The Gauss--Green formulas are of significant relevance in many areas of analysis and mathematical physics. This motivated several investigations to extend such formulas to more general classes of integration domains and weakly differentiable vector fields, and thus led to the definition of the divergence-measure fields. These are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose distributional divergence is a Radon measure, and so they form a new family of function spaces, which in a sense generalize the $BV$ fields. The divergence-measure fields were introduced at first by Anzellotti in 1983 for the case $p = \infty$, and then they have been rediscovered in the early 2000s by many authors interested in various applications.

In this talk, we shall present an overview of such researches, with a particular focus on the results concerning essentially bounded divergence-measure fields. For such a family of vector fields, Silhavy (2005), Chen, Torres and Ziemer (2009) and Comi and Payne (2017) showed that Gauss--Green formulas hold for sets of finite perimeter, by proving the existence of essentially bounded interior and exterior normal traces on the reduced boundary of the given set.

Subsequent extensions of these results and generalizations to non-Euclidean geometries will be also briefly discussed.

]]>Network dynamics preserves the sum of all incoming pairwise coupling strengths and is designed to adaptively interact with system dynamics.For adaptive couplings, we use two adaptive coupling laws for the pairwise coupling strength. Kuramoto oscillators are assumed to be on the nodes of the networks.

We present frameworks that guarantee the emergence of synchronization for various coupling feedback laws. Our results generalize earlier work on the synchronization of Kuramoto oscillators in fixed and symmetric networks.

]]>- martedì 4 aprile ore 15.30 aula 4
- mercoledì 5 aprile ore 15.30 aula 4
- giovedì 6 aprile ore 15.30 aula 5
- venerdì 7 aprile ore 10.30 aula 1

Programma:

- I Sums of squares in number theory
- II Sums of squares in real algebra
- III Basic quadratic form theory over fields
- IV Local-global principles for quadratic forms

Abstract:

In this course I want to touch on several aspects of sums of squares and their role in the development of modern algebra. The focus will be on sums of squares in commutative rings and in particular in fields.

Some basics of the theory of quadratic forms over fields will be introduced. The course will only assume some general algebraic knowledge on bachelor level.

The study of sums of squares is an old topic in number theory and algebra.

In Brahmagupta’s book an identity is given for writing a product of two sums of two squares again as a sum of two squares.

In modern terms it can be obtained by using the norm form for the ring of Gaussian integers, or more generally for the quadratic extension of a given commutative ring by adjoining a square root of -1.

Euler gave a similar identity for products of sums of four squares, which can likewise be viewed as the norm form of an algebraic structure, the quaternions.

He probably did this with the goal to prove that every positive integer is a sum of four squares, which was one of Fermat’s (correct) statements.

This way the problem had been reduced to showing the claim for prime numbers, and on this basis the proof was later completed by Lagrange.

Later Degen and Cayley found independently a similar identity for sums of eight squares and a related algebraic structure, the octonions.

On the other hand, attempts to find such an identity for sums of 16 squares failed and in the end Hurwitz showed that such an identity cannot exist.

More generally, the problem is to express a product of two quadratic forms (homogeneous polynomials of degree two) as a quadratic form applied to bilinear expressions in the variables.

If this is possible then a solution to this problem is called a composition formula. Using Clifford algebras one can explain why composition formulae do not exist for certain triples of forms, just by their dimensions.

Many problems of an analytic flavour can be described by asking whether a certain real valued function takes only non-negative values.

An obviously sufficient condition is that the function can be written as a sum of squares of other functions.

For rational functions (fractions of polynomials), Hilbert asked in his 17th problem whether the converse is true, namely whether any rational function taking only non-negative values can be written as a sum of squares of rational functions.

The positive answer was found by Artin in 1927 and it opened the way to a new research area, called real algebra.

Here different versions of the problem can be considered, for example, for a polynomial taking only non-negative values, one can ask whether it can be written as a sum of squares of polynomials.

This is generally not true. But also the number of squares in such an expression can be studied. This gives rise to the definition of the Pythagoras number of a commutative ring, the smallest number p such that every sum of squares is equal to a sum of p squares.

For a field we may ask whether -1 can be written as a sum of squares in it and if so, how many squares are needed. The smallest such number s, if it exists, is called the level of the field.

Pfister’s showed that the level of a field where -1 is a sum of squares is always a 2-power and that on the other hand every 2-power is the level of some field.

The proofs of these results give a nice insight into the topics of quadratic form theory over fields.

A main feature of 20th century algebra, number theory and geometry is the local-global analysis of problems.

It started with Hensel’s construction of the p-adic integers. It is not too difficult to see that polynomials with integer coefficients can have local solutions over every p-adic completion but no solution over the integers, but specifically for quadratic forms, by the Hasse-Minkowski Theorem, the local-global way of reasoning for finding (nontrivial) solutions works. Unfortunately, the same fails for quadratic forms over most other types of fields (when taking completions with respect to discrete valuations).

Recently, however, it was proven for certain types of function fields of arithmetic curves that such a local-global principle holds. In my own research I am studying some of these situations with the particular interest for the study of sums of squares, say, over the function field of a curve over R((t)), the field of formal Laurent series with real coefficients. I will discuss some of these results in relation to the study of valuations on these fields and give some examples.

Some basics of the theory of quadratic forms over fields will be introduced. The course will only assume some general algebraic knowledge on bachelor level.

The study of sums of squares is an old topic in number theory and algebra.

In Brahmagupta’s book an identity is given for writing a product of two sums of two squares again as a sum of two squares.

In modern terms it can be obtained by using the norm form for the ring of Gaussian integers, or more generally for the quadratic extension of a given commutative ring by adjoining a square root of -1.

Euler gave a similar identity for products of sums of four squares, which can likewise be viewed as the norm form of an algebraic structure, the quaternions.

He probably did this with the goal to prove that every positive integer is a sum of four squares, which was one of Fermat’s (correct) statements.

This way the problem had been reduced to showing the claim for prime numbers, and on this basis the proof was later completed by Lagrange.

Later Degen and Cayley found independently a similar identity for sums of eight squares and a related algebraic structure, the octonions.

On the other hand, attempts to find such an identity for sums of 16 squares failed and in the end Hurwitz showed that such an identity cannot exist.

More generally, the problem is to express a product of two quadratic forms (homogeneous polynomials of degree two) as a quadratic form applied to bilinear expressions in the variables.

If this is possible then a solution to this problem is called a composition formula. Using Clifford algebras one can explain why composition formulae do not exist for certain triples of forms, just by their dimensions.

Many problems of an analytic flavour can be described by asking whether a certain real valued function takes only non-negative values.

An obviously sufficient condition is that the function can be written as a sum of squares of other functions.

For rational functions (fractions of polynomials), Hilbert asked in his 17th problem whether the converse is true, namely whether any rational function taking only non-negative values can be written as a sum of squares of rational functions.

The positive answer was found by Artin in 1927 and it opened the way to a new research area, called real algebra.

Here different versions of the problem can be considered, for example, for a polynomial taking only non-negative values, one can ask whether it can be written as a sum of squares of polynomials.

This is generally not true. But also the number of squares in such an expression can be studied. This gives rise to the definition of the Pythagoras number of a commutative ring, the smallest number p such that every sum of squares is equal to a sum of p squares.

For a field we may ask whether -1 can be written as a sum of squares in it and if so, how many squares are needed. The smallest such number s, if it exists, is called the level of the field.

Pfister’s showed that the level of a field where -1 is a sum of squares is always a 2-power and that on the other hand every 2-power is the level of some field.

The proofs of these results give a nice insight into the topics of quadratic form theory over fields.

A main feature of 20th century algebra, number theory and geometry is the local-global analysis of problems.

It started with Hensel’s construction of the p-adic integers. It is not too difficult to see that polynomials with integer coefficients can have local solutions over every p-adic completion but no solution over the integers, but specifically for quadratic forms, by the Hasse-Minkowski Theorem, the local-global way of reasoning for finding (nontrivial) solutions works. Unfortunately, the same fails for quadratic forms over most other types of fields (when taking completions with respect to discrete valuations).

Recently, however, it was proven for certain types of function fields of arithmetic curves that such a local-global principle holds. In my own research I am studying some of these situations with the particular interest for the study of sums of squares, say, over the function field of a curve over R((t)), the field of formal Laurent series with real coefficients. I will discuss some of these results in relation to the study of valuations on these fields and give some examples.

Fra le tecniche della logica computazionale, i linguaggi di programmazione logica abduttiva e a vincoli forniscono espressività sufficiente per specificare le proprietà di un'ampia classe di sistemi aperti, mentre le loro controparti operazionali, grazie alle proprietà di correttezza rispetto alle rispettive semantiche dichiarative, costituiscono un appropriato meccanismo di verifica.

Il seminario verterà su alcune applicazioni ed estensioni di queste tecniche per la specifica e verifica di sistemi aperti.

]]>Abstract:

We briefly introduce the basic definitions of Malliavin calculus from the different perspectives of infinite dimensional analysis (following Da Prato) on the one side and of probability (following Nualart). Then we introduce, for suitable functionals of the Brownian motion, the problem of constructing a surface measure (i.e., the restriction of the reference measure on the infinite dimensional Wiener space) on the level sets defined by them. Finally, we give some interpretation of such results in terms of other stochastic processes.

]]>In particular, I will describe geometrical properties of an example of an embedding of Bir(P^2) into Bir(P^5) by Gizatullin and I will give a classification of certain embeddings of Bir(P^n) into Bir(X), where X is a variety of dimension n+1.]]>

In particular, I will discuss the higher differentiability of such minimizers under some restrictions on the exponents of anisotropy.

In the two dimensional case I will also show that minimizers are Lipschitz continuous without any limitation on the exponents of anisotropy.]]>

In recent years, the Smoluchowski equation has been considered in biomedical research to model the aggregation and diffusion of beta-amyloid peptide in the cerebral tissue, a process thought to be associated with the development of Alzheimer's disease. In this work, we study the homogenization of a Smoluchowski system of discrete diffusion-coagulation equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain through a two scale convergence technique.

]]>In the context of international nuclear safeguards, the International Atomic Energy Agency (IAEA) has recently approved passive gamma emission tomography (PGET) as a method for inspecting spent nuclear fuel assemblies (SFAs). The PGET instrument is essentially a single photon emission computed tomography (SPECT) system that allows the reconstruction of axial cross-sections of the emission map of SFA. The fuel material heavily self-attenuates its gamma-ray emissions, so that correctly accounting for the attenuation is a critical factor in producing accurate images. Due to the nature of the inspections, it is desirable to use as little a priori information as possible about the fuel, including the attenuation map, in the reconstruction process. Current reconstruction methods either do not correct for attenuation, assume a uniform attenuation throughout the fuel assembly, or assume an attenuation map based on an initial filtered back-projection reconstruction. We propose a method to simultaneously reconstruct the emission and attenuation maps by formulating the reconstruction as a constrained minimization problem with a least squares data fidelity term and regularization terms. Using simulated data, we show that our approach produces clear reconstructions which allow for a highly reliable classification of spent, missing, and fresh fuel rods.

]]>In the context of international nuclear safeguards, the International Atomic Energy Agency (IAEA) has recently approved passive gamma emission tomography (PGET) as a method for inspecting spent nuclear fuel assemblies (SFAs). The PGET instrument is essentially a single photon emission computed tomography (SPECT) system that allows the reconstruction of axial cross-sections of the emission map of SFA. The fuel material heavily self-attenuates its gamma-ray emissions, so that correctly accounting for the attenuation is a critical factor in producing accurate images. Due to the nature of the inspections, it is desirable to use as little a priori information as possible about the fuel, including the attenuation map, in the reconstruction process. Current reconstruction methods either do not correct for attenuation, assume a uniform attenuation throughout the fuel assembly, or assume an attenuation map based on an initial filtered back-projection reconstruction. We propose a method to simultaneously reconstruct the emission and attenuation maps by formulating the reconstruction as a constrained minimization problem with a least squares data fidelity term and regularization terms. Using simulated data, we show that our approach produces clear reconstructions which allow for a highly reliable classification of spent, missing, and fresh fuel rods.

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