Linear Extension Research Articles

Zero-sum game-based neuro-optimal control of modular robot manipulators with uncertain disturbance using critic only policy iteration

In this paper, a zero-sum differential game strategy-based neuro-optimal control method is presented via critic only policy iteration-adaptive dynamic programming (COPI-ADP) approach to address optimal trajectory tracking control problem of modular robot manipulators (MRMs) with uncertain disturbance. The dynamic model of modular robot manipulator systems is formulated as an integration of joint subsystems and unknown robotic model uncertainties are identified by the developed linear extension state observer. Then, the optimal control issue of the modular robot manipulator systems with uncertain disturbance is transformed into a two-player zero-sum differential game one. Based on adaptive dynamic programming and policy iteration algorithms, the Hamilton-Jacobi-Issacs (HJI) equation is approximately solved using only critic neural network and thus facilitating the feasible derivation of the approximated optimal control policy. The trajectory of tracking errors of modular robot manipulator system is guaranteed to be uniform ultimate bounded by using the Lyapunov theory. Finally, experiments are provided to demonstrate the advantage and effectiveness of the developed control method.

Coral community composition and carbonate production in an urbanized seascape

Coastal urbanization causes environmental modifications that directly and indirectly influence the distribution and functioning of coral reefs. However, the capacity of urban infrastructure to support corals and vertically accrete is less understood. Here, we investigated if coral communities on reefs and seawalls in Singapore are distinct, and examined the environmental variables influencing coral carbonate production. Surveys at 22 sites yielded 134 coral species, with richness significantly higher on reefs. Coral cover and Shannon index did not differ between habitat types. Community composition was distinct between habitat types, with seawalls supporting a higher proportion of massive and thick-plating species. ‘Distance from mainland’ was the single most important variable influencing normalized carbonate production rates (a function of species-specific linear extension rate and skeletal bulk density and site coral cover), which were higher further from the mainland where human activity and development pressures were greater. Our results indicate that environmental filtering strongly shapes coral communities and may influence ecosystem functioning in Singapore's urbanized reef system. The findings will guide the management of reefs on increasingly urbanized coastlines.

On Polyhedral Approximations of the Positive Semidefinite Cone

It is well known that state-of-the-art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes. This leads us to consider the question, how well can we approximate semidefinite programs with linear programs? In this paper, we prove lower bounds on the size of linear programs that approximate the positive semidefinite cone. Let D be the set of n × n positive semidefinite matrices of trace equal to one. We prove two results on the hardness of approximating D with polytopes. We show that if 0 < ε < 1and A is an arbitrary matrix of trace equal to one, any polytope P such that (1-ε) (D-A) ⊂ P ⊂ D-A must have linear programming extension complexity at least [Formula: see text], where c > 0 is a constant that depends on ε. Second, we show that any polytope P such that D ⊂ P and such that the Gaussian width of P is at most twice the Gaussian width of D must have extension complexity at least [Formula: see text]. Our bounds are both superpolynomial in n and demonstrate that there is no generic way of approximating semidefinite programs with compact linear programs. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.

Inverse property of nonassociative abelian extensions

Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.

Multi-level Monte Carlo computation of the hadronic vacuum polarization contribution to (gμ − 2)

The hadronic contribution to the muon anomalous magnetic moment aμ=(gμ−2)/2 has to be determined at the per-mille level for the Standard Model prediction to match the expected final uncertainty from the ongoing E989 experiment. This is 3 times better than the current precision from the dispersive approach, and 5-15 times smaller than the uncertainty on the purely theoretical determinations from lattice QCD. So far the stumbling-block is the large statistical error in the Monte Carlo evaluation of the required correlation functions which can hardly be tamed by brute force. Here we propose to solve this problem by multi-level Monte Carlo integration, a technique which reduces the variance of correlators exponentially in the distance of the fields. We test our strategy by computing the Hadronic Vacuum Polarization on a lattice with a linear extension of 3 fm, a spacing of 0.065 fm, and a pion mass of 270 MeV. Indeed the two-level integration makes the contribution to the statistical error from long-distances de-facto negligible by accelerating its inverse scaling with the cost of the simulation. These findings establish multi-level Monte Carlo as a solid and efficient method for a precise lattice determination of the hadronic contribution to aμ. As the approach is applicable to other computations affected by a signal-to-noise ratio problem, it has the potential to unlock many open problems for the nuclear and particle physics community.

De Finetti Lattices and Magog Triangles

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

Reconciling slow linear growth and equivocal competitive ability with rapid spread of peyssonnelid algae in the Caribbean

Within the past decade, a functional group of encrusting red macroalgae defined as peyssonnelid algal crusts (PAC) has been rapidly spreading on shallow Caribbean reefs, frequently encrusting corals and sponges. This study focused on PAC growing on the reefs of St. John, US Virgin Islands, to (1) expand the monitoring of PAC abundance in this location, and (2) test two mechanisms, rapid linear growth and competitive dominance, by which PAC is hypothesized to enhance its capacity to spread. In August 2019, we found that the percentage coverage of PAC reached 46.6% at Tektite (3 m depth), and 63.8% at Cabritte Horn (3 m depth). Overall, the cover of PAC at these sites in 2019 was identical to, or up to 2.4-fold higher than in August 2017, and 2.1–3.7-fold higher than in November 2017, just after two category 5 storms impacted St. John. PAC was 1.8–29.1-fold more abundant at 3 m versus 9 m depth, and 1.4–22.0-fold more abundant at sites close to the tips of headlands than sites sheltered within bays. From August 2019 to January 2020, the rate of linear extension of PAC at each site and depth ranged from 26 to 43 µm d − 1, and of 80 PAC-coral interactions that were tagged in August 2019, none involved corals that had become completely overgrown by PAC by January 2020. The slow linear growth and absence of cases in which PAC completely overgrew coral colonies in 6 months are inconsistent with the high rate at which PAC has occupied space on Caribbean reefs over the past decade. One means to reconcile these observations is the possibility that recruitment of peyssonnelid spores to benthic surfaces facilitates the rapid spread of PAC.

A polynomial time algorithm for the 2-Poset Cover Problem

Given a set ϒ of linear orders, we say that a poset Ha<b=(V,<) is a halfspace of ϒ if its set of linear extensions L(Ha<b)⊂ϒ and a<Lb for every L∈L(Ha<b) and b<La for every L∈ϒ∖L(Ha<b). In this paper, we devise an efficient algorithm to expand the halfspace and determine a maximal poset P in ϒ that supercovers it, that is, L(Ha<b)⊆L(P)⊆ϒ. Moreover, the said algorithm paves the way for the improvement of the existing exponential running time solution in literature for the 2-Poset Cover Problem to a polynomial running time solution.

Growth, calcification, and photobiology of the threatened coral Acropora cervicornis in natural versus artificial light.

Land-based coral culture is of increasing interest for conservation and educational display. Shallow water corals generate most of their energy from photosynthesis, and light is a critical abiotic factor in their husbandry. We compared growth, calcification, and photobiology in the coral Acropora cervicornis between natural and artificial (light-emitting diode; LED) light to better understand the impact of light source on coral performance. One tank of a greenhouse recirculating system at The Florida Aquarium's Center for Conservation was used to culture replicate coral colonies. Half of the tank and corals were covered to block sunlight and illuminated with a commercial reef aquarium LED fixture, while the other half was exposed to natural sunlight. Treatments were matched in terms of maximum photosynthetically active radiation and spectral measurements characterized both light regimes. Coral growth and calcification were tracked over a period of 19 weeks by repeated measurements of total linear extension (TLE) and buoyant weight. For the first 5 weeks, photosynthetic yield was measured weekly using a pulse-amplitude-modulated fluorometer. Calcification was significantly higher under LED lighting relative to natural light, but TLE did not differ. Photobiology data suggest that corals in both treatments were acclimated to the same light level, but photosynthetic efficiency was ultimately greater in the natural light treatment. More consistent light delivery and different spectral composition under LED treatment conditions may explain the incongruity between calcification and photosynthetic efficiency. This experiment informs husbandry of shallow-water scleractinian corals maintained in both natural sunlight and enclosed structures.

Influence of dimension chains on product assembly sequences

The paper considers the influence of design dimension chains on the assembly sequence of a complex product. When assembling products, the rule of connectivity of the dimension chain must be followed. For this reason, the dimension chain induces a binary partial order relation on the set of product parts. The assembly sequences for which the connectivity rule is applied are linear extensions of this order relation. The number of the linear extensions for the orders generated by the main methods of dimensioning is calculated. It is shown that the chain method of dimensioning is the most rigid, and the coordinate method is the most “tolerant”. The combined method occupies an intermediate position. The principle of matching the dimension and mechanical structures of a complex product is proposed. The results obtained can be used to generate rational dimension schemes in integrated CAD / CAM systems.

On the Extension of Whitney Ultrajets of Beurling Type

We prove a version of Whitney’s extension theorem in the ultradifferentiable Beurling setting with controlled loss of regularity. As a by-product we show the existence of continuous linear extension operators on certain spaces of Whitney ultrajets on arbitrary closed sets in $$\mathbb {R}^n$$ .

Whitney Numbers for Poset Cones

Hyperplane arrangements dissect \(\mathbb {R}^{n}\) into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the reflecting hyperplanes xi = xj inside \(\mathbb {R}^{n}\) for the symmetric group, thought of as the type An− 1 reflection group. Here, cones correspond to posets, chambers within the cone correspond to linear extensions of the poset, the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of left-to-right maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata’s theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.

Assessment of the effect of different compost materials on aggregation and mechanical properties in an Entisol

ABSTRACT It is necessary to know the effect of organic matter on soil physical properties for successful soil management. A short-term incubation study was carried out to investigate the effects of different compost materials on soil aggregation and mechanical properties. Compost materials derived from cattle, sheep, poultry manure, plant residues, and a blend of an equal amount of those residues/wastes (called as mixture) were mixed with soil at doses of 1%, 2%, 3% and 4% with control treatment and subjected to a 60-day incubation period. It was revealed that compost materials significantly increased the values of water-stable aggregate (WSA), organic carbon (OC) and moisture content at tension values of pF 0 (saturation), pF 1.7 (aeration porosity), pF 2.54 (field capacity) and pF 4.2 (permanent wilting point), as well as the hydraulic conductivity (HC) and liquid limit (LL) values; however, they decreased the bulk density (BD) and coefficient of linear extensibility (COLE) values. On the other hand, WSA, OC, saturation, BD, LL, and shrinkage limit (SL) values showed distinct changes in application doses. However, it was found that material and dose applications had no effect on the plasticity index (PI), shrinkage index (SI) or clay activity (CA). The increases in WSA, HC and OC values compared to the control due to the increasing amount of compost were more evident than other aggregation properties. Overall, the results showed that compost materials and dose applications are more effective on the soil aggregation properties compared to the mechanical properties.

Lyapunov-type inequalities for third-order linear and half-linear difference equations and extensions

We derive Lyapunov-type inequalities for third-order half-linear difference equations. The results are significant as the existing work on Lyapunov-type inequalities for difference equations is limited only to even order cases. As the techniques for even order difference equations are not applicable to odd order difference equations, the methods used here are novel. The resulting inequalities utilize and . New results for linear difference equations are obtained as a special case and are further extended to more general linear equations. Moreover, brief discussions are also given for third-order backward difference equations and dynamic equations on time scales.

Computing Linear Extensions for Polynomial Posets Subject to Algebraic Constraints

In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small "admissible" subset of these linear extensions, determined implicitly by the evaluation map, are of interest. This seemingly novel problem arises in the study of global dynamics of gene regulatory networks in which case the poset is a Boolean lattice. We provide an algorithm for solving this problem using linear programming for arbitrary partial orders of linear polynomials. This algorithm exploits this additional algebraic structure inherited from the polynomials to efficiently compute the admissible linear extensions. The biologically relevant problem involves multilinear polynomials and we provide a construction for embedding it into an instance of the linear problem.

Counting Linear Extensions of Posets with Determinants of Hook Lengths

We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entr...

Cervical cone dimension and linear CIN extension trend: 28 years' experience of a tertiary center

Cervical cone dimension and linear CIN extension trend: 28 years' experience of a tertiary center

JointAI: Joint Analysis and Imputation of Incomplete Data in R

Missing data occur in many types of studies and typically complicate the analysis. Multiple imputation, either using joint modeling or the more flexible fully conditional specification approach, are popular and work well in standard settings. In settings involving nonlinear associations or interactions, however, incompatibility of the imputation model with the analysis model is an issue often resulting in bias. Similarly, complex outcomes such as longitudinal or survival outcomes cannot be adequately handled by standard implementations. In this paper, we introduce the R package JointAI, which utilizes the Bayesian framework to perform simultaneous analysis and imputation in regression models with incomplete covariates. Using a fully Bayesian joint modeling approach it overcomes the issue of uncongeniality while retaining the attractive flexibility of fully conditional specification multiple imputation by specifying the joint distribution of analysis and imputation models as a sequence of univariate models that can be adapted to the type of variable. JointAI provides functions for Bayesian inference with generalized linear and generalized linear mixed models and extensions thereof as well as survival models and joint models for longitudinal and survival data, that take arguments analogous to the corresponding well known functions for the analysis of complete data from base R and other packages. Usage and features of JointAI are described and illustrated using various examples and the theoretical background is outlined.

Characterizing expansiveness and iron reducibility of red‐clay soils for the purpose of on‐site wastewater system placement in Arkansas

AbstractThe lack of iron‐reducing capability in some red soils poses an issue for determining the suitability for on‐site wastewater treatment systems (OSWWTS). The Arkansas Department of Health allows for placement of OSWWTS in certain red‐clay soils that have the potential for iron reduction and do not show expansiveness upon wetting, but only in the Ozark Highlands (Major Land Resource Area [MLRA] 116A), which is referred to as the red‐soil exception. The objectives of this study were (a) to confirm the nonexpansiveness and iron reducibility of select soils in the Ozark Highlands that meet the criteria for the red‐soil exception, and (b) to compare the expansiveness and iron reducibility of the residual or colluvial red‐clay soils of the Ozark Highlands with those in four other MLRA region groupings in Arkansas. Fresh and archived soils were used to create a 51‐sample dataset from the 38‐ to 50‐cm soil depth interval for analyses. The alluvial soils of the Red and Arkansas River valleys had a mean coefficient of linear extensibility (COLE) of 0.116 mm mm–1, whereas the residual or colluvial soils from the other MLRA regions evaluated had lower (P &lt; .01) COLE values. Soils from the Red and Arkansas Rivers region changed red color (∆A) the least (∆A of −8.6 in the LAB color scheme; P &lt; .01) compared with soils from the other regions. Results of this study indicate that the red‐soil exception, as used by the Arkansas Department of Health for determining OSWWTS suitability, could be expanded to include soils from several additional MLRAs with additional, site‐specific information.

Linear uncertain extensions of the minimum cost consensus model based on uncertain distance and consensus utility

Uncertainty theory adopts the belief degree and uncertainty distribution to ensure good alignment with a decision-maker’s uncertain preferences, making the final decisions obtained from the consensus-reaching process closer to the actual decision-making scenarios. Under the constraints of the uncertain distance measure and consensus utility, this article explores the minimum-cost consensus model under various linear uncertainty distribution-based preferences. First, the uncertain distance is used to measure the deviation between individual opinions and the consensus through uncertainty distributions. A nonlinear analytical formula is derived to avoid the computational complexity of integral and piecewise function operations, thus reducing the calculation cost of the uncertain distance measure. The consensus utility function defined in this article characterizes the adjustment value and degree of aggregation of individual opinions. Three new consensus models are constructed based on the consensus utility and linear uncertainty distribution. The results show that, in complex group decision-making contexts, the uncertain consensus models are more flexible than traditional minimum-cost consensus models: compared with the high volatility of the adjusted opinions in traditional deterministic consensus models with crisp number-based preferences, the variation trends of both individual adjusted opinions and the collective opinion with a linear uncertainty distribution are much smoother and the fitting range is closer to reality. The introduction of the consensus utility not only reflects the relative changes of individual opinions, but also accounts for individual psychological changes during the opinion-adjustment process. Most importantly, it reduces the cost per unit of consensus utility, facilitates the determination of the optimal threshold for the consensus utility, and improves the efficiency of resource allocation.