Letter graphs and geometric grid classes of permutations: characterization and recognition

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author

Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density

The classical monomer-dimer model in two-dimensional lattices has been shown to belong to the \emph{``#P-complete''} class, which indicates the problem is computationally ``intractable''. We use exact computational method to investigate the number of ways to arrange dimers on $m \times n$ two-dimensional rectangular lattice strips with fixed dimer density $\rho$. For any dimer density $0 < \rho < 1$, we find a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term is exactly -1/2. This logarithmic correction term is explained by the newly developed asymptotic theory of Pemantle and Wilson. The sequence of the free energy of lattice strips with cylinder boundary condition converges so fast that very accurate free energy $f_2(\rho)$ for large lattices can be obtained. For example, for a half-filled lattice, $f_2(1/2) = 0.633195588930$, while $f_2(1/4) = 0.4413453753046$ and $f_2(3/4) = 0.64039026$. For $\rho < 0.65$, $f_2(\rho)$ is accurate at least to 10 decimal digits. The function $f_2(\rho)$ reaches the maximum value $f_2(\rho^*) = 0.662798972834$ at $\rho^* = 0.6381231$, with 11 correct digits. This is also the \md constant for two-dimensional rectangular lattices. The asymptotic expressions of free energy near close packing are investigated for finite and infinite lattice widths. For lattices with finite width, dependence on the parity of the lattice width is found. For infinite lattices, the data support the functional form obtained previously through series expansions.Comment: 15 pages, 5 figures, 5 table

The Experimental Monitoring of the Water Regime in the Reka River

Reka Reka, s prispevno površino 422 km2 , ponika v Škocjanskih jamah, ki jih je UNESCO leta 1986 proglasil za svetovno dediščino. V sedemdesetih letih je bila Reka ena od najbolj onesnaženih rek v Sloveniji. V času visokih vod leta 1999 in 2000 smo izvedli meritve hitrosti, kalnosti, vrste fizikalnih in kemičnih parametrov ter teste strupenosti. Glavni cilji teh pionirskih meritev so bili preveriti mersko opremo v terenskih pogojih visokih voda, zbrati čimveč podatkov ter primerjati uporabnost opreme.The river Reka, with 422 square kilometres of drainage area sinks into the Škocijan Cave system, which was proclaimed by UNESCO as a World Heritage Site in 1986. In the seventies, the Reka river was one of most polluted rivers in Slovenia. During floods in 1999 and 2000, experimental measurements of velocity, water level, suspended sediment transport, chemical parameters and toxicity tests were conducted. The main tasks in the first stage of the investigation: check the equipment in field conditions and test the toxicity of water in particular cross sections. In the paper, the measurements and some discussion of the results and applicability of equipment are presented

Three osculating walkers

We consider three directed walkers on the square lattice, which move simultaneously at each tick of a clock and never cross. Their trajectories form a non-crossing configuration of walks. This configuration is said to be osculating if the walkers never share an edge, and vicious (or: non-intersecting) if they never meet. We give a closed form expression for the generating function of osculating configurations starting from prescribed points. This generating function turns out to be algebraic. We also relate the enumeration of osculating configurations with prescribed starting and ending points to the (better understood) enumeration of non-intersecting configurations. Our method is based on a step by step decomposition of osculating configurations, and on the solution of the functional equation provided by this decomposition

Recurrence and Polya number of general one-dimensional random walks

The recurrence properties of random walks can be characterized by P\'{o}lya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right with probabilities $l$ and $r$, or remain at the same position with probability $o$ ($l+r+o=1$). We calculate P\'{o}lya number $P$ of this model and find a simple expression for $P$ as, $P=1-\Delta$, where $\Delta$ is the absolute difference of $l$ and $r$ ($\Delta=|l-r|$). We prove this rigorous expression by the method of creative telescoping, and our result suggests that the walk is recurrent if and only if the left-moving probability $l$ equals to the right-moving probability $r$.Comment: 3 page short pape

Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

We investigate the asymptotic behavior of the Selberg-like integral $$\frac1{N!}\int_{[0,1]^N}x_1^p\prod_{i<j}(x_i-x_j)^2\prod_ix_i^{a-1}(1-x_i)^{b-1}dx_i$$, as $N\to\infty$ for different scalings of the parameters $a$ and $b$ with $N$. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting $N$ channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly

Form Sequences to Polynomials and Back, via Operator Orderings

C.M. Bender and G. V. Dunne showed that linear combinations of words $q^{k}p^{n}q^{n-k}$, where $p$ and $q$ are subject to the relation $qp - pq = \imath$, may be expressed as a polynomial in the symbol $z = \tfrac{1}{2}(qp+pq)$. Relations between such polynomials and linear combinations of the transformed coefficients are explored. In particular, examples yielding orthogonal polynomials are provided

The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions

We consider the application of the DRA method to the case of several master integrals in a given sector. We establish a connection between the homogeneous part of dimensional recurrence and maximal unitarity cuts of the corresponding integrals: a maximally cut master integral appears to be a solution of the homogeneous part of the dimensional recurrence relation. This observation allows us to make a necessary step of the DRA method, the construction of the general solution of the homogeneous equation, which, in this case, is a coupled system of difference equations.Comment: 17 pages, 2 figure

Changes in quality characteristics of fresh blueberries: Combined effect of cultivar and storage conditions

The influences of two storage conditions (regular atmosphere-RA and modified-atmosphere packaging-MAP) and different storage time on fruit textural parameters, chemical composition, and total quality index (TQI) of two blueberry cultivars were investigated. Freshly harvested fruit of mid and late season cultivars (‘Bluecrop’ and ‘Liberty’, respectively) were placed in plastic punnets, packed into low-density polyethylene bags of 25 μm thickness with two perforations of 3 mm and stored at 2 ◦C and 90% relative humidity for 30 days, either in RA or in MAP. Changes in gas composition inside the package and fruit quality characteristics were analyzed at 10- day intervals during storage: 0, 10, 20, and 30 days. ‘Liberty’ was dominant over ‘Bluecrop’ in terms of hardness (428 g and 296 g, respectively), as well as individual and total sugars (100 and 76 g⋅kg–1, respectively), organic acids (19 and 12 g⋅kg–1, respectively) and most subclasses of phenolic compounds (anthocyanins, flavonols, and hydroxycinnamic acids). In addition, a novel mathematical index of TQI was introduced to compare all evaluated parameters in order to obtain a quantitative single score, as an indicator of overall fruit quality. ‘Liberty’ had the better TQI score in RA, whereas ‘Bluecrop’ behaved better in MAP. Accordingly, for longer storage of blueberry fruit MAP should not be assumed to be uniformly helpful, since the effect of storage duration in the specific type of atmosphere substantially depends on the proper cultivar selection