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COUNTEREXAMPLE TO A *CONJECTURE* ON THE EDGE RECONSTRUCTION NUMBER OF REGULAR GRAPHS

"... In a survey by Asciak et al. [2], a conjecture on the edge reconstruction number of a regular graph was made. We have discovered a counterexample. All graphs mentioned are finite and simple. Definition 1 (Edge-Deck). Let G be a graph, and let e ∈ E(G). Then the un-labeled graph G − e is said to be a ..."

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′} ∼ = G.

*Conjecture*1. [2] If G is an r ≥ 3 regular graph, then*ern*(G) ≤ 2.### MathCheck: A Math Assistant via a Combination of Computer Algebra Systems and SAT Solvers

"... Abstract. We present a method and an associated system, calledMath-Check, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assis-tant by mathematicians to either ..."

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counterexample or finitely verify open universal

*conjectures*on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS system. Such a SAT+CAS system combines the efficient search routines of mod-*ern*SAT solvers, with the expressive power of CAS, thus### AFRICA IN

"... Conjectures about the Cretaceous and post−Cretaceous verte− brate faunas of Madagascar are generally based on the fact that these faunas display similarities to those of South America, and that Africa lacks taxa that are common to Madagascar and South America. In order to account for this distributi ..."

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*Conjectures*about the Cretaceous and post−Cretaceous verte− brate faunas of Madagascar are generally based on the fact that these faunas display similarities to those of South America, and that Africa lacks taxa that are common to Madagascar and South America. In order to account

### Ultrahigh Energy Cosmic Rays: Facts, Myths, and Legends

"... This is a written version of a series of lectures aimed at graduate students in astrophysics/particle theory/particle experiment. In the first part, we explain the important progress made in recent years towards understanding the ex-perimental data on cosmic rays with energies & 108 GeV. We begi ..."

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to discuss the

*conjectured*nearby cosmic ray sources, and emphasize some of the prospects for a new (multi-particle) astronomy. Next, we survey the state of the art regarding the ultrahigh energy cosmic neutrinos which should be produced in association with the observed cosmic rays. In the second part, we### TIME AND MATTER 2010 CONFERENCE A short introduction to Asymptotic Safety

"... Abstract: I discuss the notion of asymptotic safety and possible appli-cations to quantum field theories of gravity and matter. What is asymptotic safety? We want to discuss the high energy behavior of a quantum field theory (QFT). Assume that a “theory space ” has been defined by giving a set of fi ..."

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Abstract: I discuss the notion of asymptotic safety and possible appli-cations to quantum field theories of gravity and matter. What is asymptotic safety? We want to discuss the high energy behavior of a quantum field theory (QFT). Assume that a “theory space ” has been defined by giving a set of fields, their symmetries and a class of action functionals depending on fields φ and couplings gi. We will write gi = kdi g̃i, where k is a momentum cutoff and di is the mass dimension of gi. The real numbers g̃i are taken as coordinates in theory space. Ideally the couplings gi should be defined in terms of physical observables such as cross sections and decay rates. In any case “redundant ” couplings, i.e. couplings that can be eliminated by field redefinitions, should not be included. We also assume that a Renor-malization Group (RG) flow has been defined on theory space; it describes the dependence of the action on an energy scale k (or perhaps a “RG time” t = log k). The action is assumed to have the form Γk(φ, gi) =∑ i gi(k)Oi(φ) , (1) where Oi are typically local operators constructed with the field φ and its derivatives, which are compatible with the symmetries of the theory. We identify theories with RG trajectories. It can generically be expected that when k goes to infinity some cou-plings gi(k) also go to infinity. What we want to avoid is that the dimen-sionless couplings g̃i diverge. In fact, there are famous examples such as QED and φ4 theory where this happens even at some finite scale kmax. Such

### Université de Cergy-Pontoise

, 804

"... Abstract. We classify nonconstant entire local minimizers of the standard Ginzburg-Landau functional for maps in H 1 loc (R3; R 3) satisfying a natural energy bound. Up to translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit solution equivariant under the ..."

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Abstract. We classify nonconstant entire local minimizers of the standard Ginzburg-Landau functional for maps in H 1 loc (R3; R 3) satisfying a natural energy bound. Up to translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit solution equivariant under the action of the orthogonal group.

### 1Local Identification of Overcomplete Dictionaries

"... This paper presents the first theoretical results showing that stable identification of overcomplete µ-coherent dictionaries Φ ∈ Rd×K is locally possible from training signals with sparsity levels S up to the order O(µ−2) and signal to noise ratios up to O( d). In particular the dictionary is recove ..."

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This paper presents the first theoretical results showing that stable identification of overcomplete µ-coherent dictionaries Φ ∈ Rd×K is locally possible from training signals with sparsity levels S up to the order O(µ−2) and signal to noise ratios up to O( d). In particular the dictionary is recoverable as the local maximum of a new maximisation criterion that generalises the K-means criterion. For this maximisation criterion results for asymptotic exact recovery for sparsity levels up toO(µ−1) and stable recovery for sparsity levels up to O(µ−2) as well as signal to noise ratios up to O( d) are provided. These asymptotic results translate to finite sample size recovery results with high probability as long as the sample size N scales as O(K3dSε̃−2), where the recovery precision ε ̃ can go down to the asymptotically achievable precision. Further to actually find the local maxima of the new criterion, a very simple Iterative Thresholding & K (signed) Means algorithm (ITKM), which has complexity O(dKN) in each iteration, is presented and its local efficiency is demonstrated in several experiments. Index Terms dictionary learning, dictionary identification, sparse coding, sparse component analysis, vector quanti-sation, K-means, finite sample size, sampling complexity, maximisation criterion, sparse representation 1