm{Pos}(K)$ holds for all $t\geq 0$. We characterize these maps $T$ for any closed $K\subseteq\mathbb{R}^n$ in Theorem 4.5. We characterize the maps $A$ in Theorem 5.12 for $K=\mathrm{R}^n$ and give partial results for general $K$. In Proposition 6.1 and 6.3 we give maps $A$ such that $e^{tA}$ is a positivity preserver for all $t\geq τ$ for some $τ>0$ but not for $t\in (0,τ)$, i.e., we have an eventually positive semi-group.

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arXiv2024

On Positivity Preservers with constant Coefficients and their Generators

In this work we study positivity preservers $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ with constant coefficients and define their generators $A$ if they exist, i.e., $\exp(A) = T$. We use the theory of regular Fréchet Lie groups to show the first main result. A positivity preserver with constant coefficients has a generator if and only if it is represented by an infinitely divisible measure (Main Theorem 4.7). In the second main result (Main Theorem 4.11) we use the Lévy--Khinchin formula to fully characterize the generators of positivity preservers with constant coefficients.

math.AG↗
arXiv2024

An Introduction to T-Systems -- with a special Emphasis on Sparse Moment Problems, Sparse Positivstellensätze, and Sparse Nichtnegativstellensätze

These are the lecture notes based on [dD23] for the (upcoming) lecture "T-systems with a special emphasis on sparse moment problems and sparse Positivstellensätze" in the summer semester 2024 at the University of Konstanz. The main purpose of this lecture is to prove the sparse Positiv- and Nichtnegativstellensätze of Samuel Karlin (1963) and to apply them to the algebraic setting. That means given finitely many monomials, e.g. $1, x^2, x^3, x^6, x^7, x^9,$ how do all linear combinations of these look like which are strictly positive or non-negative on some interval $[a,b]$ or $[0,\infty)$, e.g. describe and even write down all $f(x) = a_0 + a_1 x^2 + a_2 x^3 + a_3 x^6 + a_4 x^7 + a_5 x^9$ with $f(x)>0$ or $f(x)\geq 0$ on $[a,b]$ or $[0,\infty)$, respectively. To do this we introduce the theoretical framework in which this question can be answered: T-systems. We study these T-systems to arrive at Karlin's Positiv- and Nichtnegativstellensatz but we also do not hide the limitations of the T-systems approach. The main limitation is the Curtis$-$Mairhuber$-$Sieklucki Theorem which essentially states that every T-system is only one-dimensional and hence we can only apply these results to the univariate polynomial case. This can also be understood as a lesson or even a warning that this approach has been investigated and found to fail, i.e., learning about these results and limitations shall save students and researchers from following old footpaths which lead to a dead end. We took great care finding the correct historical references where the results appeared first but are perfectly aware that like people before we not always succeed.

math.CA↗
arXiv2023

The Early History of Moment Problems and Non-Negative Polynomials with Gaps: Sparse Moment Problems, Sparse Positivstellensätze, and Sparse Nichtnegativstellensätze from a T-System Point of View

We deal with and investigate sparse univariate Positivstellensätze, Nichtnegativstellensätze, and solutions to sparse moment problems. The paper relies heavily on results on T-system by Karlin in 1963 and by Karlin and Studden in 1966. We gain complete descriptions of all sparse strictly positive and sparse non-negative algebraic polynomials on $[a,b]$ with $a\geq 0$ and $[0,\infty)$. We extend, simplify, and solve the sparse Hausdorff and Stieltjes moment problem with these results and the methods of adapted spaces and T-systems.

math.CA↗
arXiv2022

Schwartz Function Valued Solutions of the Euler and the Navier-Stokes Equations

We prove the existence of a solution for the second order system of partial differential equations $\partial_t f = ν\cdotΔf + g\cdot\nabla f + h\cdot f + k$ by a Montel space version of Arzelà--Ascoli and bound all Schwartz semi-norms. We find that for the Euler and the Navier--Stokes equations the vorticity remains a Schwartz function as long as the classical solution exists. Our approach is not affected by viscosity. It treats the hyperbolic Euler and the parabolic Navier--Stokes equation simultaneously.

math.AP↗
arXiv2021

On the periodic Navier--Stokes equation: An elementary approach to existence and smoothness for all dimensions $n\geq 2$

In this paper we study the periodic Navier--Stokes equation. From the periodic Navier--Stokes equation and the linear equation $\partial_t u = νΔu + \mathbb{P} [v\nabla u]$ we derive the corresponding equations for the time dependent Fourier coefficients $a_k(t)$. We prove the existence of a unique smooth solution $u$ of the linear equation by a Montel space version of Arzelà--Ascoli. We gain bounds on the $a_k$'s of $u$ depending on $v$. With $v = -u$ these bounds show that a unique smooth solution $u$ of the $n$-dimensional periodic Navier--Stokes equation exists for all $t\in [0,T^*)$ with $T^* \geq 2ν\cdot \|u_0\|_{\mathsf{A},0}^{-2}$. $\|u_0\|_{\mathsf{A},0}$ is the sum of the $l^2$-norms of the Fourier coefficients without $e^{i\cdot 0\cdot x}$ of the initial data $u_0\in C^\infty(\mathbb{T}^n,\mathbb{R}^n)$ with $\mathrm{div}\, u_0=0$. For $\|u_0\|_{\mathsf{A},0} \leq ν$ (small initial data) we get $T^* = \infty$. All results hold for all dimensions $n\geq 2$ and are independent on $n$.

math.AP↗
arXiv2021

The multidimensional truncated Moment Problem: Carathéodory Numbers from Hilbert Functions

In this paper we improve the bounds for the Carathéodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$, and $[0,1]^n$. We also treat moment problems with small gaps. We find that for every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq d}\rightarrow\mathbb{R}$ which needs at least $(1-\varepsilon)\cdot\left(\begin{smallmatrix} n+d\\ n\end{smallmatrix}\right)$ atoms $l_{x_i}$. Consequences and results for the Hankel matrix and flat extension are gained. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\rightarrow\mathbb{R}$ which need to be extended to the worst case degree $4d$, $\tilde{L}:\mathbb{R}[x_1,\dots,x_n]_{\leq 4d}\rightarrow\mathbb{R}$, in order to have a flat extension.

math.FA↗
arXiv2020

Transformations of Moment Functionals

In measure theory several results are known how measure spaces are transformed into each other. But since moment functionals are represented by a measure we investigate in this study the effects and implications of these measure transformations to moment funcationals. We gain characterizations of moments functionals. Among other things we show that for a compact and path connected set $K\subset\mathbb{R}^n$ there exists a measurable function $g:K\to [0,1]$ such that any linear functional $L:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}$ is a $K$-moment functional if and only if it has a continuous extension to some $\overline{L}:\mathbb{R}[x_1,\dots,x_n]+\mathbb{R}[g]\to\mathbb{R}$ such that $\tilde{L}:\mathbb{R}[t]\to\mathbb{R}$ defined by $\tilde{L}(t^d) := \overline{L}(g^d)$ for all $d\in\mathbb{N}_0$ is a $[0,1]$-moment functional (Hausdorff moment problem). Additionally, there exists a continuous function $f:[0,1]\to K$ independent on $L$ such that the representing measure $\tildeμ$ of $\tilde{L}$ provides the representing measure $\tildeμ\circ f^{-1}$ of $L$. We also show that every moment functional $L:\mathcal{V}\to\mathbb{R}$ is represented by $λ\circ f^{-1}$ for some measurable function $f:[0,1]\to\mathbb{R}^n$ where $λ$ is the Lebesgue on $[0,1]$.

math.FA↗
arXiv2019

The multidimensional truncated Moment Problem: Shape and Gaussian Mixture Reconstruction from Derivatives of Moments

In this paper we introduce the theory of derivatives of moments and (moment) functionals to represent moment functionals by Gaussian mixtures, characteristic functions of polytopes, and simple functions of polytopes. We study, among other measures, Gaussian mixtures, their reconstruction from moments and especially the number of Gaussians needed to represent moment functionals. We find that there are moment functionals $L:\mathbb{R}[x_1,\dots,x_n]_{\leq 2d}\to\mathbb{R}$ which can be represented by a sum of $\binom{n+2d}{n} - n\cdot \binom{n+d}{n} + \binom{n}{2}$ Gaussians but not less. Hence, for any $d\in\mathbb{N}$ and $\varepsilon>0$ we find an $n\in\mathbb{N}$ such that $L$ can be represented by a sum of $(1-\varepsilon)\cdot\binom{n+2d}{n}$ Gaussians but not less. An upper bound is $\binom{n+2d}{n}-1$.

math.FA↗
arXiv2018

The multidimensional truncated Moment Problem: The Moment Cone

Let $\mathsf{A}=\{a_1,\dots,a_m\}$, $m\in\mathbb{N}$, be measurable functions on a measurable space $(\mathcal{X},\mathfrak{A})$. If $μ$ is a positive measure on $(\mathcal{X},\mathfrak{A})$ such that $\int a_i dμ<\infty$ for all $i$, then the sequence $(\int a_1 dμ,\dots,\int a_m dμ)$ is called a moment sequence. By Richter's Theorem each moment sequence has a $k$-atomic representing measure with $k\leq m$. The set $\mathcal{S}_\mathsf{A}$ of all moment sequences is the moment cone. The aim of this paper is to analyze the various structures of the moment cone. The main results concern the facial structure (exposed faces, facial dimensions) and lower and upper bounds of the Carathéodory number (that is, the smallest number of atoms which suffices for all moment sequences) of the convex cone $\mathcal{S}_{\mathcal{A}}$. In the case when $\mathcal{X}\subseteq \mathbb{R}^n$ and $a_i\in C^1(\mathcal{X},\mathbb{R})$, the differential structure of the moment map and regularity/singularity properties of moment sequences are analyzed. The maximal mass problem is considered and some applications to other problems are sketched.

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