Antisymmetric Solutions Research Articles

Yang-Baxter Equations and Relative Rota-Baxter Operators for Left-Alia Algebras Associated to Invariant Theory

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation.

Thermal transport of flexural phonons in a rectangular plate

The quantum thermal transport of elastic excitations through a two-dimensional elastic waveguide between two thermal reservoirs is studied. We solve the classical Kirchhoff–Love equation for rectangular plates and explore the dispersion relation for both the symmetric and antisymmetric solutions. Then, we study the phonon transport of these modes within the second quantization framework by analyzing the mean quadratic displacement, from which the energy density current, the temperature field, and conductance are determined. We identify the relevant modes contributing to thermal transport and explore the average temperature difference to reach the high-temperature limit. We expect our results to pave the way for understanding phonon-mediated thermal transport in two-dimensional mesoscopic quantum devices.

The Anti-symmetric Solution of the Matrix Equation $$AXA^\top =B$$ on a Null Subspace

The Anti-symmetric Solution of the Matrix Equation $$AXA^\top =B$$ on a Null Subspace

Relative Poisson bialgebras and Frobenius Jacobi algebras

Jacobi algebras, as the algebraic counterparts of Jacobi manifolds, are exactly the unital relative Poisson algebras. The direct approach of constructing Frobenius Jacobi algebras in terms of Manin triples is not available due to the existence of the units, and hence alternatively we replace it by studying Manin triples of relative Poisson algebras. Such structures are equivalent to certain bialgebra structures, namely, relative Poisson bialgebras. The study of coboundary cases leads to the introduction of the relative Poisson Yang–Baxter equation (RPYBE). Antisymmetric solutions of the RPYBE give coboundary relative Poisson bialgebras. The notions of \mathcal{O} -operators of relative Poisson algebras and relative pre-Poisson algebras are introduced to give antisymmetric solutions of the RPYBE. A direct application is that relative Poisson bialgebras can be used to construct Frobenius Jacobi algebras, and in particular, there is a construction of Frobenius Jacobi algebras from relative pre-Poisson algebras.

Nonexistence of anti-symmetric solutions for fractional Hardy–Hénon system

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ for the following fractional Hardy–Hénon system: \[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), & x\in\mathbb{R}_+^n, \\ u(x)\geq 0, & v(x)\geq 0,\ x\in\mathbb{R}_+^n, \end{array}\right. \]where $0< s_1,s_2<1$, $n>2\max \{s_1,s_2\}$. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$ under some corresponding assumptions of $\alpha,\beta$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$, one of our results shows that one domain of $(p,q)$, where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropriate condition of $\alpha, \beta$.

Internal Antisymmetric Lamb Waves

A class of antisymmetric solutions of Lamb wave equations with zero strains and stresses on the surface, so-called internal Lamb waves, is studied. Two types of such solutions are found: the first corresponds to the Lamé phase velocity; the second, to phase velocities exceeding the velocity of the expansion wave in an unbounded medium. It is proved that internal waves with the same phase velocity form series, while the frequencies of members of one series are multiples of the frequency of the first member of the series. The same is true for wavenumbers. The profiles of deformed plates and distributions of the maximum stress and shear values are presented.

Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras

We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of O-operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras and the set of coherent derivations on an ASI bialgebra composes a Lie algebra which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.

Experimental and numerical investigation of bistability in rotating permanent magnet-generated electrolyte flow in a ring-shaped container

Here, the stability of a transversely magnetized rotating permanent magnet-generated flow in a concentric cylindrical ring channel is studied. Numerical calculations show that the steady-state solution becomes asymmetric through a pitchfork bifurcation at a Reynolds number (Re) of 60. The two new antisymmetric steady-state solutions become cyclic at Re = 90. Nonlinearities develop at larger Re values and the limit cycle solutions are destabilized at Re = 250, enabling random transition events between the two pitchfork branches. Such transitions have been observed in all kinds of natural phenomena, spanning from neuroscientific to astrophysical systems, which are often too complex to be directly computed. Our presented system is physical yet simple enough to be used to conduct a parametric study with full three-dimensional direct numerical simulations. It raises the possibility of numerically and experimentally analyzing transitions in more detail. Experimental measurements indicated the existence of long-lived states and suitability for the proposed system for future studies of such phenomenon. However, the experimental results did not conclusively observe bistability.

Nonexistence of anti-symmetric solutions for an elliptic system involving fractional Laplacian

ABSTRACT In this paper, we are concerned with the anti-symmetric solutions to the following elliptic system involving fractional Laplacian { ( − Δ ) s u ( x ) = u m 1 ( x ) v n 1 ( x ) , u ( x ) ≥ 0 , x ∈ R + n , ( − Δ ) s v ( x ) = u m 2 ( x ) v n 2 ( x ) , v ( x ) ≥ 0 , x ∈ R + n , u ( x ′ , − x n ) = − u ( x ′ , x n ) , x = ( x ′ , x n ) ∈ R n , v ( x ′ , − x n ) = − v ( x ′ , x n ) , x = ( x ′ , x n ) ∈ R n , where 0<s<1, 0\\ _{(i=1,2)},n>2s,\\mathbb {R}_{+}^{n} =\\{(x^{\\prime},x_{n})|x_{n}>0\\} $ ]]> m i , n i > 0 ( i = 1 , 2 ) , n > 2 s , R + n = { ( x ′ , x n ) | x n > 0 } . We first show that the solutions only depend on x n variable by the method of moving planes. Moreover, we can obtain the monotonicity of solutions with respect to x n variable (for the critical and subcritical cases m i + n i ≤ n + 2 s n − 2 s ( i = 1 , 2 ) in the L 2 s space). Furthermore, when m 1 = n 2 = p , n 1 = m 2 = q , in the cases p + q + 2 s ≥ 1 , we obtain a Liouville theorem for the cases p + q ≤ n + 2 s n − 2 s in the L 2 s space. Then, through the doubling lemma, we obtain the singularity estimates of the positive solutions on a bounded domain Ω. Using the anti-symmetric property of the solutions, one can extend the space from L 2 s to L 2 s + 1 , we can still prove the Liouville theorem in the extended space. With the extension, we prove the existence of nontrivial solutions.

Analysis of unsaturated seepage in infinite slopes by means of horizontal ground infiltration models

This paper describes a simple methodology to calculate the two-dimensional seepage across an infinite unsaturated slope using models of one-dimensional infiltration through horizontal ground. The methodology decomposes the seepage across the infinite slope into antisymmetric and symmetric parts, whose respective solutions are combined to calculate the actual flow regime. The antisymmetric solution is trivial and does not even require integration of the governing continuity equation, while the symmetric solution, albeit non-trivial, reduces to the case of one-dimensional flow through horizontal ground, for which solutions already exist. The methodology is generally applicable to the calculation of distinct seepage regimes across unsaturated slopes with different hydraulic properties under both stationary and transient conditions. The paper also defines the gradient of the piezometric head parallel to the slope, which is the Neumann boundary condition to be imposed on slope sections perpendicular to the ground surface. The rigorous definition of this gradient overcomes the need of imposing arbitrary boundary conditions in finite-element models. Finally, the paper demonstrates that all infiltrated water crosses the slope along the shortest path – namely, the path normal to the surface – while the flow parallel to the slope is entirely fed by an upstream source at infinite distance.

Symmetric and antisymmetric vector solitons for the fractional quadric-cubic coupled nonlinear Schrödinger equation

The fractional quadric-cubic coupled nonlinear Schrödinger equation is concerned, and vector symmetric and antisymmetric soliton solutions are obtained by the square operator method. The relationship between the Lévy index and the amplitudes of vector symmetric and antisymmetric solitons is investigated. Two components of vector symmetric and antisymmetric solitons show a positive and negative trend with the Lévy index, respectively. The stability intervals of these solitons and the propagation constants corresponding to the maximum and minimum instability growth rates are studied. Results indicate that vector symmetric solitons are more stable and have better interference resistance than vector antisymmetric solitons.

Classification of anti-symmetric solutions to the fractional Lane-Emden system

Classification of anti-symmetric solutions to the fractional Lane-Emden system

A New Approach to Jordan D-Bialgebras via Jordan–Manin Triples

Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.

Classification of anti-symmetric solutions to nonlinear fractional Laplace equations

Classification of anti-symmetric solutions to nonlinear fractional Laplace equations

Symmetric and Anti-Symmetric Solitons of the Fractional Second- and Third-Order Nonlinear Schrödinger EquationSupported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LR20A050001), the National Natural Science Foundation of China (Grant No. 12075210), and the Scientific Research and Developed Fund of Zhejiang A&F University (Grant No. 2021FR0009).

The fractional second- and third-order nonlinear Schrödinger equation is studied, symmetric and antisymmetric soliton solutions are derived, and the influence of the Lévy index on different solitons is analyzed. The stability and stability interval of solitons are discussed. The anti-interference ability of stable solitons to the small disturbance shows a good robustness.

Mode jumping in the lateral buckling of subsea pipelines

Unburied subsea pipelines under high-temperature conditions tend to relieve their axial compressive stress by forming localised lateral buckles. This phenomenon is traditionally studied under the assumption of a specific lateral deflection profile (mode) consisting of a fixed number of lobes. We study lateral thermal buckling as a genuinely localised buckling phenomenon by applying homoclinic (‘flat’) boundary conditions. By not having to assume a particular buckling mode we are in a position to study transitions between these traditional modes in typical loading sequences. For the lateral resistance we take a realistic nonlinear pipe-soil interaction model for partially embedded pipelines. We find that for soils with appreciable breakout resistance, i.e., nonmonotonicity of the lateral resistance characteristic, sudden jumps between modes may occur. We consider both symmetric and antisymmetric solutions. The latter turn out to require much higher temperature differences between pipe and environment for the jumps to be induced. We carry out a parameter study on the effect of various pipe-soil interaction parameters on this mode jumping. Away from the jumps post-buckling solutions are reasonably well described by the traditional modes whose analytical expressions may be used during preliminary design. • Study genuine localised buckling of pipelines under thermal loading by imposing homoclinic boundary conditions. • Demonstrate possibility of sudden jumps in post-buckling behaviour. • Jumps can be interpreted as jumps between classical assumed modes. • Soil breakout resistance of the soil plays an important role in mode jumping. • Antisymmetric solutions require much higher temperatures than symmetric solutions for mode-jumping to be induced.

On antisymmetric infinitesimal conformal bialgebras

In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as C [ ∂ ] -modules such that both of them are subalgebras and the natural conformal bilinear form is invariant. The coboundary case leads to the introduction of associative conformal Yang-Baxter equation whose antisymmetric solutions give antisymmetric infinitesimal conformal bialgebras. Moreover, the construction of antisymmetric solutions of associative conformal Yang-Baxter equation is given from O -operators of associative conformal algebras as well as dendriform conformal algebras.

Viscous vortex layers subject to more general strain and comparison to isotropic turbulence

Viscous vortex layers subject to a more general uniform strain are considered. They include Townsend's steady solution for plane strain (corresponding to a parameter a = 1), in which all the strain in the plane of the layer goes toward vorticity stretching, as well as Migdal's recent steady asymmetric solution for axisymmetric strain (a = 1/2), in which half of the strain goes into vorticity stretching. In addition to considering asymmetric, symmetric, and antisymmetric steady solutions ∀a≥0, it is shown that for a &amp;lt; 1, i.e., anything less than the Townsend case, the vorticity inherently decays in time: only boundary conditions that maintain a supply of vorticity at one or both ends lead to a non-zero steady state. For the super-Townsend case a &amp;gt; 1, steady states have a sheath of opposite sign vorticity. Comparison is made with homogeneous-isotropic turbulence, in which case the average vorticity in the strain eigenframe is layer-like, has wings of opposite vorticity, and the strain configuration is found to be super-Townsend. Only zero-integral perturbations of the a &amp;gt; 1 steady solutions are stable; otherwise, the solution grows. Finally, the appendix shows that the average flow in the strain eigenframe is (apart from an extra term) the Reynolds-averaged Navier–Stokes equation.

Stability of rectangular cantilever plates with high elasticity

The problem of cantilever plate stability has been little studied due to the difficulty of solving the corresponding boundary problem. The known approximate solutions mainly concern only the first critical load. In this paper, stability of an elastic rectangular cantilever plate under the action of uniform pressure applied to its edge opposite to the clamped edge is investigated. Under such conditions, thin canopies of buildings made of new materials can be found at sharp gusts of wind in longitudinal direction. At present, cantilever nanoplates are widely used as key components of sensors to create nanoscale transistors where they are exposed to magnetic fields in the plate plane. The aim of the study is to obtain the critical force spectrum and corresponding forms of supercritical equilibrium. The deflection function is selected as a sum of two hyperbolic trigonometric series with adding special compensating summands to the main symmetric solution for the free terms of the decomposition of the functions in the Fourier series by cosines. The fulfillment of all conditions of the boundary problem leads to an infinite homogeneous system of linear algebraic equations with regard to unknown series coefficients. The task of the study is to create a numerical algorithm that allows finding eigenvalues of the resolving system with high accuracy. The search for critical loads (eigenvalues) giving a nontrivial solution of this system is carried out by brute force search of compressive load value in combination with the method of sequential approximations. For the plates with different side ratios, the spectrum of the first three critical loads is obtained, at which new forms of equilibrium emerge. An antisymmetric solution is obtained and studied. 3D images of the corresponding forms are presented.

Classical tools for antipodal identification in Reissner–Nordström spacetime

We extend the discussion of the antipodal identification of black holes to the Reissner–Nordström (RN) spacetime by developing the classical tools necessary to define the corresponding quantum field theory (QFT). We solve the massless Klein–Gordon equation in the RN background in terms of scattering coefficients and provide a procedure for constructing a solution for an arbitrary analytic extension of RN. The behavior of the maximally extended solution is highly dependent upon the coefficients of scattering between the inner and outer horizons, so we present the low-frequency behavior of, and numerical solutions for, these quantities. We find that, for low enough frequency, field amplitudes of solutions with purely positive or negative frequency at each horizon will acquire only a phase after passing both the inner and outer horizons, while at higher frequencies the amplitudes will tend to grow exponentially either to the future or to the past, and decay exponentially in the other direction. Regardless, we can always construct a basis of globally antipodal symmetric and antisymmetric solutions for any finite analytic extension of RN. We have characterized this basis in terms of positive and negative frequency solutions for future use in constructing the corresponding QFT.