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Capacity is an important tool in decision-making under risk and uncertainty and multi-criteria decision-making. When learning a capacity-based model, it is important to be able to generate uniformly a capacity. Due to the monotonicity constraints of a capacity, this task reveals to be very difficult. The classical Random Node Generator (RNG) algorithm is a fast-running speed capacity generator, however with poor performance. In this paper, we firstly present an exact algorithm for generating a n elements' general capacity, usable when n < 5 . Then, we present an improvement of the classical RNG by studying the distribution of the value of each element of a capacity. Furthermore, we divide it into two cases, the first one is the case without any conditions, and the second one is the case when some elements have been generated. Experimental results show that the performance of this improved algorithm is much better than the classical RNG while keeping a very reasonable computation time. [ABSTRACT FROM AUTHOR]

Random generation of fuzzy measures plays a pivotal role in large-scale decision-making and optimization. Random walks ensure uniform generation and adequate coverage. The Möbius representation of set functions is a valuable tool for establishing the sparse structure of fuzzy measures, with its non-negativity closely linked to monotonicity and convexity/supermodularity checking. We propose three efficient methods for monotonicity verification and convexity/supermodularity verification applicable to random walks in Möbius representations, specifically tailored for universal sets larger than ten inputs and k-order fuzzy measures. We first present the baseline methods by directly inspecting monotonicity and convexity constraints. Building on the observation that the majority of initially generated values exhibit non-negativity, we intentionally track negative Möbius values to enhance the computational performance of these baseline approaches. Further, we introduce the more agile methods that employ insertion and merge sorting techniques for both monotonicity and convexity checks in random walks that involve small perturbations of fuzzy measures. To address sparsity in large-scale scenarios, we focus on two major types of measures: k-additive and k-interactive measures, demonstrating their effectiveness through theoretical analysis and experimental results. [ABSTRACT FROM AUTHOR]

Capacities on a finite set are sets functions vanishing on the empty set and being monotonic w.r.t. inclusion. Since the set of capacities is an order polytope, the problem of randomly generating capacities amounts to generating all linear extensions of the Boolean lattice. This problem is known to be intractable even as soon as n > 5 , therefore approximate methods have been proposed, most notably one based on Markov chains. Although quite accurate, this method is time consuming. In this paper, we propose the 2-layer approximation method, which generates a subset of linear extensions, eliminating those with very low probability. We show that our method has similar performance compared to the Markov chain but is much less time consuming. [ABSTRACT FROM AUTHOR]

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.Mathematics Subject Classifications: 05A20, 05C12, 05E16, 20F55Keywords: Linear extension, promotion, Coxeter group, convex hull, metric

We study the isolated singularities of functions satisfying (E) (− Δ) s v ± | v | p − 1 v = 0 in Ω ∖ { 0 } , v = 0 in R N ∖ Ω , where 0 < s < 1 , p > 1 and Ω is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + × R N. We emphasize the obtention of a priori estimates and analyse the set of self-similar solutions via energy methods to characterize the singularities. [ABSTRACT FROM AUTHOR]

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone $E^+$ of a Riesz space $E$ taking values in an Archimedean Riesz space $F$, to the entire space $E$. We prove that, if $E$ has the principal projection property and $F$ is Dedekind $\sigma$-complete then for every $e \in E^+$ every positive finitely additive $F$-valued measure defined on the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ has a unique positive linear extension to the ideal $E_e$ of $E$ generated by $e$. If, moreover, the measure is $\tau$-continuous then the linear extension is order continuous.

We employ the combinatorial atlas technology to prove new correlation inequalities for the number of linear extensions of finite posets. These include the approximate independence of probabilities and expectations of values of random linear extensions, closely related to Stanley's inequality. We also give applications to the numbers of standard Young tableaux and to Euler numbers. [ABSTRACT FROM AUTHOR]

Introduction: The maintenance of the coral communities depends directly on the capacity of the individuals to grow as a whole colony. Since growth shapes the basis of the physical structure of the reef, determining the growth rate of builder coral-species and their response to environmental variations will allow us to identify their particular potential to implement more effective restoration strategies. Objective: To determine the growth rate of the reef-building coral Pocillopora throughout the Mexican Central Pacific (PCM) and its relationship with fluctuations in environmental conditions. Methods: From August 2019 - October 2020, a total of 153 branched coral fragments of the genus Pocillopora were monitored and transplanted as part of the restoration program in different PCM locations: Islas Marietas National Park and in the Caleta de Cuastecomates. Coral growth (cm·year-1) and survival rate (%), was calculated and correlated with temperature, PAR and Kd490 for each locality. Results: The results show similar patterns in the parameters examined among the coral fragments were found for all sites, with a survival range of 61.6 - 68 %, whose mortality was mainly related to hurricanes and storms that affected the region. However, growth rate range from 2.0 - 8.0 cm·year-1, with no significant differences between locations. Also, no differences in environmental conditions between locations were recorded. Conclusions: The branching corals of the genus Pocillopora present a similar growth rate in insular and coastal sites of the PCM, so it could be expected that the implementation of a restoration program throughout this region of the Mexican Pacific would be potentially successful, since local environmental conditions are not limiting factors. [ABSTRACT FROM AUTHOR]

We study the permutation group BK P generated by Bender–Knuth moves on linear extensions of a poset P , an analog of the Berenstein–Kirillov group on column-strict tableaux. We explore the group relations, with an emphasis on identifying posets P for which the cactus relations hold in BK P. We also examine BK P as a subgroup of the symmetric group S L (P) on the set of linear extensions of P with the focus on analyzing posets P for which BK P = S L (P). [ABSTRACT FROM AUTHOR]

Silk extension determines the timing of silking, subsequent pollination and kernel setting, which has not been accounted in functional-structural plant (FSP) modelling for maize. In this study, a two-year field experiment with two local maize hybrids i.e. AnNong 591 (referred to as AN591) and ZhongDan 909 (referred to as ZD909) was carried out in Hefei, Anhui Province, China in both 2019 and 2020. The silks at basal, middle and top sections of a maize ear were destructively sampled daily in capturing silk extension. Silk extension prior to its senescence is described with a segmented model i.e. an exponential function driven by cell division (Phase I), and two linear functions corresponding to rapid silk extension (Phase II) and slow extension (Phase III) driven by cell expansion. Data in 2019 were used to fit silk extension on basal, middle and top ear positions, and the associated parameter values for the exponential and linear functions were extracted while the highest R2 was achieved. The model was then compared with the Logistic model by evaluating the normalized root mean square error (nRMSE) between simulations and observations in 2019 and 2020. It was shown that the ability of a segmented model in 2019 fitting was sound though a bit poorer than a Logistic model, however, the validation for the former model performed better than the latter when independent data in 2020 were applied. This indicated that the simple segmented model is competent in predicting silk extension. In addition, silk extension at basal, middle and top sections of a maize ear for both maize hybrids was visually demonstrated. In conclusion, modelling silk extension was successfully realised by the segmented model composed of exponential and linear functions, which complements FSP modelling of maize. • Silk extension is an exponential Phase I, a linear Phase II and a linear Phase III. • The segmented model in predicting silk extension did better than Logistic model. • The segmented model of silk extension complements plant modelling of maize. [ABSTRACT FROM AUTHOR]

Reported divergent responses of coral growth and skeletal microstructure to the nutrient environment complicate knowledge-based management of water quality in coral reefs. By re-evaluating published results considering the taxonomy of the studied corals and the N:P stoichiometry of their nutrient environment, we could resolve some of the major apparent contradictions. Our analysis suggests that Acroporids behave differently to several other common genera and show distinct responses to specific nutrient treatments. We hypothesised that both the concentrations of dissolved inorganic N and P in the water and their stoichiometry shape skeletal growth and microstructure. We tested this hypothesis by exposing Acropora polystoma fragments to four nutrient treatments for > 10 weeks: high nitrate/high phosphate (HNHP), high nitrate/low phosphate (HNLP), low nitrate/high phosphate (LNHP) and low nitrate/low phosphate (LNLP). HNHP corals retained high zooxanthellae densities and their linear extension and calcification rates were up to ten times higher than in the other treatments. HNLP and LNLP corals bleached through loss of symbionts. The photochemical efficiency (Fv/Fm) of residual symbionts in HNLP corals was significantly reduced, indicating P-starvation. Micro-computed tomography (µCT) of the skeletal microstructure revealed that reduced linear extension in nutrient limited or nutrient starved conditions (HNLP, LNHP, LNLP) was associated with significant thickening of skeletal elements and reduced porosity. These changes can be explained by the strongly reduced linear extension rate in combination with a smaller reduction in the calcification rate. Studies using increased skeletal density as a proxy for past thermal bleaching events should consider that such an increase in density may also be associated with temperature-independent response to the nutrient environment. Furthermore, the taxonomy of corals and seawater N:P stoichiometry should be considered when analysing and managing the impacts of nutrient pollution. [ABSTRACT FROM AUTHOR]

The classical Kantorovich theorem asserts the existence and uniqueness of a linear extension of a positive additive mapping, defined on the positive cone E+ of a Riesz space E taking values in an Archimedean Riesz space F, to the entire space E. We prove that, if E has the principal projection property and F is Dedekind σ-complete then for every e∈E+ every positive finitely additive F-valued measure defined on the Boolean algebra Fe of fragments of e has a unique positive linear extension to the ideal Ee of E generated by e. If, moreover, the measure is τ-continuous then the linear extension is order continuous. [ABSTRACT FROM AUTHOR]

We seek simple conditions on a pair of labeled posets that determine when the difference of their (P , ω) -partition enumerators is F-positive, i.e., positive in Gessel's fundamental basis. This is a quasisymmetric analogue of the extensively studied problem of finding conditions on a pair of skew shapes that determine when the difference of their skew Schur functions is Schur-positive. We determine necessary conditions and separate sufficient conditions for F-positivity, and show that a broad operation for combining posets preserves positivity properties. We conclude with classes of posets for which we have conditions that are both necessary and sufficient. [ABSTRACT FROM AUTHOR]

The theory of capacities provides powerful formal methodology to account for criteria dependencies in multiple criteria decision problems. The discrete Choquet and Sugeno integrals aggregate criteria valuations accounting for criteria synergies and redundancies. We address an important problem of randomly generating capacities of special classes for simulation studies and for capacity learning through evolutionary algorithms. We discuss two efficient methods suitable for k -interactive capacities. The results are supported by the extensive numerical evidence and provide a useful tool for large scale simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Coral reefs are globally in decline and western Atlantic reefs have experienced the greatest losses in live coral cover of any region. The Flower Garden Banks (FGB) in the Gulf of Mexico are high-latitude, remote reefs that are an outlier to this trend, as they have maintained coral cover ≥ 50% since at least 1989. Quantifying the long-term trends in coral growth of key reef-building coral species, and the underlying environmental drivers, leads to a better understanding of local sensitivities to past changes that will ultimately allow us to better predict the future of reef growth at FGB. We obtained coral cores and constructed growth records for two of the most abundant hermatypic coral species at FGB, Pseudodiploria strigosa and Orbicella faveolata. Our records cover 57 yrs of growth for P. strigosa (1957–2013) and 45 yrs for O. faveolata (1970–2014). Linear extension and calcification rates of both species have increased significantly, but skeletal density did not change over the respective time periods. Extension and calcification data of both species combined were negatively correlated with the discharge from the Atchafalaya River, but positively correlated with maximum sea surface temperatures (SST). These data provide evidence that runoff from the Atchafalaya River impacts FGB corals and is a major control on coral growth at FGB. The increase in growth at FGB can be attributed to the significant warming trend in maximum monthly SSTs. Given the warming trend and recent increase in severity of bleaching at FGB, the prognosis is that bleaching events will become more deleterious with time, which will lead to a breakdown in the positive relationship between coral growth and maximum SST. This study provides further evidence that some high-latitude, cooler reef sites have experienced a stimulation in coral growth with ocean warming. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let (P , ≤) 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 P which may be covered by k antichains (respectively, k chains.) Then the number e (P) of linear extensions of poset P is not less than ∏ a i ! and not more than n ! / ∏ c i ! . A corollary: if P is partitioned onto disjoint antichains of sizes b1,b2,..., then e (P) ≥ ∏ b i ! . [ABSTRACT FROM AUTHOR]

Sea‐level rise is predicted to cause major damage to tropical coastlines. While coral reefs can act as natural barriers for ocean waves, their protection hinges on the ability of scleractinian corals to produce enough calcium carbonate (CaCO3) to keep up with rising sea levels. As a consequence of intensifying disturbances, coral communities are changing rapidly, potentially reducing community‐level CaCO3 production. By combining colony‐level physiology and long‐term monitoring data, we show that reefs recovering from major disturbances can produce 40% more CaCO3 than currently estimated due to the disproportionate contribution of juvenile corals. However, the buffering effect of highly productive juvenile corals is compromised by recruitment failures, which have been more frequently observed after large‐scale, repeated bleaching events. While the size structure of corals can bolster a critical ecological function on reefs, climate change impacts on recruitment may undermine this buffering effect, thus further compromising the persistence of reefs and their provision of important ecosystem services. [ABSTRACT FROM AUTHOR]

We introduce a new method for ranked set sampling with multiple criteria. The method relaxes the restriction of selecting just one individual variable from each ranked set. Under the new method for ranking, units are ranked in sets based on linear extensions in partially order set theory with considering all variables simultaneously. Results will be evaluated by a relatively extensive simulation studies on Bivariate normal distribution and two real case studies on commercial and medicinal use of flowers, and the pollution of herb-layer by Lead, Cadmium, Zinc and Sulfur in some regions of the southwest of Germany. [ABSTRACT FROM AUTHOR]

A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, where a MCQ is an algebra whose axioms are motivated from handlebody-knot theory. [ABSTRACT FROM AUTHOR]

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 1 2 -greedy balanced pair. [ABSTRACT FROM AUTHOR]

We investigate the problem of computing the number of linear extensions of a given n-element poset whose cover graph has treewidth t. We present an algorithm that runs in time O ~ (n t + 3) for any constant t; the notation O ~ hides polylogarithmic factors. Our algorithm applies dynamic programming along a tree decomposition of the cover graph; the join nodes of the tree decomposition are handled by fast multiplication of multivariate polynomials. We also investigate the algorithm from a practical point of view. We observe that the running time is not well characterized by the parameters n and t alone: fixing these parameters leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. We compare two approaches to select an efficient tree decomposition: one is to include additional features of the tree decomposition to build a more accurate, heuristic cost function; the other approach is to fit a statistical regression model to collected running time data. Both approaches are shown to yield a tree decomposition that typically is significantly more efficient than a random optimal-width tree decomposition. [ABSTRACT FROM AUTHOR]

The Huffman coding algorithm is interpreted in the context of the lattice of partitions of the source alphabet. Maximal chains in the partition lattice correspond to linear extensions of tree orders, and those among the chains that exhibit a simple greedy property correspond precisely to executions of the Huffman algorithm. [ABSTRACT FROM AUTHOR]

A shared variable is an abstraction of persistent interprocess communication. Processors execute operations, often concurrently, on shared variables to exchange information among themselves. The behavior of operation executions is required to be "consistent" for effective interprocess communication. Consequently, a consistency specification of a shared variable describes some guarantees on the behavior of the operation executions. A Read/Write shared variable has two operations: a Write stores a specified value in the variable and a Read returns a value from the variable. For Read/Write variables, a consistency specification describes what values Reads may return. Using an intuitive notion of illegality of Reads, we propose a framework that facilitates specifying a large variety of Read/Write variables. [ABSTRACT FROM AUTHOR]

In this paper we present a new method for deriving a random linear extension of a poset. This new strategy combines Probability with Combinatorics and obtains a procedure where each minimal element of a sequence of subposets is selected via a probability distribution. The method consists in obtaining a weight vector on the elements of P, so that an element is selected with a probability proportional to its weight. From some properties on the graph of adjacent linear extensions, it is shown that the probability distribution can be obtained by solving a linear system. The number of equations involved in this system relies on the number of what we have called positioned antichains, that allows a reduced number of equations. Finally, we give some examples of the applicability of the algorithm. This procedure cannot be applied to every poset, but it is exact when it can be used. Moreover, the method is quick and easy to implement. Besides, it allows a simple way to derive the number of linear extensions of a given poset. [ABSTRACT FROM AUTHOR]

After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consider specific families of posets and ask how many linear extensions give rise to permutations that avoid certain patterns. We build off of two of these papers. We first consider pattern avoidance in k-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case. We then prove some conjectures that Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding linear extensions of rectangular posets. [ABSTRACT FROM AUTHOR]

A balanced pair in an ordered set P = (V, ≤) is a pair (x, y) of elements of V such that the proportion of linear extensions of P that put x before y is in the real interval [1/3, 2/3]. We define the notion of a good pair and claim any ordered set that has a good pair will satisfy the conjecture and furthermore every ordered set which is not totally ordered and has a forest as its cover graph has a good pair. [ABSTRACT FROM AUTHOR]

The Tsetlin library is a very well-studied model for the way an arrangement of books on a library shelf evolves over time. One of the most interesting properties of this Markov chain is that its spectrum can be computed exactly and that the eigenvalues are linear in the transition probabilities. In this paper, we consider a generalization which can be interpreted as a self-organizing library in which the arrangements of books on each shelf are restricted to be linear extensions of a fixed poset. The moves on the books are given by the extended promotion operators of Ayyer, Klee and Schilling while the shelves, bookcases, etc. evolve according to the move-to-back moves as in the the self-organizing library of Björner. We show that the eigenvalues of the transition matrix of this Markov chain are ± 1 integer combinations of the transition probabilities if the posets that prescribe the restrictions on the book arrangements are rooted forests or more generally, if they consist of ordinal sums of a rooted forest and so called ladders. For some of the results, we show that the monoids generated by the moves are either ℛ -trivial or, more generally, in D O (A b) and then we use the theory of left random walks on the minimal ideal of such monoids to find the eigenvalues. Moreover, in order to give a combinatorial description of the eigenvalues in the more general case, we relate the eigenvalues when the restrictions on the book arrangements change only by allowing for one additional transposition of two fixed books. [ABSTRACT FROM AUTHOR]

We conducted a 93-day experiment investigating the independent and combined effects of acidification (28023300 matm pCO2) and warming (288C and 318C) on calcification and linear extension rates of four key Caribbean coral species (Siderastrea siderea, Pseudodiploria strigosa, Porites astreoides, Undaria tenuifolia) from inshore and offshore reefs on the Belize Mesoamerican Barrier Reef System. All species exhibited nonlinear declines in calcification rate with increasing pCO2. Warming only reduced calcification in Ps. strigosa. Of the species tested, only S. siderea maintained positive calcification in the aragonite-undersaturated treatment. Temperature and pCO2 had no effect on the linear extension of S. siderea and Po. astreoides, and natal reef environment did not impact any parameter examined. Results suggest that S. siderea is the most resilient of these corals to warming and acidification owing to its ability to maintain positive calcification in all treatments, Ps. strigosa and U. tenuifolia are the least resilient, and Po. astreoides falls in the middle. These results highlight the diversity of calcification responses of Caribbean corals to projected global change. [ABSTRACT FROM AUTHOR]

One of the methods of analysing the manufacturing process of lithic tools is by "reading" the scars of removals visible on the surface of cores or bifaces. The paper aims to review the limitations of this approach, by answering the question as to how much information from the original knapping process is still preserved in the cores and bifaces. To quantify the information loss we used experimentally knapped pieces, that we then refitted to check which removals are still visible on the surface of the analysed forms. We tested different measures of information loss in the context of removals and sequences of removals, but also the chronological relationship between sequences i.e. the number of linear extensions and missing nearest connections. On average, 39.9% of the original removals and 54.1% of the sequences of removals are preserved on the surface, but only 14.1% of the original relationships between sequences can be observed. Finally, we used simple machine learning to find an algorithm for estimating the loss of the nearest chronological relationships between sequences when no refits are available. Due to the extremely high information loss for the linear order of the knapping process we: a) propose to abandon these elements of the diacritic approach which aim to reconstruct the rhythms and schemes of knapping; b) evaluate the quality of the reconstructed knapping scheme (partial order) prior to further comparative analyses. Here we propose an algorithm to estimate the number of missing connections to test the quality of the knapping process while using diacritic schemes. • The information loss refers to detached removals and the chronological relation between them. • 39.9% of detached removals are preserved on the surface of bifaces or cores. • 14.1% of the chronological relationships between sequences of removals can be observed. • The information loss during the knapping process is substantial. • It influences the chaine operatoire analyses based on tools or cores solely. [ABSTRACT FROM AUTHOR]