Extensions of Stern's congruence for Euler numbers

For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $2^{m+10}$ for $m\ge 5$. For $m\ge 5$ we also establish congruences for $U_{k\varphi{(5^m)}+b},\; E_{k\varphi{(5^m)}+b},\; S_{k\varphi{(5^m)}+b}\pmod{5^{m+5}}$ and $S_{k\varphi{(3^m)}+b}\pmod{3^{m+5}},$ where $U_{2n}=E_{2n}^{(3/2)}$, $S_n=E_n^{(2)}$ and $\varphi(n)$ is Euler's function.Comment: 16 page

Blowing Polar Skyrmion Bubbles in Oxide Superlattices

Particle-like topological structures such as skyrmions and vortices have garnered ever-increasing interests due to the rich physical insights and potential broad applications. Here we discover the reversible switching between polar skyrmion bubbles and ordered vortex arrays in ferroelectric superlattices under an electric field, reminiscent of the Plateau-Raleigh instability in fluid mechanics. Electric field phase diagram is constructed, showing wide stability window for the observed polar skyrmions. This study is a demonstration for the computational design of ferroelectric topological structures and field-induced topological phase transitions.Comment: 16 Pages 4 figure

Strong CP Problem, Neutrino Masses and the 750 GeV Diphoton Resonance

We present an $SU(3)^{}_{c}\times SU(2)^{}_{L}\times SU(2)^{}_{R}\times U(1)_{L}^{}\times U(1)_{R}^{}\rightarrow SU(3)^{}_{c}\times SU(2)^{}_{L}\times SU(2)^{}_{R}\times U(1)^{}_{B-L}$ left-right symmetric model with a discrete parity symmetry to realize a universal seesaw scenario. The model can simultaneously solve the strong CP problem without resorting to the unobserved axion and explain the 750 GeV diphoton resonance reported recently by the ATLAS and CMS collaborations at the LHC. Owing to large suppressions in the two-loop induced Dirac mass terms, the Majorana mass matrices of left- and right-handed neutrinos naturally share the same structure. That allows us to quantitatively study the neutrinoless double beta decay induced by the right-handed currents

Quantum particle confined to a thin-layer volume: Non-uniform convergence toward the curved surface

We clearly refine the fundamental framework of the thin-layer quantization procedure, and further develop the procedure by taking the proper terms of degree one in $q_3$ ($q_3$ denotes the curvilinear coordinate variable perpendicular to curved surface) back into the surface quantum equation. The well-known geometric potential and kinetic term are modified by the surface thickness. Applying the developed formalism to a toroidal system obtains the modification for the kinetic term and the modified geometric potential including the influence of the surface thickness.Comment: 9 pages, 3 figure

Algebraic Characterizations of Consensus Problems for Networked Dynamic Systems

In this paper, we study the consensus problem for networked dynamic systems with arbitrary initial states, and present some structural characterization and direct construction of consensus functions. For the consensus problem under similar transformation, we establish some necessary and sufficient conditions by exploiting the structure of consensus functions. Finally, we discuss the consensus problem for dynamic systems under switching by using the common Lyapunov function method

Unsupervised Manifold Clustering of Topological Phononics

Classification of topological phononics is challenging due to the lack of universal topological invariants and the randomness of structure patterns. Here, we show the unsupervised manifold learning for clustering topological phononics without any priori knowledge, neither topological invariants nor supervised trainings, even when systems are imperfect or disordered. This is achieved by exploiting the real-space projection operator about finite phononic lattices to describe the correlation between oscillators. We exemplify the efficient unsupervised manifold clustering in typical phononic systems, including one-dimensional Su-Schrieffer-Heeger-type phononic chain with random couplings, amorphous phononic topological insulators, higher-order phononic topological states and non-Hermitian phononic chain with random dissipations. The results would inspire more efforts on applications of unsupervised machine learning for topological phononic devices and beyond.Comment: 6 pages, 4 figure

Infinitely many small energy solutions of a semilinear Schrodinger equation

In this paper we prove the existence of infinitely many small energy solution of a semilinear Schrodinger equation via the dual form of the generalized fountain theorem. This equation is with periodic potential and concave-convex nonlinearities.Comment: 15 page

A two-level atom coupled to a controllable squeezed vacuum field reservoir

The dissipative and decoherence properties of the two-level atom interacting with the squeezed vacuum field reservoir are investigated based on the nonautonomous master equation of the atomic density matrix in the framework of algebraic dynamics. The nonautonomous master equation is converted into a Schr\"{o}dinger-like equations and its dynamical symmetry is found based on the left and right representations of the relevant algebra. The time-dependent solution and the steady solution are obtained analytically. The asymptotic behavior of the solution is examined and the approach to the equilibrium state is proved. Based on the analytic solution the response of the system to the squeezed vacuum field reservoir is studied numerically.Comment: 8 pages, 6 figure

Effect of stochastic grain heating on cold dense clouds chemistry

The temperatures of dust grains play important roles in the chemical evolution of molecular clouds. Unlike large grains, the temperature fluctuations of small grains induced by photons may be significant. Therefore, if the grain size distribution is included in astrochemical models, the temperatures of small dust grains may not be assumed to be constant. We simulate a full gas-grain reaction network with a set of dust grain radii using the classical MRN grain size distribution and include the temperature fluctuations of small dust grains. Monte Carlo method is used to simulate the real-time dust grain's temperature fluctuations which is caused by the external low energy photons and the internal cosmic ray induced secondary photons. The increase of dust grains radii as ice mantles accumulate on grain surfaces is also included in our models. We found that surface CO$_2$ abundances in models with grain size distribution and temperature fluctuations are more than one order of magnitude larger than those with single grain size. Small amounts of terrestrial complex organic molecules (COMs) can also be formed on small grains due to the temperature spikes induced by external low energy photons. However, cosmic ray induced secondary photons overheat small grains so that surface CO sublime and less radicals are formed on grains surfaces, thus the production of surface CO$_2$ and COMs decreases by about one order of magnitude. The overheating of small grains can be offset by grain growth so that the formation of surface CO$_2$ and COMs becomes more efficient.Comment: 16 pages, 10 figures, accepted for publication in MNRA

Probing Higgs Width and Top Quark Yukawa Coupling from ttˉHt\bar{t}H and ttˉttˉt\bar{t}t\bar{t} Productions

We demonstrate that four top-quark production is a powerful tool to constrain the top Yukawa coupling. The constraint is robust in the sense that it does not rely on the Higgs boson decay. Taking into account the projection of the $t\bar{t}H$ production by the ATLAS collaobration, we obtain a bound on Higgs boson width, $\Gamma_H\leq 3.1~\Gamma_H^{\rm SM}$, at the 14 TeV LHC with an integrated luminosity of $300~{\rm fb}^{-1}$. Increasing the luminosity to $500~{\rm fb}^{-1}$ yields $\Gamma_H\leq 2.1~\Gamma_H^{\rm SM}$