Complex Ball Research Articles

The Study of Ball and Plate System Based on Non-Linear PID

It is difficult to build a precise mathematical model for complex ball and plate system and many intelligent algorithms’ programming are larger, so we make use of the non-linear PID controller to control actual system and realize the point to point control and circular trajectory control of the ball. The experiment results show that the control method of non-linear PID could ensure the movement accuracy of the ball, achieve the performance of control process, and have better control effect than traditional PID.

Bloch-to-BMOA compositions on complex balls

Let φ be a holomorphic map between complex unit balls. We characterize those φ for which the composition operator f 7→ f ◦ φ maps the Bloch space into BMOA.

Hyperbolic BMOA classes

Let φ be a holomorphic mapping between complex unit balls. We obtain several characterizations of those φ for which the Möbius-invariant BMOA condition holds with respect to the Bergman metric.

Complex Surfaces with Cat(0) Metrics

We study complex surfaces with locally CAT(0) polyhedral Kähler metrics and construct such metrics on $${\mathbb{C}P^{2}}$$ with various orbifold structures. In particular, in relation to questions of Gromov and Davis–Moussong we construct such metrics on a compact quotient of the two-dimensional unit complex ball. In the course of the proof of these results we give criteria for Sasakian 3-manifolds to be globally CAT(1). We show further that for certain Kummer coverings of $${\mathbb{C}P^{2}}$$ of sufficiently high degree their desingularizations are of type K(π, 1).

Double integral characterizations of harmonic Bergman spaces

Recently Li et al. have characterized, except for a critical case, the weighted Bergman spaces over the complex ball by means of integrability conditions of double integrals associated with difference quotients of holomorphic functions. In this paper we extend those characterizations to the case of weighted harmonic Bergman spaces over the real ball and complement their results by providing a characterization for the missing critical case. We also investigate the possibility of extensions to the half-space setting. Our observations reveal an interesting half-space phenomenon caused by the unboundedness of the half-space.

POMPEIU PROBLEM FOR THE HEISENBERG BALL

The standard Pompeiu results for complex balls in the setting of the Heisenberg group $\bf H^n$ are also shown to carry over to Heisenberg balls. This extension is important because it allows for integration over sets of the same dimension as the ambient space $\bf H^n$. Several different concepts of the Heisenberg ball are considered, using differing definitions for the metric on $\bf H^n$. For purposes of this work, none of these is more natural than the others. The results for each of the spaces $L^2$, $L^p$ for $1 \leq p < \infty$, and for $L^\infty$, are directly comparable to the Pompeiu results for complex balls in $\bf H^n$, as in [2, 3, 4]. The natural expression for the Pompeiu problem in $\bf H^n$ is integration over complex balls, and the extension to the Heisenberg ball builds upon the methods for this case. The extra dimension primarily leads to extra complexity in the integrals. At the level of $L^\infty$, where the results require balls of two radii satisfying appropriate conditions, the additional dimension adds to the complexity in the functions defining the conditions for the radii. The different concepts of the Heisenberg ball lead to different forms for these arithmetic conditions defining the radii. The differences between these balls can also be seen when they are rotated with $\bf H^n$, an issue to be further considered in a later work. The standard Pompeiu results have been extended to all of the cases considered here.

Task Refinement for Autonomous Robots Using Complementary Corrective Human Feedback

A robot can perform a given task through a policy that maps its sensed state to appropriate actions. We assume that a hand-coded controller can achieve such a mapping only for the basic cases of the task. Refining the controller becomes harder and gets more tedious and error prone as the complexity of the task increases. In this paper, we present a new learning from demonstration approach to improve the robot's performance through the use of corrective human feedback as a complement to an existing hand-coded algorithm. The human teacher observes the robot as it performs the task using the hand-coded algorithm and takes over the control to correct the behavior when the robot selects a wrong action to be executed. Corrections are captured as new state-action pairs and the default controller output is replaced by the demonstrated corrections during autonomous execution when the current state of the robot is decided to be similar to a previously corrected state in the correction database. The proposed approach is applied to a complex ball dribbling task performed against stationary defender robots in a robot soccer scenario, where physical Aldebaran Nao humanoid robots are used. The results of our experiments show an improvement in the robot's performance when the default hand-coded controller is augmented with corrective human demonstration.

Inequalities for Exit times and Eigenvalues of Balls

This first result of this paper is about the Laplace transform of u(X T ) where u is harmonic on some bounded domain Ω, X t is Brownian motion and T is the exit time from Ω. The following results focus on exit times from balls and Faber–Krahn and reverse Faber–Krahn type inequalities for balls. We also study the behaviour of the first Dirichlet eigenvalue for complex balls under complex interpolation. The method of proof heavily relays on the log-concavity of gaussian measures.

Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations

In this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our method applies to show that the set of critical values must be nonempty and of codimension 1. In the equivariant setting the line of arguments extends to holomorphic mappings of maximal rank into the complex projective space or the complex Euclidean space, yielding in the latter case a lower bound on the dimension of the singular loci of certain holomorphic maps defined by integrating holomorphic 1-forms. In another direction, we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume, provided that the target complex unit ball is of dimension $m \ge 2$, giving in particular a new proof that a local biholomorphism between noncompact $m$-ball quotients of finite volume must be a covering map whenever $m \ge 2$. Finally, combining our results with Hermitian metric rigidity, we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball, provided that either the lattice is cocompact or the ball is of dimension at least 2.

Moduli of Plane Quartics, Gopel Invariants and Borcherds Products

It is known that the moduli space of plane quartic curves is birational to an arithmetic quotient of a six-dimensional complex ball [Kondō, S. “A complex hyperbolic structure of the moduli space of curves of genus three.” Journal fur die Reine und Angewandte Mathematik 525 (2000): 219–32]. In this paper, we shall show that there exists a 15-dimensional space of meromorphic automorphic forms on the complex ball, which gives a birational embedding of the moduli space of plane quartics with level-2 structure into . This map coincides with the one given by Coble [“Algebraic geometry and theta functions.” Algebraic geometry and theta functions. American Mathematical Society Colloquium Publication 10. Providence, RI: American Mathematical Society, 1929 (3rd ed., 1969)] by using Gopel invariants.

A calculation of cricket ball trajectories

While a number of past investigations have measured the aerodynamic forces on cricket balls, in general these forces have not been used to calculate the trajectories of balls between bowler and batsman. This article presents such a calculation. It begins with the full trajectory equations as developed for the study of debris flight in extreme windstorms, which are adapted for the cricket ball case. A collation of earlier experimental data is then presented for drag and side force coefficients on cricket balls. From these data, which show considerable scatter, a small number of generic force characteristics are derived for use in the trajectory equations. These characteristics have constant force coefficients in the subcritical and supercritical Reynolds number regions, and a transition between these two regions with a variable gradient. An approximate analysis of the trajectory equations results in very simple forms for the trajectories in the sub- and supercritical Reynolds number regimes, which reveal the governing dimensionless parameters of the problem, and enable the effect of crosswinds to be quantified. The predicted parabolic profiles are in agreement with some very simple calculations carried out by earlier investigators, revealing the limited range of validity of such calculations, and with the limited real bowling trajectories that have been observed. A full solution of the trajectory equations is then presented for the generic force coefficient characteristics, and a study of trajectories through the Reynolds number range is carried out. In the transition region between sub- and supercritical flows, complex ball trajectories are shown to be possible.

Imitation and Reinforcement Learning

In this article, we present both novel learning algorithms and experiments using the dynamical system MPs. As such, we describe this MP representation in a way that it is straightforward to reproduce. We review an appropriate imitation learning method, i.e., locally weighted regression, and show how this method can be used both for initializing RL tasks as well as for modifying the start-up phase in a rhythmic task. We also show our current best-suited RL algorithm for this framework, i.e., PoWER. We present two complex motor tasks, i.e., ball-in-a-cup and ball paddling, learned on a real, physical Barrett WAM, using the methods presented in this article. Of particular interest is the ball-paddling application, as it requires a combination of both rhythmic and discrete dynamical systems MPs during the start-up phase to achieve a particular task.

Integrable Lagrangians and modular forms

We investigate non-degenerate Lagrangians of the form ∫ f ( u x , u y , u t ) d x d y d t such that the corresponding Euler–Lagrange equations ( f u x ) x + ( f u y ) y + ( f u t ) t = 0 are integrable by the method of hydrodynamic reductions. We demonstrate that the integrability conditions, which constitute an involutive over-determined system of fourth order PDEs for the Lagrangian density f , are invariant under a 20-parameter group of Lie-point symmetries whose action on the moduli space of integrable Lagrangians has an open orbit. The density of the ‘master-Lagrangian’ corresponding to this orbit is shown to be a modular form in three variables defined on a complex hyperbolic ball. We demonstrate how the knowledge of the symmetry group allows one to linearize the integrability conditions.

Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy

Let α be an automorphism of the local universal deformation of a Calabi-Yau 3-manifold X, which does not act by ±id on $${H^3(X,\mathbb{C})}$$ . We show that the bundle $${F^2(\mathcal{H}^3)}$$ in the VHS of each maximal family containing X is constant in this case. Thus X cannot be a fiber of a maximal family with maximally unipotent monodromy, if such an automorphism α exists. Moreover we classify the possible actions of α on $${H^3(X,\mathbb{C})}$$ , construct examples and show that the period domain is a complex ball containing a dense set of CM points given by a Shimura datum in this case.

A group-theoretic characterization of the direct product of a ball and punctured planes

Employing the same technique as in our previous papers, we establish an intrinsic characterization of the direct product of a complex Euclidean ball and punctured planes in the category of Stein manifolds from the viewpoint of holomorphic automorphism group.

A variant of Jacobi type formula for Picard curves

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( τ ) with respect to the principal congruence subgroup Γ ( 2 ) of PSL ( 2 , Z ) has an expression by the Gauss hypergeometric function F ( 1 / 2 , 1 / 2 , 1 ; 1 - λ ) via the inverse of the period map for the family of elliptic curves E ( λ ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 2 ) , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | u | 2 < 0 } , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .

THERMAL VACUUM RADIATION IN SPONTANEOUSLY BROKEN SECOND-QUANTIZED THEORIES ON CURVED PHASE SPACES OF CONSTANT CURVATURE

We construct second-quantized (field) theories on coset spaces of pseudo-unitary groups U(N+,N-). The existence of degenerated quantum vacua (coherent states of zero modes) leads to a breakdown of the original pseudo-unitary symmetry. The action of some spontaneously broken symmetry transformations destabilize the vacuum and make it to radiate. We study the structure of this thermal radiation for curved phase spaces of constant curvature: complex projective spaces ℂPN - 1 = SU(N)/U(N - 1) and open complex balls ℂDN - 1 = SU(1,N - 1)/U(N - 1). Positive curvature is related to generalized Fermi–Dirac (FD) statistics, whereas negative curvature is connected with generalized Bose–Einstein (BE) statistics, the standard cases being recovered for N = 2. We also make some comments on the contribution of the vacuum (dark) energy to the cosmological constant and the phenomenon of inflation.

Finding the Maximum Modulus of a Polynomial on the Polydisk Using a Generalization of Steckin's Lemma

This paper is a generalization of the work of J.J. Green in Calculating the maximum modulus of a polynomial using Steckin’s Lemma. This lemma is generalized to higher dimensions and is used in an algorithm to locate the absolute global max of a polynomial on the polydisk. How to apply this algorithm to the real sphere and the complex ball is also explained.

Green's formula and the Hardy-Stein identities

This is a collection of some known and some new facts on the holomorphic and the harmonic version of the Hardy-Stein identity as well as on their extensions to the real and the complex ball. For example, we prove that if f is holomorphic on the unit disk D, then ??f ??Hp = ?f(0)?p + ?D?f'(z)? p-2 ?f'(z)?2(1-?z?) dA(z), (?) where Hp is the p-Hardy space, which improves a result of Yamashita [Proc. Amer. Math. Soc. 75 (1979), no. 1, 69-72]. An extension of (?) to the unit ball of Cn improves results of Beatrous an Burbea [Kodai Math. J. 8 (1985), 36-51], and of Stoll [J. London Math. Soc. (2) 48 (1993), no. 1, 126-136]. We also prove the analogous result for the harmonic Hardy spaces. The proofs of known results are shorter and more elementary then the existing ones, see Zhu [Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005, Ch. IV]. We correct some constants in that book and in a paper of Jevtic and Pavlovic [Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 36-52].

Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator C Φ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for C Φ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in C n (respectively R n ) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.