Symmetry Theorem Research Articles

Are almost-symmetries almost linear?

It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.

Lerner Symmetry: A Modern Treatment

Which policies are protectionist and which ones are not? The Lerner Symmetry Theorem establishes that import tariffs and export taxes are equally protectionist. In this paper we provide a modern treatment of this classical result, highlighting the importance of multinational firms, global imbalances, and imperfect competition. Under perfect competition, the result follows from the separability of consumption and production across countries, ruling out tourism and some forms of multinational firms, but not others. Though we do not require trade balance, the role of initial assets is subtle: our result rules out foreign ownership of domestic assets, but does not constrain domestic ownership of foreign assets. Under imperfect competition, our result effectively rules out all multinational firms. We conclude by discussing the implications for border adjustment taxes. (JEL D41, D43, F13, F14, F23)

Perturbation to Lie symmetry and adiabatic invariants for Birkhoffian systems on time scales

Lie symmetry of a dynamical system is the invariance of differential equations of motion under the infinitesimal transformations of a group and it can lead to invariants under certain conditions. Firstly, the Lie symmetry of Birkhoffian system on time scales is studied when there is no disturbance, the determining equations of Lie symmetry are established, and the exact invariants led by the Lie symmetry are given. Secondly, the perturbation to Lie symmetry and adiabatic invariants are studied when the system is subjected to small disturbance, and the determining equations of Lie symmetry of the disturbed system are established, and the condition of the Lie symmetry leading to adiabatic invariants and the form of adiabatic invariants are given. As an application of the results, we give the Lie symmetry theorems of Hamiltonian system on time scales. The results contain the exact invariants and adiabatic invariants of Lie symmetry for the classical continuous systems and discrete systems as their special cases. Two examples are given to illustrate the application of the results.

The Global Macroeconomics of a Trade War. The Eagle Model on the US-China Trade Conflict

We study the global macroeconomic effects of tariffs using a multiregional, general equilibrium model, EAGLE, that we extend by introducing US tariffs against Chinese imports into the US, and subsequently Chinese tariffs against US imports into China, consistent with recent trade policies by the US and the Chinese governments. We abstract from tariffs on goods exported from the euro area, focusing on a US-China trade war. A unilateral tariff from the US against China dampens US exports in line with the Lerner Symmetry theorem but global output contracts. Global output contracts even further after China retaliates. The euro area benefits from this trade war. These European trade diversion benefits are caused by cheaper imports from China and improved competitiveness in the US. As price stickiness in the export sector in each region increases, the negative effects of tariffs in the US and China are mitigated, but the positive effects in the euro area are then also dampened.

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The Global Macroeconomics of a Trade War: The EAGLE Model on the US-China Trade Conflict

We study the global macroeconomic effects of tariffs using a multiregional, general equilibrium model, EAGLE, that we extend by introducing US tariffs against Chinese imports into the US, and subsequently Chinese tariffs against US imports into China, consistent with recent trade policies by the US and the Chinese governments. We abstract from tariffs on goods exported from the euro area, focusing on a US-China trade war. A unilateral tariff from the US against China dampens US exports in line with the Lerner Symmetry theorem but global output contracts. Global output contracts even further after China retaliates. The euro area benefits from this trade war. These European trade diversion benefits are caused by cheaper imports from China and improved competitiveness in the US. As price stickiness in the export sector in each region increases, the negative effects of tariffs in the US and China are mitigated, but the positive effects in the euro area are then also dampened.

Noether’s theorem of Hamiltonian systems with generalized fractional derivative operators

In this paper, we present two “transfer formulas” for generalized fractional derivative operators, and derive a Noether type symmetry theorem of fractional Hamiltonian systems with generalized fractional derivative operators. As a result, we obtain constants of motion that are valid along Hamiltonian extremals for fractional derivatives. This theorem provides an explicit algorithmic way to compute a constants of motion for Hamiltonian systems with generalized fractional derivatives operator admitting a symmetry.

Residual symmetries,nth Bäcklund transformation and interaction solutions for (2+1)-dimensional generalized Broer–Kaup equations

Based on the residual symmetry theorem, the residual symmetry is obtained for the (2+1)-dimensional generalized Broer–Kaup (GBK) equations. The multiple residual symmetries are presented and localized by introducing prolonged systems, and then n th Bäcklund transformation for the GBK system in terms of determinant is derived. By means of the consistent tanh expansion (CTE) method, we get various interaction solutions which describe soliton interacting with other nonlinear waves including multiple resonant soliton wave, error function wave, periodic wave and rational function wave.

A Study of Wind Direction Effects on Sea Surface Specular Scattering for GNSS-R Applications

A modeling study investigating the influence of wind direction on spaceborne global navigation satellite system reflectometry (GNSS-R) near-specular observations of the sea surface is reported. The study first focuses on a purely specular geometry under plane wave incidence, for which it is shown using the theorem of reciprocity and reflection symmetry that up-down wind variations are identically zero. It is also shown that “single-scattering” approximations of rough surface scattering predict no variations with wind direction of any kind for a purely specular geometry, while higher order approximations (such as the second-order small-slope approximation) can predict up/cross wind differences. Examples of these variations are reported and found to be small. Because the delay doppler maps (DDMs) measured in GNSS-R include some nonspecular contributions even for “specular” portions of the DDM, the second part of the study performs an examination of near-specular DDM variations with wind direction using the widely used geometrical optics approximation of surface scattering for a surface described with the non-Gaussian Cox–Munk slope probability density function. Variations with wind direction of the normalized radar cross section (NRCS) mapped onto the surface are examined, and again, it is shown that these variations are small for surface portions contributing to the near-specular portion of the DDM. In addition, it is shown that the dependencies of the bistatic NRCS on wind direction are such that differing portions of the surface “glistening zone” have differing phase shifts in their dependence on wind direction, causing the wind dependencies of the final near-specular DDM to be negligible. The final results of the study suggest that any wind direction dependence in spaceborne GNSS-R should be sought only in portions of the DDM away from the specular region. These results provide information to guide analyses of the wind direction information available in current GNSS-R missions such as TDS-1 and Cyclone Global Navigation Satellite System.

A local limit theorem and loss of rotational symmetry of planar symmetric simple random walk

We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$, where $x$ is thought of as dependent on $n$, in dimensions one and two for $P(S_n=x)$, the probability that symmetric simple random walk $S$ started at the origin is at some point $x$ at time $n$, that are valid for all $x$. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where the random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S_n|)$, $S_n$ is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs.

Noether's theorem of fractional Birkhoffian systems

In this paper, we study Noether type symmetry theorem to fractional Birkhoffian systems with Riemann–Liouville derivatives. This theorem provides an explicit algorithmic way to compute a constant for any Birkhoffian systems admitting a symmetry. Finally, we extend our Noether's theorem to fractional Birkhoffian systems base on Caputo or Riesz derivatives.

Variational calculus with conformable fractional derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

Reflection symmetries of Isolated Self-consistent Stellar Systems

Isolated, steady-state galaxies correspond to equilibrium solutions of the Poisson--Vlasov system. We show that (i) all galaxies with a distribution function depending on energy alone must be spherically symmetric and (ii) all axisymmetric galaxies with a distribution function depending on energy and the angular momentum component parallel to the symmetry axis must also be reflection-symmetric about the plane $z=0$. The former result is Lichtenstein's Theorem, derived here by a method exploiting symmetries of solutions of elliptic partial differential equations, while the latter result is new. These results are subsumed into the Symmetry Theorem, which specifies how the symmetries of the distribution function in configuration or velocity space can control the planes of reflection symmetries of the ensuing stellar system.

A general theory for calculating magnetoacoustic effects in pure metals

A new experimental technique yields high-precision measurements of attenuation and dispersion of ultrasound in pure metals in the presence of a magnetic field, but existing theories have approximations which make it impossible to make accurate comparisons with the data. I have constructed a theory with few approximations to eliminate this problem. In addition, symmetry theorems are derived for the effects resulting from the reversal of the magnetic field direction. Calculations for very pure copper are in very good agreement with experimental data.

A Landau–Ginzburg mirror theorem without concavity

We provide a mirror symmetry theorem in a range of cases where the state-of-the-art techniques relying on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials named after Fan, Jarvis, Ruan, and Witten's quantum singularity theory and viewed as the counterpart of a non-convex Gromov--Witten potential via the physical LG/CY correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus zero. In the non-concave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.

A radial symmetry and Liouville theorem for systems involving fractional Laplacian

We investigate the nonnegative solutions of the system involving the fractional Laplacian: $$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$ where 0 2, fi(u), 1 ≤ i ≤ m, are real-valued nonnegative functions of homogeneous degree pi ≥ 0 and nondecreasing with respect to the independent variables u1, u2,..., um. By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point x0 if pi = (n + 2α)/(n − 2α) for each 1 ≤ i ≤ m; and the only nonnegative solution of this system is u ≡ 0 if 1 < pi < (n + 2α)/(n − 2α) for all 1 ≤ i ≤ m.

Modeling the Antipodal Connectivity Structure of Neural Communities

Recent studies support the theory of the brain being composed of modules and certain nodes establishing connections between the modules [ 1 , 2 , 3 ] . The existence of such connections can only be identified by conducting a detailed investigation with sophisticated tools. Therefore, in this manuscript we provide a new mathematical model to indicate the functional dependency, which supports the idea of information exchange between the neural modules at the highest spatial and hierarchical level of bottom-up processes using EEG (ElectroEncephaloGraphy) [ 4 ] . The developed model is to study the functional dependencies between di erent regions of the cortex is based on the Borsuk-Ulam's antipodal symmetry theorem. It is a mathematical model complemented with an innovative algorithm, called Projection based on Normalized Transformation (PNT), to show the existence of unique neural activity pattern known as the Antipodal Connectivity. For validating of the model, EEG data collected from a total of 50 experiments with the participation of 18 di erent test subjects was used to measure the e ectiveness and accuracy of method. Using the data collected from the subjects in di erent stages (active or resting) of the brain, the Antipodal Hub Neurons (AHNs) were captured and compared to determine the ratio of fluctuation under di erent conditions and whether or not the stimulus has any role in antipodal neural connectivity. Although the preliminary results are not conclusive, we have successfully identified the existence of antipodal behavioral patterns in neural activities.

Symmetry analysis for structural optimization problems involving reliability measure and bi-modulus materials

In the present work, some symmetry theorems for structural optimization problems involving reliability measure and bi-modulus materials are established. In the first part of this paper, two types of symmetric reliability-based structural optimization problems are discussed. It is proved that, for the first type symmetric RBDO problems, the existence of symmetric global optimal solutions can be guaranteed even through the probability density function (PDF) is non-symmetric. While for the second type symmetric RBDO problems, more severe conditions on PDF are required to ensure the existence of symmetric global optimal solutions. In the second part of the paper, the convexity of the compliance of truss structures involving bi-modulus materials is confirmed firstly. Based on this result, it is found that for both deterministic and robust symmetric compliance minimization of truss structures involving bi-modulus materials, symmetric global optima always exist. The theoretical results are also demonstrated through some academic examples.

On the values of logarithmic residues along curves

We consider the germ of a reduced curve, possibly reducible. F.Delgado de la Mata proved that such a curve is Gorenstein if and only if its semigroup of values is symmetrical. We extend here this symmetry property to any fractional ideal of a Gorenstein curve. We then focus on the set of values of the module of logarithmic residues along plane curves, which determines the values of the Jacobian ideal thanks to our symmetry Theorem. Moreover, we give the relation with Kahler differentials, which are used in the analytic classification of plane branches. We also study the behaviour of logarithmic residues in an equisingular deformation of a plane curve.

Tropical mirror symmetry for elliptic curves

AbstractMirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov–Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov–Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.