Julia sets of Zorich maps

The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

An improved minimal error model for the robotic kinematic calibration based on the POE formula

Summary The conventional product of exponentials $\left(\rm POE\right)$ -based methods dissatisfy the parametric minimality for the kinematic calibration of serial robots due to overlooking the magnitude and pitch constraints. Thus, the minimal kinematic model is presented to solve this problem, which can be developed further. This paper puts forward an improved algorithm for the minimal parameter calibration. An actual kinematic model with the minimal parameters $\left(\rm MP\right)$ is constructed according to the geometric properties of actual joint twists in the auxiliary frames established on the basis of the nominal joint axes. Then, the initial pose error is defined in the tool coordinate frame, which is expressed as the exponential map of the twist, and all twist descriptions are unified, so as to give a unified kinematic model in mathematics. By differentiating the kinematic model, a minimal error model is derived in explicit form. Subsequently, we propose a novel parameter identification method, which identifies the orientation error and position error parameters separately by the iterative least-squares method and updates the MP uniformly. Finally, the simulations and experiments on the different serial robots are conducted to verify the correctness and effectiveness of the proposed algorithm. The simulation results show our calibration algorithm outperforms the existing ones in the accuracy aspect, and the experiment result shows that the absolute pose accuracy of the UR5 industrial robot is upgraded about 9 times under a statistics sense after the calibration.

Radial kinetic nonholonomic trajectories are Riemannian geodesics!

Nonholonomic mechanics describes the motion of systems constrained by nonintegrable constraints. One of its most remarkable properties is that the derivation of the nonholonomic equations is not variational in nature. However, in this paper, we prove (Theorem 1.1) that for kinetic nonholonomic systems, the solutions starting from a fixed point q are true geodesics for a family of Riemannian metrics on the image submanifold $${{\mathcal {M}}}^{nh}_q$$ M q nh of the nonholonomic exponential map. This implies a surprising result: the kinetic nonholonomic trajectories with starting point q, for sufficiently small times, minimize length in $${{\mathcal {M}}}^{nh}_q$$ M q nh !

Loop Groups and QNEC

We construct and study solitonic representations of the conformal net associated to some vacuum Positive Energy Representation (PER) of a loop group LG. For the corresponding solitonic states, we prove the Quantum Null Energy Condition (QNEC) and the Bekenstein Bound. As an intermediate result, we show that a Positive Energy Representation of a loop group LG can be extended to a PER of $$H^{s}(S^1,G)$$ H s ( S 1 , G ) for $$s>3/2$$ s > 3 / 2 , where G is any compact, simple and simply connected Lie group. We also show the existence of the exponential map of the semidirect product $$LG \rtimes R$$ L G ⋊ R , with R a one-parameter subgroup of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) , and we compute the adjoint action of $$H^{s+1}(S^1,G)$$ H s + 1 ( S 1 , G ) on the stress energy tensor.

Dense images of the power maps for a disconnected real algebraic group

Let 𝐺 be a complex algebraic group defined over ℝ, which is not necessarily Zariski-connected. In this article, we study the density of the images of the power maps g → g k g\to g^{k} , k ∈ N k\in\mathbb{N} , on real points of 𝐺, i.e., G ⁢ ( R ) G(\mathbb{R}) equipped with the real topology. As a result, we extend a theorem of P. Chatterjee on surjectivity of the power map for the set of semisimple elements of G ⁢ ( R ) G(\mathbb{R}) . We also characterize surjectivity of the power map for a disconnected group G ⁢ ( R ) G(\mathbb{R}) . The results are applied in particular to describe the image of the exponential map of G ⁢ ( R ) G(\mathbb{R}) .

The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

<div>This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. </div><div><br></div>

The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

<div>This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. </div><div><br></div>

New Parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and Some Explicit Formulas for Its Logarithm

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\bbr)\cong\mathrm{SL}(2,\bbr)$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\bbr)\to\mathrm{SL}(2,\bbr)$, we have found a few explicit formulas for the logarithm of the matrices in these groups.\\ Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

Cayley Map for Symplectic Groups

Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.

An Analytic Characterization of p , q -White Noise Functionals

In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.