Lifting Problem Research Articles

RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

Tropical graph curves

We study tropical line arrangements associated to a three-regular graph $G$ that we refer to as tropical graph curves. Roughly speaking, the tropical graph curve associated to $G$ , whose genus is $g$ , is an arrangement of $2g-2$ lines in tropical projective space that contains $G$ (more precisely, the topological space associated to $G$ ) as a deformation retract. We show the existence of tropical graph curves when the underlying graph is a three-regular, three-vertex-connected planar graph. Our method involves explicitly constructing an arrangement of lines in projective space, i.e. a graph curve whose tropicalization yields the corresponding tropical graph curve and in this case, solves a topological version of the tropical lifting problem associated to canonically embedded graph curves. We also show that the set of tropical graph curves that we construct are connected via certain local operations. These local operations are inspired by Steinitz’ theorem in polytope theory.

Lifts of Brauer characters in characteristic two

A conjecture raised by Cossey in 2007 asserts that if G is a finite p-solvable group and φ is an irreducible p-Brauer character of G with vertex Q, then the number of lifts of φ is at most |Q:Q′|. This conjecture is now known to be true in several situations for p odd, but there has been little progress for p even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift might be neither linear nor conjugate. In this paper we show that if χ is a lift of an irreducible 2-Brauer character in a solvable group, then χ has a linear Navarro vertex if and only if all the vertex pairs of χ are linear, and in that case all of the twisted vertices of χ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for p=2 “one vertex at a time”.

Compact perturbations of Moore-Penrose invertible operators

Firstly, we study the stability of the Moore-Penrose invertibility of Hilbert space operators, establishing necessary and sufficient conditions for the invariance of the Moore-Penrose invertibility under small perturbations, compact perturbations and small compact perturbations. Secondly, we study the lifting problem of Moore-Penrose invertible elements of the Calkin algebra, showing that essentially Moore-Penrose invertible operators are small compact perturbations of Moore-Penrose invertible operators. Finally, we discuss the compactness of Moore-Penrose spectra, showing that each Hilbert space operator can be perturbed into an operator with nonempty compact Moore-Penrose spectrum by an arbitrarily small compact perturbation.

Density topologies and the marginal strong lifting problem for hyperstonian spaces

We investigate properties of density topologies and of topological measure spaces whose topologies are density topologies, motivated by the search for necessary as well as sufficient conditions for the canonical strong lifting of a hyperstonian space to be a marginal. For topological probability spaces we establish relations between the existence of strong liftings and the marginal property of the canonical strong lifting of their hyperstonian space, building on earlier work.

Marginals and the product strong lifting problem

We study the permanence of strong liftings for products of topological probability spaces. We get results to the positive for the 'usual product' or for the product σ-algebra obtained from it by adjoining the two-sided nil-null sets under assumptions implying that the product topology is contained in this product σ-algebras. For the construction of strong liftings in the product the application of marginals is basic as well as that of the fundamental properties of lifting topologies.

Weight Lifting Trajectory Optimization for Variable Stiffness Actuated Robot

Variable stiffness actuator is a new actuation of robot which is inspired by motor control of human arm. And it is promising way to exploit the human‐like performance and human‐like motion. However, due to the mechanical complexity and redundancy of the actuators, it is not trivial to control the variable stiffness actuated robot to perform a human‐like motion. In this paper, the weight lifting problem with variable stiffness actuated robot is studied. Then, the original problem is formulated as a constrained optimal control problem and transformed as a general nonlinear programming problem with Gauss pseudospectral method. Simulation studies demonstrate the effectiveness of the proposed method compared with the iLQR approach. Furthermore, the simulations results are presented to show the influence of stiffness variation of variable stiffness actuators on the weight lifting problem.

The local lifting problem for D4

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this paper, we characterize D4-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D4- extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D4-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D4 is a local Oort group for the prime 2.

Functors of liftings of projective schemes

A classical approach to investigate a closed projective scheme W consists of considering a general hyperplane section of W, which inherits many properties of W. The inverse problem that consists in finding a scheme W starting from a possible hyperplane section Y is called a lifting problem, and every such scheme W is called a lifting of Y. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for Y is smooth also for W.We characterize all the liftings of Y with a given Hilbert polynomial by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. We use constructive methods from Gröbner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.

Lifting of polynomial symplectomorphisms and deformation quantization

ABSTRACTWe study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory.This paper is a continuation of the study [19].

Global Oort groups

We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all Oort groups lie in a particular class of finite groups that we characterize, with equality of classes under a conjecture about local liftings. We prove this equality unconditionally if the order of G is not divisible by 2p2. We also treat the local lifting problem and relate it to the global problem.

On the semicontinuity problem of fibers and global F-regularity

ABSTRACTIn this article, we discuss the semicontinuity problem of certain properties on fibers for a morphism of schemes. One aspect of this problem is local. Namely, we consider properties of schemes at the level of local rings, in which the main results are established by solving the lifting and localization problems for local rings. In particular, we obtain the localization theorems in the case of seminormal and F-rational rings, respectively. Another aspect of this problem is global, which is often related to the vanishing problem of certain higher direct image sheaves. As a test example, we consider the deformation of the global F-regularity.

The local lifting problem for A4

We solve the local lifting problem for the alternating group A_4, thus showing that it is a local Oort group. Specifically, if k is an algebraically closed field of characteristic 2, we prove that every A_4-extension of k[[s]] lifts to characteristic zero. As a consequence, every A_4-branched cover of smooth projective curves in characteristic 2 lifts to characteristic zero.

A velocity decomposition formulation for 2D steady incompressible lifting problems

The principle of velocity decomposition is used to efficiently and accurately solve the Navier–Stokes boundary-value problem for lifting flows. The velocity vector is decomposed as the sum of irrotational and vortical components. The irrotational component is represented using a velocity potential which satisfies the Laplace equation with a modified boundary condition that is written in terms of the Navier–Stokes solution. A viscous potential can be found which satisfies the Navier–Stokes problem directly outside of the rotational regions of the fluid such that the fluid domain over which the Navier–Stokes equations must be solved numerically can be greatly reduced. The viscous potential is used as a Dirichlet boundary condition for the total velocity on the boundaries of the reduced fluid domain. The velocity decomposition approach is used to solve for the flow over a 2D NACA0012 foil in both laminar and turbulent regimes.

Transgression to loop spaces and its inverse, II: Gerbes and fusion bundles with connection

We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a fusion product with respect to triples of paths. The equivalence is established by explicit functors in both directions: transgression and regression. We describe applications to geometric lifting problems and loop group extensions.

Application of a 2D harmonic polynomial cell (HPC) method to singular flows and lifting problems

Further developments and applications of the 2D harmonic polynomial cell (HPC) method proposed by Shao and Faltinsen [22] are presented. First, a local potential flow solution coupled with the HPC method and based on the domain decomposition strategy is proposed to cope with singular potential flow characteristics at sharp corners fully submerged in a fluid. The results are verified by comparing them with the analytical added mass of a double-wedge in infinite fluid. The effect of the singular potential flow is not dominant for added mass and damping, but the error is non-negligible when calculating mean wave loads using direct pressure integration. Then, the double-layer nodes technique is used to simulate a thin free shear layer shed from lifting bodies, across which the velocity potential is discontinuous. The results are verified by comparing them with analytical results for steady and unsteady lifting problems of a flat plate in infinite fluid. The latter includes the Wagner problem and the Theodorsen functions. Satisfactory agreement with other numerical results is documented for steady linear flow past a foil and beneath the free surface.

On class groups of imaginary quadratic fields

In this paper, it is proved that one can find imaginary quadratic fields with class number not divisible by a specified prime l and with certain specified splitting conditions at a finite number of primes. Such existence theorems are useful in the arithmetic of elliptic curves and, potentially, also in certain lifting problems for reducible two-dimensional Galois representations. The methods used are a blend of geometry and the theory of modular forms, especially the trace formula.

THE DOUBLE COVER OF ODD GENERAL SPIN GROUPS, SMALL REPRESENTATIONS, AND APPLICATIONS

We construct local and global metaplectic double covers of odd general spin groups, using the cover of Matsumoto of spin groups. Following Kazhdan and Patterson, a local exceptional representation is the unique irreducible quotient of a principal series representation, induced from a certain exceptional character. The global exceptional representation is obtained as the multi-residue of an Eisenstein series: it is an automorphic representation, and it decomposes as the restricted tensor product of local exceptional representations. As in the case of the small representation of$\mathit{SO}_{2n+1}$of Bump, Friedberg, and Ginzburg, exceptional representations enjoy the vanishing of a large class of twisted Jacquet modules (locally), or Fourier coefficients (globally). Consequently they are useful in many settings, including lifting problems and Rankin–Selberg integrals. We describe one application, to a calculation of a co-period integral.

Completion, extension, factorization, and lifting of operators

The well-known results of M. G. Kreĭn concerning the description of selfadjoint contractive extensions of a hermitian contraction \(T_1\) and the characterization of all nonnegative selfadjoint extensions \({{\widetilde{A}} }\) of a nonnegative operator A via the inequalities \(A_K\le {{\widetilde{A}} } \le A_F\), where \(A_K\) and \(A_F\) are the Kreĭn–von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where \({{\widetilde{A}} }\) is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators \(I-T_1^*T_1\) and A, respectively; these conditions are automatically satisfied if \(T_1\) is contractive or A is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu. L. Shmul’yan on completions of nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M. G. Kreĭn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreĭn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.

Sequence independent lifting for mixed knapsack problems with GUB constraints

In this paper, we consider the semi-continuous knapsack problem with generalized upper bound constraints on binary variables. We prove that generalized flow cover inequalities are valid in this setting and, under mild assumptions, are facet-defining inequalities for the entire problem. We then focus on simultaneous lifting of pairs of variables. The associated lifting problem naturally induces multidimensional lifting functions, and we prove that a simple relaxation in a restricted domain is a superadditive function. Furthermore, we also prove that this approximation is, under extra requirements, the optimal lifting function. We then analyze the separation problem in two phases. First, finding a seed inequality, and second, select the inequality to be added. In the first step we evaluate both exact and heuristic methods. The second step is necessary because the proposed lifting procedure is simultaneous; from where our class of lifted inequalities might contain an exponential number of these. We choose a strategy of maximizing the resulting violation. Finally, we test this class of inequalities using instances arising from electrical planning problems. Our tests show that the proposed class of inequalities is strong in the sense that the addition of these inequalities closes, on average, 57.70 % of the root integrality gap and 97.70 % of the relative gap while adding less than three cuts on average.