Nontrivial Deformations Research Articles

Hall Conductivity in the Cosmic Defect and Dislocation Spacetime**Supported by the China Scholarship Council under Grant No 201207010002, the Hanjiang Scholar Project of Shaanxi University of Technology, the National Natural Science Foundation of China under Grant No 11147181, the Scientific Research Project of Shaanxi Province under Grant Nos 2009K01-54 and 12JK0960, the Startup Foundation of the University of Science and

Influences of topological defect and dislocation on conductivity behavior of charge carriers in external electromagnetic fields are studied. Particularly the quantum Hall effect is investigated in detail. It is found that the nontrivial deformations of spacetime due to topological defect and dislocation produce an electric current at the leading order of perturbation theory. This current then induces a deformation on the Hall conductivity. The corrections on the Hall conductivity depend on the external electric fields, the size of the sample and the momentum of the particle.

Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.

Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

A spacelike surface S immersed in \(\mathbb {S}^4_1\) is marginally trapped if its mean curvature vector is everywhere lightlike. On any oriented spacelike surface S immersed in \(\mathbb {S}^4_1\) we show that a choice of orientation of the normal bundle \(\nu (S)\) determines a smooth map \(G: S \rightarrow \mathbb {S}^3\) which we call the null Gauss map of S. If S is marginally trapped we show that G is a conformal immersion away from the zeros of certain quadratic Hopf-differential of S and so the surface G(S) is uniquely determined up to conformal transformations of \(\mathbb {S}^3\) by two invariants: the normal Hopf differential \(\kappa \) and the schwartzian derivative s. These invariants plus an additional quadratic differential \(\delta \) are related by a differential equation and determine the geometry of S up to ambient isometries of \(\mathbb {S}^4_1\). This allows us to obtain a characterization of marginally trapped surfaces S whose null Gauss image is a constrained Willmore surface in \(\mathbb {S}^3\) in the sense of Bohle et al. (Calc Var Partial Differ Equ 32:263–277, 2008). As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Application of mixed finite elements to spatially non-local model of inelastic deformations

Rock behaviour frequently does not fit the classical theory of continuum mechanics because of rock aggregated granular structure. Particularly, rock fracturing may be accompanied by zonal disintegration formation. The key to building the non-classic model of rock fracturing is the granulated structure. Deformations of solid bodies with microscopic flaws can be described within the scope of non-Euclidean geometry, and non-trivial deformation incompatibility can be referred to as a fracture parameter. The non-Euclidean continuum model used in this paper enables the prediction of the zones initializing and developing as a periodic structure. The non-Euclidean description of phenomenon initiates an appearance of two new material constants. The coupled model must comprise the fourth-order parabolic equation on disintegration thermodynamic parameter be solved with the classical hyperbolic system of equations for the dynamics of continuous media. In this paper, the mixed finite element method is applied to approximate the equations and to model the zonal disintegration phenomenon numerically. The 2D model problem of disintegration zone formation was solved numerically. The zone magnitude and site that can be described by the term ‘disintegration scale’ are determined by values of new constants. Therefore, the numerical model based on the new non-Euclidean continuum model is capable of predicting formation of a disintegration field periodic structure. The second spatial direction of disintegration parameter field propagation is ascertained that allows the model to be applied to various problems of fracture mechanics of rocks.

Algebraically rigid simplicial complexes and graphs

We call a simplicial complex algebraically rigid if its Stanley–Reisner ring admits no nontrivial infinitesimal deformations, and call it inseparable if it does not allow any deformation to other simplicial complexes. Algebraically rigid simplicial complexes are inseparable. In this paper we study inseparability and rigidity of Stanley–Reisner rings, and apply the general theory to letterplace ideals as well as to edge ideals of graphs. Classes of algebraically rigid simplicial complexes and graphs are identified.

Symmetry-forced rigidity of frameworks on surfaces

A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordan et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space—as well as for other types of surfaces—we provide some observations and conjectures.

Anti-self-dual orbifolds with cyclic quotient singularities

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank–Singer scalar-flat Kähler toric ALE spaces. A corollary of this is that, except for the Eguchi–Hanson metric, all of these spaces admit non-toric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-Kähler metric on any weighted projective space \mathbb {CP}^2_{(r,q,p)} for relatively prime integers 1 < r < q < p . A corollary of this is that, while these metrics are rigid as Bochner–Kähler metrics, infinitely many of these admit non-trivial self-dual deformations, yielding a large class of new examples of self-dual orbifold metrics on certain weighted projective spaces.

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

Integrability of S-deformable surfaces: Conservation laws, Hamiltonian structures and more

We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the shape operator).

The influence of graded inhomogeneity on the thermomagnetic properties of bismuth-antimony alloys

We have studied the influence of a longitudinal graded inhomogeneity in crystals of bismuth-antimony solid solutions on their thermomagnetic properties. It is established that the presence of a temperature and/or composition gradient leads to (i) significant and nontrivial deformation of the magnetothermoelectric potential field under the action of a transverse magnetic field and (ii) the appearance of a stationary vortex electric current. It is shown that, with a certain combination of the mutual orientation of the temperature gradient and composition inhomogeneity, it is possible to significantly reduce the vortex current.

Atomic oxygen adsorption on 3.125 at.% Ga stabilized δ-Pu (1 1 1) surface

All-electron density functional theory was used to investigate atomic oxygen adsorption on a gallium stabilized δ-plutonium (111) surface. High symmetry on-surface and interstitial adsorption sites, along with local environment (as determined by the absence or presence of gallium) were explored. The calculations comprised full structural relaxations. Spin–orbit-coupling was also taken into account to assess the complexity of absorbate–substrate interactions. We observed that O adsorbate prefers to bind strongly to a gallium deficient environment, with the most stable site being the threefold hollow fcc site and associated chemisorption energy of −5.06eV. The binding energies were least favored when gallium is a nearest neighbor to the O adsorbate, suggesting that the presence of gallium in a plutonium matrix tends to slow down the oxide layer growth. Although the oxygen coordination is the highest in the interstitial sites, the adsorption energy is less favored compared to on-surface adsorption, implying that the diffusion of oxygen from the surface layer into the subsurface layers is an activated process. The adsorption process induced non-trivial deformations of the surface. Additionally, some delocalization of the plutonium 5f and 6d partial electron density of states (PDOS) at the Fermi energy was observed. Further analysis in the PDOS indicated that gallium tends to suppress hybridization between the plutonium 5f and oxygen 2p orbitals, while the 6d orbitals hybridized with oxygen 2p orbitals.

On curves and polygons with the equiangular chord property

Let $C$ be a smooth, convex curve on either the sphere $\mathbb{S}^{2}$, the hyperbolic plane $\mathbb{H}^{2}$ or the Euclidean plane $\mathbb{E}^{2}$, with the following property: there exists $\alpha$, and parameterizations $x(t), y(t)$ of $C$ such that for each $t$, the angle between the chord connecting $x(t)$ to $y(t)$ and $C$ is $\alpha$ at both ends. Assuming that $C$ is not a circle, E. Gutkin completely characterized the angles $\alpha$ for which such a curve exists in the Euclidean case. We study the infinitesimal version of this problem in the context of the other two constant curvature geometries, and in particular we provide a complete characterization of the angles $\alpha$ for which there exists a non-trivial infinitesimal deformation of a circle through such curves with corresponding angle $\alpha$. We also consider a discrete version of this property for Euclidean polygons, and in this case we give a complete description of all non-trivial solutions.

Deformations of Constant Scalar Curvature Sasakian Metrics and K-Stability

Extending the work of G. Sz\'ekelyhidi and T. Br\"onnle to Sasakian manifolds we prove that a small deformation of the complex structure of the cone of a constant scalar curvature Sasakian manifold admits a constant scalar curvature structure if it is K-polystable. This also implies that a small deformation of the complex structure of the cone of a constant scalar curvature structure is K-semistable. As applications we give examples of constant scalar curvature Sasakian manifolds which are deformations of toric examples, and we also show that if a 3-Sasakian manifold admits a non-trivial transversal complex deformation then it admits a non-trivial Sasaki-Einstein deformation.

Hypermultiplet metric and D-instantons

We use the twistorial construction of D-instantons in Calabi-Yau compactifications of type II string theory to compute an explicit expression for the metric on the hypermultiplet moduli space affected by these non-perturbative corrections. In this way we obtain an exact quaternion-Kähler metric which is a non-trivial deformation of the local c-map. In the four-dimensional case corresponding to the universal hypermultiplet, our metric fits the Tod ansatz and provides an exact solution of the continuous Toda equation. We also analyze the fate of the curvature singularity of the perturbative metric by deriving an S-duality invariant equation which determines the singularity hypersurface after inclusion of the D(-1)-instanton effects.

Classical integrable systems and soliton equations related to eleven-vertex R-matrix

In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest case – the 11-vertex R-matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n-particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau–Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is described.

Deformation of modules of weighted densities on the superspace $${\mathbb{R}^{1|N}}$$ R 1 | N

Over the (1,N)-dimensional real superspace, $${N \geqq 3}$$ , we study non-trivial deformations of the natural action of the orthosymplectic Lie superalgebra $${\mathfrak{osp}(N|2)}$$ on the direct sum of the superspaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and prove that any formal deformation is equivalent to its infinitisemal part. Likewise we study the same problem for the Lie superalgebra $${\mathcal{K}(N)}$$ of contact vector fields instead of $${\mathfrak{osp}(N|2)}$$ getting the same results. This work is the simplest generalization of a result by I. Basdouri and M. Ben Ammar [4] and F. Ammar and K. Kammoun [3].

Superpotentials, Calabi–Yau algebras, and PBW deformations

The paper [9] by Bocklandt, Schedler and Wemyss considers path algebras with relations given by the higher derivations of a superpotential, giving a condition for such an algebra to be Calabi–Yau. In particular they show that the algebra C[V]⋊G, for V a finite dimensional C vector space and G a finite subgroup of GL(V), is Morita equivalent to a path algebra with relations given by a superpotential, and is Calabi–Yau for G<SL(V). In this paper we extend these results, giving a condition for a PBW deformation of a Calabi–Yau, Koszul path algebra with relations given by a superpotential to have relations given by a superpotential, and proving these are Calabi–Yau in certain cases.We apply our methods to symplectic reflection algebras, where we show that every symplectic reflection algebra is Morita equivalent to a path algebra whose relations are given by the higher derivations of an inhomogeneous superpotential. In particular we show these are Calabi–Yau regardless of the deformation parameter.Also, for G a finite subgroup of GL2(C) not contained in SL2(C), we consider PBW deformations of a path algebra with relations which is Morita equivalent to C[x,y]⋊G. We show there are no non-trivial PBW deformations when G is a small subgroup.

Systematic study of isovector and isoscalar giant quadrupole resonances in normal and superfluid deformed nuclei

The systematic study of isoscalar (IS) and isovector (IV) giant quadrupole responses (GQR) in normal and superfluid nuclei presented in [G. Scamps and D. Lacroix, Phys. Rev. 88, 044310 (2013)] is extended to the case of axially deformed and triaxial nuclei. The static and dynamical energy density functional based on Skyrme effective interaction are used to study static properties and dynamical response functions over the whole nuclear chart. Among the 749 nuclei that are considered, 301 and 65 are respectively found to be prolate and oblate while 54 do not present any symmetry axis. For these nuclei, the IS- and IV-GQR response functions are systematically obtained. In these nuclei, different aspects related to the interplay between deformation and collective motion are studied. We show that some aspects like the fragmentation of the response induced by deformation effects in axially symmetric and triaxial nuclei can be rather well understood using simple arguments. Besides this simplicity, more complex effects show up like the appearance of non-trivial deformation effects on the collective motion damping or the influence of hexadecapole or higher-orders effects. A specific study is made on the triaxial nuclei where the absence of symmetry axis adds further complexity to the nuclear response. The relative importance of geometric deformation effects and coupling to other vibrational modes are discussed.