Discrete Mathematics 2010-2022

Abstract:

In this talk, we discuss recent results concerning spectra of exponents of Diophantine approximation. Weighted homogeneous-inhomogeneous transferences, and joint spectra of an exponent and its uniform counterpart. Included are some recent pictures of the Himalayas.

Abstract:

We discuss the mutual behavior of the irrationality measure functions for two real numbers and some related problems. In particular we will formulate some new results about the function associated with the Minkowski diagonal continued fraction and with the functions related to the second best approximations, and introduce some multidimensional generalizations.

Abstract:

The Schatten classes $S_p$ ($0< p \leq \infty$), consisting of all compact linear operators on a Hilbert space for which the sequence of their singular values belongs to the sequence space $l_p$, are one of the most important classes of unitary operator ideals. Their analysis, particularly in the finite-dimensional setting, has a long tradition in asymptotic geometric analysis and the local theory of Banach spaces.
In [Studia Math. 80, 63--75, 1984], Saint Raymond studied the volumetric properties of unit balls in finite-dimensional real and complex Schatten $p$-classes and his results are used frequently in the literature.
He obtained an asymptotic formula for their volume, which contains an unknown limiting constant for which he provided both lower and upper bounds. We determine the exact limiting constant and as an application compute the precise asymptotic volume ratio of Schatten $p$-classes as the dimension tends to infinity. This extends Saint Raymond's estimate in the case of the nuclear norm ($p=1$) to the full regime $1\leq p \leq \infty$ with exact limiting behavior. (Joint work with Z. Kabluchko and C. Thäle)

Abstract:

Recently Marcus, Spielman and Srivastava studied certain combinatorial polynomial convolutions that preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability.
We show that with respect to these convolutions, the subset of diagonal matrices and the subset of principally balanced matrices form a ``convolvent pair''.
We also explore analogues of these types of convolutions in the setting of max-plus algebra. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots of the convolutions.
Joint work with Franz Lehner, Aljoša Peperko and Octavio Arizmendi.

Abstract:

We consider general self-adjoint polynomials in several independent
random matrices whose entries are centered and have constant
variance. Under some numerically checkable conditions, we establish
the optimal local law, i.e., we show that the empirical spectral
distribution on scales just above the eigenvalue spacing follows the
global density of states which is determined by free probability
theory. First, we give a brief introduction to the linearization
technique that allows to transform the polynomial model into a linear
one, which has simpler correlation structure but higher
dimension. After that we show that the local law holds up to the
optimal scale for the generalized resolvent of the linearized model,
which also yields the local law for polynomials. Finally, we show how
the above results can be applied to prove the optimal bulk local law
for two concrete families of polynomials: general quadratic forms in
Wigner matrices and symmetrized products of independent matrices with
i.i.d. entries. This is a joint work with Laszlo Erdös and Torben
Krüger.

Abstract:

The eigenvalue density of many large Hermitian random matrices is
well-approximated by a deterministic measure on \mathbb{R}, the self-consistent
density of states. In the case of an $N\times N$ random matrix with
nontrivial expectations of its entries or a nontrivial correlation
among them, this measure is obtained from the matrix Dyson equation on
$N\times N$ matrices. The matrix Dyson equation generalizes scalar- or
vector-valued Dyson equations that have been studied previously. In
this talk, we will show that the self-consistent density of states is
real-analytic apart from finitely many square root edges and cubic
root cusps. We will also explain how detailed information about these
singularities can be used to prove Tracy-Widom fluctuation for the
eigenvalues close to the square root edges of the associated
self-consistent density of states. This is joint work with László
Erdos, Torben Krüger and Dominik Schröder.

Abstract:

It is well-known that the Pell equation $a^2-db^2=1$ in integers has a
non-trivial solution if and only if $d$ is positive and not a perfect
square. If one considers the polynomial analog, i.e., for fixed $D \in
\mathbb{C}[X]$, the equation $A^2-DB^2=1$, the matter is more
complicated. Indeed, for the existence of a solution the clear
necessary conditions that the degree of $D$ must be even and that $D$
cannot be a perfect square are not sufficient. While the case of
degree two is analogous to the integer case, there are non-square
polynomials of degree 4 such that the corresponding Pell equation is
not solvable. On the other hand, as in the integer case, once we have
a non-trivial solution, we have infinitely many and we call minimal
solution a solution $(A,B)$ with $A$ of minimal degree.

In joint work with Laura Capuano and Umberto Zannier we showed that
there exist equations $A^2-DB^2=1$, with $(A,B)$ minimal solution, for
any choice of degrees deg$D \geq 4$ even and deg$A \geq $deg$D/2$.

Abstract:

Abstract: It goes back to Lagrange that a real quadratic irrational
has always a periodic continued fraction. Starting from decades ago,
several authors proposed different definitions of a p-adic continued
fraction, and the definition depends on the chosen system of residues
mod p. It turns out that the theory of p-adic continued fractions has
many differences with respect to the real case; in particular, no
analogue of Lagranges theorem holds, and the problem of deciding
whether the continued fraction is periodic or not seemed to be not
known until now. In recent work with F. Veneziano and U. Zannier we
investigated the expansion of quadratic irrationals, for the p-adic
continued fractions introduced by Ruban, giving an effective criterion
to establish the possible periodicity of the expansion. This
criterion, somewhat surprisingly, depends on the ‘real’ value of the
p-adic continued fraction.

Abstract:

Abstract:


Let $(R, \mathfrak{m})$ be a regular local ring of dimension $d \geq 2$. A \it local monoidal transform \rm of $R$ is a ring of the form $$R_1= R \left[ \frac{\mathfrak{p}}{x} \right]_{\mathfrak{m}_1}$$ where $ \mathfrak{p} $ is a prime ideal generated by regular parameters, $x \in \mathfrak{p}$ is a regular parameter and $ \mathfrak{m}_1 $ is a maximal ideal of $ R[\frac{\mathfrak{m}}{x}] $ lying over $ \mathfrak{m}. $ If $\mathfrak{p}= \mathfrak{m}$ the ring $R_1$ is called a \it local quadratic transform\rm.

Recently, several authors studied the rings of the form $ S= \cup_{n \geq 0}^{\infty} R_n $ obtained as infinite directed union of iterated local quadratic transforms of $R$, and call them \it quadratic Shannon extension\rm.
A directed union of local monoidal transforms of a regular local ring is said \it monoidal Shannon extension\rm.

Here we study features of monoidal Shannon extensions and more in general of directed unions of Noetherian UFDs.

(L. Guerrieri, Directed unions of local monoidal transforms and GCD domains (2018) arXiv:1808.07735 )

Abstract:

In the first part of the talk, I will speak about the normal structure of classical and classical-like groups. The description of the normal subgroups for various classes of concrete groups, and especially for classical groups over rings, has been one of the central themes of group theory in the last two centuries, right after Galois introduced the notion of normal subgroups. In the second part of the talk, I will speak about weighted Leavitt path algebras. Weighted Leavitt path algebras are algebras associated to weighted graphs. They generalise in a natural way the usual Leavitt path algebras and also Leavitt's algebras of module type $(n,k)$ where $n,k>0$.

Abstract:

This talk consists of two sections. In the first section,
we prove that if $D$ is an almost P$v$MD, then $D$ is of finite $t$-character
if and only if each nonzero $t$-locally finitely generated $w$-ideal of $D$ is of finite type.
As a corollary, we
get that if $D$ is an almost Pr\"ufer domain, then $D$ is of finite character
if and only if each locally finitely generated ideal of $D$ is finitely generated.

In the second section, we consider to
what extent conditions on the homogeneous elements or ideals of a graded integral domain $R$ carry
over to all elements or ideals of $R$.
For instance, we prove that $R$ is a gr-$t$-quasi-Pr\"ufer domain if and only if $R$ is a $t$-quasi-Pr\"ufer domain.
However, we show that homogeneously $tv$-domains and homogeneously $w$-divisorial domains are not equal to $tv$-domains and $w$-divisorial domains, respectively.


This talk is based on joint works with G. W. Chang and P. Sahandi.

Abstract:

Abstract: This talk contains two parts. In the first section of this talk, we define the set of integer-valued polynomials over the subsets of matrix rings. We do this on full matrix ring, upper triangular matrix ring and upper triangular matrix ring with constant diagonal. We present some examples to show that these sets may be not rings.
Then, we introduce some cases that the set of integer-valued polynomials over subsets of matrix ring is a ring. Furthermore, we consider some properties of these rings as Noetherian property and Krull dimension.
In the second section, we generalize the ring of integer-valued polynomials over upper triangular matrices and define the set of integer-valued polynomials over some cases of block matrices. Then, we show that this set is a ring. It solves the open problem of integer-valued polynomials on algebras in a special case of block matrix algebras.

Abstract:

\noindent Let $G$ be a group. A non-empty subset $S$ of $G$ is `product-free}' if $ab \not\in S$ for all $a, b \in S$. We call such a set `locally maximal}' in $G$ if it is not properly contained in any other product-free subset of $G$.
Locally maximal product-free sets were first studied in 1974 by Street and Whitehead, who analysed some properties of these sets, and introduced the concept of filled groups.
We say a locally maximal product-free subset $S$ of $G$ `fills}' $G$ if
$G^{\ast}\subseteq S \cup SS$ (where $G^{\ast}=G\setminus \{1\}$), and $G$ is called a `filled group}'
if every locally maximal product-free set in $G$ fills $G$.
In this talk, we shall consider questions like:
(a) for a given positive integer k, which finite groups contain a locally maximal product-free set of size k?; (b) how many finite groups are filled?.

Abstract:

\noindent
It is well known that Gauss Elimination produces a factorization into elementary matrices of any invertible matrix over a field. A classical problem, studied since the middle of the 1960's, is to characterize integral domains different from fields that satisfy the same property. As a partial answer, in 1993, Ruitenburg proved that in the class of B\'ezout domains, any invertible matrix can be written as a product of elementary matrices if and only if any singular matrix can be written as a product of idempotents.

\noindent
In this talk, after giving an overview of the classical results on these factorization properties, we will present some recent developments on the topic. In particular, we will consider products of elementary and idempotent matrices over special classes of non-Euclidean PID's and over integral domains that are not B\'ezout.

\vskip 0.5cm

\textsc{References}

\vskip 0.1cm

\begin{itemize}{\footnotesize

\item[\textnormal{[1]}] L.~Cossu, P.~Zanardo, U.~Zannier,
Products of elementary matrices and non-Euclidean principal ideal domains}, J. Algebra 501: 182–205, 2018.
\item[\textnormal{[2]}] L.~Cossu, P.~Zanardo,
Factorizations into idempotent factors of matrices over Pr\"ufer domains}, accepted for publication on Communications in Algebra, 2018.


}\end{itemize}

\vskip 0.5cm

\noindent{\footnotesize
\textsc{Department of Mathematics ``Tullio Levi Civita'', University of Padova}
{Via Trieste, 63}
{35121, Padova}
{Italy}
}

\vskip 0.2cm
\noindent{\footnotesize{E-mail address}:
lcossu@math.unipd.it}
}

Abstract:

The theory of time harmonic electromagnetic waves propagation involves always the structure of the obstacle, which have in most cases thin geometry. In the case of perfectly conducting obstacle coated with a thin dielectric layer some difficulties linked to the numerical simulation appear. To overcome this problem many authors gave approximations of an impedance operator for perfectly conducting obstacle coated by thin shell of dielectric material. In the present work, we calculated approximations until the third order of the impedance operator for perfectly conducting obstacle coated by two contrasted thin layers of dielectric materials, the approach used is that of Bendali et al. [1], when they wrote the boundary condition on the perfect conductor in terms of Taylor expansion in the thickness of the thin layer.

[1] A. Bendali, M. Fares, K. Lemrabet and S. Pernet, Recent Developments in the Scattering of an Electromagnetic Wave by a Coated Perfectly Conducting Obstacle.
Waves 09 (2009), Pau, France.

Abstract:

On a countable tree T, allowing vertices with infinite degree, we consider an arbitrary stochastic irreducible nearest neighbour transition operator $P$. We provide a boundary integral representation for general eigenfunctions of $P$ with complex eigenvalue $\lambda$. This is possible whenever $\lambda$ is in the resolvent set of $P$ as a self-adjoint operator on a suitable l(2)-space and the on-diagonal elements of the resolvent do not vanish at $\lambda$.

When $P$ is invariant under a transitive group action, the latter condition
holds for all $\lambda$ in the resolvent set except possibly 0. These results extend and complete previous results by Cartier, by Figa'-Talamanca and Steger, and by Woess.

Furthermore, for those eigenvalues, we provide an integral representation of $\lambda$-polyharmonic functions of any order n, that is, complex functions $f$ on T for which $(\lambda I - P)^n f=0$. This is a far-reaching extension of work of Cohen et al.

We can also provide an analogous result for polyharmonic functions on the unit disk with respect to the hyperbolic Laplacian, based on old results of Helgason.

This is joint work with Massimo Picardello (Rome).

Abstract:

Abstract:
Im my talk I would like to put a little footbridge between clinical environment and mathematical modelling society in the topic of aortic mechanics.
The biggest artery of our organizm has it’s own life. It may not be treated as a viscoelastic tube but we may have to take into account some physiological considerations.
Aorta by itself is a living organ with internal metabolism and high level of protein production and turnover. In modelling perspective it may be quite difficult to bridge the scale between molecular level and cellular level. I propose to avoid that by organising model into subdomains representing basic structural unit of the aorta. By degenerating properties of single components of the subunit we may simulate the pathology on the level of whole organ. It may also help in explanation of the ageing processes of the aorta and acconying
changes of mechanical properties.

Abstract:

Abstract: Unlike the real line, the $d$-dimensional space $\mathbb{R}^d$, for $d \geq 2$, is not canonically ordered. As a consequence, such fundamental and strongly order-related univariate concepts as quantile and distribution functions, and their empirical counterparts, involving ranks and signs, do not canonically extend to the multivariate context. Palliating that lack of a canonical ordering has remained an open problem for more than half a century, and has generated an abundant literature, motivating, among others, the development of statistical depth and copula-based methods. We show that, unlike the many definitions that have been proposed in the literature, the measure transportation-based ones introduced in Chernozhukov, Galichon, Hallin and Henry (2017) enjoy all the properties (distribution-freeness and the maximal invariance property that entails preservation of semiparametric efficiency) that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new center-outward} definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we establish a Glivenko-Cantelli result---the quintessential property of all distribution functions.
Our approach, based on results by McCann (1995), is geometric rather than analytical and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017) (which assumes compact supports, hence finite moments of all orders), does not require any moment assumptions. The resulting ranks and signs are shown to be strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the family of absolutely continuous distributions; that maximal invariance, in view of a general result by Hallin and Werker (2003), is the theoretical foundation of the semiparametric efficiency preservation property of ranks. The corresponding quantiles are equivariant under the same transformations.

Abstract:

Abstract:

Let $K$ be a number field with ring of integers $O_K$. Recently, Loper and Werner introduced the following ring which generalizes the classical definition of ring of integer-valued polynomials: ${\rm Int}_{\mathbb Q}(O_K)=\{f\in \mathbb Q[X] \mid f(O_K)\subseteq O_K\}$. If $K=\mathbb Q$ then we get the classical ring ${\rm Int}(\mathbb Z)$ of polynomials with rational coefficients mapping $\mathbb Z$ into $\mathbb Z$. Loper and Werner prove that ${\rm Int}_{\mathbb Q}(O_K)$ is a Pr\"ufer domain, which is stricly contained in ${\rm Int}(\mathbb Z)$ if $K$ is a proper extension of $\mathbb Q$. Here, we show that in case $K,K'$ are Galois extensions of $\mathbb Q$, then ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$ if and only if $K=K'$. We also characterize a basis for ${\rm Int}_{\mathbb Q}(O_K)$ as a $\mathbb Z$-module when $K/\mathbb Q$ is a tamely ramified Galois extension. This is a joint work with Bahar Heidaryan and Matteo Longo.

We also give the following new generalization: for any number fields $K,K'$, if ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$, then $K,K'$ are conjugated over $\mathbb Q$.

Abstract:

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Let $K$ be a number field with ring of integers $O_K$. Recently, Loper and Werner introduced the following ring which generalizes the classical definition of ring of integer-valued polynomials: ${\rm Int}_{\mathbb Q}(O_K)=\{f\in \mathbb Q[X] \mid f(O_K)\subseteq O_K\}$. If $K=\mathbb Q$ then we get the classical ring ${\rm Int}(\mathbb Z)$ of polynomials with rational coefficients mapping $\mathbb Z$ into $\mathbb Z$. Loper and Werner prove that ${\rm Int}_{\mathbb Q}(O_K)$ is a Pr\"ufer domain, which is stricly contained in ${\rm Int}(\mathbb Z)$ if $K$ is a proper extension of $\mathbb Q$. Here, we show that in case $K,K'$ are Galois extensions of $\mathbb Q$, then ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$ if and only if $K=K'$. We also characterize a basis for ${\rm Int}_{\mathbb Q}(O_K)$ as a $\mathbb Z$-module when $K/\mathbb Q$ is a tamely ramified Galois extension. This is a joint work with Bahar Heidaryan and Matteo Longo.

We also give the following new generalization: for any number fields $K,K'$, if ${\rm Int}_{\mathbb Q}(O_K)={\rm Int}_{\mathbb Q}(O_{K'})$, then $K,K'$ are conjugated over $\mathbb Q$.

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%\addcontentsline{toc}{section}{Bibliography}
%\begin{thebibliography}{99}
%\bibliographystyle{plain}
%\bibitem{ChabPolCloVal} J.-L. Chabert, On the polynomial closure in a valued field}, J. Number Theory 130 (2010), 458-468.
%\bibitem{LW} K. A. Loper, N. Werner, Pseudo-convergent sequences and Pr\"ufer domains of integer-valued polynomials}, J. Commut. Algebra 8 (2016), no. 3, 411-429.
%\bibitem{PerPrufer} G. Peruginelli, Pr\"ufer intersection of valuation domains of a field of rational functions}, sottomesso (2017), Arxiv: \href{https://arxiv.org/abs/1711.05485}{https://arxiv.org/abs/1711.05485}.
%\end{thebibliography}

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Abstract:

Abstract:

Spherical designs are discrete point sets on the sphere which provide exact equal-weight cubature formulas for polynomials up to a certain degree, i.e. the average of any polynomial over the points of a design is equal to the average over the whole sphere. Several years ago, Bondarenko, Radchenko, and Viazovska solved a long-standing conjecture of Korevaar and Meyers, showing that there exist designs of degree t on the d-dimensional sphere, which have $t^d$ points. We shall discuss properties of spherical designs and present the proof of the aforementioned conjecture.

This is the fourth lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subjects involved in the talk will be assumed.

Abstract:

Unit norm tight frames are objects that play an important role in functional analysis, discrete geometry, signal processing, and even quantum physics. These are collections of unit vectors, which behave like orthonormal bases: they satisfy an analog of Parseval's identity and every vector can be exactly reconstructed from its projections onto the elements of the frame. Tight frames have a number of interesting properties -- in particular, they minimize a certain discrete energy. This object is closely intertwined with the following interesting question in discrete geometry: can one draw N lines in a d-dimensional Euclidean space, so that the angle between any two is the same? For which values of N, d, and the angle is this possible? What is the maximal cardinality of a set of equiangular lines for a given dimension? We shall survey the main results and conjectures on these topics.

This is the third lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subjects involved in the talk will be assumed.

Abstract:

Two most standard ways to measure the quality of a point set on the sphere are discrepancy and energy. In the former, one compares the proportion of points in certain subdomains to their area, while in the latter one views points as electrons that repel according to a certain force. We shall talk about various energy minimization problems on the sphere, when minimal energy induces uniform distribution, how the structure of the function affects minimizers, special point sets that arise as minimizers (tight frames, spherical designs), connections to spherical harmonics, Gegenbauer polynomials, and positive definite functions etc. Then we shall discus discrepancy on the sphere: known bounds, methods, constructions, as well as relations to energy.

This is the second lecture in a mini-series of four lectures. All lectures are independent and fully self-contained. No prior knowledge of any of the subject involved in the talk will be assumed.

Abstract:

We investigate the model selection properties of the Lasso estimator in finite samples with no conditions on the regressor matrix X. We show that which covariates the Lasso estimator may potentially choose in high dimensions (where the number of explanatory variables p exceeds sample size n) depends only on X and the given penalization weights. This set of potential covariates can be determined through a geometric condition on X and may be small enough (less than or equal to n in cardinality) so that the Lasso estimator acts as a low-dimensional procedure also in high dimensions. Related to the geometric conditions in our considerations, we also provide a necessary and sufficient condition for uniqueness of the Lasso solutions.