Linear Extension Research Articles

The Diversity of Linear Conjugated Polyenes and Colours in Nature: Raman Spectroscopy as a Diagnostic Tool.

This review is centered on the linear conjugated polyenes, which encompasses chromatic biomolecules, such as carotenoids, polyunsaturated aldehydes and polyolefinic fatty acids. The linear extension of the conjugated double bonds in these molecules is the main feature that determines the spectroscopic properties as light-absorbing. These classes of compounds are responsible for the yellow, orange, red and purple colors which are observed in their parent flora and fauna in nature. Raman spectroscopy has been used as analytical tool for the characterization of these molecules, mainly due to the strong light scattering produced by the delocalized pi electrons in the carbon chain. In addition, conjugated polyenes are one of the main target molecular species for astrobiology, and we also present a brief discussion of the use of Raman spectroscopy as one of the main analytical tools for the detection of polyenes extra-terrestrially.

A linear hypergraph extension of the bipartite Turán problem

An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of r-uniform hypergraphs F, the linear Turán number exrlin(n,F) is the maximum number of edges of a linear r-uniform hypergraph on n vertices that does not contain any member of F as a subhypergraph.Given a graph F and a positive integer r≥2, the r-expansion of F is the r-graph F+ obtained from F by enlarging each edge of F with r−2 new vertices disjoint from V(F) such that distinct edges of F are enlarged by distinct vertices. For t≥s≥2, we prove the following extension of Kővári–Sós–Turán’s theorem exrlin(n,Ks,t+)≤(t−1)1sr(r−1)⋅n2−1s+O(n2−2s). Specially, for s=2, r=3, we prove that ex3lin(n,K2,t+)=1+ot(1)16t−1⋅n3∕2, which is an improvement of Gerbner, Methuku and Vizer’s result (Gerbner et al., 2019). Moreover, we also prove some sharp bounds for the linear Turán number of Ks,t+.

Affine extensions of multiple conjugation quandles and augmented MCQ Alexander pairs

A multiple conjugation quandle is an algebra whose axioms are motivated from handlebody-knot theory. Any linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair. In this paper, we show that any affine extension of a multiple conjugation quandle can be described by using a quadruple of maps, called an augmented MCQ Alexander pair.

Counting Linear Extensions of Restricted Posets

We prove the 1991 conjecture by Brightwell and Winkler that counting the number of linear extensions for posets of height two is #P-complete. We further extend this result to incidence posets of graphs.

Volumetric and linear evaluation of the sphenoidal sinus of a Brazilian population, in cone beam computed tomography

The determination of sex from skeletal remains has been widely used in biological profile reconstruction since these are some of the last structures to perish after death. The sphenoid sinus is located deeply in the skull and is enveloped by diverse structures within the sphenoid body. It is, therefore, less predisposed to injuries and/or pathological changes. The aim of this study is to evaluate the possibility of determining sex in a Brazilian population by means of linear and volumetric measurements of the sphenoid sinus, in addition to identifying and quantifying the presence of extensions of this sinus in cone beam computed tomography (CBCT) scans. In total 268 CBCT scans were analysed, from patients aged between 22 and 70 years old, of which 145 were female, and 123 were male. The images, representative of a Brazilian population, were selected by a dental radiologist from a CBCT image bank. The volumetric measurements were made by means of the segmentation software ITK-SNAP 3.0® and the linear inferio-superior, latero-lateral and anteroposterior measurements, as well as the presence of extensions of the sphenoidal sinus, were made using the CS Imaging Software®. The results of this study suggested that when comparing men and women, there were no statistical differences in the linear measurements and extensions of the sphenoidal sinus, although the values were considerable in the extensions for the pterygoid processes and greater wings of the sphenoid bone, particularly in women. However, there was a statistically significant difference when the sphenoidal sinus volume was compared, as it was larger for men (11.364 mm³ ± 4.229 mm³) than for women (10.000 mm³ ± 3.615 mm³). In summary, the sphenoidal sinus volumetric measurements in CBCT exams are useful for determining the sex of unknown individuals.

Improving the $$\frac{1}{3} - \frac{2}{3}$$ Conjecture for Width Two Posets

Extending results of Linial (1984) and Aigner (1985), we prove a uniform lower bound on the balance constant of a poset P of width 2. This constant is defined as δ(P) = max(x,y)∈P2 min{ℙ(x ≺ y), ℙ(y ≺ x)}, where ℙ(x≺y) is the probability x is less than y in a uniformly random linear extension of P. In particular, we show that if P is a width 2 poset that cannot be formed from the singleton poset and the three element poset with one relation using the operation of direct sum, then $$\delta \left( P \right) \ge \frac{{ - 3 + 5\sqrt {17} }}{{52}} \approx 0.33876....$$ This partially answers a question of Brightwell (1999); a full resolution would require a proof of the $$\frac{1}{3} - \frac{2}{3}$$ Conjecture that if P is not totally ordered, then $$\delta \left( P \right) \ge \frac{1}{3}$$ . Furthermore, we construct a sequence of posets Tn of width 2 with δ(Tn) → β ≈ 0.348843…, giving an improvement over a construction of Chen (2017) and over the finite posets found by Peczarski (2017). Numerical work on small posets by Peczarski suggests the constant β may be optimal.

Linear Lipschitz and C1 extension operators through random projection

We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and C1 functions. This way we prove more directly a result by Lee and Naor [5] and we generalize the C1 extension theorem by Whitney [8] to Banach spaces.

A mixed-integer programming formulation of the double row layout problem based on a linear extension of a partial order

The double row layout problem (DRLP) occurs in automated manufacturing environments, where a material-handling device transports materials among machines arranged in a double-row layout, i.e. a layout in which the machines are located on either side of a straight line corridor. The DRLP is how to minimize the total cost of transporting materials between machines. The problem is NP-Hard and one great challenge nowadays is how to solve problem instances in reasonable computational times. In this paper, we give a new mixed-integer programming model of the DRLP, which is based on a linear extension of a partial order. In addition, we propose a reformulation of this model, which yields stronger results. The new models have the least number of 0–1 variables in comparison with previous models in the literature. Computational experiments demonstrate that the proposed models obtain optimal solutions faster than previously published ones.

Effects of feeding on the physiological performance of the stony coral Pocillopora acuta

Reef-building corals rely on both heterotrophy and endosymbiotic dinoflagellate autotrophy to meet their metabolic needs. Those looking to culture these organisms for scientific or industrial purposes must therefore consider both feeding regimes and the light environment. Herein the effects of three photosynthetically active radiation (PAR) levels were assessed in fed and unfed specimens of the model coral Pocillopora acuta that were cultured in a recirculating aquaculture system (RAS). Half of the corals were fed Artemia sp. brine shrimp in a separate feeding tank to prevent biofouling, and fragments were exposed to PAR levels of 105, 157, or 250 μmol quanta m−2 s−1 over a 12-h period each day. All cultured corals survived the 140-day treatment, and the physiological response variables assessed-buoyant weight, specific growth rate, linear extension, color, and Fv/Fm-were significantly influenced by feeding, and, to a lesser extent, light. Specifically, fed corals grew faster and larger, and presented darker pigmentation; corals fed at the highest light levels grew at the fastest rate (6 cm year−1 or 175 mg g−1 week−1). Given the high physiological performance observed, we advocate the active feeding of brine shrimp in RAS by those looking to cultivate P. acuta, and likely other corals, over long-term timescales.

Path-Dependent Morphological Evolution of Se-Te Mesostructures Prepared by Inorganic Phototropic Growth.

We describe herein a path-dependent "history" effect wherein the film morphology generated in the second step of a two-step inorganic phototropic growth process depends on a preexisting structure that has been first grown under different optical stimulation conditions. Se-Te generated with static illumination exhibited a highly anisotropic lamellar morphology with a characteristic feature pitch proportional to the input wavelength. Growth using first a short wavelength of light, followed by growth using a longer wavelength, resulted in the second-stage morphology exhibiting termination of lamellae formed during the first growth step. The lamellar pitch at the end of the second growth step was larger than that effected in the first step. In contrast, use of the same input wavelengths but in the opposite order produced no change in the feature pitch but rather only linear feature extension. Analysis of light absorption in simulated structures, in tandem with the empirical data, indicated that the history effect and asymmetric path dependence are a result of emergent nanophotonic processes at the growth interface that dynamically shape the optical field and direct morphological evolution of the photodeposit in a continuous feedback loop.

A geometric lower bound on the extension complexity of polytopes based on the [formula omitted]-vector

A linear extension of a polytope Q is any polytope which can be mapped onto Q via an affine transformation. The extension complexity of a polytope is the minimum number of facets of any linear extension of this polytope. In general, computing the extension complexity of a given polytope is difficult. The extension complexity is also equal to the nonnegative rank of any slack matrix of the polytope.In this paper, we introduce a new geometric lower bound on the extension complexity of a polytope, i.e., which relies only on the knowledge of some of its geometric characteristics. It is based on the monotone behaviour of the f-vector of polytopes under affine maps. We present numerical results showing that this bound can improve upon existing geometric lower bounds, and provide a generalization of this lower bound for the nonnegative rank of any matrix.

Comparative Circulation Dynamics of the Five Main HIV Types in China.

The HIV epidemic in China accounts for 3% of the global HIV incidence. We compared the patterns and determinants of interprovincial spread of the five most prevalent circulating types. HIV pol sequences sampled across China were used to identify relevant transmission networks of the five most relevant HIV-1 types (B and circulating recombinant forms [CRFs] CRF01_AE, CRF07_BC, CRF08_BC, and CRF55_01B) in China. From these, the dispersal history across provinces was inferred. A generalized linear model (GLM) was used to test the association between migration rates among provinces and several measures of human mobility. A total of 10,707 sequences were collected between 2004 and 2017 across 26 provinces, among which 1,962 are newly reported here. A mean of 18 (minimum and maximum, 1 and 54) independent transmission networks involving up to 17 provinces were identified. Discrete phylogeographic analysis largely recapitulates the documented spread of the HIV types, which in turn, mirrors within-China population migration flows to a large extent. In line with the different spatiotemporal spread dynamics, the identified drivers thereof were also heterogeneous but are consistent with a central role of human mobility. The comparative analysis of the dispersal dynamics of the five main HIV types circulating in China suggests a key role of large population centers and developed transportation infrastructures as hubs of HIV dispersal. This advocates for coordinated public health efforts in addition to local targeted interventions.IMPORTANCE While traditional epidemiological studies are of great interest in describing the dynamics of epidemics, they struggle to fully capture the geospatial dynamics and factors driving the dispersal of pathogens like HIV as they have difficulties capturing linkages between infections. To overcome this, we used a discrete phylogeographic approach coupled to a generalized linear model extension to characterize the dynamics and drivers of the across-province spread of the five main HIV types circulating in China. Our results indicate that large urbanized areas with dense populations and developed transportation infrastructures are facilitators of HIV dispersal throughout China and highlight the need to consider harmonized country-wide public policies to control local HIV epidemics.

Greedy balanced pairs in [formula omitted]-free ordered sets

An α-greedy balanced pair in an ordered set P=(V,≤) is a pair (x,y) of elements of V such that the proportion of greedy linear extensions of P that put x before y among all greedy linear extensions is in the real interval [α,1−α]. We prove that every N-free ordered set which is not totally ordered has a 12-greedy balanced pair.

Differential survival of nursery‐reared Acropora cervicornis outplants along the Florida reef tract

In recent decades, the Florida reef tract has lost over 95% of its coral cover. Although isolated coral assemblages persist, coral restoration programs are attempting to recover local coral populations. Listed as threatened under the Endangered Species Act, Acropora cervicornis is the most widely targeted coral species for restoration in Florida. Yet strategies are still maturing to enhance the survival of nursery‐reared outplants of A. cervicornis colonies on natural reefs. This study examined the survival of 22,634 A. cervicornis colonies raised in nurseries along the Florida reef tract and outplanted to six reef habitats in seven geographical subregions between 2012 and 2018. A Cox proportional hazards regression was used within a Bayesian framework to examine the effects of seven variables: (1) coral‐colony size at outplanting, (2) coral‐colony attachment method, (3) genotypic diversity of outplanted A. cervicornis clusters, (4) reef habitat, (5) geographical subregion, (6) latitude, and (7) the year of monitoring. The best models included coral‐colony size at outplanting, reef habitat, geographical subregion, and the year of monitoring. Survival was highest when colonies were larger than 15 cm (total linear extension), when outplanted to back‐reef and fore‐reef habitats, and when outplanted in Biscayne Bay and Broward–Miami subregions, in the higher latitudes of the Florida reef tract. This study points to several variables that influence the survival of outplanted A. cervicornis colonies and highlights a need to refine restoration strategies to help restore their population along the Florida reef tract.

The Bounds for the Number of Linear Extensions Via Chain and Antichain Coverings

Let $(\mathcal {P},\leqslant )$ be a finite poset. Define the numbers a1,a2,… (respectively, c1,c2,…) so that a1 + … + ak (respectively, c1 + … + ck) is the maximal number of elements of $\mathcal {P}$ which may be covered by k antichains (respectively, k chains.) Then the number $e(\mathcal {P})$ of linear extensions of poset $\mathcal {P}$ is not less than $\prod a_{i}!$ and not more than $n!/\prod c_{i}!$ . A corollary: if $\mathcal {P}$ is partitioned onto disjoint antichains of sizes b1,b2,…, then $e(\mathcal {P})\geqslant \prod b_{i}!$ .

Acyclic Cucurbit[n]uril-Type Receptors: Aromatic Wall Extension Enhances Binding Affinity, Delivers Helical Chirality, and Enables Fluorescence Sensing.

We report the linear extension from M1 to M2 to anthracene walled M3 which adopts a helical conformation (X-ray) to avoid unfavorable interactions between sidewalls. M3 is water soluble (=30 mm) and displays enhanced optical properties (ϵ=1.28×105 m-1 cm-1 , λmax =370 nm) relative to M2. The binding properties of M3 toward guests 1-29 were examined by 1 H NMR and ITC. The M3⋅guest complexes are stronger than the analogous complexes of M2 and M1. The enhanced binding of M3 toward neuromuscular blockers 25, 27-29 suggests that M3 holds significant promise as an in vivo reversal agent. The changes in fluorescence observed for M3⋅guest complexes are a function of the relative orientation of the anthracene sidewalls, guest concentration, Ka , and guest electronics which rendered M3 a superb component of a fluorescence sensing array. The work establishes M3 as a next generation sequestering agent and a versatile component of fluorescence sensors.

Approximation and Extension of Functions of Vanishing Mean Oscillation

We consider various definitions of functions of vanishing mean oscillation on a domain $$\Omega \subset {{{\mathbb {R}}}^n}$$ . If the domain is uniform, we show that there is a single extension operator which extends functions in these spaces to functions in the corresponding spaces on $${{{\mathbb {R}}}^n}$$ , and also extends $$\mathrm{BMO}(\Omega )$$ to $$\mathrm{BMO}({{{\mathbb {R}}}^n})$$ , generalizing the result of Jones. Moreover, this extension maps Lipschitz functions to Lipschitz functions. Conversely, if there is a linear extension map taking Lipschitz functions with compact support in $$\Omega $$ to functions in $$\mathrm{BMO}({{{\mathbb {R}}}^n})$$ , which is bounded in the $$\mathrm{BMO}$$ norm, then the domain must be uniform. In connection with these results we investigate the approximation of functions of vanishing mean oscillation by Lipschitz functions on unbounded domains.

Mechanisms of lateral and linear extension of gullies (dongas) in a subhumid grassland of South Africa

AbstractThe formation of deep gullies (called ‘dongas’ locally) in rangeland in KwaZulu‐Natal Province in South Africa is a natural phenomenon. These U‐shaped, very wide gullies have considerable lateral expansion due to the episodic collapse of sidewalls.The dongas have developed in duplex soils such as Luvisols and Lixisols formed on Permian sedimentary rocks or unconsolidated Quaternary colluvium. This study combined morphological, mineralogical and chemical characterization with measurements of grain‐size content, structural stability and the complete shrinkage curve to detect changes in soil properties of the different horizons located in the gully banks.The different soil horizons present clear and sharp differences in physical and mineralogical properties. The topsoil with complete grass cover is very resistant to soil detachment. However, the leached E horizon and the BC horizon have low structural stability. The soil profile down to and including the Bt horizon contains exclusively illite in the clay fraction, while the BC colluvial layer and the C horizon (mudstone) contain expandable interstratified illite–smectite. The Bt horizon has a high water content at saturation and high shrinkage, while the BC and C horizons have a high residual shrinkage and a very low water content at saturation.Because this type of gully expansion is not significantly linked to slope value or the stream power index (SPI) at the gully head, to land‐use change, high rainfall intensities or the threshold of concentrated runoff being exceeded at the gully head, other causes were investigated. It was concluded that the heterogeneity between horizons with different mineralogical properties and structural stabilities, soil types and parent material, anisotropic water‐saturation and shrink‐swell properties are of major importance. This heterogeneity between different soil horizon morphologies and their physical properties can explain why the relationship between the critical slope and the drainage area for gully initiation showed a threshold for gullying much lower than that found elsewhere. © 2020 John Wiley & Sons, Ltd.

On an order-based multivariate median

Although the order-based definition of the univariate median is ubiquitous in statistics, the same order-based definition is typically abandoned when extending the univariate median to higher dimensions. In this paper, an example of order-based multivariate median based on the use of a linear extension of the product order is brought to the attention. Symmetry and internality properties are fulfilled by such order-based multivariate median, however, it turns out that this function fails to fulfil in general other appealing properties such as equivariance to different geometrical transformations, monotonicity properties and continuity. Interestingly, translation and scale equivariance and some monotonicity properties can be guaranteed if the linear extension of the product order is carefully chosen. Finally, it is proved that, unlike in the univariate setting, the finite-sample breakdown point of the order-based multivariate median is 1/n, thus implying that it is not robust in the presence of outliers.

Acclimatization Drives Differences in Reef-Building Coral Calcification Rates

Coral reefs are susceptible to climate change, anthropogenic influence, and environmental stressors. However, corals in Kāneʻohe Bay, Hawaiʻi have repeatedly shown resilience and acclimatization to anthropogenically-induced rising temperatures and increased frequencies of bleaching events. Variations in coral and algae cover at two sites—just 600 m apart—at Malaukaʻa fringing reef suggest genetic or environmental differences in coral resilience between sites. A reciprocal transplant experiment was conducted to determine if calcification (linear extension and dry skeletal weight) for dominant reef-building species, Montipora capitata and Porites compressa, varied between the two sites and whether or not parent colony or environmental factors were responsible for the differences. Despite the two sites representing distinct environmental conditions with significant differences between temperature, salinity, and aragonite saturation, M. capitata growth rates remained the same between sites and treatments. However, dry skeletal weight increases in P. compressa were significantly different between sites, but not across treatments, with linear mixed effects model results suggesting heterogeneity driven by environmental differences between sites and the parent colonies. These results provide evidence of resilience and acclimatization for M. capitata and P. compressa. Variability of resilience may be driven by local adaptations at a small, reef-level scale for P. compressa in Kāneʻohe Bay.