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A novel mode-localized acceleration sensor employing an electrostatically coupled resonator and integrating a lever with proof mass is micromachined using standard silicon on insulator (SOI) technology. In order to determine the linear dynamic range of the sensor, a reduced order model is developed while assuming that the resonators vibrate below the critical amplitude. Then, open-loop and closed-loop testing platforms are established to measure the performance of the linearly operating accelerometer in a vacuum environment (less than 5 Pa). Moreover, the corresponding amplifier circuit based on the capacitive detection principle is designed in order to extract and amplify the current signal from the resonators. The obtained results show that the accelerometer sensitivity can be increased by three orders of magnitude when using the relative shift of amplitude ratio as the output metric instead of the relative shift of frequency, and the experimental measurements are consistent with the theoretical predictions. Remarkably, the Allan standard deviation of the mode-localized acceleration sensor obtained from the closed-loop testing circuit is around 5.03 μg. [ABSTRACT FROM AUTHOR]

Estimating ranks, quantiles, and distributions over streaming data is a central task in data analysis and monitoring. Given a stream of n items from a data universe equipped with a total order, the task is to compute a sketch (data structure) of size polylogarithmic in n. Given the sketch and a query item y, one should be able to approximate its rank in the stream, i.e., the number of stream elements smaller than or equal to y. Most works to date focused on additive εn error approximation, culminating in the KLL sketch that achieved optimal asymptotic behavior. This article investigates multiplicative (1±ε )-error approximations to the rank. Practical motivation for multiplicative error stems from demands to understand the tails of distributions, and hence for sketches to be more accurate near extreme values. The most space-efficient algorithms due to prior work store either O(log(ε²n)/ε²) or O(log³ (εn)/ε ) universe items. We present a randomized sketch storing O(log1.5 (εn)/ε ) items that can (1 ± ε )-approximate the rank of each universe item with high constant probability; this space bound is within an O(√log(εn)) factor of optimal. Our algorithm does not require prior knowledge of the stream length and is fully mergeable, rendering it suitable for parallel and distributed computing environments. [ABSTRACT FROM AUTHOR]

While recounting an experience, one can employ multiple strategies to transition from one part to the next. For instance, if the event was learned out of linear order, one can recall events according to the time they were learned (temporal), similar events (semantic), events occurring nearby in time (chronological), or events produced by the current event (causal). To disentangle the importance of these factors, we had participants watch the nonlinear narrative, Memento, under different task instructions and presentation orders. For each scene of the film, we also separately computed semantic and causal networks. We then contrasted the evidence for temporal, semantic, chronological, or causal strategies during recall. Critically, there was stronger evidence for the causal and chronological strategies than semantic or temporal strategies. Moreover, the causal and chronological strategies outperformed the temporal one even when we asked participants to recall the film in the presented order, underscoring the fundamental nature of causal structure in scaffolding understanding and organizing recall. Nevertheless, time still marginally predicted recall transitions, suggesting it operates as a weak signal in the presence of more salient forms of structure. In addition, semantic and causal network properties predicted scene memorability, including a stronger role for incoming causes to an event than its outgoing effects. In summary, these findings highlight the importance of accounting for complex, causal networks in knowledge building and memory. [ABSTRACT FROM AUTHOR]

Due to the increasing applications of dual quaternion and their matrices in recent years, as well as the significance of LU decomposition as a matrix decomposition technique, in this paper, we propose dual quaternion Gaussian transformation and obtain dual quaternion LU decomposition by using Gaussian transformation. We also use the total order of dual numbers to obtain the partial pivoting dual quaternion LU decomposition. Based on the real structure-preserving algorithm of quaternion matrix, we propose the real structure-preserving algorithms of LU decomposition and partial pivoting LU decomposition for dual quaternion matrix. Numerical experiments have verified the effectiveness of the new real structure-preserving approaches. [ABSTRACT FROM AUTHOR]

Quantification, i.e., the task of predicting the class prevalence values in bags of unlabeled data items, has received increased attention in recent years. However, most quantification research has concentrated on developing algorithms for binary and multi-class problems in which the classes are not ordered. Here, we study the ordinal case, i.e., the case in which a total order is defined on the set of n > 2 classes. We give three main contributions to this field. First, we create and make available two datasets for ordinal quantification (OQ) research that overcome the inadequacies of the previously available ones. Second, we experimentally compare the most important OQ algorithms proposed in the literature so far. To this end, we bring together algorithms proposed by authors from very different research fields, such as data mining and astrophysics, who were unaware of each others' developments. Third, we propose a novel class of regularized OQ algorithms, which outperforms existing algorithms in our experiments. The key to this gain in performance is that our regularization prevents ordinally implausible estimates, assuming that ordinal distributions tend to be smooth in practice. We informally verify this assumption for several real-world applications. [ABSTRACT FROM AUTHOR]

Background: The family Lauraceae is subdivided into six main lineages: Caryodaphnopsideae, Cassytheae, Cryptocaryeae, Hypodaphnideae, Laureae, and Neocinnamomeae. However, phylogenetic relationships among these lineages have been debatable due to incongruence between trees constructed using nuclear ribosomal DNA (nrDNA) sequences and chloroplast (cp) genomes. As with cp DNA, the mitochondrial (mt) DNA of most flowering plants is maternally inherited, so the phylogenetic relationships recovered with mt genomes are expected to be consistent with that from cp genomes, rather than nrDNA sequences. Results: The mitogenome of Machilus yunnanensis, with a length of 735,392 bp, has a very different genome size and gene linear order from previously published magnoliid mitogenomes. Phylogenomic reconstructions based on 41 mt genes from 92 Lauraceae mitogenomes resulted in highly supported relationships: sisterhood of the Laureae and a group containing Neocinnamomeae and Caryodaphnopsideae, with Cassytheae being the next sister group, followed by Cryptocaryeae. However, we found significant incongruence among the mitochondrial, chloroplast, and nuclear phylogenies, especially for the species within the Caryodaphnopsideae and Neocinnamomeae lineages. Time-calibrated phylogenetic analyses showed that the split between Caryodaphnopsideae and Neocinnamomeae dated to the later Eocene, around 38.5 Ma, Laureae originated in the Late Cretaceous, around 84.9 Ma, Cassytheae originated in the mid-Cretaceous around 102 Ma, and Cryptocaryeae originated in the Early Cretaceous around 116 Ma. From the Late Cretaceous to the Paleocene, net diversification rates significantly increased across extant clades of major lineages, and both speciation rates and net diversification rates continued steady growth towards the present. Conclusions: The topology obtained here for the first time shows that mt genes can be used to support relationships among lineages of Lauraceae. Our results highlight that both Caryodaphnopsideae and Neocinnamomeae lineages are younger than previously thought, likely first diversifying in the Eocene, and species in the other extant lineages of Lauraceae dates in a long-time span from the Early Cretaceous to the Eocene, and the climate of a period of about 90 million years was relatively warm, while the extant species of Lauraceae then continuously diversified with global cooling from the Eocene to the present day. [ABSTRACT FROM AUTHOR]

We consider a school choice matching model where priorities for schools are represented by binary relations that may not be linear orders. Even in that case, it is necessary to construct linear orders from the original priority relations to execute several mechanisms. We focus on the (linear order) extensions of the priority relations, because a matching that is stable for an extension profile is also stable for the profile of priority relations. We show that if the priority relations are partial orders, then for each stable matching for the original profile of priority relations, an extension profile for which it is also stable exists. Furthermore, if there are multiple stable matchings that are ranked by Pareto dominance, then there is an extension for which all these matchings are stable. We apply the result to a version of efficiency adjusted deferred acceptance mechanisms. • We focus on the school choice model where the original priority relations are partial orders. • We consider the extensions of the original priority relations and the stable matchings for the original profile. • For each stable matching for the original profile, an extension profile for which it is also stable exists. • If multiple stable matchings that are ranked by Pareto dominance exist, then an extension for which all these matchings are stable exists. • The result is applied to the efficiency adjusted deferred acceptance mechanisms. [ABSTRACT FROM AUTHOR]

The study of longest increasing subsequences (LIS) in permutations led to that of Young diagrams via Robinson–Schensted's (RS) correspondence. In a celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams and found that the LIS of a uniform permutation of size n$$ n $$ behaves as 2n$$ 2\sqrt{n} $$. Independently and much later, Hoppen et al. introduced the theory of permutons as a scaling limit of permutations. In this paper, we extend in some sense the RS correspondence of permutations to the space of permutons. When the "RS‐tableaux" of a permuton are non‐trivial, we show that the RS‐tableaux of random permutations sampled from this permuton exhibit a linear behavior, in the sense that their first rows and columns have lengths of linear order. In particular, the LIS of such permutations behaves as a multiple of n$$ n $$. We also prove some large deviation results for these convergences. Finally, by studying asymptotic properties of Fomin's algorithm for permutations, we show that the RS‐tableaux of a permuton satisfy a partial differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the action of a countable group Γ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group, the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner–Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on SL3. Z corresponding to the semigroup generated by the standard unipotents cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended. [ABSTRACT FROM AUTHOR]

We revisit Brenner's seminal work on the Stokes resistance of a slightly deformed sphere (Chem. Engng Sci. , vol. 19, 1964, p. 519), evaluate its range of validity and extend its applicability to higher deformations for axisymmetric particles, using hydrodynamic radius as the measure of Stokes resistance. Brenner's method solves the flow around a slightly deformed sphere through two mapping steps: the first mapping translates the surface velocity on the deformed sphere to that over a reference sphere of arbitrary radius using an asymptotic expansion of the flow field in terms of deformation amplitude and a Taylor expansion of the velocity field around the surface of the reference sphere. Subsequently, the second mapping extrapolates the velocity field from the surface of the reference sphere to any point in the fluid using Lamb's general solution for Stokes flow. While the original work addresses slightly deformed spheres to a linear order in deformation amplitude, we demonstrate that the first mapping, in combination with axisymmetric spectral modes (J. Fluid Mech. , vol. 936, 2022, R1), can accommodate significant deformations to arbitrary orders of perturbation, and thus is not limited to slightly deformed spheres. Also, while first-order analysis is suitable for nearly spherical particles, second-order terms can provide a reasonable range for significantly higher deformations. [ABSTRACT FROM AUTHOR]

Let Λ be a finite-dimensional algebra over a field. We describe how Buan and Marsh’s τ-exceptional sequences can be used to give a “brick labeling” of a certain poset of wide subcategories of finitely generated Λ-modules. When Λ is representation-directed, we prove that there exists a total order on the set of bricks which makes this into an EL-labeling. Motivated by the connection between classical exceptional sequences and noncrossing partitions, we then turn toward the study of (well-separated) completely semidistributive lattices. Such lattices come equipped with a bijection between their completely join-irreducible and completely meet-irreducible elements, known as rowmotion or the “κ-map.” Generalizing known results for finite semidistributive lattices, we show that the κ-map determines exactly when a set of completely join-irreducible elements forms a “canonical join representation.” A consequence is that the corresponding “canonical join complex” is a flag simplicial complex, as has been shown for finite semidistributive lattices and lattices of torsion classes. Finally, we demonstrate how Jasso’s τ-tilting reduction of finite-dimensional algebras can be encoded using the κ-map. We use this to define κd-exceptional sequences for finite semidistributive lattices. These are distinguished sequences of completely join-irreducible elements which we prove specialize to τ-exceptional sequences in the algebra setting. [ABSTRACT FROM AUTHOR]

This article explores the mathematical background of a magic trick involving a sequence of cards. The trick involves choosing a card, performing a walk on a circle of cards, and revealing a hidden card that matches a card from another deck. The article focuses on sequences with nontrivial sums, which are sequences of distinct elements that have specific mathematical properties. The article provides definitions, results, and proofs related to these sequences. It also mentions other articles and books by the author that explore the intersection of mathematics and magic tricks. [Extracted from the article]

The article explores the concept of division by a natural number without the axiom of choice in mathematics. It introduces the principles of "shoe division" and "sock division" as alternative versions of division by n. The text discusses the relationship between sock division and multiplication, presenting a theorem that proves their equivalence. It also explores the implications of sock division for bundles and the different interpretations of multiplication without the axiom of choice. The article concludes with a question about the relationship between divisibility and sock division in ZF set theory. [Extracted from the article]

We call an order type inscribable if it is realized by a point configuration where all extreme points are all on a circle. In this paper, we investigate inscribability of order types. We first construct an infinite family of minimally uninscribable order types. The proof of uninscribability mainly uses Möbius transformations and the Frantz ellipse. We further show that every simple order type with at most two interior points is inscribable, and that the number of such order types is Θ (4 n n 3 / 2 ) . We also suggest open problems around inscribability. [ABSTRACT FROM AUTHOR]

The Linear Layout of Graphs problem asks, given a graph G and a positive integer k , whether G admits a layout consisting of a linear ordering of its vertices and a partition of its edges into k sets such that the edges in each set meet some special requirements. Specific linear layouts include k -Stack Layout, k -Queue Layout, k -Arch Layout, Mixed s -Stack q -Queue Layout and others. In this paper, we present a unified approach for kernelization of these linear layout problems parameterized by the vertex cover number τ of the input graph. The key point underlying our approach is to partition each set of related vertices into two distinct subsets with respect to the specific layouts, which immediately leads to some efficient reduction rules. We first apply this approach to the Mixed s -Stack q -Queue Layout problem and show that it admits a kernel of size 2 (τ) , which results in an algorithm running in time (2 2 (τ) + τ ⋅ | V |) , where | V | denotes the size of the input graph. Our work does not only confirm the existence of a fixed-parameter tractable algorithm for this problem mentioned by Bhore et al. (J. Graph Algorithms Appl. 2022), but also derives new results for the k -Stack Layout problem and for the k -Queue Layout problem respectively. We also employ this approach to the Upward k -Stack Layout problem and obtain a new result improving that presented by Bhore et al. (GD 2021). Last but not least, we use this approach to the k -Arch Layout problem and obtain a similar result. [ABSTRACT FROM AUTHOR]

We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and Hölder in the 1890s.) The groups satisfying this condition are PSL2(8), PSL2(9) and PSL2(p) for primes p such that p² -- 1 is a product of six primes. The conjecture suggests that there are infinitely many such primes p, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, the construction of highly symmetric block designs, and the possible existence of infinitely many Kn groups for each n ≥ 5. [ABSTRACT FROM AUTHOR]

For 0<λ<1$$ 0<\lambda <1 $$ and n→∞$$ n\to \infty $$ pick uniformly at random λn$$ \lambda n $$ vectors in {0,1} n$$ {\left\{0,1\right\}}^n $$ and let C$$ C $$ be the orthogonal complement of their span. Given 0<γ<12$$ 0<\gamma <\frac{1}{2} $$ with 0<λ

This paper presents exact derivations of the first two moments of the total order picking time in a warehouse for three routing heuristics, under the assumption of random storage. The analysis is done for general order size distributions and provides exact closed expressions in terms of the probability generating function of the order size distribution. We also indicate how the methods and insights in this paper extend to different storage policies and multi-block warehouses. The exact results derived in this paper are used to investigate effects of routing heuristics, order size distributions and layouts on warehouse efficiency. As illustration, we model the warehouse as a queueing system. By using approximations of the average order-lead time in terms of the first two moments of the order picking time, we are able to find optimal warehouse layouts and batch pick sizes. [ABSTRACT FROM AUTHOR]

In polaritons, the properties of matter are modified by mixing the molecular transitions with light modes inside a cavity. Resultant hybrid light–matter states exhibit energy level shifts, are delocalized over many molecular units, and have a different excited-state potential energy landscape, which leads to modified exciton dynamics. Previously, non-Hermitian Hamiltonians have been derived to describe the excited states of molecules coupled to surface plasmons (i.e., plexcitons), and these operators have been successfully used in the description of linear and third order optical response. In this article, we rigorously derive non-Hermitian Hamiltonians in the response function formalism of nonlinear spectroscopy by means of Feshbach operators and apply them to explore spectroscopic signatures of plexcitons. In particular, we analyze the optical response below and above the exceptional point that arises for matching transition energies for plasmon and molecular components and study their decomposition using double-sided Feynman diagrams. We find a clear distinction between interference and Rabi splitting in linear spectroscopy and a qualitative change in the symmetry of the line shape of the nonlinear signal when crossing the exceptional point. This change corresponds to one in the symmetry of the eigenvalues of the Hamiltonian. Our work presents an approach for simulating the optical response of sublevels within an electronic system and opens new applications of nonlinear spectroscopy to examine the different regimes of the spectrum of non-Hermitian Hamiltonians. [ABSTRACT FROM AUTHOR]

In this work, we have studied how incorporating viscous fluids leads to exact bounce cosmological solutions in general relativity (GR) framework. Specifically, we propose a novel parameterization of bulk viscosity coefficient of the form ζ = ζ0(t−t0)−2nH, where ζ0, with n being some positive constants and t0 being the bounce epoch. We investigate how this form of bulk viscosity may assist in explaining the early universe’s behavior, with a particular focus on non-singular bounce scenario by studying the various energy conditions and other related cosmological observables and how the model parameters affect the evolution of the Universe. We demonstrate that the null energy conditions (NEC) and strong energy conditions (SEC) violation occurs at the bounce point while dominant energy conditions (DEC) is satisfied. Finally, we carried out a stability check based on linear order perturbation to the Hubble parameter. We found that the perturbation vanishes asymptotically at later times, which indicates a stable behavior of the bounce scenario. [ABSTRACT FROM AUTHOR]

We answer some questions about two cardinal invariants associated with separating and almost disjoint families and a partition relation involving indecomposable countable linear orderings. [ABSTRACT FROM AUTHOR]

In this paper, we study on the ⪯ U -partial order induced by uninorms. We also investigate some properties direct product of two uninorms on bounded lattices. We investigate some properties of the set of incomparable elements with respect to ⪯ U -partial. Then, we investigate the relation between the set of comparable elements and the distributivity property for uninorms on the unit interval [0, 1]. Finally, we define a total order induced by internal uninorms. [ABSTRACT FROM AUTHOR]

The article discusses a correction to a previous construction of the Salvetti complex of a real hyperplane arrangement. It is noted that the chosen linear extension in the previous construction cannot be arbitrary, and the results remain valid with the use of an appropriate total ordering of the chambers. The article also presents counterexamples to a claim made in the previous construction and discusses open questions regarding nonrealizable oriented matroids and Euclidean orderings. The implications of the correction on further results are also discussed. [Extracted from the article]

Wheeler (Cladistics 2023, 39, 475) recently suggested that the issues with inapplicable characters in phylogenetic analysis can be dealt with directly by treating observed absences of a feature not in a separate absence/presence character but as insertion/deletion events in a complex character that describes the feature in all its variation; and that this dynamic homology view can be achieved by imposing a sequence or linear order on a set of characters and by analysing the resulting sequence character using custom alphabet tree alignment algorithms. As Wheeler observed, this approach can lead to considering inappropriate character states (such as a head state and a foot state) homologous. We show that it is also sensitive to the specific ordering assumption used and that such different character orders can lead to a preference for different trees. We present a simple four‐taxon dataset with observations of absence, but no inapplicable characters or other kinds of character dependence, for which the dynamic homology framework gives different results to classic algorithms for independent characters, including an optimal tree with biologically impossible reconstructions at inner nodes (every terminal has a head but the inner nodes are headless). We show how these issues can be solved by removing the character ordering assumption that the approach requires. Doing so, the dynamic homology framework reduces in general to Maddison's (Syst. Biol. 1993, 42, 576) well‐known proposal to deal with inapplicability using step matrix analysis of complex characters. If in addition costs are interpreted in terms of homology, it reduces to Goloboff et al.'s (Cladistics 2021, 37, 596) step matrix implementation for maximization of homology as applied to inapplicable characters. However, if used with homogeneous costs, as Wheeler suggested, it reduces to unordered analysis of such complex characters, which is known to treat tails that may share many observed features as irrelevant for establishing kinship when they differ in just one feature, e.g. colour. [ABSTRACT FROM AUTHOR]

In this paper, we define a QLY total order $ \overset {Q}\leq $ ≤ Q over $ \mathbb {D}_m $ D m to compare the magnitude of dual vectors. Then we consider the QLY least-squares problem and give its compact formula. Meanwhile, by comparing with a least-squares and the least-squares minimal-norm solutions, we can investigate a QLY least-squares and the QLY least-squares minimal-norm of linear dual least-squares problems. In particular, in the presence of a least-squares solution, we can get a QLY least-squares solution to be more accurate than a least-squares solution under the QLY total order. [ABSTRACT FROM AUTHOR]

Dynamical system models typically involve numerous input parameters whose "effects" and orthogonality need to be quantified through sensitivity analysis, to identify inputs contributing the greatest uncertainty. Whilst prior art has compared total-order estimators' role in recovering "true" effects, assessing their ability to recover robust parameter orthogonality for use in identifiability metrics has not been investigated. In this paper, we perform: (i) an assessment using a different class of numerical models representing the cardiovascular system, (ii) a wider evaluation of sampling methodologies and their interactions with estimators, (iii) an investigation of the consequences of permuting estimators and sampling methodologies on input parameter orthogonality, (iv) a study of sample convergence through resampling, and (v) an assessment of whether positive outcomes are sustained when model input dimensionality increases. Our results indicate that Jansen or Janon estimators display efficient convergence with minimum uncertainty when coupled with Sobol and the lattice rule sampling methods, making them prime choices for calculating parameter orthogonality and influence. This study reveals that global sensitivity analysis is convergence driven. Unconverged indices are subject to error and therefore the true influence or orthogonality of the input parameters are not recovered. This investigation importantly clarifies the interactions of the estimator and the sampling methodology by reducing the associated ambiguities, defining novel practices for modelling in the life sciences. Author summary: In order to gain new insight into a biological system, one often uses mathematical models to predict possible responses from the system. One vital step when using such models is to gain knowledge of the uncertainty associated with the model responses, for any input changes. Utilising two non-linear and stiff cardiovascular models as test cases, we investigate the effects of different choices made when quantifying the uncertainty of mathematical models. Leveraging efficient solving of the mathematical model, we show that in order to truly quantify the effects of inputs on a set of outputs, one must ensure converged estimates of the inputs' influence. Our detailed study provides a robust workflow of good modelling practice for biological systems, thus ensuring a true interpretation of the uncertainty associated with model inputs. [ABSTRACT FROM AUTHOR]

Lexicographic methods to rank fuzzy numbers present the advantages of simplicity, consistency with human intuition, and power of discrimination. In this paper, we tackle the problem of finding the conditions for these methods to produce a rank in specific steps. Our main results are twofold. First, we prove that a necessary and sufficient condition for a ranking function to be a total order is that this function is either injective, surjective, or bijective. Second, we provide further insight into the required steps for a lexicographic order to rank same-type and different-type fuzzy numbers. A counterexample refutes a conjecture in the literature about the maximum number of steps needed to rank different-type fuzzy numbers. [ABSTRACT FROM AUTHOR]

Lexicographic orders produce a total order over a set of fuzzy numbers by comparing several real-valued functions of each fuzzy number under consideration following a strictly hierarchical method. Lexicographic orders present advantages such as simplicity, consistency with human intuition, and power of discrimination. In this paper, we move one step forward in the search for simplicity by proposing a simplified version of lexicographic orders that focuses on the defining parameters of fuzzy numbers instead of using more complex functions as is common in the existing methodologies. More precisely, we show that two lexicographic methods described in the literature for trapezoidal and triangular fuzzy numbers are equivalent to the lexicographic order of a reorganization of the vector of defining parameters, reducing the complexity of the methods. We illustrate our approach through numerical examples and a case study about capital adequacy in the banking sector. [ABSTRACT FROM AUTHOR]

Z-numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as a measure of certainty or sureness of A. From the initial proposal, several other approaches have been introduced in order to reduce the computational cost of the involved operations. One of such approaches is called discrete Z-numbers where A and B are modelled as discrete fuzzy numbers. In this paper, the construction of total orders on the set of discrete Z-numbers is investigated for the first time. Specifically, the total order is designed for discrete Z-numbers where the second component has membership values belonging to a finite and prefixed set of values. The method relies on solid and coherent linguistic criteria and several linguistic properties are analyzed. The order involves the transformation of the first components of the discrete Z-numbers by using the credibility of the second components in the sense that a lower credibility enlarges in a greater extent the uncertainty of the first component. Then a total order on the set of discrete fuzzy numbers is applied. Finally, a practical example on how to order discrete Z-numbers is presented and a comparison with other ranking methods is performed from which the strengths of our method are stressed. [ABSTRACT FROM AUTHOR]

The diversity of consumer demand for take-out food has led to the characteristic structure of one order with multiple items, where the different food items in a single order are provided by two or more merchants. In the context of multi-item delivery for take-out orders with time windows, this study investigates vehicle routing for order delivery. This research aims to improve the service level of merchants and the efficiency of delivery vehicles. Food vendors receive orders from consumers through online platforms, then package the food items according to the orders. This method is a preliminary exploration of the issue of fulfilment of takeout orders on online platforms, and can provide preliminary theoretical support for decision-making on the delivery process of takeout orders on online platforms. In the context of online catering sales platforms and offline food sales merchants, this paper studies the delivery problem of takeaway orders with a time window and the characteristics of one order and multiple items, and constructs a method that takes the order delivery time requirements into account and minimizes the total order fulfilment cost. Taking into account the time window constraints of both physical restaurants and consumers, genetic algorithms are utilized to solve the order delivery problem. Finally, through case studies and experiments, the effectiveness and feasibility of the mathematical model are validated. Practical recommendations and insights are provided from the perspective of management and route planning. [ABSTRACT FROM AUTHOR]

We review some of the main goals of theoretical linguistics in the tradition of Generative Grammar: description, evolvability and learnability. We evaluate recent efforts to address these goals, culminating with the Minimalist Program. We suggest that the most prominent versions of the Minimalist Program represent just one possible approach to addressing these goals, and not a particularly illuminating one in many respects. Some desirable features of an alternative minimalist theory are the dissociation between syntax and linear order, the emphasis on representational economy (i.e. Simpler Syntax) and an extra-grammatical account of non-local constraints (e.g. islands). We conclude with the outline of an alternative minimalist perspective that we believe points to more satisfactory accounts of the observed phenomena. [ABSTRACT FROM AUTHOR]

This article proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector Z, independent of X and Y, the scale parameter of this class of distributions is mixed with a function α (Z) and its skew parameter is mixed with another function β (Z). Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this class of distributions by means of several stochastic orders including the usual stochastic order, convex order, increasing convex order, supermodular order, and some related linear orders. Two insightful assumptions for the density generators of elliptical distributions, aiming to control the generators' tail, are provided to make stochastic comparisons among mixed-elliptical vectors. Some applications in applied probability and actuarial science are also provided as illustrations on the main findings. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a new approach on 'distributed systems': the processes are represented through total orders and the communications are characterized by means of biorders. The resulting distributed systems capture situations met in various fields (such as computer science, economics and decision theory). We investigate questions associated to the numerical representability of order structures, relating concepts of economics and computing to each other. The concept of 'quasi-finite partial orders' is introduced as a finite family of chains with a communication between them. The representability of this kind of structure is studied, achieving a construction method for a finite (continuous) Richter–Peleg multi-utility representation. [ABSTRACT FROM AUTHOR]

Although declarations of the death of God seem to be provocations announcing the end of the era of theology, this announcement is actually central to the Christian revelation in its most classic forms, as well as to its reworkings in contemporary religious thought. Indeed provocative new possibilities for thinking theologically open up precisely in the wake of the death of God. Already Hegel envisaged a revolutionary new realization of divinity emerging in and with the secular world through its establishment of a total order of immanence. However, in postmodern times this comprehensive order aspired to by modern secularism implodes or cracks open towards the wholly Other. A hitherto repressed demand for the absolute difference of the religious, or for "transcendence", returns with a vengeance. This difference is what could not be stated in terms of the Hegelian System, for reasons that poststructuralist writers particularly have insisted on: all representations of God are indeed dead. Yet this does not mean that they cannot still be powerful, but only that they cannot assign God any stable identity. Nietzsche's sense of foreboding concerning the death of God is coupled with his intimations of the demise of representation and "grammar" as epistemologically bankrupt, but also with his vision of a positive potential for creating value in the wake of this collapse of all linguistically articulated culture. He points the way towards the emergence of a post-secular religious thinking of what exceeds thought and representation. [ABSTRACT FROM AUTHOR]

For partially ordered sets , we consider the square matrices with rows and columns indexed by linear extensions of the partial order on . Each entry is a formal variable defined by a pedestal of the linear order with respect to linear order . We show that all eigenvalues of any such matrix are -linear combinations of those variables. [ABSTRACT FROM AUTHOR]

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3): (a) The number of extreme points in an n-point order type, chosen uniformly at random from all such order types, is on average 4 + o(1). For labeled order types, this number has average 4 − 8/n²−n+2 and variance at most 3. (b) The (labeled) order types read off a set of n points sampled independently from the uniform measure on a convex planar domain, smooth or polygonal, or from a Gaussian distribution are concentrated, i.e., such sampling typically encounters only a vanishingly small fraction of all order types of the given size. Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points 2d +o(1) and constant variance. We also discuss to what extent our methods generalize to the abstract setting of uniform acyclic oriented matroids. Moreover, our methods show the following relative of the Erdős- Szekeres theorem: for any fixed k, as n →∞, a proportion 1−O(1/n) of the n-point simple order types contain a triangle enclosing a convex k-chain over an edge. For the unlabeled case in (a), we prove that for any antipodal, finite subset of the two-dimensional sphere, the group of orientation preserving bijections is cyclic, dihedral, or one of A4, S4, or A5 (and each case is possible). These are the finite subgroups of SO(3) and our proof follows the lines of their characterization by Felix Klein. [ABSTRACT FROM AUTHOR]

In this work, we present an extension of popular selected configuration interaction (SCI) algorithms to the Transcorrelated (TC) framework. Although we used in this work the recently introduced one-parameter correlation factor [E. Giner, J. Chem. Phys. 154, 084119 (2021)], the theory presented here is valid for any correlation factor. Thanks to the formalization of the non-Hermitian TC eigenvalue problem as a search of stationary points for a specific functional depending on both left- and right-functions, we obtain a general framework, allowing for different choices for both the selection criterion in SCI and the second order perturbative correction to the energy. After numerical investigations on different second-row atomic and molecular systems in increasingly large basis sets, we found that taking into account the non-Hermitian character of the TC Hamiltonian in the selection criterion is mandatory to obtain a fast convergence of the TC energy. In addition, selection criteria based on either the first order coefficient or the second order energy lead to significantly different convergence rates, which is typically not the case in the usual Hermitian SCI. Regarding the convergence of the total second order perturbation energy, we find that the quality of the left-function used in the equations strongly affects the quality of the results. Within the near-optimal algorithm proposed here, we find that the SCI expansion in the TC framework converges faster than the usual SCI in terms of both the basis set and the number of Slater determinants. [ABSTRACT FROM AUTHOR]

Close to equilibrium, the excess heat governs the static fluctuations. We study the heat capacity in that McLennan regime, i.e. in linear order around equilibrium, using an expression in terms of the average energy that extends the equilibrium formula in the canonical ensemble. It is derivable from an entropy and it always vanishes at zero temperature. Any violation of an extended Third Law is therefore a nonlinear effect. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in detail in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T , we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M C H T and find explicitly what operators T have the property that M C H T = C. [ABSTRACT FROM AUTHOR]

This erratum corrects the proof of the main result of a mathematical research paper. The proof contained errors, specifically in Lemmas 5.1 and 5.3. The authors provide a corrected proof for Theorem 5.2, which involves calculating a Gröbner basis for an ideal. The corrected proof is based on the concept of Motzkin paths and introduces the notion of exterior algebra elements. The authors conclude with a corollary stating that a specific set of elements forms a Gröbner basis for the ideal. [Extracted from the article]

Ramscar, Yarlett, Dye, Denny, and Thorpe (2010) showed how, consistent with the predictions of error‐driven learning models, the order in which stimuli are presented in training can affect category learning. Specifically, learners exposed to artificial language input where objects preceded their labels learned the discriminating features of categories better than learners exposed to input where labels preceded objects. We sought to replicate this finding in two online experiments employing the same tests used originally: A four pictures test (match a label to one of four pictures) and a four labels test (match a picture to one of four labels). In our study, only findings from the four pictures test were consistent with the original result. Additionally, the effect sizes observed were smaller, and participants over‐generalized high‐frequency category labels more than in the original study. We suggest that although Ramscar, Yarlett, Dye, Denny, and Thorpe (2010) feature‐label order predictions were derived from error‐driven learning, they failed to consider that this mechanism also predicts that performance in any training paradigm must inevitably be influenced by participant prior experience. We consider our findings in light of these factors, and discuss implications for the generalizability and replication of training studies. [ABSTRACT FROM AUTHOR]

This paper addresses the optimal control problem with sparse controls of a Timoshenko beam, its numerical approximation using the finite element method, and the numerical solution via nonsmooth methods. Incorporating sparsity‐promoting terms in the cost function is practically useful for beam vibration models and results in the localization of the control action that facilitates the placement of actuators or control devices. We consider two types of sparsity‐inducing penalizers: the L1$$ {L}^1 $$‐norm and the L0$$ {L}^0 $$‐penalizer, which measures function support. We analyze discretized problems utilizing linear finite elements with a locking‐free scheme to approximate the states and adjoint states. We confirm that this approximation has the looking‐free property required to achieve a linear convergence linear order of approximation for L1$$ {L}^1 $$ control case and depending on the set of switching points in the L0$$ {L}^0 $$ controls. This is similar to the purely L2$$ {L}^2 $$‐norm penalized optimal control, where the order of approximation is independent of the thickness of the beam. [ABSTRACT FROM AUTHOR]

Wolves have returned to Germany since 2000. Numbers have grown to 209 territorial pairs in 2021. XGBoost machine learning, combined with SHAP analysis is applied to predict German wolf pair presence in 2022 for 10 × 10 km grid cells. Model input consisted of 38 variables from open sources, covering the period 2000 to 2021. The XGBoost model predicted well, with 0.91 as the AUC. SHAP analysis ranked the variables: distance to the closest neighboring wolf pair was the main driver for a grid cell to become occupied by a wolf pair. The clustering tendency of related wolves seems to be an important explanatory factor here. Second was the percentage of wooded area. The next eight variables related to wolf presence in the preceding year, except at fifth, eighth and tenth position in the total order: human density (square root) in the grid, percentage arable land and road density respectively. Other variables including the occurrence of wild prey were the weakest predictors. The SHAP analysis also provided crucial added value in identifying a variable that had threshold values where its contribution to the prediction changed from positive to negative or vice versa. For instance, low density of people increased the probability of wolf pair presence, whereas a high density decreased this probability. Cumulative lift techniques showed that the model performed almost four times better than random prediction. The combination of XGBoost, SHAP and cumulative lift techniques is new in wolf management and conservation, allowing for the focusing of educational and financial resources. [ABSTRACT FROM AUTHOR]

We consider bond percolation on high-dimensional product graphs $G=\square _{i=1}^tG^{(i)}$ , where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev (J. Graph Theory , 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$ , where $d$ is the average degree of the host graph. In the supercritical regime, we strengthen Lichev's result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$ , and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory , 99(4):651–670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722 , 2022). [ABSTRACT FROM AUTHOR]

This article explores the issue of auditability in centralized school choice systems, which use rankings and priority scores to assign students to schools. The authors propose an auditability index to measure the ease of detecting deviations from the intended rules. They find that some assignment rules have a low auditability index, meaning that only a small group of applicants can detect deviations, while others have a high index. The authors argue that the Immediate Acceptance algorithm performs better in terms of auditability compared to the commonly used Deferred Acceptance algorithm. The study emphasizes the importance of trust and transparency in school choice mechanisms. [Extracted from the article]

The spotlight of this research is on the application of the fixed‐pole interpolation, sometimes used in the analysis of three‐dimensional (3D) geometrically non‐linear beams, but for which no attempts have been made to apply it to linear analysis so far. Particular attention is given to the correlation between the linearised forms of the fixed‐pole and helicoidal interpolation with the linked interpolation. Knowledge of this interdependence is crucial for identifying paths for possible enhancement and extension from the Timoshenko beam of arbitrary order to the new hexahedral finite element of arbitrary order for linear analysis of 3D micropolar continuum. After ensuring the convergence of the newly developed micropolar element through a set of patch tests, three numerical examples of a 3D micropolar continuum in static equilibrium and free vibration of 3D micropolar plates with different geometric properties and boundary conditions have been analysed. Based on these results, the newly proposed finite elements have been critically assessed against the conventional elements. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity. [ABSTRACT FROM AUTHOR]