We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$, the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.

This paper presents a comprehensive study of the forward and inverse kinematics of a six-degrees-of-freedom (DoF) spatial manipulator with a novel architecture. Developed by Systemantics India Pvt. Ltd., Bangalore, and designated as the H6A (i.e., Hybrid 6-Axis), this manipulator consists of two arm-like branches, which are attached to a rigid waist at the proximal end and are coupled together via a wrist assembly at the other. Kinematics of the manipulator is challenging due to the presence of two multi-DoF passive joints: a spherical joint in the right arm and a universal in the left. The forward kinematic problem has eight solutions, which are derived analytically in the closed form. The inverse kinematic problem leads to $160$ solutions and involves the derivation of a $40$-degree polynomial equation, whose coefficients are obtained as closed-form symbolic expressions of the pose parameters of the end-effector, thus ensuring the generality of the results over all possible inputs. Furthermore, the analyses performed lead naturally to the conditions for various singularities involved, including certain non-trivial architecture singularities. The results are illustrated via numerical examples which are validated extensively.

The time-optimal path following (OPF) problem is to find a time evolution along a prescribed path in task space with shortest time duration. Numerical solution algorithms rely on an algorithm-specific (usually equidistant) sampling of the path parameter. This does not account for the dynamics in joint space, that is, the actual motion of the robot, however. Moreover, a well-known problem is that large joint velocities are obtained when approaching singularities, even for slow task space motions. This can be avoided by a sampling in joint space, where the path parameter is replaced by the arc length. Such discretization in task space leads to an adaptive refinement according to the nonlinear forward kinematics and guarantees bounded joint velocities. The adaptive refinement is also beneficial for the numerical solution of the problem. It is shown that this yields trajectories with improved continuity compared to an equidistant sampling. The OPF is reformulated as a second-order cone programming and solved numerically. The approach is demonstrated for a 6-DOF industrial robot following various paths in task space.

We classify all $\mathbb{Q}$-factorial Fano intrinsic quadrics of dimension three and Picard number one having at most canonical singularities.

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver.

This paper presents the computation of the safe working zone (SWZ) of a parallel manipulator having three degrees of freedom. The SWZ is defined as a continuous subset of the workspace, wherein the manipulator does not suffer any singularity, and is also free from the issues of link interference and physical limits on its joints. The proposed theory is illustrated via application to two parallel manipulators: a planar 3-R̲RR manipulator and a spatial manipulator, namely, MaPaMan-I. It is also shown how the analyses can be applied to any parallel manipulator having three degrees of freedom, planar or spatial.

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

This paper deals with the kinematic analysis and enumeration of singularities of the six degree-of-freedom 3-RPS-3-SPR series–parallel manipulator (S–PM). The characteristic tetrahedron of the S–PM is established, whose degeneracy is bijectively mapped to the serial singularities of the S–PM. Study parametrization is used to determine six independent parameters that characterize the S–PM and the direct kinematics problem is solved by mapping the transformation matrix between the base and the end-effector to a point in ℙ7. The inverse kinematics problem of the 3-RPS-3-SPR S–PM amounts to find the location of three points on three lines. This problem leads to a minimal octic univariate polynomial with four quadratic factors.

We show that the anti-canonical volume of an $n$-dimensional Kähler–Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.

If the joint clearances of the joints of a manipulator are considered, an unconstrained motion of the end-effector can be computed. This is true for all poses of the manipulator, even with all actuators locked.

This paper presents how this unconstrained motion can be determined for a planar 3-RPR manipulator. The singularities are then studied. It is shown that when clearances are considered, the singularity curves normally found in the workspace of such a manipulator become singular zones. These zones can be significant and greatly reduce the usable workspace of a manipulator. Since a prescribed configuration that would not, in theory, corresponds to a singular pose can become singular due to the unconstrained motion, the results of this paper are relevant to manipulator design and trajectory planning.

Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial.

We classify and analyze the orbits of the Kepler problem on surfaces of constant curvature (both positive and negative, ${{\mathbb{S}}^{2}}$ and ${{\mathbb{H}}^{2}}$, respectively) as functions of the angular momentum and the energy. Hill's regions are characterized, and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, depending on the constant angular momentum, and we describe the orbits of the regularized vector field. The phase portraits both for ${{\mathbb{S}}^{2}}$ and ${{\mathbb{H}}^{2}}$ are pointed out.

Making use of Gruson–Raynaud’s technique of ‘platification par éclatement’, Kerz and Strunk proved that the negative homotopy $K$-theory groups of a Noetherian scheme $X$ of Krull dimension $d$ vanish below $-d$. In this article, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy $K$-theory groups vanish below $-1$. Furthermore, in the case of cyclic quotient singularities, we provide an explicit ‘upper bound’ for the first negative homotopy $K$-theory group.

We study the Minkowski symmetry set of a closed smooth curve γ in the Minkowski plane. We answer the following question, which is analogous to one concerning curves in the Euclidean plane that was treated by Giblin and O’Shea (1990): given a point p on γ, does there exist a bi-tangent pseudo-circle that is tangent to γ both at p and at some other point q on γ? The answer is yes, but as pseudo-circles with non-zero radii have two branches (connected components) it is possible to refine the above question to the following one: given a point p on γ, does there exist a branch of a pseudo-circle that is tangent to γ both at p and at some other point q on γ? This question is motivated by the earlier quest of Reeve and Tari (2014) to define the Minkowski Blum medial axis, a counterpart of the Blum medial axis of curves in the Euclidean plane.