Non-trivial Extension Research Articles

Dynamics of the Collision of Two Nearly Equal Solitary Waves for the Zakharov–Kuznetsov Equation

AbstractWe study the dynamics of the collision of two solitary waves for the 2 and 3-dimensional Zakharov–Kuznetsov equation, a high-dimensional non-integrable version of the Korteweg-de Vries equation that appears as an asymptotic model in plasma physics. We describe the evolution of the solution behaving as a sum of 2-solitary waves of nearly equal speeds at time $$t=-\infty $$ t = - ∞ up to time $$t=+\infty $$ t = + ∞ . We show that this solution behaves as the sum of two modulated solitary waves and an error term which is small in $$H^1$$ H 1 for all time $$t \in {\mathbb {R}}$$ t ∈ R . Finally, we also prove the stability of this solution for large times around the collision. The proofs are a non-trivial extension of the ones of Martel and Merle for the quartic generalized Korteweg-de Vries equation to higher dimensions. First, despite the non-explicit nature of the solitary wave, we construct an approximate solution in an intrinsic way by canceling the error to the equation only in the natural directions of scaling and translation. Then, to control the difference between a solution and the approximate solution, we use a modified energy functional and a refined modulation estimate in the transverse variable. Moreover, we rely on the Hamiltonian structure of the ODE governing the distance between the waves, which cannot be approximated by explicit solutions, to close the bootstrap estimates on the parameters. We hope that the techniques introduced here are robust and will prove useful in studying the collision phenomena for other focusing non-linear dispersive equations with non-explicit solitary waves.

Some formulas for Bessel functions related to diag(1, −1, −1)-matrices and an intertwining operator

In this paper, we have developed novel proofs of two known formulas for the Meijer integral transform and obtained non-trivial extensions of these formulas (in one of them, the Bessel function is replaced by the Coulomb wave function). Also we have derived a new formula for a series involving product of Whittaker functions and modified Bessel functions of the second kind. Our proofs are based on a group theoretical approach and involves the use of an intertwining operator for two representations of one matrix group.

The Relative Brauer Group of K(1)-Local Spectra

Abstract Using profinite Galois descent, we compute the Brauer group of the $K(1)$-local category relative to Morava E-theory. At odd primes this group is generated by a cyclic algebra formed using any primitive $(p-1)$st root of unity, but at the prime two is a group of order $32$ with nontrivial extensions; we give explicit descriptions of the generators, and consider their images in the Brauer group of $KO$. Along the way, we compute the relative Brauer group of completed $KO$, using the étale locally trivial Brauer group of Antieau, Meier and Stojanoska.

K± → π±a at next-to-leading order in chiral perturbation theory and updated bounds on ALP couplings

The weak decays K± → π±a offer a powerful probe of axion-like particles (ALPs). In this work, we provide a comprehensive analysis of these processes within chiral perturbation theory, extending existing calculations by including complete next-to-leading order (NLO) contributions and isospin-breaking corrections at first order in (md – mu). We show that the consistent incorporation of ALPs in the QCD and weak chiral Lagrangians requires a non-trivial extension of the corresponding operator bases, which we describe in detail. Furthermore, we show that in the presence of an ALP the so-called “weak mass term”, which is unobservable in the Standard Model, is non-redundant already at leading order. We find that NLO corrections associated with flavor-violating ALP couplings modify the leading-order result by a few percent, with negligible uncertainties. NLO corrections proportional to flavor-conserving ALP couplings lead to potentially larger corrections, which, however, are accompanied by sizable uncertainties mainly due to the currently limited knowledge of various low-energy constants. We study how these corrections impact bounds on the ALP couplings, first model independently, and then specializing to the case of an ALP with flavor-universal couplings in the UV. Our findings confirm that the decays K± → π±a provide the strongest particle-physics constraints for ma ≲ 300 MeV. In addition, we point out that these bounds have interesting implications for the ALP couplings to nucleons, which were so far only constrained by astrophysical measurements and non-accelerator experiments.

The extended evolutionary synthesis: An integrated historical and philosophical examination

AbstractAmong biologists and philosophers, there is an ongoing debate over the Modern Synthesis and the Extended Evolutionary Synthesis. Some argue that our current evolutionary biology is in need of (at least) some substantial revision or nontrivial extension, while others maintain that the Modern Synthesis remains the foundational framework for evolutionary biology. It has been widely debated whether the Extended Evolutionary Synthesis provides a more promising framework than the Modern Synthesis. The nature and methodological implications of the Extended Evolutionary Synthesis were also examined. This paper offers an integrated historical and philosophical examination of the debate over the Extended Evolutionary Synthesis. It reviews the development of evolutionary biology of the twentieth century. It argues that there are substantial conceptual and theoretical differences between the Modern Synthesis and the Extended Evolutionary Synthesis, but they are not incommensurable paradigms in the Kuhnian sense. It also argues for a functional approach to the debate over these two frameworks of evolutionary theory.

Matrix recovery from nonconvex regularized least absolute deviations

In this paper, we consider the low-rank matrix recovery problem. We propose the nonconvex regularized least absolute deviations model via ℓ1−αℓ2 (0<α<1) minimization. We establish the theoretical analysis of the proposed model and obtain a stable error estimation. Our result is a nontrivial extension of some previous work. Different from most of the state-of-the-art methods, our method does not need any knowledge of standard deviation or any moment assumption of the noise. Numerical experiments show that our method is effective for many types of noise distributions.

New lower bounds on scattering amplitudes: non-locality constraints

Under reasonable working assumptions including the polynomial boundedness, one proves the well-known Cerulus-Martin lower bound on how fast an elastic scattering amplitude can decrease in the hard-scattering regime. In this paper we consider two non-trivial extensions of the previous bound. (i) We generalize the assumption of polynomial boundedness by allowing amplitudes to exponentially grow for some complex momenta and prove a more general lower bound in the hard-scattering regime. (ii) We prove a new lower bound on elastic scattering amplitudes in the Regge regime, in both cases of polynomial and exponential boundedness. A bound on the Regge trajectory for negative momentum transfer squared is also derived. We discuss the relevance of our results for understanding gravitational scattering at the non-perturbative level and for constraining ultraviolet completions. In particular, we use the new bounds as probes of non-locality in black-hole formation, perturbative string theory, classicalization, Galileons, and infinite-derivative field theories, where both the polynomial boundedness and the Cerulus-Martin bound are violated.

A tensor based varying-coefficient model for multi-modal neuroimaging data analysis.

All neuroimaging modalities have their own strengths and limitations. A current trend is toward interdisciplinary approaches that use multiple imaging methods to overcome limitations of each method in isolation. At the same time neuroimaging data is increasingly being combined with other non-imaging modalities, such as behavioral and genetic data. The data structure of many of these modalities can be expressed as time-varying multidimensional arrays (tensors), collected at different time-points on multiple subjects. Here, we consider a new approach for the study of neural correlates in the presence of tensor-valued brain images and tensor-valued covariates, where both data types are collected over the same set of time points. We propose a time-varying tensor regression model with an inherent structural composition of responses and covariates. Regression coefficients are expressed using the B-spline technique, and the basis function coefficients are estimated using CP-decomposition by minimizing a penalized loss function. We develop a varying-coefficient model for the tensor-valued regression model, where both covariates and responses are modeled as tensors. This development is a non-trivial extension of function-on-function concurrent linear models for complex and large structural data, where the inherent structures are preserved. In addition to the methodological and theoretical development, the efficacy of the proposed method based on both simulated and real data analysis (e.g., the combination of eye-tracking data and functional magnetic resonance imaging (fMRI) data) is also discussed.

Hitting all maximum stable sets in P5-free graphs

We prove that every P5-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where Pt denotes the t-vertex path, and for graphs G,H, we say G is H-free if no induced subgraph of G is isomorphic to H).More generally, let us say a class C of graphs is η-bounded if there exists a function h:N→N such that η(G)≤h(ω(G)) for every graph G∈C, where η(G) denotes smallest cardinality of a hitting set of all maximum stable sets in G, and ω(G) is the clique number of G. Also, C is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial.We introduce η-boundedness inspired by a question of Alon (asking how large η(G) can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,•given a graph G, we have η(H)≤ω(H) for every induced subgraph H of G if and only if G is perfect;•there are graphs G with both η(G) and the girth of G arbitrarily large; and•if C is a hereditary class of graphs which is polynomially η-bounded, then C satisfies the Erdős-Hajnal conjecture. The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all H-free graphs is η-bounded if (and only if) H is a forest. Like χ-boundedness, the case where H is a star is easy to verify, and we prove two non-trivial extensions of this: H-free graphs are η-bounded if (1) H has a vertex incident with all edges of H, or (2) H can be obtained from a star by subdividing at most one edge, exactly once.Unlike χ-boundedness, the case where H is a path is surprisingly hard. Our main result mentioned at the beginning shows that P5-free graphs are η-bounded. The proof is rather involved compared to the classical “Gyárfás path” argument which establishes, for all t, the χ-boundedness of Pt-free graphs. It remains open whether Pt-free graphs are η-bounded for t≥6.It also remains open whether P5-free graphs are polynomially η-bounded, which, if true, would imply the Erdős-Hajnal conjecture for P5-free graphs. But we prove that H-free graphs are polynomially η-bounded if H is a proper induced subgraph of P5. We further generalize the case where H is a 1-regular graph on four vertices, showing that H-free graphs are polynomially η-bounded if H is a forest with no vertex of degree more than one and at most four vertices of degree one.

Multi-Pseudo Peakons in the b-Family Fifth-Order Camassa–Holm Model

The b-family fifth-order Camassa–Holm model is a nontrivial extension of the celebrated Camassa–Holm model. This work investigates single-pseudo and multi-pseudo peakon solutions of this model via analytical calculations and numerical simulations. Some intriguing phenomena of multi-pseudo peakon which do not appear in the classical Camassa–Holm model interactions are observed, such as two-pseudo peakon collapses, three-pseudo peakon resonance, and multi-pseudo peakon inelastic collisions. The present work will inspire further studies on the higher-dimensional integrable Camassa–Holm systems which may have high value in investigating the related higher-dimensional physical problems.

On the modeling of thin moving sound sources using a cutFEM approach

While for vibro-acoustic analysis it is often assumed that the source is stationary, in many situations it is moving. Thus, to accurately determine the acoustic response of these systems, a time-varying system has to be solved. This leads to additional challenges, such as how to deal with the varying geometrical configuration and choosing an appropriate time-stepper. Recently, it was shown that cut finite elements (cutFEM) might be a viable solution, since this approach uses a stationary background mesh and cuts the elements wherever the moving source is located, thus omitting remeshing and mesh morphing steps. In this contribution, the approach is extended to also include the possibility to model thin sources (plates/shells), which are a common cause for noise radiation in structures. This is a non-trivial extension, since an immersed thin source leads to a discontinuity in the pressure response among the cut and a local doubling of the elements to capture the pressure on both sides of the cut. This paper describes how to perform numerical integration on the cut elements, how to apply stabilization, and shows a stable time-integration scheme to calculate the solution. It concludes by showing an example of a moving line source.

Higgs inflation at the pole

We propose a novel possibility for Higgs inflation where the perturbative unitarity below the Planck scale is ensured by construction and the successful predictions for inflation are accommodated. The conformal gravity coupling for the Higgs field leads to the proximity of the effective Planck mass to zero in the Jordan frame during inflation, corresponding to a pole in the Higgs kinetic term in the Einstein frame. Requiring the Higgs potential to vanish at the conformal pole in the effective theory in the Jordan frame, we make a robust prediction of the successful Higgs inflation. For a successful Higgs inflation at the pole, we take the running quartic coupling for the Higgs field to be small enough at the inflation scale, being consistent with the low-energy data, but we need a nontrivial extension of the SM with extra scalar or gauge fields in order to keep the running Higgs quartic coupling small during inflation. Performing the perturbative analysis of reheating with the known couplings of the SM particles to the Higgs boson, we show that a concrete realization of the Higgs pole inflation can be pinned down by the reheating processes with a general equation of state for the Higgs inflaton. We illustrate some extensions of the simple Higgs pole inflation to the general pole expansions, the running Higgs quartic coupling in the Standard Model and its extension with a singlet scalar field, a supergravity embedding of the Higgs pole inflation.

Multistate Reaction Coordinate Model for Charge and Energy Transfer Dynamics in the Condensed Phase.

Constructing multistate model Hamiltonians from all-atom electronic structure calculations and molecular dynamics simulations is crucial for understanding charge and energy transfer dynamics in complex condensed phases. The most popular two-level system model is the spin-boson Hamiltonian, where the nuclear degrees of freedom are represented as shifted normal modes. Recently, we proposed the general multistate nontrivial extension of the spin-boson model, i.e., the multistate harmonic (MSH) model, which is constructed by extending the spatial dimensions of each nuclear mode so as to satisfy the all-atom reorganization energy restrictions for all pairs of electronic states. In this work, we propose the multistate reaction coordinate (MRC) model with a primary reaction coordinate and secondary bath modes as in the Caldeira-Leggett form but in extended spatial dimensions. The MRC model is proven to be equivalent to the MSH model and offers an intuitive physical picture of the nuclear-electronic feedback in nonadiabatic processes such as the inherent trajectory of the reaction coordinate. The reaction coordinate is represented in extended dimensions, carrying the entire reorganization energies and bilinearly coupled to the secondary bath modes. We demonstrate the MRC model construction for photoinduced charge transfer in an organic photovoltaic caroteniod-porphyrin-C60 molecular triad dissolved in tetrahydrofuran as well as excitation energy transfer in a photosynthetic light-harvesting Fenna-Matthews-Olson complex. The MRC model provides an effective and robust platform for investigating quantum dissipative dynamics in complex condensed-phase systems since it allows a consistent description of realistic spectral density, state-dependent system-bath couplings, and heterogeneous environments due to static disorder in reorganization energies.

Nonlinear PDE Models in Semi-relativistic Quantum Physics

Abstract We present the self-consistent Pauli equation, a semi-relativistic model for charged spin- 1 / 2 1/2 particles with self-interaction with the electromagnetic field. The Pauli equation arises as the O ⁢ ( 1 / c ) O(1/c) approximation of the relativistic Dirac equation. The fully relativistic self-consistent model is the Dirac–Maxwell equation where the description of spin and the magnetic field arises naturally. In the non-relativistic setting, the correct self-consistent equation is the Schrödinger–Poisson equation which does not describe spin and the magnetic field and where the self-interaction is with the electric field only. The Schrödinger–Poisson equation also arises as the mean field limit of the 𝑁-body Schrödinger equation with Coulomb interaction. We propose that the Pauli–Poisson equation arises as the mean field limit N → ∞ N\to\infty of the linear 𝑁-body Pauli equation with Coulomb interaction where one has to pay extra attention to the fermionic nature of the Pauli equation. We present the semiclassical limit of the Pauli–Poisson equation by the Wigner method to the Vlasov equation with Lorentz force coupled to the Poisson equation which is also consistent with the hierarchy in 1 / c 1/c of the self-consistent Vlasov equation. This is a non-trivial extension of the groundbreaking works by Lions &amp; Paul and Markowich &amp; Mauser, where we need methods like magnetic Lieb–Thirring estimates.

A mixed integration point (MIP) formulation for hyperelastic Kirchhoff–Love shells for nonlinear static and dynamic analysis

We present a mixed integration point (MIP) formulation for hyperelastic isogeometric Kirchhoff–Love shells. While previous works have proposed mixed integration point schemes for different structural formulations in the context of geometric nonlinearity, we extend this concept to the large strain regime in this paper. The non-trivial extension to the nonlinear dynamic analysis for these materials based on the one-step energy-conserving method is also proposed. We present a general, consistent derivation of the formulation, which is not restricted to Kirchhoff–Love shells, hyperelastic materials, or isogeometric analysis, but can be applied to any structural problem involving geometric and material nonlinearities. Several numerical benchmark examples demonstrate the applicability and efficiency of the method.

Robust Subgroup Multisignature with One-Time Public Keys in Order

Robust subgroup multisignature allows any subgroup of signers from a global set to sign a given message on behalf of the whole group, and the individual signatures should be verified before the combination process, which resists poison signature attacks. An emerging application of robust subgroup multisignatures in blockchain is that a qualified subgroup of a global set of users has reached agreement. In the integrated blockchain and edge computing system, the edge server can naturally act as a combiner in multisignatures and help other end devices produce the final aggregate signature. In this paper, we propose a robust subgroup multisignature with one-time public keys in order that has two advantages for solving the signers ordering problem and one-time public key problem simultaneously. Our scheme is a nontrivial extension of Galindo et al.’s robust subgroup multisignature scheme and can be proven unforgeable, robust and chronological in random oracles. Our scheme can also be suitable for the consortium blockchain by adding a noninteractive zero-knowledge (NIZK) proof system for certifying the one-time public keys.

Kac-Moody symmetry in the light front of gauge theories

We discuss the emergence of a new symmetry generator in a Hamiltonian realisation of four-dimensional gauge theories in the flat space foliated by retarded (advanced) time. It generates an asymptotic symmetry that acts on the asymptotic fields in a way different from the usual large gauge transformations. The improved canonical generators, corresponding to gauge and asymptotic symmetries, form a classical Kac-Moody charge algebra with a non-trivial central extension. In particular, we describe the case of electromagnetism, where the charge algebra is the U(1) current algebra with a level proportional to the coupling constant of the theory, κ = 4π2/e2. We construct bilinear generators yielding Virasoro algebras on the null boundary. We also provide a non-Abelian generalization of the previous symmetries by analysing the evolution of Yang-Mills theory in Bondi coordinates.

Historic and physical wandering domains for wild blender-horseshoes

We present diffeomorphisms of wild blender-horseshoes which belong to closures of two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure supported by saddle periodic orbit. It is a non-trivial extension of Colli–Vargas’ model (Colli and Vargas 2001 Ergod. Theory Dyn. Syst. 21 1657–81) to the higher dimensional dynamics with the use of wild blender-horseshoes.

Superlinear stochastic heat equation on ℝ^{𝕕}

In this paper, we study the stochastic heat equation (SHE) on R d \mathbb {R}^d subject to a centered Gaussian noise that is white in time and colored in space. We establish the existence and uniqueness of the random field solution in the presence of locally Lipschitz drift and diffusion coefficients, which can have certain superlinear growth. This is a nontrivial extension of the recent work by Dalang, Khoshnevisan and Zhang [Ann. Probab. 47 (2019), pp. 519–559], where the one-dimensional SHE on [ 0 , 1 ] [0,1] subject to space-time white noise has been studied.

Topological gauging and double current deformations

We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved U(1) currents Ja. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the S-matrix of the system, similarly as what happens in the case of a textrm{T}overline{textrm{T}} deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents Ja are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the textrm{T}overline{textrm{T}} deformation to the central extension of the two-dimensional Poincaré algebra.