I.9. Open problems 23 restore a finite conductivity without thermal activation. The question remained un- solved until the work of Basko, Aleiner and Altshuler [ 48 ] **in** 2006: the authors showed that, if the temperature is finite but small enough, electron-electron interaction alone can not cause finite conductivity. Yet, there exist a critical temperature T c such that for T > T c the conductivity is finite. For such temperature the system of interacting electrons undergoes thus a phase transition, called Many Body **Localization** Transition. The analysis **in** [ 48 ] consists **in** taking into account the interactions **in** perturbation the- ory, studying the inelastic quasiparticle relaxation, represented by the imaginary part of the sigle-particle self-energy. **In** a pictorial view, MBL can be thought of as local- ization **in** the Fock space of Slater determinants, which plays the role of lattice sites **in** a disordered one-particle **Anderson** tight-binding model. The problem of N 1 interacting particles **in** a finite **dimensional** lattice is thus interpreted as a one-particle **localization** problem on a very **high** **dimensional** lattice, which for spinless electrons consists **in** an N -**dimensional** hyper-cube of 2 N sites. This makes the study of single- particle **Anderson** **Localization** **in** very **high** dimension (and consequently the problem on the Bethe Lattice) an interesting issue for the comprehension of the problem with interactions. The idea of interpreting a many particle problem as a single-particle one **in** a very **high** **dimensional** space appeared **in** the context of the study of vibrational degrees of freedom of very big molecules [ 29 ], and was then applied **in** [ 47 ] to study the problem of electron-electron lifetime **in** a quantum dot. The work of Oganesyan and Huse [ 136 ], already cited **in** section ( I.6 ), points out instead the link of MBL with Random Matrix Theory, and offer, at least numerically, a way to study the transi- tion **in** relation to the level spacing distribution and

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(Dated: August 13, 2009)
We study **Anderson** **localization** of ultracold atoms **in** weak, one-**dimensional** speckle potentials, using perturbation theory beyond Born approximation. We show the existence of a series of sharp crossovers (effective mobility edges) between energy regions where **localization** lengths differ by orders of magnitude. We also point out that the correction to the Born term explicitly depends on the sign of the potential. Our results are **in** agreement with numerical calculations **in** a regime relevant for experiments. Finally, we analyze our findings **in** the light of a diagrammatic approach.

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• **Anderson** **localization** and rarefied conduct- ing paths. The fact that the d → ∞ limit pro- vides a very good starting point to quantitatively describe AL suggests that the solution of AL on Bethe **lattices** is a good starting point to get a physical picture of AL on finite **dimensional** lat- tices. Recently the delocalized phase of the An- derson model on tree-like structures (and on re- lated d → ∞ random matrix models with long- range hopping [27]) has attracted a lot of atten- tion [30–35]. Although it is still debated whether before the AL transition there is a non-ergodic de- localised phase or very strong cross-over regime, it is clear that localisation is related to the rarefaction of paths over which electrons can travel, as antici- pated **in** [39, 44]. Some authors advocates that this leads to a bona-fide multi-fractal intermediate non- ergodic but delocalised phase [31, 32, 39], others that this picture is valid below a certain scale that diverges (extremely fast) approaching the transi- tion [33–35]. Although we do not see numerical ev- idences of an intermediate non-ergodic delocalised phase for large d, the fact that **in** the scaling vari- ables that govern finite size scaling the linear size of the system, L = N 1/d , enters raised to powers that remain finite for d → ∞ suggests that (1) quasi one- **dimensional** paths are indeed the relevants geomet- rical objects for AL **in** **high** dimensions, (2) scaling becomes logarithmic **in** N for d → ∞ as found for tree-like structures [27] and Bethe **lattices** [33, 35]. **In** summary the idea of non-ergodic transport along rarified paths is relevant even **in** finite dimension even though possibly only on finite but very large length-scales (on larger ones transport would be **in**- stead described by standard diffusion).

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Control Protocol) SYN flooding type. This attack consists **in** exploiting the TCP three-way hand- shake mechanism and its limitation **in** maintaining half-open connections. More precisely, when a server receives a SYN packet, it returns a SYN/ACK packet to the client. Until the SYN/ACK packet is acknowledged by the client, the connection remains half-opened for a period of at most the TCP connection timeout. A backlog queue is built up **in** the system memory of the server to maintain all half-open connections, this leading to a saturation of the server. **In** Siris and Papagalou ( 2006 ), the authors use the CUSUM algorithm to detect a change-point **in** the time series corresponding to the aggregation of the SYN packets received by all the requested destination IP addresses. With such an approach, it is only possible to set off an alarm when a massive change occurs **in** the aggregated series; it is moreover impossible to identify the attacked IP addresses.

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tribution is peaked around 0 and π when the mode is localized (n=2), while it is more uniformly distributed **in** the extended case. Note that for values of the refractive index as low as 1.05, scattering is weak and the field is rather concentrated at the edges of the system. **In** that case, residual reflection either at the boundaries or from the PML layers may not be negligible and may result **in** periodic patterns, similar to those of a Fabry-Perrot cavity, as seen **in** the bottom-right frame of Fig. 1. We checked that above n=1.10, this effect is insignifi- cant and lasing is solely due to multiple scattering within the system.

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Tunneling has been known since early quantum me- chanics as a striking example of a purely quantum effect that is classically forbidden. However, the simplest pre- sentation based on tunneling through a one-**dimensional** barrier does not readily generalize to more generic situ- ations. Indeed, **in** dimension two or higher or **in** time- dependent systems, the dynamics becomes more com- plex with various degrees of chaos, and the tunneling effect can become markedly different [1]. An especially spectacular effect of chaos **in** this context is known as chaos-assisted tunneling [2]: **in** this case, tunneling is mediated by ergodic states **in** a chaotic sea, and tunnel- ing amplitudes have reproducible fluctuations by orders of magnitude over small changes of a parameter. This is reminiscent of universal conductance fluctuations which arise **in** condensed matter disordered systems. There, re- producible fluctuations of the conductance when e.g. a magnetic field is varied are an interferential signature of the disorder configuration [3].

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ξ .
Insulating regime. **In** the insulating regime L x ≥ ξ,
we expect a Log-normal conductance statistical distribu- tion [11]. However **in** the region g = 1 and for hgi . 1, we find a non-analytical behavior of P (g) **in** agreement with [13, 14, 15, 16] as shown for instance **in** fig. 2. **In** this figure we plot the distribution P (g) for similar values of hgi for GOE (J = 0) and GUE (J = 0.2). The shapes of these distributions are highly similar if hgi ≪ 1, showing that both distributions tend to become Log-normal with the same cumulants. **In** the intermediate regime, shapes are symmetry dependent. Moreover the non-analyticity appears for different values of conductance (close to 1) and the rate of the exponential decay [13] **in** the metallic

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Ultracold atomic gases are now widely considered to revisit standard problems of condensed matter physics under unique control possibilities. Dilute atomic Bose- Einstein condensates (BEC) [26, 27, 28, 29] and degenerate Fermi gases (DFG) [30, 31, 32, 33, 34] are produced routinely taking advantage of the recent progress **in** cooling and trapping of neutral atoms [35, 36, 37]. **In** addition, controlled potentials with no defects, for instance periodic potentials (optical **lattices**), can be designed **in** a large variety of geometries [38]. **In** periodic optical **lattices**, transport has been widely investigated, showing lattice-induced reduction of mobility [39, 40, 41] and interaction-induced self-trapping [42, 43]. Controlled disordered potentials can also be produced optically as demonstrated **in** several recent experiments [44, 45, 46, 47, 48], for instance using speckle patterns [49, 50]. Other techniques can be employed to produce controlled disorder such as the use of magnetic traps designed on atomic chips with rough wires [51, 52, 53, 54, 55], the use of localized impurity atoms [56, 57], or the use of radio-frequency fields [58]. However, the use of speckle potentials has unprecedented advantages from both practical and fundamental points of view. First, they are created using simple optical devices and their statistical properties are very well known [59, 60]. Second, they have finite-range correlations which offers richer situations than theoretical δ-correlated potentials (i.e. uncorrelated disorder) and the correlation functions can be designed almost at will by changing the geometry of the optical devices [59, 60]. Finally, both the amplitude and the correlation length (down to fractions of micrometers) can be controlled accurately and calibrated using ultracold atoms [48].

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Calculations performed at 1K depict linear scaling of the thermal conductivity for lengths up to 560 nm for various levels of nanoparticle densities and justify the transition from ballistic to diffusive transport as T increases. !(!), shown **in** Fig. 3, is calculated for T = 10K and T = 100K and ErAs interfacial area density of 2.38% and 23.8%. At 10K, both levels of disorder obey similar scaling behaviors, implying transport governed predominantly by ballistic and diffusive phonons. Here, the diffusive scattering is caused by momentum randomization due to elastic scattering, since phase breaking anharmonic processes have much larger mean free paths. At 100K there is a clear departure **in** the behavior of the low nanoparticle concentration and the **high** nanoparticle concentration superlattices. While the 2.38% sample monotonically increases to a bulk thermal conductivity, the 23.8% sample exhibits a maximum thermal conductivity at L = 112 nm. The ratio of the maximum thermal conductivity to the value calculated for L = 560 nm is ~2.1. Although the simulations are limited to small transverse dimensions, increasing the size of the ErAs nanoparticles should enhance this ratio. Larger nanoparticles should be able to reduce the frequency where ! !"# saturates to the approximate lower bound of

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factor” from the accurate TMM for our corrected FT. **In** this way, we avoid directly assuming Bragg resonant behavior which may be more unpredictable and complicated than Debye–Waller approximation model **in** terms of aperiodic gratings. Simulations reveal that the phase-accumulation errors are hugely minimized, and the peak of each main mode is adjusted to the correct wavelength location. Finally, we believe that the proposed correction to the FT of strongly scattering **lattices** is helpful for algorithm development of aperiodic lattice design under **high**-contrast refractive index conditions so as to create a fast Fourier transform relationship between the aperiodic pitch distribution and the corresponding reflective spectrum.

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n X T
X = I p×p or when X is a matrix with i.i.d. standard normal
entries. Moreover, their result has not only the optimal minimax rate, but also the exact optimal constant. General isotropic random designs are explored by Lecué and Mendelson [93]. For non-isotropic random designs and deterministic designs under conditions close to the Restricted Eigenvalue, the behavior of the Slope estimator is studied **in** [12]. The Slope estimator is adaptive only to s, and requires knowledge of σ, which is not available **in** practice. **In** order to have an estimator which is adaptive both to s and σ, we will use the Square-Root Slope, introduced by Stucky and van de Geer [131]. They give oracle inequalities for a large group of square-root estimators, including the new Square-Root Slope, but still following the scheme where (i) and (ii) cannot be avoided. The square-root estimators are also members of a more general family of penalized estimators defined by Owen [109, equations (8)-(9)] ; using their notation, these estimators correspond to the case where H M is the squared loss and B M is a norm

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Cambridge, MA 02139
Abstract—Among today’s robotics applications, exploration
missions **in** dynamic, **high** clutter and uncertain environmental conditions is quite common. Autonomous multi-vehicle systems come **in** handy for such exploration missions since a team of autonomous vehicels can explore an environment more efﬁciently and reliably than a single autonomous vehicle (AV). **In** order to improve the navigation accuracy, especially **in** the absence of a priori feature maps, various simultaneous **localization** and mapping (SLAM) algorithms are widely used **in** such applica- tions. As for multi-vehicle scenarios, collaborative multi-vehicle simultaneous **localization** and mapping algorithm (CSLAM) is an effective strategy. However use of multiple AVs poses additional scaling problems such as inter-vehicle map fusion, and data association which needs to be addressed. Although existing CSLAM algorithms are shown to perform quite adequately **in** simulations, their performance is much less to be desired **in** **high** clutter scenarios that is inevitable **in** actual environments. **In** this paper, we present an approach to improve the performance of a CSLAM algorithm **in** the presence of **high** clutter, by combining an effective clutter ﬁlter framework based on Random Finite Sets (RFS). The performance of the improved CSLAM algorithm is evaluated using simulations under varying clutter conditions.

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2.1. **Lattices**
A lattice L is a structure that is defined differently depending on the domain. **In** order theory, L is a partially ordered set (E, ≤) **in** which every pair of elements
e 1 , e 2 ∈ E has both a least upper bound and a greatest lower bound. **In** algebra, L = (E, ∧, ∨) with ∧ and ∨ being binary operations respecting the commutative, associative and absorption laws. **In** this work, we will prefer the order theoretic approach and consider our **lattices** to be complete - i.e. having a finite set of elements including a greatest and a least.

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d for the L 1 (Manhattan) metric and remains stable **in** distribution for the L 2 (Eu-
clidean) metric. It tends to infinity **in** probability for p < 2 and to zero for p > 2. This observation is **in** line with the conclusions of Hinneburg et al. (2000), who ar- gue that nearest neighbor search **in** a **high**-**dimensional** space tends to be meaning- less for norms with larger exponents, since the maximum observed distance tends towards the minimum one. It should be noted, however, that the variance of the limiting distribution depends on the value of p.

We denote the d-**dimensional** Apollonian network after t iterations by A(d, t), d ≥ 2, t ≥ 0. Then the d-**dimensional** Apollonian network at step t is constructed as follows: For t = 0, A(d, 0) is the complete graph K d+1 (or (d + 1)-clique), and A(d, 0)
has d + 1 vertices and (d+1)d 2 edges. For t ≥ 1, A(d, t) is obtained from A(d, t − 1) by adding for each of its existing subgraphs isomorphic to a (d + 1)-clique and created at step t − 1 a new vertex and joining it to all the vertices of this subgraph (see Fig. 1(b) for the case d = 2). Then, at t = 1, we add one new vertex and d + 1 new edges to the graph, creating d + 1 new K d+1 cliques and resulting **in** the complete graph with d + 2

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3.8. Conclusion 85
3.8 Conclusion
**In** this paper we introduced the use of low discrepancy sequences **in** approximate Bayesian computation. We found that from both a theoretical and practical perspec- tive the use of (R)QMC **in** ABC can yield substantial variance reduction of estimators based on the approximate posterior distribution. However, care must be taken when using (R)QMC sequences. First, the transformation of uniform sequences to the distri- bution of interest must preserve the low discrepancy properties of the point set. This is of major importance for a sequential version of the ABC algorithm that is based on adaptive importance sampling. Second, the advantage of using (R)QMC tends to diminish with increasing dimension. **In** the setting of ABC, however, the parameter space is often of low dimension such that the gains from using (R)QMC are visible **in** most applications, as shown by our simulations. From a practical perspective we recommend to use RQMC point sets instead of QMC as these allow the assessment of the error via repeated simulation and are unbiased.

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• Filamentary structures and stratified spaces: 1-**dimensional** filamen- tary structures appear **in** many domains (road networks, network of blood vessels, astronomy, etc.) and can be modeled as 1-**dimensional** stratified sets, or (geometric) graphs. Various methods, motivated and driven by spe- cific applications have been developed to reconstruct such structures from point cloud data. From a general perspective the (relatively) simple structure of graphs allows to propose new approaches to design metric graph recon- struction algorithms coming with various topological guarantees, e.g. home- omorphy or homotopy type and closeness **in** the Gromov-Hausdorf metric [GSBW11, ACC + 12, CHS15]. Despite a few attempts [BCSE + 07, BWM12],

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Towards understanding computational-statistical gaps, and specifically identifying the funda- mentally hard region for various inference problems, a couple of approaches have been considered. One of the approaches seeks to identify the algorithmic limit "from above", **in** the sense of iden- tifying the fundamental limits **in** the statistical performance of various families of known com- putationally efficient algorithms. Some of the families that have been analyzed are (1) the Sum of Squares (SOS) hierarchy, which is a family of convex relaxation methods [Par00], [Las01] (2) the family of local algorithms inspired by the Belief Propagation with the celebrated example of Approximate Message Passing [DMM09], [DJM13]), (3) the family of statistical query algorithms [Kea98] and (4) several Markov Chain Monte Carlo algorithms such as Metropolis Hasting and Glauber Dynamics [LPW06]. Another approach offers an average-case complexity-theory point of view [BR13], [CLR17], [WBP16], [BBH18]. **In** this line of work, the hard regimes of the var- ious inference problems are linked by showing that solving certain **high** **dimensional** statistical problems **in** their hard regime reduces **in** a polynomial time to solving other **high** **dimensional** statistical problems **in** their own hard regime.

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