Characterizing 2-crossing-critical graphs

It is very well-known that there are precisely two minimal non-planar graphs: $K_5$ and $K_{3,3}$ (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: they each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the M\"obius Ladder $V_{10}$; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of $V_{10}$; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of $V_{8}$.Comment: 176 pages, 28 figure

The metastate approach to thermodynamic chaos

In realistic disordered systems, such as the Edwards-Anderson (EA) spin glass, no order parameter, such as the Parisi overlap distribution, can be both translation-invariant and non-self-averaging. The standard mean-field picture of the EA spin glass phase can therefore not be valid in any dimension and at any temperature. Further analysis shows that, in general, when systems have many competing (pure) thermodynamic states, a single state which is a mixture of many of them (as in the standard mean-field picture) contains insufficient information to reveal the full thermodynamic structure. We propose a different approach, in which an appropriate thermodynamic description of such a system is instead based on a metastate, which is an ensemble of (possibly mixed) thermodynamic states. This approach, modelled on chaotic dynamical systems, is needed when chaotic size dependence (of finite volume correlations) is present. Here replicas arise in a natural way, when a metastate is specified by its (meta)correlations. The metastate approach explains, connects, and unifies such concepts as replica symmetry breaking, chaotic size dependence and replica non-independence. Furthermore, it replaces the older idea of non-self-averaging as dependence on the bulk couplings with the concept of dependence on the state within the metastate at fixed coupling realization. We use these ideas to classify possible metastates for the EA model, and discuss two scenarios introduced by us earlier --- a nonstandard mean-field picture and a picture intermediate between that and the usual scaling/droplet picture.Comment: LaTeX file, 49 page

Multi-View Image Compositions

The geometry of single-viewpoint panoramas is well understood: multiple pictures taken from the same viewpoint may be stitched together into a consistent panorama mosaic. By contrast, when the point of view changes or when the scene changes (e.g., due to objects moving) no consistent mosaic may be obtained, unless the structure of the scene is very special. Artists have explored this problem and demonstrated that geometrical consistency is not the only criterion for success: incorporating multiple view points in space and time into the same panorama may produce compelling and informative pictures. We explore this avenue and suggest an approach to automating the construction of mosaics from images taken from multiple view points into a single panorama. Rather than looking at 3D scene consistency we look at image consistency. Our approach is based on optimizing a cost function that keeps into account image-to-image consistency which is measured on point-features and along picture boundaries. The optimization explicitly considers occlusion between pictures. We illustrate our ideas with a number of experiments on collections of images of objects and outdoor scenes

Simplicity of State and Overlap Structure in Finite-Volume Realistic Spin Glasses

We present a combination of heuristic and rigorous arguments indicating that both the pure state structure and the overlap structure of realistic spin glasses should be relatively simple: in a large finite volume with coupling-independent boundary conditions, such as periodic, at most a pair of flip-related (or the appropriate number of symmetry-related in the non-Ising case) states appear, and the Parisi overlap distribution correspondingly exhibits at most a pair of delta-functions at plus/minus the self-overlap. This rules out the nonstandard SK picture introduced by us earlier, and when combined with our previous elimination of more standard versions of the mean field picture, argues against the possibility of even limited versions of mean field ordering in realistic spin glasses. If broken spin flip symmetry should occur, this leaves open two main possibilities for ordering in the spin glass phase: the droplet/scaling two-state picture, and the chaotic pairs many-state picture introduced by us earlier. We present scaling arguments which provide a possible physical basis for the latter picture, and discuss possible reasons behind numerical observations of more complicated overlap structures in finite volumes.Comment: 22 pages (LaTeX; needs revtex), 1 figure (PostScript); to appear in Physical Review

Elastic energy of proteins and the stages of protein folding

We propose a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. It is constructed using physical arguments based on the hydrophobic effect and hydrogen bonding. Adjustable parameters are fitted to data from the computer simulation of the folding of a set of proteins using the CSAW (conditioned self-avoiding walk) model. The elastic energy gives rise to scaling relations of the form $R_{g}\sim N^{\nu}$ in different regions. It shows three folding stages characterized by the progression with exponents $\nu = 3/5, 3/7, 2/5$, which we identify as the unfolded stage, pre-globule, and molten globule, respectively. The pre-globule goes over to the molten globule via a break in behavior akin to a first-order phase transition, which is initiated by a sudden acceleration of hydrogen bonding

Steric constraints in model proteins

A simple lattice model for proteins that allows for distinct sizes of the amino acids is presented. The model is found to lead to a significant number of conformations that are the unique ground state of one or more sequences or encodable. Furthermore, several of the encodable structures are highly designable and are the non-degenerate ground state of several sequences. Even though the native state conformations are typically compact, not all compact conformations are encodable. The incorporation of the hydrophobic and polar nature of amino acids further enhances the attractive features of the model.Comment: RevTex, 5 pages, 3 postscript figure