Symmetric Rules Research Articles

Bribe-proofness for single-peaked preferences: characterizations and maximality-of-domains results

This paper considers the problem of allocating an amount of a perfectly divisible resource among agents. We are interested in rules eliminating the possibility that an agent can compensate another to misrepresent her preferences, making both agents strictly better off. Such rules are said to be bribe-proof (Schummer in J Econ Theory 91:180–198, 2000). We first provide necessary and sufficient conditions for any rule defined on the single-peaked domain to be bribe-proof. By invoking this and Ching’s (Soc Choice Welf 11:131–136, 1994) result, we obtain the uniform rule as the unique bribe-proof and symmetric rule on the single-peaked domain. Furthermore, we examine how large a domain can be to allow for the existence of bribe-proof and symmetric rules and show that the convex domain is such a unique maximal domain.

Robust path planning by propagating rhythmic spiking activity in a hippocampal network model

It is believed that humans and animals like rodents and bats navigate in a familiar environment using a cognitive map. Yet, how maps are previously learned when exploring a novel environment and then used to plan routes to specific goals remains unclear. We propose here a biologically inspired method, called Multiwave, with two main ingredients: (i) a symmetric spike-timing dependent plasticity rule that compels the connectivity to map the environment, and (ii) periodic wavefronts propagating through the network to direct the agent towards the goal and increase robustness to noise. The modeling involves a detailed neural network for biological plausibility and a simplified equivalent network, better suited for robotic implementation. Using both simulations and robotic experiments, we show the efficiency of Multiwave for path planning. Multiwave is relevant not only to neuroscience, as the detailed model is biologically grounded, but also to robotic applications, as the simplified model is suitable for analog implementation on neuromorphic hardware.

A Novel Fifth-Degree Cubature Kalman Filter for Real-Time Orbit Determination by Radar

A novel fifth-degree cubature Kalman filter is proposed to improve the accuracy of real-time orbit determination by ground-based radar. A fully symmetric cubature rule, approaching the Gaussian weighted integral of a nonlinear function in general form, is introduced, and the specific points and weights are calculated by matching the monomials of degree not greater than five with the exact values. On the basis of the above rule, a novel fifth-degree cubature Kalman filter, which can achieve a higher accuracy than UKF and CKF, is derived under the Bayesian filtering framework. Then, to describe the nonlinear system more accurately, the orbital dynamics equation with J2 perturbation is used as the state equation, and the nonlinear relationship between the radar measurement elements and orbital states is built as the measurement equation. The simulation results show that, compared with the traditional third-degree algorithm, the proposed fifth-degree algorithm has a higher accuracy of orbit determination.

Event-Triggered State Estimation: An Iterative Algorithm and Optimality Properties

This paper investigates the optimal design of event-triggered estimation for linear systems. The synthesis approach is posed as a team decision problem where the decision makers are given by the event trigger and the estimator. The event-trigger decides upon its available measurements whether the estimator shall obtain the current state information by transmitting it through a resource constrained channel. The objective is to find the optimal tradeoff between the mean square estimation error and the expected number of transmissions over a finite horizon. After deriving basic characteristics of the optimal solution, we propose an iterative algorithm that alternates between optimizing one decision maker while fixing the other and vice versa. By analyzing the dynamical behavior of the iterative method, it is shown that the algorithm converges to a symmetric threshold policy for first-order systems if the statistics of the uncertainties are even and unimodal. In the case of bimodal distributions, we show numerically that the iterative method may find asymmetric threshold policies that outperform symmetric rules.

On Extension-Shearing Bending-Twisting coupled laminates

This article presents details of the development of a special class of laminate, possessing Extension-Shearing Bending-Twisting coupling, necessary for optimised passive-adaptive flexible wing-box structures. The possibility of achieving a measurable drag reduction in cruise flight, without the cost or reliability issues associated with active control mechanisms, is of significant interest for achieving increased fuel burn efficiency, and meeting associated emissions targets. The introduction of passive Bending-Twisting coupling at the wing-box level has been previously demonstrated through laminate level tailoring with Extension-Shearing coupling only, but the limited design space and the possibility for ply terminations (to produce tapered thickness) effectively rule out this special class of laminate for practical construction. The study is now broadened to consider laminates with Extension-Shearing Bending-Twisting coupling, beyond the less well-known un-balanced and symmetric design rule or indeed balanced and symmetric designs with off-axis alignment. Results reveal a vast laminate design space with Extension-Shearing coupling that can be maximised without the unfavourable strength characteristics associated with off-axis alignment. Results also reveal that shear buckling strength can be maximised through Bending-Twisting coupling when load reversal is not a design constraint.

Proof Analysis of Peirce’s Alpha System of Graphs

Charles Peirce's alpha system $$\mathfrak {S}_\alpha $$S? is reformulated into a deep inference system where the rules are given in terms of deep graphical structures and each rule has its symmetrical rule in the system. The proof analysis of $$\mathfrak {S}_\alpha $$S? is given in terms of two embedding theorems: the system $$\mathfrak {S}_\alpha $$S? and Brunnler's deep inference system for classical propositional logic can be embedded into each other; and the system $$\mathfrak {S}_\alpha $$S? and Gentzen sequent calculus $$\mathbf {G3c}^*$$G3c? can be embedded into each other.

On Bending-Twisting coupled laminates

The definitive list of stacking sequences is presented for Bending-Twisting coupled (ASB0DF) laminates, with up to 21 plies. This class of laminate arises from the ubiquitous balanced and symmetric design rule, but symmetry is shown to be a sufficient rather than a necessary constraint. Each stacking sequence configuration is derived in symbolic form together with dimensionless parameters from which the extensional and bending stiffness terms are readily calculated for any fibre/matrix system and angle-ply orientation. Expressions for ply orientation dependent lamination parameters are also given, together with graphical representations, which demonstrate the extent of the design space. Pseudo Quasi-Homogeneous (ASB0DF) laminates are introduced as an important laminate sub-set, since such laminates have concomitant orthotropic properties, i.e. matching orthotropic or isotropic stiffnesses in extension and bending, from which the isolated effects of Bending-Twisting coupling can be studied. These coupling effects are quantified for compression buckling of Angle-ply and Quasi-Isotropic laminated plates with simply supported and clamped edges.

Bribe-Proofness for Single-Peaked Preferences: Characterizations and Maximality-of-Domains Results

This paper considers the problem of allocating an amount of a perfectly divisible resource among agents. We are interested in rules eliminating the possibility that an agent can compensate another to misrepresent her preferences, making both agents strictly better off. Such rules are said to be bribe-proof (Schummer, 2000). We first provide necessary and sufficient conditions for any rule defined on the single-peaked domain to be bribe-proof. By invoking this and Ching's (1994) result, we obtain the uniform rule as the unique bribe-proof and symmetric rule on the single-peaked domain. Furthermore, we examine how large a domain can be to allow for the existence of bribe-proof and symmetric rules and show that the convex domain is such a unique maximal domain.

Maximum Entropy Derivation of Quasi-Newton Methods

This paper presents a maximum-entropy (MaxEnt) derivation of many commonly used quasi-Newton rules. (i) This derivation interprets the elements of the Jacobian or Hessian as means of a multivariate probability distribution; (ii) the variance is chosen to represent the uncertainty about the mean. This interpretation is more intuitive than previous maximum-entropy formulations of quasi-Newton rules, in which the Jacobian or Hessian elements are variances while the mean is fixed to zero. The current formulation also gives an alternative theoretical foundation for quasi-Newton methods based on the combinatorial and axiomatic justifications of the MaxEnt method. We also present an equivalent linear system to the MaxEnt solution for a Gaussian prior distribution and diagonal covariance matrix. A new family of symmetrical quasi-Newton rules are derived using additional constraints which guarantee Jacobian or Hessian symmetry, even if the prior means are not symmetric. The analysis gives new insights into quasi-Newton methods, including guidance for the relative change of elements in each row of the Jacobian or Hessian.

Characterizing minimal impartial rules for awarding prizes

We study the problem of selecting prize winners from a group of experts when each expert nominates another expert for the prize. A nomination rule determines the set of winners on the basis of the profile of nominations; the rule is impartial if one's nomination never influences one's own chance of winning the prize. In this paper, we consider impartial, anonymous, symmetric, and monotonic nomination rules and characterize the set of all minimal such rules. We show that the set consists of exactly one nomination rule: a natural variant of the plurality correspondence called plurality with runners-up.

Mirrored STDP Implements Autoencoder Learning in a Network of Spiking Neurons.

The autoencoder algorithm is a simple but powerful unsupervised method for training neural networks. Autoencoder networks can learn sparse distributed codes similar to those seen in cortical sensory areas such as visual area V1, but they can also be stacked to learn increasingly abstract representations. Several computational neuroscience models of sensory areas, including Olshausen & Field’s Sparse Coding algorithm, can be seen as autoencoder variants, and autoencoders have seen extensive use in the machine learning community. Despite their power and versatility, autoencoders have been difficult to implement in a biologically realistic fashion. The challenges include their need to calculate differences between two neuronal activities and their requirement for learning rules which lead to identical changes at feedforward and feedback connections. Here, we study a biologically realistic network of integrate-and-fire neurons with anatomical connectivity and synaptic plasticity that closely matches that observed in cortical sensory areas. Our choice of synaptic plasticity rules is inspired by recent experimental and theoretical results suggesting that learning at feedback connections may have a different form from learning at feedforward connections, and our results depend critically on this novel choice of plasticity rules. Specifically, we propose that plasticity rules at feedforward versus feedback connections are temporally opposed versions of spike-timing dependent plasticity (STDP), leading to a symmetric combined rule we call Mirrored STDP (mSTDP). We show that with mSTDP, our network follows a learning rule that approximately minimizes an autoencoder loss function. When trained with whitened natural image patches, the learned synaptic weights resemble the receptive fields seen in V1. Our results use realistic synaptic plasticity rules to show that the powerful autoencoder learning algorithm could be within the reach of real biological networks.

Functional consequences of non-equilibrium dynamics caused by antisymmetric and symmetric learning rules

Connectivity in the brain is asymmetric, which is evident by Dale's principle of excitatory and inhibitory neurons. As a consequence, biologically realistic neuronal networks cannot be in thermodynamic equilibrium [1]. Even in a stationary state, probability fluxes perpetuate, leading to non-equilibrium steady states [2,3]. We here investigate the computational consequences of non-equilibrium dynamics for synaptic plasticity and learning. Formulating the stochastic dynamics in sparsely connected networks of non-linear Langevin equations in terms of path integrals [4], we show that biologically plausible correlation-sensitive plasticity rules follow from first principles. We investigate two different rules: 1.) Maximizing a measure of irreversibility [2] by gradient descent with respect to the weights, we obtain a local learning rule sensitive to the derivative of the covariance between the pre- and postsynaptic neuron. The obtained rule can be interpreted as spike-timing dependent plasticity (STDP) [5,6] with a narrow antisymmetric learning window. We show that the learning rule increases synaptic weights in the direction of the (direct or indirect) causal influence between a pair of neurons. In this way indirect causal relationships are transformed into strengthened direct connections. An instability of the state with vanishing weights manifests itself as spontaneous symmetry breaking and the divergence of connection weights. 2.) We investigate a rule that is sensitive to the zero-time-lag covariance. It can be considered as a complement to the first rule approximating STDP with a symmetric narrow learning window. This rule results in the strengthening of loops, of mutual connections between neurons getting common input, and of indirect (also backward) connections between neurons a and b if a influences b. We derive analytical expressions that describe these effects quantitatively. We show how nonlinearities in the neuronal transmission naturally stabilize synaptic weights and how the limited dynamic range of neuronal activity mediates competition between synapses. The work elucidates the tight connection between measures of reversible and irreversible dynamics and the structures resulting from learning rules that optimize either of the two.

New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases

Cubature rules on the triangle have been extensively studied, as they are of great practical interest in numerical analysis. In most cases, the process by which new rules are obtained does not preclude the existence of similar rules with better characteristics. There is therefore clear interest in searching for better cubature rules.Here we present a number of new cubature rules on the triangle, exhibiting full or rotational symmetry, that improve on those available in the literature either in terms of number of points or in terms of quality. These rules were obtained by determining and implementing minimal orthonormal polynomial bases that can express the symmetries of the cubature rules. As shown in specific benchmark examples, this results in significantly better performance of the employed algorithm.

A symmetric Super Bowl stock market predictor model

Krueger and Kennedy (J Fin 45:691–697, 1990) were the first to empirically document the remarkable stock market predictive power of the winner of the Super Bowl. The original model had investors go “long” in the market when the Super Bowl was won by a team from the old NFL, but park their money in T-Bills when the Super Bowl was won by a team from the old AFL—a non-symmetric trading rule. We create a symmetric rule (go “long” in the market when the old NFL wins; go “short” when they lose) and compare its efficacy to the original formulation. The symmetric rule outperforms the original KK specification in the period covered by their study (1967–1988), but performs worse than the original specification (and the naive buy-and-hold strategy) since 1988.

Symmetric majority rules

In the standard arrovian framework and under the assumption that individual preferences and social outcomes are linear orders on the set of alternatives, we suppose that individuals and alternatives have been exogenously partitioned into subcommittees and subclasses, and we study the rules that satisfy suitable symmetries and obey the majority principle. In particular, we provide necessary and sufficient conditions for the existence of reversal symmetric majority rules that are anonymous and neutral with respect to the considered partitions. We also determine a general method for constructing and counting those rules and we explicitly apply it to some simple cases.

On the identification of symmetric quadrature rules for finite element methods

In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad v1.0 we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.

Computation of moderate-degree fully-symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving

A novel method is presented for expressing the moment equations involved in computing fully symmetric cubature rules on the triangle, by using symmetric polynomials to represent the two kinds of invariance inherent in these rules. This method results in a system of polynomial equations that is amenable to solution using algebraic solving techniques; using Gröbner bases, rules of degree up to 15 are computed and presented, some of them new and with all their points inside the triangle.Since all solutions to the polynomial system are computed, it is for the first time possible to prove whether a given rule type results in specific rules of a given quality; it is thus proved that for degrees up to 14 there are no non-fortuitous rules that can improve on the presented results. For degree 10, an example is also provided showing how the proposed method can be used to exclude the existence of better fortuitous rules as well.

A modified two-lane traffic model considering drivers’ personality

Based on the two-lane traffic model proposed by Chowdhury et al., a modified traffic model (R-STCA model, for short) is presented, in which the new symmetric lane changing rules are introduced by considering driving behavioral difference and dynamic headway. After the numerical simulation, a broad scattering of simulated points is exhibited in the moderate density region on the flow-density plane. The synchronized flow phase accompanied with the wide moving jam phase is reproduced. The spatial–temporal profiles indicate that the vehicles move according to the R-STCA model can change lane more easily and more realistically. Then vehicles are convenient to get rid of the slow vehicles that turn into plugs ahead, and hence the capacity increases. Furthermore the phenomenon of the high speed car-following is discovered by using the R-STCA model, which has been already observed in the traffic measured data. All these results indicate that the presented model is reasonable and more realistic.

Polymer-ceramic composite filler selection using mixing rules

In this work, classical Bruggeman symmetric and Looyenga mixing rules were used together with a general mixing model to estimate the permittivities and loss tangents of polymer composites in order to make rational filler selections. Eight different fillers were used and compared with theoretical polymer composites. Permittivity levels of the composites were between 2.1 and 70 in the frequency range of 2.5−8 GHz. Results showed that rational selections of fillers also provided better properties of composites when using low permittivity instead of high permittivity filler. However, in certain permittivity targets, better properties were achieved with high permittivity filler. The selections were based on the required volume ratios of fillers and total loss tangents of the composites. Theoretical results correlated well with previously reported values.

An indirectly coupled open-ended resonator applied to characterize dielectric properties of MgTiO3–CaTiO3 powders

The effect of CaTiO3 addition on the complex permittivity of MgTiO3 powder was characterized with an open-ended coaxial cavity resonator in the range of 2.12–3.66 GHz. Permittivities and loss tangents of (1 − x)MgTiO3-xCaTiO3 composite powders with x of 0, 2, 5, and 10 mol. % were measured and compared to theoretical values. Inclusion permittivities and dielectric loss tangents were determined by using Bruggeman symmetric and Looyenga mixing rules and a general mixing model. Additions of CaTiO3 resulted in a clear increase in inclusion permittivities from 13.4 up to 14.9 and in loss tangents from 7.1 × 10−3 up to 8.5 × 10−3. Comparison with the theoretical loss tangent values and quantitative determination of CaTiO3 molar ratios by using measured loss tangents and a general mixing model gave a good correlation. The characterization method was proved to be capable of detecting dielectric changes of MgTiO3–CaTiO3 composite powder and of quantifying the amount of additional substances. This information can be exploited, for example, in the analysis and quality control of different composite powders.