Symmetric Case Research Articles

Harmonic price targeting

This paper studies third-degree price discrimination in a classical model of price competition with differentiated products. Firms charge different prices to different consumers, based on their estimate of their price sensitivity. If the market is fully covered and information is symmetric, more accurate information has a pure redistributive effect, leading to higher profits but lower consumer welfare. If the market is not covered, information always benefit firms but the welfare effects are ambiguous. If information is asymmetric, firms benefit from more information, but less so than in the symmetric case, and total welfare depends on the extent of this asymmetry. I conclude that firms have strong incentives to share information about consumer tastes.

Temporally stable rotational coherent states for molecular simulations. II. Symmetric rotor case.

Following our preceding work on spherical and linear rotors [C. Stopera and J. A. Morales, J. Chem. Phys. 152, 134112 (2020)], we reformulate an earlier rotational coherent state (CS) set to obtain a temporally stable (TS) CS set for symmetric rotors. Being TS, the new CSs remain within its own set during dynamics by evolving exclusively through their parameters. The TS CS set is now appropriate to reconstruct quantum rotational properties from classical-mechanics simulations of chemical reactions. Following literature precedents, we enforce temporal stability by incorporating action-angle-related phase factors into two parameters of the original CS set. Proofs and final expressions of the symmetric-rotor CS turn out more intricate than those of its spherical-rotor counterpart. We demonstrate and examine the key properties of the new CS set: continuity, resolution of unity, temporal stability, action identity, minimum uncertainty relationships, and quasi-classical behavior. Regarding the last property, we demonstrate that the body-fixed z-component of the CS angular momentum average evolves exactly as its classical counterpart, and that the x- and y-components display an astonishing analogy with their classical counterparts in terms of functional form, precession angular velocities, amplitudes, and phases. We elucidate some of these properties via computer simulations of a rotating benzene molecule represented with the CS set. We discuss the utilization of this CS set to reconstruct quantum rotational properties of symmetric-rotor molecules from classical-mechanics simulations. The new CS set is appropriate to establish quantum-classical connections for rotational properties in chemical dynamics, statistical mechanics, spectroscopy, nuclear physics, and quantum computing.

On the Average Age-of-Information for Hybrid Multiple Access Protocols

We consider age of information of data transmitted over a single-hop wireless network using a hybrid multiple access (MAC) protocol. We characterize the average age of information (AAoI) for this hybrid MAC using analytical upper and lower bounds as well as an approximation for symmetric and asymmetric scenarios. Simulations show that the upper and lower bounds and the approximation provide a good characterization of the AAoI. We compare the AAoI performance of hybrid MACs with EZMAC and a centralized policy. We observe that the hybrid MAC performance is close to that of the centralized policy in the symmetric case.

Explosion of a Minimum-Mass Neutron Star within Relativistic Hydrodynamics

The relativistic hydrodynamics equations are adapted for the spherically symmetric case and the Lagrangian form. They are used to model the explosive disruption of a minimum-mass neutron star: a key ingredient of the stripping model for short gamma-ray bursts. The shock breakout from the neutron star surface accompanied by the acceleration of matter to ultrarelativistic velocities is studied. A comparison with the results of previously published nonrelativistic calculations is made.

Local available quantum correlations of X states: The symmetric and anti-symmetric cases

Local available quantum correlations (LAQC), as defined by Mundarain et al., are analyzed for two-qubit X states with local Bloch vectors of equal magnitude. Symmetric X-states are invariant under the exchange of subsystems, hence having the same Bloch vector. On the other hand, anti-symmetric X states have Bloch vectors with an equal magnitude but opposite direction. In both cases, we obtain exact analytical expressions for their LAQC quantifier. We present some examples and compare this quantum correlation to concurrence and quantum discord. We have also included Markovian decoherence, with Werner states under amplitude damping decoherence. As is the case for depolarization and phase damping, no sudden death behavior occurs for the LAQC of these states with this quantum channel.

Shift invariance and reflexivity of compressions of multiplication operators

Abstract The properties of being shift invariant and being reflexive or transitive in the case of the space of (asymmetric) truncated Toeplitz operators, and the space of (asymmetric) dual truncated operators are investigated. Most of the results obtained are new even for the symmetric case. A characterization of asymmetric dual truncated Toeplitz operators is also given.

Critical properties of quantum three- and four-state Potts models with boundaries polarized along the transverse field

By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we show that boundaries polarized in the direction of the transverse field lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of the excitation spectra we phenomenologically establish the duality between the transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point. Finally, for completeness, we verify that in the quantum three-state Potts model the "new" boundary conditions dual to the mixed ones can be realized by polarizing edge spins along the transverse field.

Equivariant bifurcation, quadratic equivariants, and symmetry breaking for the standard representation of S k

Motivated by questions originating from the study of a class of shallow student-teacher neural networks, methods are developed for the analysis of spurious minima in classes of gradient equivariant dynamics related to neural networks. In the symmetric case, methods depend on the generic equivariant bifurcation theory of irreducible representations of the symmetric group on k symbols, S k ; in particular, the standard representation of S k . It is shown that spurious minima (non-global local minima) do not arise from spontaneous symmetry breaking but rather through a complex deformation of the landscape geometry that can be encoded by a generic S k -equivariant bifurcation. We describe minimal models for forced symmetry breaking that give a lower bound on the dynamic complexity involved in the creation of spurious minima when there is no symmetry. Results on generic bifurcation when there are quadratic equivariants are also proved; this work extends and clarifies results of Ihrig & Golubitsky and Chossat, Lauterbach & Melbourne on the instability of solutions when there are quadratic equivariants.

A Hierarchical Universal Algorithm for Geometric Objects’ Reflection Symmetry Detection

A new algorithm is presented for detecting the global reflection symmetry of geometric objects. The algorithm works for 2D and 3D objects which may be open or closed and may or may not contain holes. The algorithm accepts a point cloud obtained by sampling the object’s surface at the input. The points are inserted into a uniform grid and so-called boundary cells are identified. The centroid of the boundary cells is determined, and a testing symmetry axis/plane is set through it. In this way, the boundary cells are split into two parts and they are faced with the symmetry estimation function. If the function estimates the symmetric case, the boundary cells are further split until a given threshold is reached or a non-symmetric result is obtained. The new testing axis/plane is then derived and tested by rotation around the centroid. This paper introduces three techniques to accelerate the computation. Competitive results were obtained when the algorithm was compared against the state of the art.

Dynamic behaviors in a Cournot duopoly model with knowledge spillover effect based on constant conjectural variation

On the basis of constant conjectural variation, a static Cournot duopoly model is built first in this paper, where the effects of knowledge spillover and product differentiation have been parameterized. Through the first order conditions, we can obtain the static conjectural variation equilibrium. By using the gradient adjustment mechanism, a dynamic conjectural variation model, represented by a two-dimensional difference equation, is established, in which two firms have bounded rationality. Then, the local stability of the Nash equilibrium is analyzed. From further studies on the effects of conjectural variation to the dynamics of the established model, we conclude that the more intense the market competition, the bigger the size of the stability region. In general case, the two axes are invariant sets, while the diagonal line is also an invariant set in symmetric case. Through the method of numerical simulation, we can deeply understand the complex characteristics of dynamic behaviors and the means for economic activities. Finally, synchronized behaviors are also studied by the method of critical curves and basins of attraction.

The effect of hydrogen enrichment, flame-flame interaction, confinement, and asymmetry on the acoustic response of a model can combustor

To maximise power density practical gas turbine combustion systems have several injectors which can lead to complex interactions between flames. However, our knowledge about the effect of flame-flame interactions on the flame response, the essential element to predict the stability of a combustor, is still limited. The present study investigates the effect of hydrogen enrichment, flame-flame interaction, confinement, and asymmetries on the linear and non-linear acoustic response of three premixed flames in a simple can combustor. A parametric study of the linear response, characterised by the flame transfer function (FTF), is performed for swirling and non-swirling flames. Flame-flame interactions were achieved by changing the injector spacing and the level of hydrogen enrichment by power from 10 to 50%. It was found that the latter had the most significant effect on the flame response. Asymmetry effects were investigated by changing one of the flames by using a different bluff-body to alter both the flame shape and flow field. The global flame response showed that the asymmetric cases can be reconstructed using a superposition of the two symmetric cases where all three bluff-bodies and flames are the same. Overall, the linear response characterised by the flame transfer function (FTF) showed that the effect of increasing the level of hydrogen enrichment is more pronounced than the effect of the injector spacing. Increasing hydrogen enrichment results in more compact flames which minimises flame-flame interactions. More compact flames increase the cut-off frequency which can lead to self-excited modes at higher frequencies. Finally, the non-linear response was characterised by measuring the flame describing function (FDF) at a frequency close to a self-excited mode of the combustor for different injector spacings and levels of hydrogen enrichment. It is shown that increasing the hydrogen enrichment leads to higher saturation amplitude whereas the effect of injector spacing has a comparably smaller effect.

Asymptotic efficiency of some nonparametric tests for location on hyperspheres

In the present paper, we show that several classical nonparametric tests for multivariate location in the Euclidean case can be adapted to nonparametric tests for the location problem on hyperspheres. The tests we consider are spatial signed-rank tests for location on hyperspheres. We compute the asymptotic powers of the latter tests in the classical rotationally symmetric case. In particular, we show that the spatial signed-rank test uniformly dominates the spatial sign test and has performances that are extremely close to the asymptotically optimal test in the well-known von Mises–Fisher case. Monte-Carlo simulations confirm our asymptotic results.

Dynamic disorder in a coupled two-lane exclusion process with Langmuir kinetics

In intracellular transport, the motor proteins may jump between the filaments due to structural deformity where the motion is hindered by presence of bound proteins. This motivates us to study a system of two parallel disordered exclusion processes with Langmuir kinetics under symmetric and asymmetric coupling conditions. Theoretically, continuum mean-field approximation is employed to obtain steady-state equations which are further solved numerically. The steady-state system dynamics are examined for different values of coupling constant [Formula: see text], which indicates ratio of lane-changing rates. The effect of binding constant [Formula: see text] and defect parameters is also explored. Under the symmetric coupling case, the topological structure of phase diagram turns out to be similar to that in single-lane model, which differs significantly under fully asymmetric coupling. The asymmetric coupling increases magnitude of difference between densities in two lanes. At a very large attachment/detachment rate, the LK dynamics dominate the effect of asymmetric coupling in the lanes. The defect parameters are found to reduce particle flux in the steady state. The numerical results are validated through extensive Monte Carlo simulation.

Heat transfer of couple stress fluid from a vertical funnel: Impedance boundary conditions

AbstractThe present level of literature on the subject matter indicates that nothing is known on the heat transfer across the couple stress rheological fluid flowing over a vertical avenue with Robin (mixed) wall conditions. The obtained conservation equations of the model are solved through DTM (differential transform method) and RPM (regular perturbation method). The nondimensional parameters obtained are a couple stress parameter, Brinkman number, mixed convection parameter, and Biot number. The computations reveal that flow acceleration and thermal enhancement is induced with increasing mixed convection parameter and Brinkman number for symmetric and asymmetric conditions. Increasing couple stress parameters dwindle the velocity and temperature for symmetric and asymmetric cases. The large values of the mixed convection parameter and Brinkman number increase the Nusselt values at the left wall and reduces at the right wall. The mass flow rate is augmented with the mixed convection parameter and Brinkman number but it is reduced with the couple stress parameter. The DTM, RKSM, and RPM solutions are in good agreement.

On the non-symmetric semidefinite Procrustes problem

In this paper, we consider the non-symmetric positive semidefinite Procrustes (NSPSDP) problem: Given two matrices X,B∈Rn,m, find the matrix A∈Rn,n that minimizes the Frobenius norm of AX−B and which is such that A+AT is positive semidefinite. We generalize the semi-analytical approach for the symmetric positive semidefinite Procrustes problem, where A is required to be positive semidefinite, that was proposed by Gillis and Sharma (A semi-analytical approach for the positive semidefinite Procrustes problem, Linear Algebra Appl. 540, 112-137, 2018). As for the symmetric case, we first show that the NSPSDP problem can be reduced to a smaller NSPSDP problem that always has a unique solution and where the matrix X is diagonal and has full rank. Then, an efficient semi-analytical algorithm to solve the NSPSDP problem is proposed, solving the smaller and well-posed problem with a fast gradient method which guarantees a linear rate of convergence. This algorithm is also applicable to solve the complex NSPSDP problem, where X,B∈Cn,m, as we show that the complex NSPSDP problem can be written as an overparametrized real NSPSDP problem. The efficiency of the proposed algorithm is illustrated on several numerical examples.

Study on the magneto-Rayleigh–Taylor instability of inverse exploding planar foils

Experiments of the explosion of thin planar foils have been carried on the QiangGuang-I facility (∼1.4 MA peak current, ∼110 ns rise time) to study the evolution of the plasma instability. An inverse-field configuration was utilized with the current return post placed in the center and two parallel planar aluminum foils of 20 μm thickness, 1 cm width built on each side. The foil was ablated into a plasma slab with a particular width, and the inner surface and outer surface expanded in opposite directions and suffered from unequal magnetic pressure, resulting in divergent instability evolution. To alter plasma acceleration, multiple load configurations, including symmetric and asymmetric cases, were used to change the distance between the foil and back-post. The diagnostic system was fielded to provide a side-on view of exploding foils, including laser shadowgraphy, laser interferometry, and an optical framing camera. The characteristic structure, wavelength spectrum, and growth rate of the instability were compared for both sides. During the early magnetohydrodynamics instability developing stage, the inner side had a similar feature to the outer side surface, but with a larger ratio of the long wavelength in the spectrum. The anti-correlated relationship between the correlation coefficients of two surfaces appeared to be a kink instability mode. When the interface was subjected to deceleration, a transition from an early instability mode to a Magneto-Rayleigh–Taylor (MRT) instability mode was observed in studies, with the growth rate and characteristic wavelength growing fast. Because of the higher magnetic pressure and earlier retardation, the inner edge was more severely affected by MRT instability than the outer edge. A theoretical model was developed to interpret the experimental data.

A novel difference and derivative of linearly correlated fuzzy number-valued functions

In the present paper, the notion of the linearly correlated difference for linearly correlated fuzzy numbers is introduced. Especially, the linearly correlated difference and the generalized Hukuhara difference are coincident for interval numbers or even symmetric fuzzy numbers. Accordingly, an appropriate metric is induced by using the norm and the linearly correlated difference in the set of linearly correlated fuzzy numbers. Based on the symmetry of the basic fuzzy number, the linearly correlated derivative is proposed by the linearly correlated difference of linearly correlated fuzzy number-valued functions. In both non-symmetric and symmetric cases, the equivalent characterizations of the linearly correlated differentiability of a linearly correlated fuzzy number-valued function are established, respectively. Moreover, it is shown that the linearly correlated derivative is consistent with the generalized Hukuhara derivative for interval-valued functions.

Efficiency at optimal performance: A unified perspective based on coupled autonomous thermal machines.

We show that coupled autonomous thermal machines, in the presence of three heat reservoirs and following a global linear-irreversible description, can have efficiency at maximum power (EMP) which is analogous in form to the EMP of models with two (hot and cold) reservoirs. In particular, the temperature dependence of EMP in the coupled model is via only the ratio of hot and cold temperatures if the intermediate reservoir temperature is expressed as an algebraic mean of these temperatures. Many popular expressions of EMP in the literature can be recovered by making a choice of some standard mean. Further, the universal properties of EMP near equilibrium can be explained in terms of the properties of symmetric means. For the case of broken time-reversal symmetry, a universal second-order coefficient of 6/49 is predicted in the series expansion of EMP, analogous to the 1/8 coefficient in the time-reversal symmetric case.

Duopoly price competition with quality improvement spillover

AbstractWhen competing firms embark and explore a new market, two salient features are often observed. On the one hand, the firms need to improve their quality by accumulating more experience and climbing up the learning curve. On the other hand, their quality may jointly expand the brand awareness of all competing products. In this article, we study a two‐period duopoly price competition where firms can improve their quality based on the accumulated demand (learn‐by‐doing effect) and their potential market size is positively affected by both firms' quality levels (quality spillover effect). In addition, we investigate two pricing schemes, namely, committed pricing and dynamic pricing, and their impact on the equilibrium outcomes. Assuming the two firms are symmetric in every aspect, our main findings include the following. First, we establish the existence and uniqueness of the pure Nash equilibrium for the dynamic game under either pricing scheme, and show that firms always set a low price in the first period to leverage quality improvement. As the quality spillover effect gets stronger, firms tend to raise their first‐period price, leading to a lower individual quality improvement and a non‐monotonic impact on firms' profit. Moreover, we find that committed pricing scheme benefits the duopoly when the spillover effect is strong, otherwise dynamic pricing scheme brings more profits. Finally, we examine two asymmetric cases where the firms are different in certain attributes pertaining to their learning speed and the quality spillover strength. Our analysis shows that the findings in the symmetric case still hold qualitatively. Useful managerial insights are derived from these studies.

Symmetries of conformal correlation functions

A program of wide interest in modern conformal bootstrap studies is to numerically solve general conformal field theories, based on a critical assumption that the dynamics is encoded in the conformal four-point crossing equations and positivity condition. In this paper we propose and verify a novel algebraic property of the crossing equations which provides strong restriction for this program. We show for various types of symmetries $\mathcal{G}$, the crossing equations can be linearly converted into the $SO(N)$ vector crossing equations associated with the $SO(N)\ensuremath{\rightarrow}\mathcal{G}$ branching rules and the transformations satisfy positivity condition. The dynamics constrained by the $\mathcal{G}$-symmetric crossing equations combined with the positivity condition degenerates to the $SO(N)$ symmetric cases, while the non-$SO(N)$ symmetric theories are not directly solvable without introducing the $SO(N)$ symmetry breaking assumptions on the spectrum.