A generalization of heterochromatic graphs

In 2006, Suzuki, and Akbari & Alipour independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. In this paper, we propose $f$-chromatic graphs as a generalization of heterochromatic graphs. An edge-colored graph is $f$-chromatic if each color $c$ appears on at most $f(c)$ edges. We also present a necessary and sufficient condition for edge-colored graphs to have an $f$-chromatic spanning forest with exactly $m$ components. Moreover, using this criterion, we show that a $g$-chromatic graph $G$ of order $n$ with $|E(G)|>\binom{n-m}{2}$ has an $f$-chromatic spanning forest with exactly $m$ ($1 \le m \le n-1$) components if $g(c) \le \frac{|E(G)|}{n-m}f(c)$ for any color $c$.Comment: 14 pages, 4 figure

Triviality of the ground-state metastate in long-range Ising spin glasses in one dimension

We consider the one-dimensional model of a spin glass with independent Gaussian-distributed random interactions, that have mean zero and variance $1/|i-j|^{2\sigma}$, between the spins at sites $i$ and $j$ for all $i\neq j$. It is known that, for $\sigma>1$, there is no phase transition at any non-zero temperature in this model. We prove rigorously that, for $\sigma>3/2$, any Newman-Stein metastate for the ground states (i.e.\ the frequencies with which distinct ground states are observed in finite size samples in the limit of infinite size, for given disorder) is trivial and unique. In other words, for given disorder and asymptotically at large sizes, the same ground state, or its global spin flip, is obtained (almost) always. The proof consists of two parts: one is a theorem (based on one by Newman and Stein for short-range two-dimensional models), valid for all $\sigma>1$, that establishes triviality under a convergence hypothesis on something similar to the energies of domain walls, and the other (based on older results for the one-dimensional model) establishes that the hypothesis is true for $\sigma>3/2$. In addition, we derive heuristic scaling arguments and rigorous exponent inequalities which tend to support the validity of the hypothesis under broader conditions. The constructions of various metastates are extended to all values $\sigma>1/2$. Triviality of the metastate in bond-diluted power-law models for $\sigma>1$ is proved directly.Comment: 18 pages. v2: subsection on bond-diluted models added, few extra references. 19 pages. v3: published version; a few changes; 20 page

On the convergence of chemical reaction optimization for combinatorial optimization

A novel general-purpose optimization method, chemical reaction optimization (CRO), is a population-based metaheuristic inspired by the phenomenon of interactions between molecules in a chemical reaction process. CRO has demonstrated its competitive edge over existing methods in solving many real-world problems. However, all studies concerning CRO have been empirical in nature and no theoretical analysis has been conducted to study its convergence properties. In this paper, we present some convergence results for several generic versions of CRO, each of which adopts different combinations of elementary reactions. We investigate the limiting behavior of CRO. By modeling CRO as a finite absorbing Markov chain, we show that CRO converges to a global optimum solution with a probability arbitrarily close to one when time tends to infinity. Our results also show that the convergence of CRO is determined by both the elementary reactions and the total energy of the system. Moreover, we also study and discuss the finite time behavior of CRO. © 1997-2012 IEEE.published_or_final_versio

Position sensing systems including magnetoresistive elements

The present invention provides a position-sensing system which employs sensors incorporating magnetoresistive materials. The position of a magnetic information input member is determined through the resistance change of the magnetoresistive sensor in response to the magnetic field from the magnetic information input member. Exemplary magnetoresistive materials are lanthanum manganites having high magnetoresistive ratios. Two-dimensional position sensing systems for graphics tablets are also described.Published versio

Large area metal nanowire arrays with submicron pitch and tunable sub-20 nm nanogaps

We present a new top-down nanofabrication technology to realize large area metal nanowire (m-NW) arrays with tunable sub-20 nm separation nanogaps without the use of chemical etching or milling of the metal layer. The nanofabrication technology is based on a self-regulating metal deposition process that is facilitated by closely spaced and isolated heterogeneous template surfaces that confines the metal deposition into two dimensions. Electrically isolated parallel arrays of m-NW can be realized with uniform and controllable nanogaps. Au-NW arrays are presented with high-density ~105 NWs cm-1, variable NW diameters down to 50 nm, variable nanogaps down to 5 nm, and very large nanogap length density ~1 km cm-2. A spatially averaged surface enhanced Raman scattering (SERS) analytical enhancement factor of (1.5±0.2)×107 is demonstrated from a benzenethiol monolayer chemisorbed on a Au-NW array substrat

A graphic method for depicting horizontal direction data on vertical outcrop photographs

Outcrop photographs which show two-dimensional representations of three-dimensionally dipping surfaces (e.g., bedding planes, cross-bed foresets) are commonly utilized in the description of sedimentary strata. In many instances, accurate depiction of the dip direction of such features is paramount for understanding their interpretation, and for visualizing the true form of three-dimensional bodies (e.g., conceptualizing the form of an architectural element in a cliff-face, preserved as a vertical slice that has been cut oblique to paleocurrent direction). However, as an outcrop photograph often presents information on a vertical plane and directional data refers to a horizontal plane, the accurate co-depiction of both sets of information may be challenging. There is presently no universal method for illustrating such measurements on outcrop photographs: techniques in common usage are often imprecise, and the lack of uniformity hinders comparison between different images. Here we present a method for accurately depicting horizontal direction data on vertical outcrop photographs which permits instant visualization of dip relative to the illustrated outcrop geometry. The method is simple to apply, does not compromise primary data, and is unobtrusive to other visual information within images; thus having utility across a broad spectrum of geological investigations