arXiv subjects
Publications and source records attributed to Giuseppe De Nittis.
In this paper we investigate the problem of the cohomological classification of "Quaternionic" vector bundles in low-dimension ($d\leqslant 3$). We show that there exists a characteristic classes $κ$, called the FKMM-invariant, which takes value in the relative equivariant Borel cohomology and completely classifies "Quaternionic" vector bundles in low-dimension. The main subject of the paper concerns a discussion about the surjectivity of $κ$.
Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. This monograph presents a thoroughly modern framework to make linear response theory rigorous for a wide array of systems, that is suitable for novel applications such as periodic and random light conductors not yet covered in the literature. Our analytic-algebraic approach, based on von Neumann algebras and associated non-commutative $L^p$-spaces, can deal with discrete and continuous models alike, and include effects of disorder. First, we explain the mathematical setting, give a complete list of our hypotheses and state the main results, which include Kubo and Kubo-Streda formulas. To make our book accessible to a wide audience, we spend Chapters 3 and 4 explaining the mathematical underpinnings such as non-commutative $L^p$- and Sobolev spaces, and generalized commutators. Furthermore, we show how to construct a von Neumann algebras from a topological dynamical system and a 2-cocycle, a procedure which applies to discrete and continuous quantum systems. We dedicate Chapters 5 and 6 to the proofs of our main results. We close the book by sketching a novel application, linear response theory for periodic and random light conductors. This monograph is aimed at advanced students in mathematical physics and researchers.
In this work we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. Our main results, Theorems 3.3 and 4.1, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all sub-leading order terms, some of which are also new to the physics literature. While the ray optics limit we prove initially (Theorem 3.3) applies to photonic crystals of any topological class, we also consider the ray optics limit for real electromagnetic fields propagating in non-gyrotropic photonic crystals. Such an extension is non-trivial, because the ray optic limit for real fields is necessarily a multiband problem.
The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources-corresponding to the team members-that cannot communicate. It is known that such equilibrium always exists and it is unique (unless degeneracy) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in)efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances.
The present survey aims at presenting the current machine learning techniques employed in security games domains. Specifically, we focused on papers and works developed by the Teamcore of University of Southern California, which deepened different directions in this field. After a brief introduction on Stackelberg Security Games (SSGs) and the poaching setting, the rest of the work presents how to model a boundedly rational attacker taking into account her human behavior, then describes how to face the problem of having attacker's payoffs not defined and how to estimate them and, finally, presents how online learning techniques have been exploited to learn a model of the attacker.
Security Games employ game theoretical tools to derive resource allocation strategies in security domains. Recent works considered the presence of alarm systems, even suffering various forms of uncertainty, and showed that disregarding alarm signals may lead to arbitrarily bad strategies. The central problem with an alarm system, unexplored in other Security Games, is finding the best strategy to respond to alarm signals for each mobile defensive resource. The literature provides results for the basic single-resource case, showing that even in that case the problem is computationally hard. In this paper, we focus on the challenging problem of designing algorithms scaling with multiple resources. First, we focus on finding the minimum number of resources assuring non-null protection to every target. Then, we deal with the computation of multi-resource strategies with different degrees of coordination among resources. For each considered problem, we provide a computational analysis and propose algorithmic methods.
Topological quantum systems subjected to an even (resp. odd) time-reversal symmetry can be classified by looking at the related "Real" (resp. "Quaternionic") Bloch-bundles. If from one side the topological classification of these time-reversal vector bundle theories has been completely described in [DG1] for the "Real" case and in [DG2] for the "Quaternionic" case, from the other side it seems that a classification in terms of differential geometric invariants is still missing in the literature. With this article (and its companion [DG3]) we want to cover this gap. More precisely, we extend in an equivariant way the theory of connections on principal bundles and vector bundles endowed with a time-reversal symmetry. In the "Real" case we generalize the Chern-Weil theory and we showed that the assignment of a "Real" connection, along with the related differential Chern class and its holonomy, suffices for the classification of "Real" vector bundles in low dimensions.
When securing complex infrastructures or large environments, constant surveillance of every area is not affordable. To cope with this issue, a common countermeasure is the usage of cheap but wide-ranged sensors, able to detect suspicious events that occur in large areas, supporting patrollers to improve the effectiveness of their strategies. However, such sensors are commonly affected by uncertainty. In the present paper, we focus on spatially uncertain alarm signals. That is, the alarm system is able to detect an attack but it is uncertain on the exact position where the attack is taking place. This is common when the area to be secured is wide such as in border patrolling and fair site surveillance. We propose, to the best of our knowledge, the first Patrolling Security Game model where a Defender is supported by a spatially uncertain alarm system which non-deterministically generates signals once a target is under attack. We show that finding the optimal strategy in arbitrary graphs is APX-hard even in zero-sum games and we provide two (exponential time) exact algorithms and two (polynomial time) approximation algorithms. Furthermore, we analyse what happens in environments with special topologies, showing that in linear and cycle graphs the optimal patrolling strategy can be found in polynomial time, de facto allowing our algorithms to be used in real-life scenarios, while in trees the problem is NP-hard. Finally, we show that without false positives and missed detections, the best patrolling strategy reduces to stay in a place, wait for a signal, and respond to it at best. This strategy is optimal even with non-negligible missed detection rates, which, unfortunately, affect every commercial alarm system. We evaluate our methods in simulation, assessing both quantitative and qualitative aspects.
We provide a classification of type AII topological quantum systems in dimension d=1,2,3,4. Our analysis is based on the construction of a topological invariant, the FKMM-invariant, which completely classifies "Quaternionic" vector bundles (a.k.a. "symplectic" vector bundles) in dimension d<4. This invariant takes value in a proper equivariant cohomology theory and, in the case of examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d=4 the classification requires a combined use of the FKMM-invariant and the second Chern class. Among the other things, we prove that the FKMM-invariant is a bona fide characteristic class for the category of "Quaternionic" vector bundles in the sense that it can be realized as the pullback of a universal topological invariant.
When a flux quantum is pushed through a gapped two-dimensional tight-binding operator, there is an associated spectral flow through the gap which is shown to be equal to the index of a Fredholm operator encoding the topology of the Fermi projection. This is a natural mathematical formulation of Laughlin's Gedankenexperiment. It is used to provide yet another proof of the bulk-edge correspondence. Furthermore, when applied to systems with time reversal symmetry, the spectral flow has a characteristic $Z_2$ signature, while for particle-hole symmetric systems it leads to a criterion for the existence of zero energy modes attached to half-flux tubes. Combined with other results, this allows to explain all strong invariants of two-dimensional topological insulators in terms of a single Fredholm operator.
Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow which is shown to be equal to the index of the operator. This purely operator theoretic result is interpreted in terms of the $K$-theory of an associated mapping cone. It is then extended to connect $Z_2$ indices of odd symmetric Fredholm operators to a $Z_2$-valued spectral flow.
We provide a classification of type AI topological quantum systems in dimension d=1,2,3,4 which is based on the equivariant homotopy properties of "Real" vector bundles. This allows us to produce a fine classification able to take care also of the non stable regime which is usually not accessible via K-theoretic techniques. We prove the absence of non-trivial phases for one-band AI free or periodic quantum particle systems in each spatial dimension by inspecting the second equivariant cohomology group which classifies "Real" line bundles. We also show that the classification of "Real" line bundles suffices for the complete classification of AI topological quantum systems in dimension d<4. In dimension d=4 the determination of different topological phases (for free or periodic systems) is fixed by the second "Real Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide explicit realizations of non trivial 4-dimensional free models for each given topological degree.
We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan-Altland-Zirnbauer scheme. Of particular importance are complex conjugation C and time-reversal T, but we identify also other significant symmetries. Borrowing the jargon of the classification theory of topological insulators, we show that C is a particle-hole-type symmetry rather than a time-reversal symmetry if one consider the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schrödinger equation; The symmetry which implements physical time-reversal is a chiral-type symmetry. We justify by an analysis of the band structure why the first-order formalism seems to be more advantageous than the second-order formalism. Moreover, based on the Schrödinger formalism, we introduce a class of effective (tight-binding) models called Maxwell-Harper operators. Some considerations about the breaking of the particle-hole-type symmetry in the case of gyrotropic crystals are added at the end of this paper.
After a short discussion of various random Bogoliubov-de Gennes (BdG) model operators and the associated physics, the Aizenman-Molchanov method is applied to prove Anderson localization in the weak disorder regime for the spectrum in the central gap. This allows to construct random BdG operators which have localized states in an interval centered at zero energy. Furthermore, techniques for the calculation of Chern numbers are reviewed and applied to two non-trivial BdG operators, the p+ip wave and d+id wave superconductors.
In this work, we rigorously derive effective dynamics for light from within a limited frequency range propagating in a photonic crystal that is modulated on the macroscopic level; the perturbation parameter $λ\ll 1$ quantifies the separation of spatial scales. We do that by rewriting the dynamical Maxwell equations as a Schrödinger-type equation and adapting space-adiabatic perturbation theory. Just like in the case of the Bloch electron, we obtain a simpler, effective Maxwell operator for states from within a relevant almost invariant subspace. A correct physical interpretation for the effective dynamics requires to establish two additional facts about the almost invariant subspace: (1) The source-free condition has to be verified and (2) it has to support real states. The second point also forces one to consider a multiband problem even in the simplest possible setting; This turns out to be a major difficulty for the extension of semiclassical methods to the domain of photonic crystals.
As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell operator M_0. In particular, we characterize the behavior of M_0 and the physical initial states at small crystal momenta $k$ and small frequencies |ω|. Among other things, we prove that generically the band spectrum is symmetric with respect to inversions at k = 0 and that there are exactly 4 ground state bands with approximately linear dispersion near k = 0.
We derive a class of equations of state for a multi-phase thermodynamic system associated with a finite set of order parameters that satisfy an integrable system of hydrodynamic type. As particular examples, we discuss one-phase systems such as the van der Waals gas and the effective molecular field model. The case of N-phase systems is also discussed in detail in connection with entropies depending on the order parameter according to Tsallis' composition rule.
In this article we investigate the possibility of generating piezoelectric orbital polarization in graphene-like systems which are deformed periodically. We start with discrete two-level models which depend on control parameters; in this setting, time-dependent model hamiltonians are described by loops in parameter space. Then, the gap structure at a given Fermi energy generates a non-trivial topology on parameter space which then leads to possibly non-trivial polarizations. More precisely, we show the polarization, as given by the *King-Smith--Vanderbilt formula*, depends only on the homotopy class of the loop; hence, a necessary condition for non-trivial piezo effects is that the fundamental group of the gapped parameter space must not be trivial. The use of the framework of non-commutative geometry implies our results extend to systems with weak disorder. We then apply this analysis to the uniaxial strain model for graphene which includes nearest-neighbor hopping and a stagger potential, and show that it supports non-trivial piezo effects; this is in agreement with recent physics literature.