Singular Spaces Research Articles

On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O’Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O’Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford–Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.

GEODESICS ON RIEMANNIAN STACKS

Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky geodesics. Our main results show that the length of stacky curves measure distances on the orbit space, characterize stacky geodesics as locally minimizing curves, and establish a stacky version of Hopf-Rinow Theorem. We include a concise overview that bypasses nonessential technicalities, and we lay stress on the examples of orbit spaces of isometric actions and leaf spaces of Riemannian foliations.

From “planning in section” to “semi-duplex”: The New Humanism and the alternate-corridor technique

Most of the pretending innovative modern housing projects show a common element that can be perceived when looking at the cross-section, evidencing the use of alternate corridors, which was recognized as a projective technique under Wells Coates expression “Planning in Section”. Contrasting virtues concerning the resulting space were observed at different historical moments. The late 19th Century used it to create privacy. The modern architecture used it to increase efficiency, where flexibility was included. However, during the '60s, the alternative corridor technique started to be used to attend social issues, popularizing the split-level term, and echoing 19th Century privacy interests. Keywords like Smithson's “doorstep” and Candilis “semi-duplex” section reveal their approaches. Well-differentiated areas (public/intimate/collective) and socially controlled areas were obtained in domestic interiors with a few steps, creating singular spaces where children can play while being monitored by adults. Semi-duplex geometry showed new adaptability to different profiles: a small kitchenless apartment for a bachelor could be inserted aside from a larger two-level family apartment. This paper shows an overview of this projective technique by contrasting different housing projects. Since similar split-level geometries seem to have been coming back recently, questions about it are necessary.

Documenting a cultural landscape using point-cloud 3d models obtained withgeomatic integration techniques. The case of the El Encín atomic garden, Madrid (Spain).

A country’s cultural landscapes are an important part of its heritage. The growing need to identify, catalogue and preserve these resources has led to a rapid change in the management and inventorying of heritage in general and of cultural landscapes in particular. The main aim of this work is to develop and apply an updated and integrated methodology for capturing and processing geo-information for the digital documentation of cultural heritage. The proposed case study is the atomic garden in the Finca El Encín (Madrid), a singular space with unique biogeographical features created over 60 years ago. The results of the case study validate the method, consisting of an unmanned aerial platform equipped with sensors to obtain point clouds and aerial images in conjunction with point clouds and images captured with a terrestrial laser scanner.

On the Finiteness of Quantum K-Theory of a Homogeneous Space

Abstract We show that the product in the quantum K-ring of a generalized flag manifold $G/P$ involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the $J$-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

Geometrothermodynamics of black hole binary systems

We study a stationary and axisymmetric binary system composed of two identical Kerr black holes, whose physical parameters satisfy the Smarr thermodynamic formula. Then, we use the formalism of geometrothermodynamics to show that the spatial distance between the black holes must be considered as a thermodynamic variable. We investigate the main thermodynamic properties of the system by using the contact structure of the phase-space, which generates the first law of thermodynamics and the equilibrium conditions. The phase transition structure of the system is investigated through the curvature singularities of the equilibrium space. It is shown that the thermodynamic and stability properties and the phase transition structure of the binary system strongly depend on the distance between the black holes.

Numerical Loop-Tree Duality: contour deformation and subtraction

We introduce a novel construction of a contour deformation within the framework of Loop-Tree Duality for the numerical computation of loop integrals featuring threshold singularities in momentum space. The functional form of our contour deformation automatically satisfies all constraints without the need for fine-tuning. We demonstrate that our construction is systematic and efficient by applying it to more than 100 examples of finite scalar integrals featuring up to six loops. We also showcase a first step towards handling non-integrable singularities by applying our work to one-loop infrared divergent scalar integrals and to the one-loop amplitude for the ordered production of two and three photons. This requires the combination of our contour deformation with local counterterms that regulate soft, collinear and ultraviolet divergences. This work is an important step towards computing higher-order corrections to relevant scattering cross-sections in a fully numerical fashion.

On the Construction of Strange Space in Stage Sets

In this article, I will explain the definition of the singular space in a stage setting as I understand, and explain the importance of the singular effect produced by the stage setting construction space in the field of stage design. The dramatic style of the theater,and the singular effects of the construction space may bring practical significance to the theater.

Involution surface bundles over surfaces

We construct models of involution surface bundles over algebraic surfaces, degenerating over normal crossing divisors, and with controlled singularities of the total space.

Monotonicity and rigidity of the $${\mathcal {W}}$$-entropy on $${\mathsf {RCD}} (0,N)$$ spaces

By means of a space-time Wasserstein control, we show the monotonicity of the $${\mathcal {W}}$$ -entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the $${\mathcal {W}}$$ -entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

A New Type of Ankle-Foot Rehabilitation Robot Based on Muscle Motor Characteristics

Ankle injury and dysfunction are always accompanied by abnormal peripheral tissues. In order to achieve a more comprehensive rehabilitation training, this article firstly studies the movement characteristics of the surrounding muscles of the ankle-foot. The human musculoskeletal model in OpenSim is used for human exercise experiment. And the relationship between different muscles around the ankle-foot and the direction of the ankle-foot movement are measured respectively, so as to formulate ankle-foot static/dynamic rehabilitation strategies related to the auxiliary training of the ankle-foot muscle. In order to implement the established rehabilitation exercise strategy and to reduce the control difficulty of rehabilitation robot, a new type of decoupling series-parallel mechanism with three rotations and one movement is proposed. After analyzing the degree of freedom and the positive kinematics solution, it is proved that it is completely decoupled in kinematics, can realize the independence of motion control, and there is no singular space. Then, under the premise of meeting the space required for the implementation of the rehabilitation strategy, the proposed new ankle-foot rehabilitation robot is optimized in terms of structure and component size, which improves its space utilization and optimizes the overall mechanical performance. Finally, the dynamic simulation experiment is carried out with the dynamic rehabilitation strategy as the goal, which proves that the robot can realize the normal gait simulation movement in sitting posture, and the motion control of the robot is independent of each other and shows good dynamic performance. Through investigation, the ankle-foot rehabilitation robot designed in this article is cost-effective.

Sensitivity Analysis of Burgers' Equation with Shocks

The generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but this is what hyperbolic type problems cannot provide. We provide a counterargument in this paper and show that even though the solution profile develops singularities in the random space, which destroys the spectral accuracy of gPC, the physical quantities (such as the shock emergence time, the shock location, and the shock strength) are all smooth functions of the uncertainties coming from both initial data and the wave speed. With proper shifting, the solution's polynomial interpolation approximates the real solution accurately, and the error decays as the order of the polynomial increases. Therefore this work provides a new perspective to “quantify uncertainties" and significantly improves the accuracy of the gPC method with a slight reformulation. We use the Burgers' equation as an example for thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.

Normal Forms for Flat Two-input Control Systems Linearizable via a Two-fold Prolongation

We present normal forms for nonlinear two-input control systems that become static feedback linearizable after a two-fold prolongation of a suitably chosen control, which is one of the simplest dynamic feedback. They form a particular class of flat systems, namely those of differential weight n + 4, where n is the number of states. We also show that the dynamic feedback creates singularities in the control space depending on the state and we discuss them.

A Hybrid Model Based on Multi-Stage Principal Component Extraction, GRU Network and KELM for Multi-Step Short-Term Wind Speed Forecasting

Accurate wind speed forecasting exerts a critical role in energy conversion and management of wind power. In term of this purpose, a hybrid model based on multi-stage principal component extraction, kernel extreme learning machine (KELM) and gated recurrent unit (GRU) network is developed in this paper, where the multi-stage principal component extraction combines complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), singular spectrum analysis (SSA) and phase space reconstruction (PSR). Firstly, CEEMDAN is employed to decompose the raw wind speed data into a sequence of intrinsic mode functions (IMFs) and a residual component. Then the principal components and residual components of all IMFs are captured by SSA. Meanwhile, all residual components obtained by CEEMDAN decomposition and SSA processing are added to form a new component. Subsequently, PSR is utilized to construct each forecasting component obtained by CEEMDAN-SSA into the input and output of training set and testing set for the prediction model. Later, KELM and GRU neural network are conducted to predict the high-frequency and low-frequency components, respectively. Eventually, the prediction values of each component are accumulated to acquire the final prediction result. To evaluate the performance of the proposed model, four datasets from Sotavento Galicia wind farm are adopted to conduct experimental research. The experimental results manifest that the proposed model achieves higher accuracy of multi-step prediction than other comparative models.

Generalized nonlinear Schrödinger equations describing the Second Harmonic Generation of femtosecond pulse, containing a few cycles, and their integrals of motion.

An interaction of laser pulse, containing a few cycles, with substance is a modern problem, attracting attention of many researches. The frequency conversion is a key problem for a generation of such pulses in various ranges of frequencies. Adequate description of such pulse interaction with a medium is based on a slowly evolving wave approximation (SEWA), which has been proposed earlier for a description of propagation of the laser pulse, containing a few cycles, in a medium with cubic nonlinear response. Despite widely applicability of the frequency conversion for various nonlinear optics problems solutions, SEWA has not been applied and developed for a theoretical investigation of the frequency doubling process until present time. In this study the set of generalized nonlinear Schrödinger equations describing a second harmonic generation of the super-short femtosecond pulse is derived. The equations set contains terms, describing the pulses self-steepening, and the second order dispersion (SOD) of the pulse, a diffraction of the beam as well as mixed derivatives. We propose the transform of the equations set to a type, which does not contain both the mixed derivatives and time derivatives of the nonlinear terms. This transform allows us to derive the integrals of motion of the problem: energy, spectral invariants and Hamiltonian. We show the existence of two specific frequencies (singularities in the Fourier space) inherent to the problem. They may cause an appearance of non-physical absolute instability of the problem solution if the spectral invariants are not taken into account. Moreover, we claim that the energy preservation at the laser pulses propagation may not occur if these invariants do not preserve. Developed conservation laws, in particular, have to be used for developing of the conservative finite-difference schemes, preserving the conservation laws difference analogues, and for developing of adequate theory of the modulation instability of the laser pulses, containing a few cycles.

Phase space of static wormholes sustained by an isotropic perfect fluid

A phase space is built that allows to study, classify and compare easily large classes of static spherically symmetric wormholes solutions, sustained by an isotropic perfect fluid in General Relativity. We determine the possible locations of equilibrium points, throats and curvature singularities in this phase space. Throats locations show that the spatial variation of the gravitational redshift at the throat of a static spherically symmetric wormhole sustained by an isotropic perfect fluid is always diverging, generalising the result that there is no such wormhole with zero-tidal force. Several specific static spherically symmetric wormholes models are studied. A vanishing density model leads to an exact solution of the field equation allowing to test our dynamical system formalism. It also shows how to extend it to the description of static black holes. Hence, the trajectory of the Schwarzschild black hole is determined. The static spherically symmetric wormhole solutions of several usual isotropic dark energy (generalised Chaplygin gas, constant, linear and Chevallier-Polarski-Linder equations of state) and dark matter (Navarro-Frenk-White profile) models are considered. They show various behaviours far from the throat: singularities, spatial flatness, cyclic behaviours, etc. None of them is asymptotically Minkowski flat. This discards the natural formation of static spherically symmetric and isotropic wormholes from these dark fluids. Last we consider a toy model of an asymptotically Minkowski flat wormhole that is a counterexample to a recent theorem claiming that a static wormhole sustained by an isotropic fluid cannot be asymptotically flat on both sides of its throat.

The {bar{partial }}-Equation, Duality, and Holomorphic Forms on a Reduced Complex Space

We solve the {bar{partial }}-equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the {bar{partial }}-equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces.

Classifying complete mathbb {C}-subalgebras of mathbb {C}[[t

We address the problem of classifying complete mathbb {C}-subalgebras of mathbb {C}[[t]]. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given mathbb {C}-subalgebra. Hence we can define the space mathcal {R}_{Gamma } of all mathbb {C}-subalgebras of mathbb {C}[[t]] with semigroup Gamma . After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that mathcal {R}_{Gamma } is an affine variety by describing its defining equations in an ambient affine space in terms of an explicit algorithm. Moreover, we identify certain types of semigroups Gamma for which mathcal {R}_{Gamma } is always an affine space, and for general Gamma we describe the stratification of mathcal {R}_{Gamma } by embedding dimension. We also describe the natural map from mathcal {R}_{Gamma } to the Zariski moduli space in some special cases. Explicit examples are provided throughout.

A Preliminary Study on the Shaping of Outdoor Space in Future Urban Community

Nowadays, the singularity of outdoor space in urban community has been unable to meet the needs of people. The concept of home, the smallest unit of society, is changing with the change of family structure. The huge living community consists of a family of loose relationship or fragmented families, each of which is an island. In the past, it is said that a neighbor that is near is better than a brother far off. However, at present, a close neighbor is like a stranger. The walls of the building have become a barrier for people to communicate. People’s activities should not be restricted by the ”box” of home. Can we open the walls and re-design the outdoor space? Therefore, we begin to think about the connection between people, can we achieve it through space? Can there be more public space between neighborhoods? Can such a public space be used as part of the family space again? In this paper, the outdoor space of urban community is taken as the research object to strive to break the boundaries of traditional indoor and outdoor space. How to optimize the outdoor public space in urban communities to promote residents’ communication is taken as the purpose of research to carry out a new design of this kind of space. The outdoor space outside the traditional wall is combined with the living needs of present people to discuss the possibility shaping of outdoor space in urban community.

Spatiotemporal Patterns in a Delayed Reaction–Diffusion Mussel–Algae Model

We study the spatiotemporal patterns of a delayed reaction–diffusion mussel–algae system subject to Neumann boundary conditions. This paper is a continuation of our previous studies on the mussel–algae model. We prove the global existence and positivity of solutions. By analyzing the distribution of eigenvalues, we obtain the stability conditions for the positive constant steady state, the existence of Hopf bifurcation and the Turing instability. We show the dynamic classification near the Turing–Hopf singularity in the dimensionless parameter space and observe a transiently spatially nonhomogeneous periodic solution in simulations. Both theoretical and numerical results reveal that the Turing–Hopf bifurcation can enrich the diversity of the spatial distribution of populations.