Treatment of Linear and Nonlinear Dielectric Property of Molecular Monolayer and Submonolayer with Microscopic Dipole Lattice Model: I. Second Harmonic Generation and Sum-Frequency Generation

In the currently accepted models of the nonlinear optics, the nonlinear radiation was treated as the result of an infinitesimally thin polarization sheet layer, and a three layer model was generally employed. The direct consequence of this approach is that an apriori dielectric constant, which still does not have a clear definition, has to be assigned to this polarization layer. Because the Second Harmonic Generation (SHG) and the Sum-Frequency Generation vibrational Spectroscopy (SFG-VS) have been proven as the sensitive probes for interfaces with the submonolayer coverage, the treatment based on the more realistic discrete induced dipole model needs to be developed. Here we show that following the molecular optics theory approach the SHG, as well as the SFG-VS, radiation from the monolayer or submonolayer at an interface can be rigorously treated as the radiation from an induced dipole lattice at the interface. In this approach, the introduction of the polarization sheet is no longer necessary. Therefore, the ambiguity of the unaccounted dielectric constant of the polarization layer is no longer an issue. Moreover, the anisotropic two dimensional microscopic local field factors can be explicitly expressed with the linear polarizability tensors of the interfacial molecules. Based on the planewise dipole sum rule in the molecular monolayer, crucial experimental tests of this microscopic treatment with SHG and SFG-VS are discussed. Many puzzles in the literature of surface SHG and SFG spectroscopy studies can also be understood or resolved in this framework. This new treatment may provide a solid basis for the quantitative analysis in the surface SHG and SFG studies.Comment: 23 pages, 3 figure

The Lambek calculus with iteration: two variants

Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omega-rule for introducing iteration to the antecedent. We also consider a version with infinite (but finitely branching) derivations and prove equivalence of these two versions. In Kleene algebras, this line of calculi corresponds to the *-continuous case. For the second line, we restrict our infinite derivations to cyclic (regular) ones. We show that this system is equivalent to a variant of action logic that corresponds to general residuated Kleene algebras, not necessarily *-continuous. Finally, we show that, in contrast with the case without division operations (considered by Kozen), the first system is strictly stronger than the second one. To prove this, we use a complexity argument. Namely, we show, using methods of Buszkowski and Palka, that the first system is $\Pi_1^0$-hard, and therefore is not recursively enumerable and cannot be described by a calculus with finite derivations

Capacitated Center Problems with Two-Sided Bounds and Outliers

In recent years, the capacitated center problems have attracted a lot of research interest. Given a set of vertices $V$, we want to find a subset of vertices $S$, called centers, such that the maximum cluster radius is minimized. Moreover, each center in $S$ should satisfy some capacity constraint, which could be an upper or lower bound on the number of vertices it can serve. Capacitated $k$-center problems with one-sided bounds (upper or lower) have been well studied in previous work, and a constant factor approximation was obtained. We are the first to study the capacitated center problem with both capacity lower and upper bounds (with or without outliers). We assume each vertex has a uniform lower bound and a non-uniform upper bound. For the case of opening exactly $k$ centers, we note that a generalization of a recent LP approach can achieve constant factor approximation algorithms for our problems. Our main contribution is a simple combinatorial algorithm for the case where there is no cardinality constraint on the number of open centers. Our combinatorial algorithm is simpler and achieves better constant approximation factor compared to the LP approach

On the Number of Synchronizing Colorings of Digraphs

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have different colors. Such a coloring corresponds naturally to an automaton. The road coloring theorem states that every primitive digraph has a synchronizing coloring. In the present paper we study how many synchronizing colorings can exist for a digraph with $n$ vertices. We performed an extensive experimental investigation of digraphs with small number of vertices. This was done by using our dedicated algorithm exhaustively enumerating all small digraphs. We also present a series of digraphs whose fraction of synchronizing colorings is equal to $1-1/k^d$, for every $d \ge 1$ and the number of vertices large enough. On the basis of our results we state several conjectures and open problems. In particular, we conjecture that $1-1/k$ is the smallest possible fraction of synchronizing colorings, except for a single exceptional example on 6 vertices for $k=2$.Comment: CIAA 2015. The final publication is available at http://link.springer.com/chapter/10.1007/978-3-319-22360-5_1

An ISS Small-Gain Theorem for General Networks

We provide a generalized version of the nonlinear small-gain theorem for the case of more than two coupled input-to-state stable (ISS) systems. For this result the interconnection gains are described in a nonlinear gain matrix and the small-gain condition requires bounds on the image of this gain matrix. The condition may be interpreted as a nonlinear generalization of the requirement that the spectral radius of the gain matrix is less than one. We give some interpretations of the condition in special cases covering two subsystems, linear gains, linear systems and an associated artificial dynamical system.Comment: 26 pages, 3 figures, submitted to Mathematics of Control, Signals, and Systems (MCSS

Elevated CO₂ does not increase eucalypt forest productivity on a low-phosphorus soil

Rising atmospheric CO2 stimulates photosynthesis and productivity of forests, offsetting CO2 emissions. Elevated CO2 experiments in temperate planted forests yielded ~23% increases in productivity over the initial years. Whether similar CO2 stimulation occurs in mature evergreen broadleaved forests on low-phosphorus (P) soils is unknown, largely due to lack of experimental evidence. This knowledge gap creates major uncertainties in future climate projections as a large part of the tropics is P-limited. Here,we increased atmospheric CO2 concentration in a mature broadleaved evergreen eucalypt forest for three years, in the first large-scale experiment on a P-limited site. We show that tree growth and other aboveground productivity components did not significantly increase in response to elevated CO2 in three years, despite a sustained 19% increase in leaf photosynthesis. Moreover, tree growth in ambient CO2 was strongly P-limited and increased by ~35% with added phosphorus. The findings suggest that P availability may potentially constrain CO2-enhanced productivity in P-limited forests; hence, future atmospheric CO2 trajectories may be higher than predicted by some models. As a result, coupled climate-carbon models should incorporate both nitrogen and phosphorus limitations to vegetation productivity in estimating future carbon sinks

Solutions of Several Coupled Discrete Models in terms of Lame Polynomials of Order One and Two

Coupled discrete models abound in several areas of physics. Here we provide an extensive set of exact quasiperiodic solutions of a number of coupled discrete models in terms of Lame polynomials of order one and two. Some of the models discussed are (i) coupled Salerno model, (ii) coupled Ablowitz-Ladik model, (iii) coupled saturated discrete nonlinear Schrodinger equation, (iv) coupled phi4 model, and (v) coupled phi6 model. Furthermore, we show that most of these coupled models in fact also possess an even broader class of exact solutions.Comment: 31 pages, to appear in Pramana (Journal of Physics) 201

Classification of non-Riemannian doubled-yet-gauged spacetime

Assuming $\mathbf{O}(D,D)$ covariant fields as the `fundamental' variables, Double Field Theory can accommodate novel geometries where a Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, $(n,\bar{n})$, $0\leq n+\bar{n}\leq D$. Upon these backgrounds, strings become chiral and anti-chiral over $n$ and $\bar{n}$ directions respectively, while particles and strings are frozen over the $n+\bar{n}$ directions. In particular, we identify $(0,0)$ as Riemannian manifolds, $(1,0)$ as non-relativistic spacetime, $(1,1)$ as Gomis-Ooguri non-relativistic string, $(D{-1},0)$ as ultra-relativistic Carroll geometry, and $(D,0)$ as Siegel's chiral string. Combined with a covariant Kaluza-Klein ansatz which we further spell, $(0,1)$ leads to Newton-Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as $D=10$, $(3,3)$ may open a new scheme of the dimensional reduction from ten to four.Comment: 1+41 pages; v2) Refs added; v3) Published version; v4) Sign error in (2.51) correcte

Anomaly Cancelation in Field Theory and F-theory on a Circle

We study the manifestation of local gauge anomalies of four- and six-dimensional field theories in the lower-dimensional Kaluza-Klein theory obtained after circle compactification. We identify a convenient set of transformations acting on the whole tower of massless and massive states and investigate their action on the low-energy effective theories in the Coulomb branch. The maps employ higher-dimensional large gauge transformations and precisely yield the anomaly cancelation conditions when acting on the one-loop induced Chern-Simons terms in the three- and five-dimensional effective theory. The arising symmetries are argued to play a key role in the study of the M-theory to F-theory limit on Calabi-Yau manifolds. For example, using the fact that all fully resolved F-theory geometries inducing multiple Abelian gauge groups or non-Abelian groups admit a certain set of symmetries, we are able to generally show the cancelation of pure Abelian or pure non-Abelian anomalies in these models.Comment: 48 pages, 2 figures; v2: typos corrected, comments on circle fluxes adde

Random walk with barriers: Diffusion restricted by permeable membranes

Restrictions to molecular motion by barriers (membranes) are ubiquitous in biological tissues, porous media and composite materials. A major challenge is to characterize the microstructure of a material or an organism nondestructively using a bulk transport measurement. Here we demonstrate how the long-range structural correlations introduced by permeable membranes give rise to distinct features of transport. We consider Brownian motion restricted by randomly placed and oriented permeable membranes and focus on the disorder-averaged diffusion propagator using a scattering approach. The renormalization group solution reveals a scaling behavior of the diffusion coefficient for large times, with a characteristically slow inverse square root time dependence. The predicted time dependence of the diffusion coefficient agrees well with Monte Carlo simulations in two dimensions. Our results can be used to identify permeable membranes as restrictions to transport in disordered materials and in biological tissues, and to quantify their permeability and surface area.Comment: 8 pages, 3 figures; origin of dispersion clarified, refs adde