On the number of two-dimensional threshold functions

A two-dimensional threshold function of k-valued logic can be viewed as coloring of the points of a k x k square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.Comment: 17 pages, 2 figure

Rosenblatt's first theorem and frugality of deep learning

First Rosenblatt's theorem about omnipotence of shallow networks states that elementary perceptrons can solve any classification problem if there are no discrepancies in the training set. Minsky and Papert considered elementary perceptrons with restrictions on the neural inputs: a bounded number of connections or a relatively small diameter of the receptive field for each neuron at the hidden layer. They proved that under these constraints, an elementary perceptron cannot solve some problems, such as the connectivity of input images or the parity of pixels in them. In this note, we demonstrated first Rosenblatt's theorem at work, showed how an elementary perceptron can solve a version of the travel maze problem, and analysed the complexity of that solution. We constructed also a deep network algorithm for the same problem. It is much more efficient. The shallow network uses an exponentially large number of neurons on the hidden layer (Rosenblatt's $A$-elements), whereas for the deep network the second order polynomial complexity is sufficient. We demonstrated that for the same complex problem deep network can be much smaller and reveal a heuristic behind this effect

Epidemiological Risk of Dirofilariasis in the Voronezh Region

Background. Dirofilariasis is a vector-borne helminthiasis that tends to expand spatial boundaries in areas with temperate and cold climates. In the Voronezh oblast, located in the Central Black Earth Region of the Russian Federation there were previously isolated reports of cases of dirofilariasis in humans and dogs, but this issue was not specifically studied.Aims. To determine the degree of infection of dogs with Dirofilaria immitis and Dirofilaria repens in the Voronezh oblast in order to determine the level of epidemiological risk.Materials and methods. For seven years, targeted work was carried out to study the spread of different types of dirofilariae in dogs in the Voronezh region. Blood from 3 498 dogs of different breeds, ages, conditions of keeping and household use was examined. Dirofilaria larvae were counted using the Fuchs-Rosenthal camera. The type of dirofilaria larvae isolated from the blood of dogs was determined by the histochemical method. The antigen of the imaginal form of D. immitis was detected using a chromatographic test system the ImmunoRun Antigen Detection Kit Caninae Heartworm Biogal chromatographic test system (Israel). Statistical processing of the material was carried out in the Microsoft Excel 2010 program. Studies were carried out with the consent of animal owners and in accordance with the European Convention for the Protection of Vertebral Animals used for experimental and other scientific purposes (Strasbourg, 1986).Results. As a result of the conducted studies, it was found that in the Voronezh oblast, the average infection of domestic dogs (Canis familiaris) with dirofilariasis is 14.14 ± 0.31 %. The indicator depends on the conditions of keeping animals and makes 27.15 % in stray dogs, 25.26 % – in service dogs and 10.54 % – in apartment dogs. In service dogs, the species D. repens occurs in 52.39 % of cases, D. immitis – in 42.95 %, mixtinvasia – in 4.80 % of cases. In stray dogs, the species D. repens prevails (80.10 %). The second type of D. immitis was established in 10.0 % of cases. At the same time, both types were also established in 10.0 % of cases. The maximum number of cases of dirofilariasis occurs in urban dogs (65.87 %), which is due to the high density of their population and close contact with the vector of transmission in urbanized areas. The seasonal dynamics of infestation is determined by the year-round detection of cases with a peak in October (46.77 %). The number of cases of dirofilariasis increases with age, in dogs from one to six years old it reaches maximum of 63.26 %. There is an epidemiological risk of dirofilariasis in the Voronezh oblast. This is due to the existence and active functioning of synanthropic and natural foci of invasion on the territory of the Voronezh oblast

Monitoring the wear of the refractory lining in the blast-furnace hearth

The Razgar Gorna computer program is developed for calculating two-dimensional temperature fields in any vertical and horizontal cross section of the blast-furnace hearth. In the calculations, the heat-conduction equations are solved by means of readings from many temperature sensors (up to 1000, depending on the volume) installed within the furnace lining between the refractory modules. Continuous temperature monitoring at each point permits the determination of the remaining lining thickness and prediction of the onset of lining wear, as necessary. A mathematical model is employed in continuous temperature monitoring of the lining. The database of the Razgar Gorna program relies on the collection, analysis, and transmission of information from the temperature or heat-flux sensors. The program is in use at blast furnaces in Chinese steelworks at Jinan (two furnaces), Jiyuan, and Liuzhou. © 2013 Allerton Press, Inc