In this article we use immersions for representing the motions of massive composite membranes in background Lorentzian spacetimes. We construct an action for deriving the equation of motions by variation in modern differential geometric terms. By analyzing the top form line bundle structures we investigate and comment on the divisibility of a membrane in this setup. We also give as an example the motion of a membrane with spherical topology in Schwarzschild spacetime.
We find two Lax representations with a non-removable parameter for the 2D Boussinesq equations in vorticity form. Then we construct a local variational Poisson structure (a Hamiltonian operator) and describe the actions of this operator and its inverse on the linear space of the second-order cosymmetries and the Lie algebra of the infinitesimal contact symmetries of the equations. Finally, we employ the Lax representations to find nonlocal shadows of cosymmetries and nonlocal conservation laws.
Dynamical systems are usually described by differential equations and difference equations, and the time-scale theory can unify and expand them, which can not only avoid repeated proofs, but also lay a theoretical foundation for the construction of more complex models, thus further enriching the research content. Therefore, based on the theory of time-scale calculus, the perturbations of symmetries of general holonomic systems and nonholonomic systems in the Hamiltonian framework are first studied in this paper. Firstly, Hamilton principle is extended, and the differential equations of the corresponding systems are obtained. The definition of Noether symmetric transformation and the determining equations of Lie symmetry are proposed, and conserved quantities caused by the two symmetries are obtained. Secondly, based on this, the adiabatic invariants caused by the perturbations of the two symmetries of the systems are further discussed. Finally, examples for both systems are given to illustrate the application of the conclusions, and numerical simulations of the obtained conserved quantities are performed to verify the correctness of the conclusions.
We study quasi-derivations of the deformed Heisenberg-Virasoro algebra. It is shown that every quasi-derivation is a direct sum of a derivation, a 12-derivation, a central linear map, and certain explicit quasi-derivations. Utilizing the obtained results, we also describe all δ-(bi)derivations of the deformed Heisenberg-Virasoro algebra, leading to a classification of the transposed δ-Poisson structures.
In this paper, we extend the notion of tangent vectors at a point of a smooth manifold (also known as derivations) by introducing biderivations on manifolds. This is a geometric counterpart of the algebraic notion of biderivation, which has been extensively studied in the beginning in rings, and subsequently in the context of non-associative algebra. Our construction leads to the definition of a bitangent space at a pair of points. When the construction is carried out on a single manifold and the two points coincide, this object reduces to a contravariant 2-tensor. We then introduce the bidifferential of a pair of smooth maps as a linear map between bitangent spaces, and we describe its associated matrix explicitly. Finally, we illustrate the theory through applications to some partial differential equations. Indeed, we discuss connections with the eikonal equation, a classical problem arising in the study of wave propagation, and elliptic operators.
We explore the properties of the derivative nonlinear Schrödinger-type systems that are associated to irreducible compact Hermitian symmetric spaces. We derive the derivative nonlinear Schrödinger systems using a consistent algebraic framework of Loop group factorization the give a unified construction of Darboux transformations for these derivative nonlinear Schrödinger-type systems.
We extend Newton's problem of minimal resistance to Riemannian surfaces endowed with a warped-product metric. This setting includes the two-dimensional space forms, namely the sphere and the hyperbolic plane. Assuming that the fluid particles flow along radial geodesics, we derive the resistance functional and prove that its smooth extremals are the loxodromes of the surface. Furthermore, we analyze the constrained minimization problem, establishing the absence of strong local minima for smooth extremals, and characterizing their global minimizers.
Motivated by the geometric interpretation of spatially homogeneous cosmological models, we formulate a spacetime closedness theorem directly at the Lorentzian level. A classical space-form classification theorem organizes homogeneous and isotropic spatial geometries, but by itself it does not yield a Lorentzian statement about the ambient spacetime. The main technical step is to derive finite slice-volume from genuinely Lorentzian control hypotheses on the foliation. This is achieved in a strong version, for maximally regular globally hyperbolic (n+1)–spacetimes with a finite-time Big Bang and homogeneous complete spacelike slices, and in a weaker version in which maximal regularity is replaced by time-integrability of the accumulated expansion rate. In both cases, the argument separates an analytic step, deriving finite slice-volume from the Big Bang and temporal control, from a geometric step, upgrading finite volume to compactness by homogeneity and completeness. The theorem is stated in arbitrary spacetime dimension and is accompanied by a Lean 4 formalization of the strong and weak abstract statements.
We consider classical integrable systems associated with K-twisted non-skew-symmetric elliptic r-matrix constructed with the help of general three parametric boundary matrix K(u). For a special two-parametric subfamily of the boundary matrices K(u) we construct variables of separation for all integrable models associated with the considered elliptic r-matrix. The case of the classical elliptic Gaudin-type models in external magnetic field is considered in details.
The construction of triply orthogonal surface systems is classically governed by the Lamé compatibility equations, a coupled nonlinear system whose explicit resolution remains highly nontrivial. In this work, we present an analytical reformulation of this problem based on a scalar parametrization of tangent directions and Frobenius integrability conditions.We show that the classical Lamé-type system can be reduced to a single quadratic algebraic condition coupled with a first-order partial differential equation. This reduction provides a constructive framework for generating orthogonal surface families and enables the explicit recovery of the associated coordinate systems.Furthermore, the proposed formulation yields a complete classification of solutions in terms of existence and multiplicity, distinguishing between non-existence, uniqueness, and infinite families. We establish that the resulting compatibility condition is equivalent to the classical Darboux–Cayley criterion, thereby ensuring consistency with the traditional geometric theory.Finally, the framework allows the explicit construction of triply orthogonal systems associated with non-separable and non-symmetric implicit surfaces, extending beyond the classical families obtainable by symmetry-based methods. The results provide a unified analytical perspective on integrability, curvature structure, and orthogonal coordinate systems.
In this article, we explore thermodynamic processes and their controls using the geometry of jet spaces, instead of contact geometry. The equations of state are now defined by 1-jets of submanifolds of codimension one in the space of intensive quantities, while thermodynamic processes are described by vector fields in this space. This allows us to describe and study high-order phase transitions that occur in both thermodynamic processes and their controls.
In this paper, we establish the dispersive estimates and Strichartz inequalities for solutions to the Schr & ouml;dinger equation associated with the full Laplacian on H-type groups. Our work extends the results of Furioli and Veneruso (2004) [21] on the Heisenberg group. Furthermore, it can be compared with an earlier study by Del Hierro (2005) [15], which focuses on solutions to the Schr & ouml;dinger equation involving the sub-Laplacian. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let E be the graded Frechet space of vector fifields on R-n with coordinate values in the Schwartzian space F-n and supports contained in a small ball of R-n centered at the origin. Let D(R-n) denote the group of smooth diffeomorphisms f of R-n such that f - id(R)n has all derivatives globally bounded to infifinite order. Let Gbe a closed connected subgroup of D(R-n) modeled on E, Xa suitable hyperbolic vector fifield. In this paper we show via Nash-Moser inverse functions theorem that there exists a neighborhood V of the identity in G such that any diffeomorphism from phi o V (neighborhood of phi = exp X in the shift phi o G) embeds in a C-infinity-flow. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We degenerate the solutions to the (KP) equation from the general formulation given in terms of Riemann functions when gaps tend to points to get solutions given in terms of Fredholm determinants. For this we compute the limits of all the terms in the argument of the Riemann theta function and establish a link between Riemann theta functions and Fredholm determinants. So we get effective multi-parametric solutions to the (KPII) equation. We deduce easily solution to the (KPI) equation. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Sheng and Wang show in [13] that there is a one-to-one correspondence between isomorphic classes of nonabelian extensions of associative algebras and homotopy classes of associative 2-algebra homomorphisms. We show in this paper that this correspondence extends to nonabelian extensions of AssDer pairs and homomorphisms of Ass2Der pairs. Various other results of [13] are also carrived over to the derivation setting. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We study MA-positivity, a notion of positivity relevant to a vector bundle version of the complex Monge–Ampère equation introduced in an earlier work, and show that for rank-two holomorphic bundles over complex surfaces, MA-semi-positive solutions of the vector bundle Monge–Ampère equation are also MA-positive. For vector bundles of rank-three and higher, over complex manifolds of dimension greater than one, we show that this positivity-preservation property need not hold for an algebraic solution of the vbMA equation treated as a purely algebraic equation at a given point. This result has ramifications on higher-rank versions of the deformed Hermitian Yang-Mills equation. Finally, we set up a continuity path for certain classes of highly symmetric rank-two vector bundles over complex three-folds and prove a restricted version of positivity preservation which is nevertheless sufficient to prove openness along this continuity path.
Every Lichnerowicz-harmonic map phi: M Sm from a compact Riemannian manifold M into a sphere Sm is shown to either meet (i.e. phi(M) f E =/ 0) or link (i.e. phi: M Sm \ E is not null-homotopic) a codimension two totally geodesic submanifold E C Sm. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In the present paper we consider an ohm-stable 3-diffeomorphism with a solid or thickened surfaced non-trivial basic set. Such basic sets include, for instance, all one-dimensional expanding attractors and those two-dimensional basic sets that are not expanding. We prove that the chain recurrent set of every such a diffeomorphism necessarily contains at least two non-trivial basic sets. Also, we have collected few results on expanding attractors that are generally considered well-known but have not been formulated in the literature in a way that allows for direct citation. This concerns the path connected components of such m-dimensional attractors for n-diffeomorphism 1 <= m < n, as well as the topology of trapping neighborhood of one-dimensional expanding attractor of a 3-diffeomorphism. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, Al training, and similar technologies.
