SISSA OpenScience
SISSA Open Science is a digital repository providing free, open access to SISSA academic scientific production before it is refereed, according to SISSA Regulation on open access (approved in December, 2016).
This repository includes SISSA preprints, unpublished proceedings of conferences held in SISSA, lecture notes and presentations by SISSA professors.
If you want to include one or more of the aforementioned documents in the SISSA Open Science repository, please send your pdf file to library@sissa.it.
Before posting your preprint, remember to check your publisher's policy in the SHERPA/RoMEO database.
Communities in DSpace
Select a community to browse its collections.
- This collection holds the lecture notes of the SISSA members
- SISSA authors' not-referred manuscripts of the research output
- SISSA workshops: proceedings of the workshops organized and held in SISSA
Recent Submissions
Book of Abstracts: Fractional Calculus Seminar Series - 2024
(SISSA, International School of Advanced Studies, 2024) Pranjivan Mehta, Pavan ; Fernandez, Arran; Rozza, Gianluigi
Preface:
Fractional calculus is the field of research that studies fractional derivatives and fractional integrals, which are variously defined as derivatives and integrals to noninteger orders. This is a generalised form of integer-order calculus, with more richness and variety since fractional derivatives and integrals can be defined in many different ways which are not equivalent to each other: there is no single unique answer to a question like “what is the derivative to order one-half of the identity function?”
It is well known that integer-order derivatives are local operators and integerorder integrals are non-local operators. In fractional calculus, since the derivative operators are defined using the integral operators, both fractional integrals and fractional derivatives are non-local operators. Depending on the type of fractional
derivative used, these operators may depend on values of the function in a finite region, or in a one-way region modelling a memory effect, or in its entire domain. The non-locality property is one of the reasons why fractional calculus has found manyapplications: real-world applications of non-local models can be found in turbulence, viscoelasticity, fracture mechanics, economic models, diffusion processes, electrical circuits, and plasma physics. The full range of applications is not yet understood, and new research is ongoing in many of these domains.
The theory and applications of fractional integro-differential operators and equations has not received much attention in the wider scientific community, beyond a
few specialists developing the field, so that many fundamental questions remain unanswered and the field is ripe for ongoing research in many directions. Currently, however, there is not a single united research community in fractional calculus, but rather many different groups working on it in different ways. Some research is undertaken without awareness of the established fundamentals of the field or of what
other research groups are doing.
Thus, the Fractional Calculus seminar series grew out of the necessity to connect different research communities and to touch on as many aspects of fractional calculus as possible. This seminar series is intended to provide deep knowledge on all aspects of fractional calculus, from analytical mathematics to numerical simulations to modelling applications. Some of the presentations are from long-standing experts who have been working in the field for decades, while some reflect new developments in particular research directions. Some of the topics are of broad interest to anyone working in fractional calculus, but there were also some focus sessions (listed below) which allowed us to drill further into particular broad topics of research in fractional calculus.
Special focus sessions:
• 24 May to 21 June 2024 (7 talks): fractional/nonlocal modelling of turbulence.
• 05 July to 02 August 2024 (6 talks): stochastic processes and probability
theory for fractional PDEs.
• 09 August to 06 September 2024 (5 talks): general fractional-calculus
operators.
• 13 September to 11 October 2024 (5 talks): fractional inverse problems.
• 18 October to 06 December 2024 (10 talks): numerical analysis, methods,
and singular integral computation.
Homogenisation of vectorial free-discontinuity functionals with cohesive type surface terms
(2024-09-12) Dal Maso, Gianni; Donati, Davide
The results on Γ-limits of sequences of free-discontinuity functionals with bounded
cohesive surface terms are extended to the case of vector-valued functions. In this framework, we
prove an integral representation result for the Γ-limit, which is then used to study deterministic
and stochastic homogenisation problems for this type of functionals.
On the Symmetry TFT of Yang–Mills–Chern–Simons theory
(2024-04-04) Riccardo Argurio, Francesco Benini, Matteo Bertolini, Giovanni Galati, Pierluigi Niro
Three-dimensional Yang–Mills–Chern–Simons theory has the peculiar property that its one-form symmetry defects have non-trivial braiding, namely they are charged under the same symmetry they generate, which is then anomalous. This poses a few puzzles in describing the corresponding Symmetry TFT in a four-dimensional bulk. First, the braiding between lines at the boundary seems to be ill-defined when such lines are pulled into the bulk. Second, the Symmetry TFT appears to be too trivial to allow for topological boundary conditions encoding all the different global variants.
We show that both of these puzzles can be solved by including endable (tubular) surfaces in the lass of bulk topological operators one has to consider. In this way, we are able to reproduce all global variants of the theory, with their symmetries and their anomalies. We check the validity ofour proposal also against a top-down holographic realization of the same class of theories.
A closure theorem for gAMMA-convergence and H-convergence with applications to non-periodic homogenization
(2024-02-29) Braides, Andrea; Dal Maso, Gianni; Le Bris, Claude
In this work we examine the stability of some classes of integrals, and in particular
with respect to homogenization. The prototypical case is the homogenization of
quadratic energies with periodic coe cients perturbed by a term vanishing at in nity,
which has been recently examined in the framework of elliptic PDE.We use localization
techniques and higher-integrability Meyers-type results to provide a closure theorem by
gamma-convergence within a large class of integral functionals. From such result we derive
stability theorems in homogenization which comprise the case of perturbations with
zero average on the whole space. The results are also extended to the stochastic case,
and specialized to the G-convergence of operators corresponding to quadratic forms. A
corresponding analysis is also carried on for non-symmetric operators using the localization
properties of H-convergence. Finally, we treat the case of perforated domains
with Neumann boundary condition, and their stability.
EXISTENCE AND BLOW-UP FOR NON-AUTONOMOUS SCALAR CONSERVATION LAWS WITH VISCOSITY
(2023-11-23) Bianchini, Stefano; Leccese, Giacomo Maria
We consider a question posed in [1], namely the blow-up of the PDE
ut + (b(t, x)u1+k)x = uxx
when b is uniformly bounded, Lipschitz and k = 2. We give a complete answer to the behavior of
solutions when b belongs to the Lorentz spaces b ∈ Lp,∞, p ∈ (2,∞], or bx ∈ Lp,∞, p ∈ (1,∞].