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Despite its high significance in nonlinear elasticity, the neo-Hookean energy is still not known to admit minimizers in some appropriate admissible class. Using ideas from relaxation theory, we propose a larger minimization space and a modified functional that coincides with the neo-Hookean energy on the original space. This modified energy is the sum of the neo-Hookean energy and a term penalizing the singularities of the inverse deformation. The new functional attains its minimum in the larger space, so the initial question of existence of minimizers of the neo-Hookean energy is thus transformed into a question of regularity of minimizers of this new energy. [ABSTRACT FROM AUTHOR]

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$. [ABSTRACT FROM AUTHOR]

Plastoelastohydrodynamic lubrication of rough surfaces (R-PEHL) is a cutting-edge area of research in interface fluid-structure coupling analysis. The existing R-PEHL model calculates the elastic-plastic deformation of rough surface by the Love equation in a semi-infinite space smooth surface, which deviates from the actual surface. Therefore, it is an innovative work to study the exact solution of elastic-plastic deformation of rough surface and its influence on the solution results of R-PEHL model. In this paper, a new contact calculation model of plastoelastohydrodynamic lubrication (PEHL) with three-dimensional (3D) rough surface is proposed by integrating numerical method of EHL and finite element method. The new model eliminates an original error introduced by the assumption of semi-infinite space in contact calculation, providing wide applicability and high accuracy. Under the given rough surfaces and working conditions, the study reveals that: (1) the oil film pressure calculated by the new model is lower than that of the smooth surface in semi-infinite space by 200–800 MPa; (2) the Mises stress of the new model is 2.5%–26.6% higher than that of the smooth surface in semi-infinite space; (3) compared with the semi-infinite space assumption, the rough surface plastic deformation of the new model is increased by 71%–173%, and the local plastic deformation singularity may appear under the semi-infinite space assumption; (4) the plastic deformation caused by the first contact cycle on the rough surface of the new model accounts for 66.7%–92.9% of the total plastic deformation, and the plastic deformation of the semi-infinite space accounts for 50%–83.3%. This study resolves the contradiction between the smooth surface assumption and the rough surface in the existing R-PEHL model, establishing a solid logic foundation for the accurate solution of R-PEHL model. [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized isomonodromic deformations of rank-two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank-two irregular connections by apparent singularities. Using the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In this paper, we studied the Picard–Fuchs systems and equations which appear in the theory of Gauss–Manin systems and connections associated with deformations of isolated singularities. Among other things, we describe some interesting properties of such systems and relationships between them. Then we show how to calculate the fundamental solutions to the Gauss–Manin system for A μ -singularities and to the corresponding generalized Legendre equations in terms of the multidimensional Horn's hypergeometric functions. In conclusion, some important questions concerning basic properties of the local and global Picard–Fuchs systems of Pfaffian type, involving integrability conditions and commuting relations, are discussed in some detail. [ABSTRACT FROM AUTHOR]

This paper is a continuation of two recent publications on crack growth in viscoelastic media. It provides further theoretical results for large strains that enable prediction of crack opening displacement for comparison with experimental data in the region of the singularity. In order to achieve good agreement with experiment it was necessary to account for far-field viscoelasticity. Additionally, it is found that with large deformation throughout the singularity, the deformation consists of simple shearing and stretching normal to the crack plane. Thus, there is no significant displacement parallel to the crack plane; such simplicity exists for materials that stiffen or soften at high strains if the stress obeys a power law in strain at high strains. This finding means that, despite the frame-dependence of the theory, there is no local rotation in the singularity to affect the stress. [ABSTRACT FROM AUTHOR]

The notion of a Harish-Chandra bimodule, that is, finitely generated |$U(\mathfrak {g})$| -bimodule with locally finite adjoint action, was generalized to any filtered algebra in a work of Losev [ 9 ]. Similarly to the classical case we can define the notion of a unitarizable bimodule. We investigate a question when the regular bimodule, that is, the algebra itself, for a deformation of Kleinian singularity of type |$A$| is unitarizable. We obtain a partial classification of unitarizable regular bimodules. [ABSTRACT FROM AUTHOR]

This paper presents the reverse priority impedance control of manipulators with reference to redundant robots of a given task. The reverse priority kinematic control of redundant manipulators is first expressed in detail. The motion in the joint space is derived following the opposite order compared with the classical task priority–based solution. Then the Cartesian impedance control is combined with the reverse priority impedance control to solve the reverse hierarchical impedance controlled, so that the Cartesian impedance behavior can be divided into the primary priority impedance control and the secondary priority impedance control. Furthermore, the secondary impedance control task will not disturb the primary impedance control task. The motion in the joint space is affected following the opposite order and working in the corresponding projection operators. The primary impedance control tasks are implemented at the end, so as to avoid the possible deformations caused by the singularities occurring in the secondary impedance control tasks. Hence, the proposed reverse priority impedance control of manipulator can achieve the desired impedance control tasks with proper hierarchy. In this paper, the simulation experiments of the manipulator will verify the proposed reverse priority control algorithm. [ABSTRACT FROM AUTHOR]

Let $ {\mathbb {C}}^{n+1}_o$ denote the germ of $ {\mathbb {C}}^{n+1}$ at the origin. Let $V$ be a hypersurface germ in $ {\mathbb {C}}^{n+1}_o$ and $W$ a deformation of $V$ over $ {\mathbb {C}}_{o}^{m}$. Under the hypothesis that $W$ is a Newton non-degenerate deformation, in this article we prove that $W$ is a $\mu$ -constant deformation if and only if $W$ admits a simultaneous embedded resolution. This result gives a lot of information about $W$ , for example, the topological triviality of the family $W$ and the fact that the natural morphism $(\operatorname {W({\mathbb {C}}_{o})}_{m})_{{\rm red}}\rightarrow {\mathbb {C}}_{o}$ is flat, where $\operatorname {W({\mathbb {C}}_{o})}_{m}$ is the relative space of $m$ -jets. On the way to the proof of our main result, we give a complete answer to a question of Arnold on the monotonicity of Newton numbers in the case of convenient Newton polyhedra. [ABSTRACT FROM AUTHOR]

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits. [ABSTRACT FROM AUTHOR]

Compactification of M-theory and of IIB string theory on threefold canonical singularities gives rise to superconformal field theories (SCFTs) in 5d and 4d, respectively. The resolutions and deformations of the singularities encode salient features of the SCFTs and of their moduli spaces. In this paper, we build on Part 0 of this series [1] and further explore the physics of SCFTs arising from isolated hypersurface singularities. We study in detail these canonical isolated hypersurface singularities that admit a smooth Calabi-Yau (crepant) resolution. Their 5d and 4d physics is discussed and their 3d reduction and mirrors (the magnetic quivers) are determined in many cases. As an explorative tool, we provide a Mathematica code which computes key quantities for any canonical isolated hypersurface singularity, including the 5d rank, the 4d Coulomb branch spectrum and central charges, higher-form symmetries in 4d and 5d, and crepant resolutions. [ABSTRACT FROM AUTHOR]

The aims of this study present a three-dimensional finite element method for the analysis of fractures in a mixed mode of two materials the Functionally Graded (FGM) and the composite materials. The modeling was carried out on a hull containing an inclined crack, provided by a fine mesh around these crack points due to singular elements. The structure is examined under the effect of transient thermal stresses. The singularity of the deformations around the crack front is simulated by singular elements. The stress intensity factors of modes I, II are calculated by applying the displacement extrapolation technique (DET) and presented as a function of time. The different types of FGM are examined in parametric analyzes which are rich in metals, rich in ceramics. The results provided illustrate the influences of the inclination angle of the cracks on the transient behavior of the stress intensity factors in mixed mode. The mechanical behavior of this material has been described by the exponential function The numerical evaluation of this factor is determined using the Numerical code. [ABSTRACT FROM AUTHOR]

In constitutive modelling of rubber-like materials, the strain-hardening effect at large deformations has traditionally been captured successfully by non-Gaussian statistical molecular-based models involving the inverse Langevin function, as well as the phenomenological limiting chain extensibility models. A new model proposed by Anssari-Benam and Bucchi (Int. J. Non Linear Mech. 2021; 128; 103626. DOI: 10.1016/j.ijnonlinmec.2020.103626), however, has both a direct molecular structural basis and the functional simplicity of the limiting chain extensibility models. Therefore, this model enjoys the benefits of both approaches: mathematical versatility, structural objectivity of the model parameters, and preserving the physical features of the network deformation such as the singularity point. In this paper we present a systematic approach to constructing the general class of this type of model. It will be shown that the response function of this class of models is defined as the [1/1] rational function of I 1 , the first principal invariant of the Cauchy–Green deformation tensor. It will be further demonstrated that the model by Anssari-Benam and Bucchi is a special case within this class as a rounded [3/2] Padé approximant in λ c (the chain stretch) of the inverse Langevin function. A similar approach for devising a general I 2 term as an adjunct to the I 1 part of the model will also be presented, for applications where the addition of an I 2 term to the strain energy function improves the fits or is otherwise required. It is concluded that compared with the Gent model, which is a [0/1] rational approximation in I 1 and has no direct connection to Padé approximations of any order in λ c , the presented new class of the molecular-based limiting chain extensibility models in general, and the proposed model by Anssari-Benam and Bucchi in specific, are more accurate representations for modelling the strain-hardening behaviour of rubber-like materials in large deformations. [ABSTRACT FROM AUTHOR]

We extend to a degenerate case a result by B. Malgrange on integrable deformations of irregular singularities, inspired by Cotti, Dubrovin and Guzzetti (Local moduli of semisimple Frobenius coalescent structures, SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), paper no. 040). We give an application to integrable deformations of some meromorphic connections in Birkhoff normal form and to the construction of Frobenius manifolds. [ABSTRACT FROM AUTHOR]

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory. [ABSTRACT FROM AUTHOR]

This article is a summary of the author's unpublished Ph.D thesis (Caradot 2017). Its purpose is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of type Ar, Dr, E6, E7 and E8 to the inhomogeneous simple singularities of type Br, Cr, F4 and G2. To a homogeneous simple singularity, one can associate the representation space of a particular quiver. This space is endowed with an action of the symmetry group of the Dynkin diagram associated to the simple singularity. From this we will construct and compute explicitly the semiuniversal deformations of the inhomogeneous simple singularities. By quotienting such maps, we obtain deformations of other simple singularities. In some cases, the discriminants of these last deformations will be computed. [ABSTRACT FROM AUTHOR]

This is a first introduction to unprojection methods, and more specifically to Tom and Jerry unprojections. These two harmless tricks deserve to be better known, since they answer many practical questions about constructing codimension 4 Gorenstein subschemes. In particular, we discuss here the two smoothing components of the anticanonical cone over P(1, 2, 3). [ABSTRACT FROM AUTHOR]

We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums. [ABSTRACT FROM AUTHOR]

In this short note, we prove that there are finitely many indecomposable filtrable bundles with trivial determinant and vanishing second Chern class on Oeljeklaus-Toma manifolds that satisfy certain conditions. With few precise exceptions, they are simple bundles. [ABSTRACT FROM AUTHOR]

We construct a differential graded Lie algebra g controlling the Poisson deformations of an affine Poisson variety. We analyze g in the case of affine Gorenstein toric Poisson varieties. Moreover, explicit description of the second and third Hochschild cohomology groups is given for three-dimensional affine Gorenstein toric varieties. [ABSTRACT FROM AUTHOR]

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney's approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest 2 -component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by (k + 1 , k + 2) curves which are of cusp type for k = 2 n + 1 , n = 1 , 2 , 3 , .... Regularization of these singularities by deformation of plane into plane mappings into surface S 2 + k (⊂ ℝ 2 + k) to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems. [ABSTRACT FROM AUTHOR]

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities and gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings. [ABSTRACT FROM AUTHOR]

The reproducing kernel particle methods (RKPM) are meshfree methods arising in mechanics, especially in dealing with problems involving large deformation and singularities. We provide a theoretical analysis of super-convergence in Sobolev norms for reproducing kernel (RK) approximations when the interpolation order p is even. Super-convergence phenomenon means the convergence rate is higher than the order that is generally expected. We distinguish the continuous RK approximation and the discrete RKP approximation. While the continuous RK approximations are proven to be super-convergent when p is even, its discrete counterpart has super-convergence only with uniform particle distribution and special choices of RK kernel functions and support sizes. Moreover, super-convergence does not exist for the discrete RKP approximation with general RK support sizes. The concept of pseudo-super-convergence is then introduced to explain why in practice the super-convergence phenomenon is sometimes observed for general cases although in theory it is not true. Our analysis is general for multi-dimensional RK approximations. [ABSTRACT FROM AUTHOR]

Abstract We prove that irreducible residual crystalline representations of the absolute Galois group of an unramified extension of Q p have smooth representable crystalline framed deformation problems, provided that the Hodge–Tate weights lie in the Fontaine–Laffaille range. We then extend this result to the flagged lifting problem associated to any Fontaine–Laffaille upper triangular representation whose flag is of maximal length. We calculate the relative dimension of these various crystalline lifting functors in terms of the underlying Hodge–Tate weight structures, and also apply these results to give an alternative proof of the fact that every such residual representation admits a so-called "universally twistable lift". Finally we give some brief indications as to the various directions in which these results might be generalised. [ABSTRACT FROM AUTHOR]

Abstract We describe semiuniversal deformation spaces for the noncompact surfaces Z k : = Tot (O P 1 (− k)) and prove that any nontrivial deformation Z k (τ) of Z k is affine. It is known that the moduli spaces of instantons of charge j on Z k are quasi-projective varieties of dimension 2 j − k − 2. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation Z k (τ) are empty. [ABSTRACT FROM AUTHOR]

Abstract In this article we study quotients of deformations of simple singularities, and attempt to characterise them using subsystems of simple root systems. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type B r (r ≥ 2), C r (r ≥ 3), F 4 or G 2 by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type X = D s , E 6 or E 7 , but not semiuniversal anymore. Therefore not all subdiagrams of X appear as singular configurations of the fibres of the deformation. We propose a conjecture for the types of singular configurations in terms of sub-root systems of a root system of type X and prove it for types B 2 , B 3 , C 3 , F 4 and G 2. [ABSTRACT FROM AUTHOR]

Abstract For each pair of complex symmetric matrices (A , B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (A ˜ , B ˜) , close to (A , B) can be reduced by congruence transformation that smoothly depends on the entries of A ˜ and B ˜. Such a normal form is called a miniversal deformation of (A , B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A , B) to its miniversal deformation. [ABSTRACT FROM AUTHOR]

Three-fold quasi-homogeneous isolated rational singularity is argued to define a four dimensional N = 2 SCFT. The Seiberg–Witten geometry is built on the mini-versal deformation of the singularity. We argue in this paper that the corresponding Seiberg–Witten differential is given by the Gelfand–Leray form of K. Saito's primitive form. Our result also extends the Seiberg–Witten solution to include irrelevant deformations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularity f. Assuming that f is irreducible, one can write f = ∑ q α + p β ≥ p q c α β x α y β where c p 0 c 0 q ≠ 0 , 2 ≤ p < q and p, q are coprime. We show that as Milnor numbers of deformations of f one can attain all numbers from μ (f) to μ (f) - r (p - r) , where q ≡ r (mod p) . Moreover, we provide an algorithm which produces the desired deformations. [ABSTRACT FROM AUTHOR]

An odd deformation of a super Riemann surface S is a deformation of S by variables of odd parity. In this article, we study the obstruction theory of these odd deformations X of S. We view X here as a complex supermanifold in its own right. Our objective in this article is to show, when X is a deformation of second order of S with genus g>1: if the primary obstruction class to splitting X vanishes, then X is in fact split. This result leads naturally to a conjectural characterisation of odd deformations of S of any order. [ABSTRACT FROM AUTHOR]

Abstract Let G be the F -points of a connected reductive group over a non-archimedean local field F of residue characteristic p and R be a commutative ring. Let P = L U be a parabolic subgroup of G and Q be a parabolic subgroup of G containing P. We study the functor St Q G taking a smooth R -representation σ of L which extends to a representation e G (σ) of G trivial on U to the smooth R -representation e G (σ) ⊗ R St Q G (R) of G where St Q G (R) is the generalised Steinberg representation. [ABSTRACT FROM AUTHOR]

Abstract For a representation of the absolute Galois group of the rationals over a finite field of characteristic p , we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine–Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at p arising from Fontaine–Laffaille theory. [ABSTRACT FROM AUTHOR]

Abstract Let G be a real reductive Lie group, and H C the complexification of its maximal compact subgroup H ⊂ G. We consider classes of semistable G -Higgs bundles over a Riemann surface X of genus g ⩾ 2 whose underlying H C -principal bundle is unstable. This allows us to find obstructions to a deformation retract from the moduli space of G -Higgs bundles over X to the moduli space of H C -bundles over X , in contrast with the situation when g = 1 , and to show reducibility of the nilpotent cone of the moduli space of G -Higgs bundles, for G complex. [ABSTRACT FROM AUTHOR]

Abstract A revised Yau's Curvature Difference Flow is considered to deform one convex curve X 0 to another one X ˜. It is proved that this flow exists globally on time interval [ 0 , + ∞) and the evolving curve, preserving its convexity and bounded area A , converges to a fixed limiting curve X ∞ (congruent to A / A ˜ X ˜) as time tends to infinity, where A ˜ is the area bounded by the target curve X ˜. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a family of transverse invariants arising from the deformations of Khovanov homology. This family includes the invariants introduced by Plamenevskaya and by Lipshitz, Ng, and Sarkar. Then, we investigate the invariants arising from Bar-Natan's deformation. These invariants, called β -invariants, are essentially equivalent to Lipshitz, Ng, and Sarkar's invariants ψ ± . From the β -invariants, we extract two non-negative integers which are transverse invariants (the c -invariants). Finally, we give several conditions which imply the non-effectiveness of the c -invariants, and use them to prove several vanishing criteria for the Plamenevskaya invariant [ ψ ] , and the non-effectiveness of the vanishing of [ ψ ] , for all prime knots with less than 12 crossings. [ABSTRACT FROM AUTHOR]

Over the (1 , 2) -dimensional supercircle S 1 | 2 , we investigate the first 𝔬 𝔰 𝔭 (2 | 2) -relative cohomology space associated with the embedding of the Lie superalgebra 𝒦 (2) of contact vector fields in the Lie superalgebra 𝒮 Ψ 𝒟 𝒪 (S 1 | 2) of superpseudodifferential operators with smooth coefficients, where 𝔬 𝔰 𝔭 (2 | 2) is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra 𝔞 𝔣 𝔣 (2 | 1) instead of 𝔬 𝔰 𝔭 (2 | 2). We classify generic formal 𝔬 𝔰 𝔭 (2 | 2) -trivial deformations of the 𝒦 (2) -module structure on the superspace of the supercommutative algebra 𝒮 𝒫 (2) of pseudodifferential symbols on S 1 | 2 . [ABSTRACT FROM AUTHOR]

Over the (1 , 1) -dimensional supercircle S 1 | 1 , we investigate the first 𝔬 𝔰 𝔭 (1 | 2) -relative cohomology space associated with the embedding of the Lie superalgebra 𝒦 (1) of contact vector fields in the Lie superalgebra 𝒮 Ψ 𝒟 𝒪 (S 1 | 1) of superpseudodifferential operators with smooth coefficients, where 𝔬 𝔰 𝔭 (1 | 2) is the orthosymplectic Lie superalgebra. Likewise, we study the same problem for the affine Lie superalgebra 𝔞 𝔣 𝔣 (1 | 1) instead of 𝔬 𝔰 𝔭 (1 | 2). We classify 𝔞 𝔣 𝔣 (1 | 1) -trivial deformations of the standard embedding of the Lie superalgebra 𝒦 (1) into the Lie superalgebra 𝒮 Ψ 𝒟 𝒪 (S 1 | 1). This approach leads to the deformations of the central charge induced on 𝒦 (1) by the canonical central extension of 𝒮 Ψ 𝒟 𝒪 (S 1 | 1). [ABSTRACT FROM AUTHOR]

In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary. [ABSTRACT FROM AUTHOR]

Using homological perturbation theory, we develop a formal version of the miniversal deformation associated with a deformation problem controlled by a differential graded Lie algebra over a field of characteristic zero. Our approach includes a formal version of the Kuranishi method in the theory of deformations of complex manifolds. [ABSTRACT FROM AUTHOR]

Abstract Existence of Maxwell–Chern–Simons–Higgs (MCSH) vortices in a hermitian line bundle L over a general compact Riemann surface Σ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern–Simons deformation parameter κ , and exist for all | κ | < κ ∗ , where κ ∗ depends, in principle, on the geometry of Σ , the degree n of L , which may be interpreted as the vortex number, and the vortex positions. A simple upper bound on κ ∗ , depending only on n and the volume of Σ , is found. Further, it is proved that a positive lower bound on κ ∗ , depending on Σ and n , but independent of vortex positions, exists. A detailed numerical study of rotationally equivariant vortices on round two-spheres is performed. We find that κ ∗ in general does depend on vortex positions, and, for fixed n and radius, tends to be larger the more evenly vortices are distributed between the North and South poles. A generalization of the MCSH model to compact Kähler domains Σ of complex dimension k ≥ 1 is formulated. The Chern–Simons term is replaced by the integral over spacetime of A ∧ F ∧ ω k − 1 , where ω is the Kähler form on Σ. A topological lower bound on energy is found, attained by solutions of a deformed version of the usual vortex equations on Σ. Existence, uniqueness and smoothness of vortex solutions of these generalized equations is proved, for | κ | < κ ∗ , and an upper bound on κ ∗ depending only on the Kähler class of Σ and the first Chern class of L is obtained. [ABSTRACT FROM AUTHOR]

We study several classes of non-associative algebras as possible candidates for deformation quantization in the direction of a Poisson bracket that does not satisfy Jacobi identities. We show that in fact alternative deformation quantization algebras require the Jacobi identities on the Poisson bracket and, under very general assumptions, are associative. At the same time, flexible deformation quantization algebras exist for any Poisson bracket. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic zero, and V a hypersurface defined by an irreducible polynomial f with coefficients in K . In this article we prove that an Embedded Deformation of V which admits a Simultaneous Embedded Resolution induces, under certain conditions, a deformation of the space of m -jets V m , m ≥ 0 . An example of an Embedded Deformation of V which admits a Simultaneous Embedded Resolution is a Γ ( f ) -deformation of V , where V has at most one isolated singularity, and f is non-degenerate with respect to the Newton Boundary Γ ( f ) . [ABSTRACT FROM AUTHOR]

In this paper, first we study infinitesimal deformations of a Lie algebra with a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Lie algebra with a representation. Then we introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a representation, which is a generalization of the r − n structure (r-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam, and Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we define a Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a Lie algebra with respect to the adjoint representation and study the relation between Rota-Baxter-Nijenhuis structures and r-matrix-Nijenhuis structures. [ABSTRACT FROM AUTHOR]

Deformations of compact Riemann surfaces are considered using a Čech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. Deformations fixing the periods of a differential and deformations splitting zeros are considered. A second order deformation expansion is presented for the Riemann period matrix. A complete deformation expansion is presented for Abelian differentials. Schiffer's kernel function approach for deformations of a Green's function is followed. [ABSTRACT FROM AUTHOR]

The notions of a Q-Gorenstein scheme and a Q-Gorenstein morphism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of a Q-Gorenstein algebraic variety and a Q-Gorenstein deformation satisfying the Kollár condition, over a field. By studying the relative S2-condition and base change properties, valuable results are proved for Q-Gorenstein morphisms, which include the infinitesimal criteria, the valuative criterion, and Q-Gorenstein refinements. [ABSTRACT FROM AUTHOR]

We prove that a weak -Fano -fold with terminal singularities has unobstructed deformations. By using this result and computing some invariants of a terminal singularity, we provide two results on global deformation of a weak -Fano -fold. We also treat a stacky proof of the unobstructedness of deformations of a -Fano -fold. [ABSTRACT FROM AUTHOR]