Noether Theory Research Articles

Herglotz-type vakonomic dynamics and its Noether symmetry for nonholonomic constrained systems

In this paper, Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems are studied. Firstly, Herglotz-type vakonomic dynamical equations for nonholonomic systems are derived on the premise of Herglotz variational principle. Secondly, in terms of the Herglotz-type vakonomic dynamical equations, the Noether symmetry of Herglotz-type vakonomic dynamics is explored, and the Herglotz-type vakonomic dynamical Noether theorems and their inverse theorems are deduced. Finally, the conservation laws of Appell–Hamel case with non-conservative forces are analyzed to show the validity of our results.

Brill–Noether Theory of Stable Vector Bundles on Ruled Surfaces

Let X be a ruled surface over a nonsingular curve C of genus g≥0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 0.$$\\end{document} Let MH:=MX,H(2;c1,c2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H:=M_{X,H}(2;c_1,c_2)$$\\end{document} be the moduli space of H-stable rank 2 vector bundles E on X with fixed Chern classes ci:=ci(E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_i:=c_i(E)$$\\end{document} for i=1,2.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,2.$$\\end{document} The main goal of this paper is to contribute to a better understanding of the geometry of the moduli space MH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H$$\\end{document} in terms of its Brill–Noether locus WHk(2;c1,c2),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_H^k(2;c_1,c_2),$$\\end{document} whose points correspond to stable vector bundles in MH\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_H$$\\end{document} having at least k independent sections. We deal with the non-emptiness of this Brill–Noether locus, getting in most of the cases sharp bounds for the values of k such that WHk(2;c1,c2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_H^k(2;c_1,c_2)$$\\end{document} is non-empty.

Herglotz-type vakonomic dynamics and Noether theory of nonholonomic systems with delayed arguments

The Herglotz-type vakonomic dynamics of nonholonomic constrained systems with delayed arguments and its Noether theory are studied in this paper. First of all, the Herglotz-type equations of time-delayed vakonomic dynamics for nonholonomic systems are established, and the Herglotz-type local extremal equations are given. Secondly, on the basis of derivation of the variational formulas of Hamilton–Herglotz action with time delay, the Herglotz-type Noether symmetry criteria for time-delayed vakonomic dynamics are investigated. Thirdly, the Herglotz-type Noether’s theorems and inverse theorems for time-delayed vakonomic dynamics of nonholonomic systems are deduced. Finally, an example is presented to demonstrate the application of the results.

Towards Brill–Noether theory for cuspidal curves

Towards Brill–Noether theory for cuspidal curves

Brill–Noether Theory of Hilbert Schemes of Points on Surfaces

Abstract We show that Brill–Noether loci in Hilbert scheme of points on a smooth connected surface $S$ are non-empty whenever their expected dimension is positive and that they are irreducible and have expected dimensions. More precisely, we consider the loci of pairs $(I, s)$, where $I$ is an ideal that locally at the point $s$ of $S$ needs a given number of generators. We give two proofs. The first uses Iarrobino’s description [9] of the Hilbert–Samuel stratification of local punctual Hilbert schemes, and the second is based on induction via birational relationships between different Brill–Noether loci given by nested Hilbert schemes.

Linear series with ρ < 0 \rho<0 via thrifty Lego building

Abstract The moduli space G g , d r → M g \mathcal{G}^{r}_{g,\smash{d}}\to\mathcal{M}_{g} parameterizing algebraic curves with a linear series of degree 𝑑 and rank 𝑟 has expected relative dimension ρ = g − ( r + 1 ) ⁢ ( g − d + r ) \rho=g-(r+1)(g-d+r) . Classical Brill–Noether theory concerns the case ρ ≥ 0 \rho\geq 0 ; we consider the non-surjective case ρ < 0 \rho<0 . We prove the existence of components of this moduli space with the expected relative dimension when 0 > ρ ≥ − g + 3 0>\rho\geq-g+3 , or 0 > ρ ≥ − C r ⁢ g + O ⁢ ( g 5 / 6 ) 0>\rho\geq-C_{r}g+\mathcal{O}(g^{5/6}) , where C r C_{r} is a constant depending on the rank of the linear series such that C r → 3 C_{r}\to 3 as r → ∞ r\to\infty . These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.

On An Illustrative Examples of a Studied Noetherity Dirac-Delta Extensions for a Noether Operator

The purpose of this work is to illustrate by clear examples the noetherity nature of a finite Dirac-delta Extensions of a studied noether operator. Previously in our published papers, we have investigated in different two cases, the noetherization of a Dirac-delta extensions of a noether linear integro-differential operator defined by a third kind integral equation in some specific well chosen functional spaces. Our various already published researches were connected with such topic widely studied and clearly presenting different specific approaches, applied when establishing fundamentaly noether theory for some kind of integro-differential operators to reach the noetherization. The initial considered noether operator <I>A</I> has been extended with some finite dimensional spaces of Dirac-delta functions, and the noetherization of the two cases of extensions has been established depending with the parameters of the third kind integral equation defining <I>A</I>. The previous lead us to set the problem of the construction of practical examples clearly illustrating the relationship between theory and practise. For this aim, we based on an established wellknown noether theory and, we construct in this work step by step, two illustrative examples to show the interconnexion between the theory and pratise related to the investigation of the construction of noether theory for the considered extended noether operator denoted <img width="8" height="10" src="http://article.sciencepublishinggroup.com/journal/347/3471192/image001.png" /> defined by a third kind linear singular integral equation in some generalized functional spaces. The extended operator <I>A</I> of the initial noether operator <I>A</I> is verified being also noether and therefore we deduce the index of the extended operator <img width="8" height="10" src="http://article.sciencepublishinggroup.com/journal/347/3471192/image001.png" />.

Discrete Lagrange problems with constraints valued in a Lie group

The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the discrete field theory, concluding as an illustration with the remarkable example of the harmonic maps of the discrete plane in the Lie group SO(n).

Equivariant Brill–Noether theory for elliptic operators and superrigidity of J-holomorphic maps

Abstract The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of J-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.

Refined Brill–Noether theory for all trigonal curves

Trigonal curves provide an example of Brill–Noether special curves. Theorem 1.3 of Larson (Invent Math 224(3):767–790, 2021) characterizes the Brill–Noether theory of general trigonal curves and the refined stratification by Brill–Noether splitting loci, which parametrize line bundles whose push-forward to \(\mathbb {P}^1\) has a specified splitting type. This note describes the refined stratification for all trigonal curves. Given the Maroni invariant of a trigonal curve, we determine the dimensions of all Brill–Noether splitting loci and describe their irreducible components. When the dimension is positive, these loci are connected, and if furthermore the Maroni invariant is 0 or 1, they are irreducible.

On the Segre invariant for rank two vector bundles on ℙ2

Abstract We extend the concept of the Segre invariant to vector bundles on a surface X. For X = ℙ2 we determine what numbers can appear as the Segre invariant of a rank 2 vector bundle with given Chern classes. The irreducibility of strata with fixed Segre invariant is proved and their dimensions are computed. Finally, we present applications to the Brill–Noether Theory for rank 2 vector bundles on ℙ2.

The Strong Maximal Rank conjecture and higher rank Brill–Noether theory

In this paper, we compute the cohomology class of certain "special maximal-rank loci" originally defined by Aprodu and Farkas. By showing that such classes are nonzero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher-rank Brill--Noether theory.

The zilch electromagnetic conservation law revisited

It is shown that the zilch conservation law arises as the Noether current corresponding to a variational symmetry of a duality-symmetric Maxwell Lagrangian. The action of the corresponding symmetry generator on the duality-symmetric Lagrangian, while non-vanishing, is a total divergence as required by the Noether theory. The variational nature of the zilch conservation law was previously known only for some of the components of the zilch tensor, notably the optical chirality. By contrast, our analysis is fully covariant and is, therefore, valid for all components of the zilch tensor. The analysis is presented here for both the real and complex versions of duality-symmetric Maxwell Lagrangians.

A refined Brill–Noether theory over Hurwitz spaces

Let $$f:C \rightarrow \mathbb {P}^1$$ be a degree k genus g cover. The stratification of line bundles $$L \in {{\,\mathrm{Pic}\,}}^d(C)$$ by the splitting type of $$f_*L$$ is a refinement of the stratification by Brill–Noether loci $$W^r_d(C)$$ . We prove that for general degree k covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of $$W^r_d(C)$$ for a general k-gonal curve (there are often components of different dimensions), extending results of Pflueger (Adv Math 312:46–63, 2017) and Jensen and Ranganathan (Brill–Noether theory for curves of a fixed gonality, arXiv:1701.06579 , 2017). The results here apply over any algebraically closed field.

The Abel map for surface singularities II. Generic analytic structure

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants includes: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line bundles, the multivariable Hilbert and Poincaré series associated with the divisorial filtration, the analytic semigroup, the maximal ideal cycle.The first part contains the definition of ‘generic structure’ based on the work of Laufer [14]. The second technical ingredient is the Abel map developed in [21].The results can be compared with certain parallel statements from the Brill–Noether theory and from the theory of Abel map associated with projective smooth curves (see e.g. [1] and [6]), though the tools and machineries are very different.

Hox Gene Collinearity May Be Related to Noether Theory on Symmetry and Its Linked Conserved Quantity

Hox Gene Collinearity (HGC) is a fundamental property that controls the development of many animal species, including vertebrates. In the Hox gene clusters, the genes are located in a sequential order Hox1, Hox2, Hox3, etc., along the 3’ to 5’ direction of the cluster in the chromosome. During Hox cluster activation, the Hox genes are expressed sequentially in the ontogenetic units D1, D2, D3, etc., along the anterior–posterior axis (A-P) of the early embryo. This collinearity, first observed by E.B. Lewis, is surprising because the spatial collinearity of these structures (Hox clusters and embryos) correlates entities that differ by about four orders of magnitude. Biomolecular mechanisms alone cannot explain such correlations. Long-range physical interactions, such as diffusion or electric attractions, should be involved. A biophysical model (BM) was formulated, which, in alignment with the biomolecular processes, successfully describes the existing vertebrate genetic engineering data. One hundred years ago, Emmy Noether made a fundamental discovery in mathematics and physics. She proved, rigorously, that a physical system obeying a symmetry law (e.g., rotations or self-similarity) is followed by a conserved physical quantity. It is argued here that HGC obeys a ‘primitive’ self-similarity symmetry. In this case, the associated primitive conserved quantity is the irreversibly increasing ‘ratchet’-like Hoxgene ordering where some genes may be missing. The genes of a vertebrate Hox clusterare located along a finite straight line. The same order follows the ontogenetic unitsof the vertebrate embryo. Therefore, HGC is a manifestation of a primitive Noether Theory (NT). NT may be applied to other than the vertebrate case, for instance, to animals with a circular topological symmetry. For example, the observed abnormal Hox gene ordering of the echinoderm Hox clusters may be reproduced by a double-strand break of the circular Hox gene ordering and its subsequent incorporation in the flanking chromosome.

Non-normal type singular integral-differential equations by Riemann-Hilbert approach

This article deals with one class of singular integral-differential equations of non-normal type with convolution and Cauchy principal value integral in class {0}. By using Fourier transform, this classes of equations are transformed into a Riemann boundary value problem with nodal point, and we prove the existence of solutions and the Noether theory for the equations. For such equations, we propose one method different from classical one, and we obtain the general solutions and the conditions of solvability. All cases about the behaviors of the solution are considered at nodal points. Therefore, the theory of integral equations and the classical Riemann boundary value problems is extended further.

The problem of Lagrange in discrete field theory

A discrete Lagrange problem in a fibered manifold on a cellular complex is defined as a Lagrangian density and a constraint submanifold in the 1-jets space of the manifold. After defining the concepts of admissible section and infinitesimal admissible variation, the aim of these problems is to find admissible sections that are critical for the action functional of the Lagrangian density with respect to the infinitesimal admissible variations. For admissible sections satisfying a certain regularity condition, we prove that such critical sections are the solutions of an extended unconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established for the discrete Lagrange problems considered. Finally, the whole theory is illustrated with an example of geometric interest: the discrete harmonic 1-forms on the elliptic or hyperbolic discrete plane, for which we obtain two explicit variational integrators: one with initial conditions (Cauchy problem) and other with boundary conditions (Dirichlet problem).

The Abel map for surface singularities I: generalities and examples

Let (X, o) be a complex normal surface singularity. We fix one of its good resolutions widetilde{X}rightarrow X, an effective cycle Z supported on the reduced exceptional curve, and any possible (first Chern) class l'in H^2(widetilde{X},{mathbb {Z}}). With these data we define the variety mathrm{ECa}^{l'}(Z) of those effective Cartier divisors D supported on Z, which determine a line bundles {{mathcal {O}}}_Z(D) with first Chern class l'. Furthermore, we consider the affine space mathrm{Pic}^{l'}(Z)subset H^1({{mathcal {O}}}_Z^*) of isomorphism classes of holomorphic line bundles with Chern class l' and the Abel map c^{l'}(Z):mathrm{ECa}^{l'}(Z)rightarrow mathrm{Pic}^{l'}(Z). The present manuscript develops the major properties of this map, and links them with the determination of the cohomology groups H^1(Z,{mathcal {L}}), where we might vary the analytic structure (X, o) (supported on a fixed topological type/resolution graph) and we also vary the possible line bundles {mathcal {L}}in mathrm{Pic}^{l'}(Z). The case of generic line bundles of mathrm{Pic}^{l'}(Z) and generic line bundles of the image of the Abel map will have priority roles. Rewriting the Abel map via Laufer duality based on integration of forms on divisors, we can make explicit the Abel map and its tangent map. The case of superisolated and weighted homogeneous singularities are exemplified with several details. The theory has similar goals (but rather different techniques) as the theory of Abel map or Brill–Noether theory of reduced smooth projective curves.

Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches

Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches