The Critical Strips of the Sums 1 + 2z + ... + nz

We give a partition of the critical strip, associated with each partial sum 1 + 2z + ... + nz of the Riemann zeta function for Re z < −1, formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.

On some solutions of a functional equation related to the partial sums of the Riemann zeta function

In this paper, we prove that infinite-dimensional vector spaces of α-dense curves are generated by means of the functional equations f(x)+f(2x)+⋯+f(nx)=0, with n≥2, which are related to the partial sums of the Riemann zeta function. These curves α-densify a large class of compact sets of the plane for arbitrary small α, extending the known result that this holds for the cases n=2,3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the nth power of the density approaches the Jordan content of the compact set which the curve densifies.

Geometric 3D point cloud compression

The use of 3D data in mobile robotics applications provides valuable information about the robot’s environment but usually the huge amount of 3D information is unmanageable by the robot storage and computing capabilities. A data compression is necessary to store and manage this information but preserving as much information as possible. In this paper, we propose a 3D lossy compression system based on plane extraction which represent the points of each scene plane as a Delaunay triangulation and a set of points/area information. The compression system can be customized to achieve different data compression or accuracy ratios. It also supports a color segmentation stage to preserve original scene color information and provides a realistic scene reconstruction. The design of the method provides a fast scene reconstruction useful for further visualization or processing tasks.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

Technologies for Anatomical and Geometric Characterization of the Corneal Structure and Anterior Segment: A Review

Corneal and anterior segment imaging techniques have become a crucial tool in the clinical practice of ophthalmology, with a great variety of applications, such as corneal curvature and pachymetric analysis, detection of ectatic corneal conditions, anatomical study of the anterior segment prior to phakic intraocular lens implantation, or densitometric analysis of the crystalline lens. From the Placido-based systems that allow only a characterization of the geometry of the anterior corneal surface to the Scheimpflug photography-based systems that provide a characterization of the cornea, anterior chamber, and crystalline lens, there is a great variety of devices with the capability of analyzing different anatomical parameters with very high precision. To date, Scheimpflug photography-based systems are the devices providing the more complete analysis of the anterior segment in a non-invasive way. More developments are required in anterior segment imaging technologies in order to improve the analysis of the crystalline lens structure as well as the ocular structures behind the iris in a non-invasive way when the pupil is not dilated.

A note on the real projection of the zeros of partial sums of Riemann zeta function

This paper proves that the real projection of each simple zero of any partial sum of the Riemann zeta function ζn(s):=∑nk=11ks,n>2 , is an accumulation point of the set {Res : ζ n (s) = 0}.

Computing the zeros of the partial sums of the Riemann zeta function

In this paper, we introduce a formula for the exact number of zeros of every partial sum of the Riemann zeta function inside infinitely many rectangles of the critical strips where they are situated.

Visual perception for the 3D recognition of geometric pieces in robotic manipulation

During grasping and intelligent robotic manipulation tasks, the camera position relative to the scene changes dramatically because the robot is moving to adapt its path and correctly grasp objects. This is because the camera is mounted at the robot effector. For this reason, in this type of environment, a visual recognition system must be implemented to recognize and “automatically and autonomously” obtain the positions of objects in the scene. Furthermore, in industrial environments, all objects that are manipulated by robots are made of the same material and cannot be differentiated by features such as texture or color. In this work, first, a study and analysis of 3D recognition descriptors has been completed for application in these environments. Second, a visual recognition system designed from specific distributed client-server architecture has been proposed to be applied in the recognition process of industrial objects without these appearance features. Our system has been implemented to overcome problems of recognition when the objects can only be recognized by geometric shape and the simplicity of shapes could create ambiguity. Finally, some real tests are performed and illustrated to verify the satisfactory performance of the proposed system.