Mathematical retroreflectors

Retroreflectors are optical devices that reverse the direction of incident beams of light. Here we present a collection of billiard type retroreflectors consisting of four objects; three of them are asymptotically perfect retroreflectors, and the fourth one is a retroreflector which is very close to perfect. Three objects of the collection have recently been discovered and published or submitted for publication. The fourth object - notched angle - is a new one; a proof of its retroreflectivity is given.Comment: 32 pages, 19 figure

On the two-dimensional rotational body of maximal Newtonian resistance

We investigate, by means of computer simulations, shapes of nonconvex bodies that maximize resistance to their motion through a rarefied medium, considering that bodies are moving forward and at the same time slowly rotating. A two-dimensional geometric shape that confers to the body a resistance very close to the theoretical supremum value is obtained, improving previous results.Comment: This is a preprint version of the paper published in J. Math. Sci. (N. Y.), Vol. 161, no. 6, 2009, 811--819. DOI:10.1007/s10958-009-9602-

Invisibility in billiards

The question of invisibility for bodies with mirror surface is studied in the framework of geometrical optics. We construct bodies that are invisible/have zero resistance in two mutually orthogonal directions, and prove that there do not exist bodies which are invisible/have zero resistance in all possible directions of incidence

High frequency limit of the Transport Cross Section and boundedness of the Total Cross Section in scattering by an obstacle with impedance boundary conditions

The scalar scattering of the plane wave by a strictly convex obstacle with impedance boundary conditions is considered. The uniform boundedness of the Total Cross Section for all values of frequencies is proved. The high frequency limit of the Transport Cross Section is founded and presented as a classical functional of the variational theory

Force acting on a rough disk spinning in a flow of noninteracting particles

Pressure force exerted on a rough disk spinning in a flow of noninteracting particles is determined by considering that a flow of point particles impinges on a body spinning around a fixed point. The rough disk is identical with the sequence of sets and thus the sets can be viewed as successive approximations of the rough disk. A proper choice of sequence of sets shows that the characteristic of billiard scattering is independent of n, and the billiard scattering on the rough set is defined. The pressure force exerted on the disk is independent of its angular velocity and that the characteristic of the interaction that is the moment of the pressure force slows down the rotation of the rough disk. The transverse force aligned with the instantaneous velocity of the front point of the body results in Magnus effect

Changes in Indicators of Microcirculation in the Early Postoperative Period in the Treatment of Diaphyseal Fractures of the Shin Bones Using a Plate with Limited Contact

Background. Unsatisfactory results of treatment, such as delayed consolidation and non-fusion of fractures, the formation of false joints and limb bone defects, have no tendency to decrease. We can assume that one of the leading factors of complications in traumatology is a violation of microcirculation in the affected segment of the limb.Aims. To identify patterns of changes in the parameters of the microcirculatory bed of the damaged segment of the lower limb when fixing bone fragments with a plate with limited contact in the early period after surgery.Materials and methods. In 25 patients, we studied four parameters of microcirculation of the lower limb segment with application of laser Doppler flowmetry. The control group consisted of 25 healthy volunteers, comparable in age and sex with the study group.Results. We found that in the early postoperative period (from the first to the 10th day after the surgery) in patients with diaphyseal fractures of the tibia operated with metal plate with limited contact there was an increase in microcirculation by 75.69 %, an increase in the proportion of the nutritive component of microcirculation compared to the shunt fraction by 24.64 %, as well as an increase in more than one ratio of the amplitude of the heart and respiratory range. All of that indicates a local circulatory disorder in the nutritive arterial hyperemia. We note that the increase in the amplitude of the respiratory component by 17.22 % and the equality of the amplitude of the cardiac range compared with the control group indicate violations of local blood circulation by the type of venous stagnation.Conclusion. On the basis of the results obtained, we note that patients with diaphyseal fractures of the shin bones treated with metal osteosynthesis with a plate with limited contact in the early postoperative period develop a violation of local blood circulation in the stagnant-hyperemic type

Microcirculation parameters of the damaged segment of the lower extremity after treatment of diaphyseal fractures using a locked intramedullary nail

Background. An in-depth analysis of the scientific works of scientists and medical practitioners allows us to conclude that locked nail intramedullary osteosynthesis is  the optimal and the most effective method of treating closed diaphyseal fractures of the lower leg bones, which is caused by the high stability of osteosynthesis and minimal damage to soft tissues during surgery. The processes of microcirculation changes in the early postoperative period by various metal structures, including a locked intramedullary nail, still remain unexplored. In particular, there is insufficient data on the use of a locked intramedullary nail. The aim of the study. To identify the features of changes in microcirculation indices of injured lower leg bones during fixation of fragments with a locked intramedullary nail in the early postoperative period. Materials and methods. The microcirculation of the lower limb segment was studied in 25 patients using laser Doppler flowmetry. Data from 25 healthy volunteers were used as a comparison group. Results. It was found that in the early postoperative period, from day 1 to day 10, in patients with diaphyseal fractures of the lower leg bones operated with locked nail intramedullary osteosynthesis, there is a decrease in the cardiac range, an increase in the share of the shunt component of microcirculation compared to the nutritional share, as well as an increase in more than 1 ratio of the cardiac and respiratory range amplitude, which indicates an ischemia type of local circulatory disorder. Compensation of ischemia is done by anastomoses, since the bypass rate is increased. Conclusion. In case of surgical treatment with locked nail intramedullary osteosynthesis, in the early postoperative period, an ischemic type of compensated local circulatory disorder develops. The regeneration process takes place under conditions of reduced arterial microcirculation blood flow and stable venous outflow, as well as the inclusion of anastomoses to compensate for destroyed vessels, which is associated with nail damage to the internal blood flow of the bone endosteum and intraosseous nutrient artery during the surgery

Digital receivers for low-frequency radio telescopes UTR-2, URAN, GURT

This paper describes digital radio astronomical receivers used for decameter and meter wavelength observations. This paper describes digital radio astronomical receivers used for decameter and meter wavelength observations. Since 1998, digital receivers performing on-the-fly dynamic spectrum calculations or waveform data recording without data loss have been used at the UTR-2 radio telescope, the URAN VLBI system, and the GURT new generation radio telescope. Here we detail these receivers developed for operation in the strong interference environment that prevails in the decameter wavelength range. Data collected with these receivers allowed us to discover numerous radio astronomical objects and phenomena at low frequencies, a summary of which is also presented.Comment: 24 pages, 15 figure

ELECTROCHEMICAL CHARACTERISTICS OF BIMETALLIC IMPLANT FOR OSTEOSYNTHESIS

The article presents experimental results of electrochemical characteristics of bimetallic implants in experiment