Symmetric Set Research Articles

Classification of singular sets of solutions to elliptic equations

In this paper, we mainly investigate the classification of singular sets of solutions to elliptic equations. Firstly, we define the $ j $-symmetric singular set $ S^j(u) $ of solution $ u $, and show that the Hausdorff dimension of the $ j $-symmetric singular set $ S^j(u) $ is not more than $ j $. Then we prove the generalized $ \varepsilon $-regularity lemma for $ j $-symmetric homogeneous harmonic polynomial $ P $ with origin $ 0 $ as the isolated critical point in $ \mathbb{R}^{n-j} $, and by the generalized $ \varepsilon $-regularity lemma, we show the Hausdorff measure estimate of the $ j $-symmetric singular set $ S^j(u) $. Moreover, we study the geometric structure of interior singular points of solutions $ u $ in a planar bounded domain.

Information Design in Optimal Auctions

We study the information design problem in a single-unit auction setting. The information designer controls independent private signals according to which the buyers infer their binary private value. Assuming that the seller adopts Myerson (1981) optimal auction in response, we characterize both the buyer-optimal information structure which maximizes the buyers' surplus and the seller-worst information structure which minimizes the seller's revenue. We translate both information design problems into finite-dimensional constrained optimization problems for which the optimal information structures can be explicitly solved. In contrast to the case with one buyer (Roesler and Szentes, 2017 and Du, 2018), we show that with two or more buyers, the buyer-optimal information structure is different from the seller-worst information structure: the good is always sold under the seller-worst information structure but not under the buyer-optimal information structure. Nevertheless, as the number of buyers goes to infinity, both information structures converge to no disclosure. We also show that in an ex ante symmetric setting, an asymmetric information structure is never seller-worst but for some prior can generate a strictly higher surplus for the buyers than the symmetric buyer-optimal information structure.

From partial to complete neutralization of 2,5-dihydroxyterephthalic acid in the Li–Na system: crystal chemistry and electrochemical behavior of Na2Li2C8H2O6vs.Li

The 2,5-dihydroxyterephthalic acid (H4-p-DHT) is of special interest in the field of materials science because of the two symmetric sets of oxygen donor functional groups (i.e., β-hydroxy acid moieties).

Midwinter Suppression of Storm Tracks in an Idealized Zonally Symmetric Setting

AbstractThe midwinter suppression of eddy activity in the North Pacific storm track is a phenomenon that has resisted reproduction in idealized models that are initialized independently of the observed atmosphere. Attempts at explaining it have often focused on local mechanisms that depend on zonal asymmetries, such as effects of topography on the mean flow and eddies. Here an idealized aquaplanet GCM is used to demonstrate that a midwinter suppression can also occur in the activity of a statistically zonally symmetric storm track. For a midwinter suppression to occur, it is necessary that parameters, such as the thermal inertia of the upper ocean and the strength of tropical ocean energy transport, are chosen suitably to produce a pronounced seasonal cycle of the subtropical jet characteristics. If the subtropical jet is sufficiently strong and located close to the midlatitude storm track during midwinter, it dominates the upper-level flow and guides eddies equatorward, away from the low-level area of eddy generation. This inhibits the baroclinic interaction between upper and lower levels within the storm track and weakens eddy activity. However, as the subtropical jet continues to move poleward during late winter in the idealized GCM (and unlike what is observed), eddy activity picks up again, showing that the properties of the subtropical jet that give rise to the midwinter suppression are subtle. The idealized GCM simulations provide a framework within which possible mechanisms giving rise to a midwinter suppression of storm tracks can be investigated systematically.

Borsuk Number for Planar Convex Bodies

By using some simple tools from graph theory, we obtain a characterization of the compact sets in $$\mathbb {R}^n$$ with Borsuk number equal to two. This result allows to give some examples of planar (convex) compact sets with Borsuk number equal to three. Moreover, we also prove that the unique centrally symmetric planar convex compact sets with Borsuk number equal to three are the Euclidean balls.

On Dynamic Aggregation Systems

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭr (A) on a set of alternatives A, |A| ≥ 3: on each step, the participants are subject to intermediate collective decisions on some subsets B of the set A and transform their a priori preferences according to an adaptation function 𝒜. The sequence of intermediate decisions is determined by a lot J, i.e., an increasing (with respect to inclusion) sequence of subsets B of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set 𝔇 ⊆ ℭr (A) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ2 (A) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function f, lot J, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function fJ,A that imitates a consecutive aggregation procesure. If f dynamically preserves a set 𝔇 ⊆ ℭr (A) with respect to a set of lots 𝒥, then the aggregation function fJ,A preserves the set 𝔇 for each lot J ∈ 𝒥. If 𝔇 = ℜ2(A), then the adaptation function can be chosen in such a way that in any profile c ∈ (ℜ2(A))n, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences fJ,A (c) for each maximal lot J and f that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

Asymptotic convergence of the angular discretization error in the scalar flux computed from the particle transport equation with the method of discrete ordinates

The asymptotic convergence of the angular discretization error in the scalar flux solution of the particle transport equation computed with the Discrete Ordinates (SN) method with increasing quadrature order is examined. Five selected angular quadrature types are considered: Level Symmetric (LS), Legendre-Chebyshev Quadrangular (LCQ), Legendre-Chebyshev Triangular(LCT), Quadruple Range (QR) and Quadruple Range Spence-type (QRS) quadrature sets. We relate the SN angular discretization error to the quadrature error, and split the total flux into the uncollided flux and the fully collided flux, and then we verify the uncollided scalar flux error and the fully collided scalar flux error separately. After developing the theoretical basis for the relationship between true solution regularity and quadrature rule error, we employ a two-dimensional problem to verify the observed and theoretical convergence orders for the region-averaged uncollided and fully collided scalar flux errors. Numerical results show that the angular discretization errors in the region-averaged scalar flux obtained by different quadrature types converge with different rates, that are commensurate with the regularity order of the exact angular flux within the quadrature-designed integration interval. The angular discretization error in the uncollided region-averaged scalar flux obtained by LC class quadratures converges linearly, and the error obtained by QR class quadrature sets converges quadratically. The angular discretization error in the fully collided region-averaged scalar flux converges linearly for LC class quadratures, and faster-than-second order for QR class quadrature sets.

Information Design

A designer commits to a signal distribution that is informative about a payoff-relevant state. Conditional upon the privately observed signals, agents take actions that affect their payoffs as well as those of the designer. We show how to derive the (designer) optimal information structure in static finite environments. We fully characterize it in a symmetric binary setting for a parameterized game. In this environment, conditionally independent private signals are never strictly optimal. (JEL C72, D78, D82, D83)

Symmetry of minimizers of a Gaussian isoperimetric problem

We study an isoperimetric problem described by a functional that consists of the standard Gaussian perimeter and the norm of the barycenter. The second term is in competition with the perimeter, balancing the mass with respect to the origin, and because of that the solution is not always the half-space. We characterize all the minimizers of this functional, when the volume is close to one, by proving that the minimizer is either the half-space or the symmetric strip, depending on the strength of the barycenter term. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one. As another corollary we obtain the optimal constant in the quantitative Gaussian isoperimetric inequality.

Compatible observable decompositions for coarse-grained representations of real molecular systems.

Coarse-grained (CG) observable expressions, such as pressure or potential energy, are generally different than their fine-grained (FG, e.g., atomistic) counterparts. Recently, we analyzed this so-called "representability problem" in Wagner et al. [J. Chem. Phys. 145, 044108 (2016)]. While the issue of representability was clearly and mathematically stated in that work, it was not made clear how to actually determine CG observable expressions from the underlying FG systems that can only be simulated numerically. In this work, we propose minimization targets for the CG observables of such systems. These CG observables are compatible with each other and with structural observables. Also, these CG observables are systematically improvable since they are variationally minimized. Our methods are local and data efficient because we decompose the observable contributions. Hence, our approaches are called the multiscale compatible observable decomposition (MS-CODE) and the relative entropy compatible observable decomposition (RE-CODE), which reflect two main approaches to the "bottom-up" coarse-graining of real FG systems. The parameterization of these CG observable expressions requires the introduction of new, symmetric basis sets and one-body terms. We apply MS-CODE and RE-CODE to 1-site and 2-site CG models of methanol for the case of pressure, as well as to 1-site methanol and acetonitrile models for potential energy.

Lexically dependent estimation of acoustic information in speech III: Cross-splicing verification of cue weights

In two prior studies (Redmon and Jongman, 2018, 2019, JASA), results of open- and closed-class word recognition tasks on items drawn from a large single-speaker database (Tucker et al., 2018) were used to train a model where acoustic cue weights were optimized to distinguish words in the lexicon, rather than a balanced inventory of phones. From that work, cues were identified that had a greater weight when considering the lexicon as a whole than when studying a symmetric set of contrasts in controlled syllable productions. To verify the causal role of such cues in word recognition, two new cross-splicing versions of the open- and closed-class tasks were run with a subset of items in each prior experiment. In each, an enhancement condition was created by cross-splicing diphones from items in the database with more distinct values on a given cue dimension, and consequently greater model-predicted accuracy. For each such item, a parallel reduction condition was created wherein accuracy was predicted to decrease due to the cross-splicing of a diphone with a more ambiguous value of a given cue. Results will serve to validate the relative role of the different cues in a manner that is external to the model-fitting procedure.In two prior studies (Redmon and Jongman, 2018, 2019, JASA), results of open- and closed-class word recognition tasks on items drawn from a large single-speaker database (Tucker et al., 2018) were used to train a model where acoustic cue weights were optimized to distinguish words in the lexicon, rather than a balanced inventory of phones. From that work, cues were identified that had a greater weight when considering the lexicon as a whole than when studying a symmetric set of contrasts in controlled syllable productions. To verify the causal role of such cues in word recognition, two new cross-splicing versions of the open- and closed-class tasks were run with a subset of items in each prior experiment. In each, an enhancement condition was created by cross-splicing diphones from items in the database with more distinct values on a given cue dimension, and consequently greater model-predicted accuracy. For each such item, a parallel reduction condition was created wherein accuracy was predicted to decre...

Homopolar Bearingless Slice Motor With Flux-Biasing Halbach Arrays

In this article, we present a new configuration of bearingless slice motor that levitates and rotates a ring-shaped solid steel reluctance rotor. The rotor is 50 mm in diameter and has salient features on the outer surface. Symmetric sets of Halbach magnet arrays, mounted on the tips of stator teeth, establish a homopolar bias flux around the rotor. The bias flux passively stabilizes the rotor in the out-of-plane tilts and axial translation, whereas the rotor's radial translations are actively stabilized by feedback control. The rotor saliencies spatially modulate the air-gap bias flux such that the resulting torque-current relationship is similar to that of permanent-magnet synchronous machines. We have designed, built, and tested a prototype bearingless motor and control system. The prototype system achieves a torque constant of 14.9 mNm/A, maximum speed of 5500 r/min, and suspension bandwidth of 84 Hz with a phase margin of 11.3 $^{\circ }$ . The rated torque and speed are 26.8 mNm and 3486 r/min, respectively, and the axial and tilting passive stiffnesses are 15.3 N/mm and 34.4 mNm/ $^{\circ }$ , respectively.

Coherent-Price Systems and Uncertainty-Neutral Valuation

This paper considers fundamental questions of arbitrage pricing that arises when the uncertainty model incorporates ambiguity about risk. This additional ambiguity motivates a new principle of risk- and ambiguity-neutral valuation as an extension of the paper by Ross (1976) (Ross, Stephen A. 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13: 341–60). In the spirit of Harrison and Kreps (1979) (Harrison, J. Michael, and David M. Kreps. 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20: 381–408), the paper establishes a micro-economic foundation of viability in which ambiguity-neutrality imposes a fair-pricing principle via symmetric multiple prior martingales. The resulting equivalent symmetric martingale measure set exists if the uncertain volatility in asset prices is driven by an ambiguous Brownian motion.

Symmetry exploits for Bayesian cubature methods

Bayesian cubature provides a flexible framework for numerical integration, in which a priori knowledge on the integrand can be encoded and exploited. This additional flexibility, compared to many classical cubature methods, comes at a computational cost which is cubic in the number of evaluations of the integrand. It has been recently observed that fully symmetric point sets can be exploited in order to reduce—in some cases substantially—the computational cost of the standard Bayesian cubature method. This work identifies several additional symmetry exploits within the Bayesian cubature framework. In particular, we go beyond earlier work in considering non-symmetric measures and, in addition to the standard Bayesian cubature method, present exploits for the Bayes–Sard cubature method and the multi-output Bayesian cubature method.

Farsighted stability in patent licensing: An abstract game approach

This paper analyzes the negotiations made by an external patent holder and potential licensee firms in a new model of patent licensing, assuming that they are all farsighted, and characterizes the symmetric farsighted stable sets. Given a net profit of each licensee firm, a set of outcomes is a symmetric farsighted stable set if and only if, at any outcome in the set, each licensee firm receives the net profit and the number of licensee firms maximizes the patent holder's profit provided that licensee firms obtain the net profits. We also show the close relationship between the symmetric farsighted stable sets and the relative interior of the core. Further, we confirm that the symmetric farsighted stable sets are the absolutely maximal farsighted stable sets (Ray and Vohra, forthcoming) as well as the history dependent strongly rational expectation farsighted stable sets (Dutta and Vartiainen, forthcoming).

Generalized Degrees of Freedom of the Symmetric Cache-Aided MISO Broadcast Channel With Partial CSIT

We consider the cache-aided MISO broadcast channel (BC) in which a multi-antenna transmitter serves $K$ single-antenna receivers, each equipped with a cache memory. The transmitter has access to partial knowledge of the channel state information. For a symmetric setting, in terms of channel strength levels, partial channel knowledge levels and cache sizes, we characterize the generalized degrees of freedom (GDoF) up to a constant multiplicative factor. The achievability scheme exploits the interplay between spatial multiplexing gains and coded-multicasting gain. On the other hand, a cut-set-based argument in conjunction with a GDoF outer bound for a parallel MISO BC under channel uncertainty is used for the converse. We further show that the characterized order-optimal GDoF is also attained in a decentralized setting, where no coordination is required for content placement in the caches.

Nonlinear scalar field equations with general nonlinearity

Consider the nonlinear scalar field equation (0.1)−Δu=f(u)inRN,u∈H1(RN),where N≥3 and f satisfies the general Berestycki–Lions conditions. We are interested in the existence of positive ground states, of nonradial solutions and in the multiplicity of radial and nonradial solutions. Very recently Mederski (2017) made a major advance in that direction through the development, in an abstract setting, of a new critical point theory for constrained functionals. In this paper we propose an alternative, more elementary approach, which permits to recover Mederski’s results on (0.1). The keys to our approach are an extension to the symmetric mountain pass setting of the monotonicity trick, and a new decomposition result for bounded Palais–Smale sequences.

Experimental study of the effects of turbine solidity, blade profile, pitch angle, surface roughness, and aspect ratio on the H‐Darrieus wind turbine self‐starting and overall performance

AbstractNew and comprehensive time‐accurate, experimental data from an H‐Darrieus wind turbine are presented to further develop our understanding of the performance of these turbines with a particular focus on self‐starting. The impact of turbine solidity, blade profile, surface roughness, pitch angle, and aspect ratio on the turbine's performance is investigated, parameters that are thought to be critical for small‐scale VAWT operation, particularly when operating in the built environment. It is demonstrated clearly that high turbine solidity is beneficial for turbine self‐starting and that the selection of a thick, symmetrical aerofoil set at a low, negative pitch angle () is better than a cambered foil. Increased blade surface roughness is also shown to improve a turbine's self‐starting capability at low tip speed ratios and with high turbine solidity and the associated flow physics are discussed. Finally, it was confirmed that blade span has a significant impact on turbine starting. This paper contributes to the understanding of the turbine characteristics during the starting period and provides clear guidance and validation cases for future design and research in order to promote and justify the wider application of this wind turbine configuration.

X-DMM: Fast and Scalable Model Based Text Clustering

Text clustering is a widely studied problem in the text mining domain. The Dirichlet Multinomial Mixture (DMM) model based clustering algorithms have shown good performance to cope with high dimensional sparse text data, obtaining reasonable results in both clustering accuracy and computational efficiency. However, the time complexity of DMM model training is proportional to the average document length and the number of clusters, making it inefficient for scaling up to long text and large corpora, which is common in realworld applications such as documents organization, retrieval and recommendation. In this paper, we leverage a symmetric prior setting for Dirichlet distribution, and build indices to decrease the time complexity of the sampling-based training for DMM from O(K∗L) to O(K∗U), where K is the number of clusters, L the average length of document, and U the average number of unique words in each document. We introduce a Metropolis-Hastings sampling algorithm, which further reduces the sampling time complexity from O(K∗U) to O(U) in the nearly-to-convergence training stages. Moreover, we also parallelize the DMM model training to obtain a further acceleration by using an uncollapsed Gibbs sampler. We combine all these optimizations into a highly efficient implementation, called X-DMM, which enables the DMM model to scale up for long and large-scale text clustering. We evaluate the performance of X-DMM on several real world datasets, and the experimental results show that XDMM achieves substantial speed up compared with existing state-of-the-art algorithms without clustering accuracy degradation.

A hierarchy of entanglement criteria for four-qubit symmetric Greenberger–Horne–Zeilinger diagonal states

With a two step optimization method of entanglement witness, we analytically propose a set of necessary and sufficient entanglement criteria for four qubit symmetric Greenberger-Horne-Zeilinger (GHZ) diagonal states. The criterion set contains four criteria. Two of them are linear with density matrix elements. The other two criteria are nonlinear with density matrix elements. The criterion set has a nest structure. A proper subset of the criteria is necessary and sufficient for the entanglement of a proper subset of the states. We illustrate the nest structure of criterion set with the general Werner state set and its superset the highly symmetric GHZ diagonal state set, they are subsets of the symmetric GHZ diagonal state set.