Combinatorial Type Research Articles

A Time-Dependent Vehicle Routing Problem with Time Windows for E-Commerce Supplier Site Pickups Using Genetic Algorithm

The VRP is classified as an NP-hard problem. Hence exact optimization methods may be difficult to solve these problems in acceptable CPU times, when the problem involves real-world data sets that are very large. To get solutions in determining routes which are realistic and very close to the actual solution, we use heuristics and metaheuristics which are of the combinatorial optimization type. A literature review of VRPTW, TDVRP, and a metaheuristic such as the genetic algorithm was conducted. In this paper, the implementation of the VRPTW and its extension, the time-dependent VRPTW (TDVRPTW) has been carried out using the model as well as metaheuristics such as the genetic algorithm (GA). The algorithms were implemented, using Matlab and HeuristicLab optimization software. A plugin was developed using Visual C# and DOT NET framework 4.5. Results were tested using Solomon’s 56 benchmark instances classified into groups such as C1, C2, R1, R2, RC1, RC2, with 100 customer nodes, 25 vehicles and each vehicle capacity of 200. The results were comparable to the earlier algorithms developed and in some cases the current algorithm yielded better results in terms of total distance travelled and the average number of vehicles used.

Tropical lower bounds for extended formulations

The tropical arithmetic operations on $$\mathbb {R}$$R, defined as $$\oplus :(a,b)\rightarrow \min \{a,b\}$$?:(a,b)?min{a,b} and $$\otimes :(a,b)\rightarrow a+b$$?:(a,b)?a+b, arise from studying the geometry over non-Archimedean fields. We present an application of tropical methods to the study of extended formulations for convex polytopes. We propose a non-Archimedean generalization of the well known Boolean rank bound for the extension complexity. We show how to construct a real polytope with the same extension complexity and combinatorial type as a given non-Archimedean polytope. Our results allow us to develop a method of constructing real polytopes with large extension complexity.

Using a 2-variable method in radionuclide shuntography to predict shunt patency.

Radionuclide shuntography interpretation is uncertain when the tracer fails to enter the ventricles but quickly drains distally or when the tracer enters the ventricles but takes longer than 15 minutes to drain distally. The purpose of this study was to aid in the clinical interpretation of a variety of shuntography results and to determine the applicability of shuntography in different patient populations. The results of 259 shuntograms were reviewed. Chi-square analysis was performed to evaluate the relationship between clinical variables and shuntography results. Two-by-two binary classification analyses were performed to determine the sensitivity, specificity, positive predictive value, and negative predictive value for 4 different combinatorial types of shuntography results based on 2 variables: ventricular tracer entry and distal tracer drainage. Median patient age was 19 years, and 51% of patients were male. The most common presentation in patients undergoing shuntography was headache (169/254, 66.5%) with radiographically stable ventricle size. Of 227 patients with available imaging data, 163 (71.8%) presented with the same ventricle size as shown on a previous asymptomatic scan, 43 (18.9%) had larger ventricles, and 21 (9.2%) had smaller ventricles. Within 30 days of shuntography, 74 of 259 patients (28.6%) underwent surgical shunt exploration: 65 were found to have an obstructed shunt and 9 were found to have a patent shunt. Of those patients not undergoing surgery, the median length of benign clinical follow-up was 1051 days. Clinical variables were not significantly associated with shuntography results, including valve type (p = 0.180), ventricle size (p = 0.556), age (p = 0.549), distal drainage site (p = 0.098), and hydrocephalus etiology (p = 0.937). Shuntography results of patients with myelomeningocele were not dissociable from those of the group as a whole. Sensitivity to diagnose shunt failure was lowest (37.5%) but specificity was highest (97.2%) when the definition of a "normal" shuntogram included any tracer movement into the distal site within 45 minutes. Conversely, sensitivity was highest (87.5%) and specificity was lowest (51.4%) when the definition was limited exclusively to tracer entry into the ventricles and distal drainage within 15 minutes. Even with a stringent definition of a "normal" shuntogram, sensitivity and specificity were relatively low for a diagnostic test. Clinical variables such as valve type, ventricle size, patient age, distal drainage site, and etiology of hydrocephalus were not associated with shuntography results.

Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

In 1996 Sabitov proved that the volume $$V$$ V of an arbitrary simplicial polyhedron $$P$$ P in the $$3$$ 3 -dimensional Euclidean space $$\mathbb {R}^3$$ R 3 satisfies a monic (with respect to $$V$$ V ) polynomial relation $$F(V,\ell )=0$$ F ( V , l ) = 0 , where $$\ell $$ l denotes the set of the squares of edge lengths of $$P$$ P . In 2011 the author proved the same assertion for polyhedra in $$\mathbb {R}^4$$ R 4 . In this paper, we prove that the same result is true in arbitrary dimension $$n\ge 3$$ n ? 3 . Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular $$2$$ 2 -faces. As a corollary, we obtain the proof in arbitrary dimension of the well-known Bellows Conjecture posed by Connelly in 1978. This conjecture claims that the volume of any flexible polyhedron is constant. Moreover, we obtain the following stronger result. If $$P_t, t\in [0,1]$$ P t , t ? [ 0 , 1 ] , is a continuous deformation of a polyhedron such that the combinatorial type of $$P_t$$ P t does not change and every $$2$$ 2 -face of $$P_t$$ P t remains congruent to the corresponding face of $$P_0$$ P 0 , then the volume of $$P_t$$ P t is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in $$\mathbb {C}^n$$ C n from their orthogonal edge lengths.

Arsenate induced differential response in rice genotypes

To study the differential response in two rice genotypes (PB1 and IR-64), hydroponically grown 14 days old plants were exposed to 50, 150 and 300µM As(V) for 24 and 96h. Accumulation of As was not significantly higher in PB1 variety except at higher concentration (300μM) and duration (96h), but up regulation of gene transcripts were higher as compared to IR-64. Inhibition in seed germination, root–shoot length, chlorophyll and protein content was observed in both varieties with increasing concentration and exposure time. PB1 variety was found more capable to detoxify As(V) through induction of antioxidant defense system and other stress related parameters (cysteine, proline content). SDS-PAGE and semi quantitative RT-PCR analysis showed significant changes in protein profile and gene expression analysis. The results suggests that various studied parameters and transcripts accumulation showed a combinatorial type of tolerance mechanism in PB1 variety to provide better protection against As(V) stress.

A class of affinely equivalent Voronoi parallelohedra

Given any parallelohedron P, its affine class A (P), i.e., the set of all parallelohedra affinely equivalent to it, is considered. Does this affine class contain at least one Voronoi parallelohedron, i.e., a parallelohedron which is a Dirichlet domain for some lattice? This question, more commonly known as Voronoi’s conjecture, has remained unanswered for more than a hundred years. It is shown that, in the case where the subset of Voronoi parallelohedra in A (P) is nonempty, this subset is an orbifold, and its dimension (as a real manifold with singularities) is completely determined by its combinatorial type; namely, it is equal to the number of connected components of the so-called Venkov subgraph of the given parallelohedron. Nevertheless, the structure of this orbifold depends not only on the combinatorial properties of the parallelohedron but also on its affine properties.

The Brill–Noether rank of a tropical curve

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill-Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill-Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill-Noether rank of a tropical curve with the dimension of the Brill-Noether locus of an algerbaic curve.

Many neighborly inscribed polytopes and Delaunay triangulations

We present a very simple explicit technique to generate a large family of point configurations with neighborly Delaunay triangulations. This proves that there are superexponentially many combinatorially distinct neighborly $d$-polytopes with $n$ vertices that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (and thus also of Delaunay triangulations). It coincides with the current best lower bound for the number of combinatorial types of polytopes. Nous présentons une technique explicite simple pour générer une large famille de configurations de points dont les triangulations de Delaunay sontneighborly. Cela prouve que le nombre de $d$-polytopes combinatoirement distincts avec $n$ sommets et admettant une réalisation inscrite sur la sphère est surexponentiel. Ce sont les premiers exemples de polytopes inscriptibles neighborly qui ne sont pas des polytopes cycliques et ils donnent la meilleure borne inférieure actuelle pour le nombre de types combinatoires de polytopes inscriptibles (et donc aussi de triangulations de Delaunay). Cette borne coïncide avec la meilleure borne inférieure actuelle pour le nombre de types combinatoires de polytopes.

Sabitov polynomials for volumes of polyhedra in four dimensions

In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain monic polynomial with coefficients depending on the combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.

On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero.

Link invariants via counting surfaces

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. We then identify the resulting invariants with certain derivatives of the HOMFLYPT polynomial.

Many Neighborly Polytopes and Oriented Matroids

In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of $${\big (( r+d ) ^{( \frac{r}{2}+\frac{d}{2} )^{2}}\big )}\big /{\big ({r}^{{(\frac{r}{2})}^{2}} {d}^{{(\frac{d}{2})}^{2}}{\mathrm{e}^{3\frac{r}{2}\frac{d}{2}}}\big )}$$((r+d)(r2+d2)2)/(r(r2)2d(d2)2e3r2d2) for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d with $$r+d+1$$r+d+1 vertices. This improves current bounds on the number of combinatorial types of polytopes. The previous best lower bounds for the number of neighborly polytopes were found by Shemer in 1982 using a technique called the Sewing Construction. We provide a new simple proof that sewing works, and generalize it to oriented matroids in two ways: to Extended Sewing and to Gale Sewing. Our lower bound is obtained by estimating the number of polytopes that can be constructed via Gale Sewing. Combining both new techniques, we are also able to construct many non-realizable neighborly oriented matroids.

Inscribable stacked polytopes

Abstract We characterize the combinatorial types of stacked d-polytopes that are inscribable. Equivalently, we identify the triangulations of a simplex by stellar subdivisions that can be realized as Delaunay triangulations

Exposure of Brassica juncea (L) to arsenic species in hydroponic medium: comparative analysis in accumulation and biochemical and transcriptional alterations

Arsenic (As) contamination in the environment has attracted considerable attention worldwide. The objective of the present study was to see the comparative effect of As species As(III) and As(V) on accumulation, biochemical responses, and gene expression analysis in Brassica juncea var. Pusa Jaganath (PJn). Hydroponically grown 14-day-old seedlings of B. juncea were treated with different concentrations of As(III) and As(V). Accumulation of total As increased with increasing concentration of both As species and exposure time, mainly in roots. Reduction in seed germination, root-shoot length, chlorophyll, and protein content were observed with increasing concentration and exposure time of both As species, being more in As(III)-treated leaves. PJn variety showed that antioxidant enzymes (superoxide dismutase (SOD), catalase (CAT), and ascorbate peroxidase (APX)) and stress-related parameters (cysteine, proline, and malondialdehyde (MDA)) were stimulated and allows plant to tolerate both As species. Sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) analysis in leaves showed significant changes in protein profile with more stringent effect with As(III) stress. Semiquantitative RT-PCR analysis showed regulation in expression of phytochelatin synthase (PCS), metallothionine-2 (MT-2), glutathione reductase (GR), and glutathione synthetase (GS) genes under both As(III) and As(V) stresses. Results suggested that accumulation and inhibition on physiological parameters differ according to the As species, while molecular and biochemical parameters showed a combinatorial type of tolerance mechanism against As(III) and As(V) stresses.

On the enumeration of the combinatorial types of primitive parallelohedra inEd, 2 ≤d≤ 6

The cone of positive-definite quadratic forms is subdivided into subcones of combinatorial types of primitive parallelohedra in Ed, 2 ≤ d ≤ 6. A new algorithm is described and recent results on the enumeration are given.

Regular languages and partial commutations

The closure of a regular language under a [partial] commutation I has been extensively studied. We present new advances on two problems of this area: (1) When is the closure of a regular language under [partial] commutation still regular? (2) Are there any robust classes of languages closed under [partial] commutation? We show that the class Pol(G) of polynomials of group languages is closed under commutation, and under partial commutation when the complement of I in A2 is a transitive relation. We also give a sufficient graph theoretic condition on I to ensure that the closure of a language of Pol(G) under I-commutation is regular. We exhibit a very robust class of languages W which is closed under commutation. This class contains Pol(G), is decidable and can be defined as the largest positive variety of languages not containing (ab)⁎. It is also closed under intersection, union, shuffle, concatenation, quotients, length-decreasing morphisms and inverses of morphisms. If I is transitive, we show that the closure of a language of W under I-commutation is regular. The proofs are nontrivial and combine several advanced techniques, including combinatorial Ramsey type arguments, algebraic properties of the syntactic monoid, finiteness conditions on semigroups and properties of insertion systems.

INVARIANTS OF CLOSED BRAIDS VIA COUNTING SURFACES

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.

Hyperbolic Coxeter Pyramids

Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkin’s methods and find the remaining hyperbolic Coxeter pyramids.

Por Uma Leitura Corsária de os “Pobres” de Georg Simmel

In this text we propose an allegoric reading of Georg Simmel’s text “The Poor”, concentrating our analysis in the interactional, dialogical and reciprocal dimensions that constitute the poor as a social category and impose certain character traits and social roles to them. Signaling the continuities between alms giving and the modern Welfare State, we propose a cynical look at the social processes that defined the poor as a non-existence or as a pretext for a moral enrichment of the givers and helpers. We mark a long process that started with the definition of poverty as pathology and culminated in the buildup of an identity of the poor according to statistical categories and rational classificatory systems. The poor is characterized as a combinatorial type, receiving its meaning from the association with other concepts and as an absence of a value in itself.

Complete Enumeration of Small Realizable Oriented Matroids

Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (2001) published the first database of oriented matroids including degenerate (i.e. non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points and 5-dimensional configurations of 9 points. We could also determine all possible combinatorial types of 5-polytopes with 9 vertices.