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Invariants of Finite Metric Spaces
Understanding Distance (Invariants of Finite Metric Spaces): Metric spaces are one of the key notions in mathematics. A metric space is a set (whose elements are ``points’’) equipped with a notion of distance, that is to every pair of points we assign a nonnegative real number, the distance between them. This geometric notion has found many application, not only in geometry itself but also in algebra and number theory, where we can use the relation of division by a given prime number to define distance between integers. The reason for usefulness of metric spaces lies within the definition, which is simple enough to capture a broad class of examples, while expressive enough to build a rich theory. My paper approaches the study of (finite) metric spaces by assigning to them various invariants which are less complicated than the spaces themselves, but allow us to distinguish between spaces in a simple way. These invariants are twofold: algebraic and combinatorial. The algebraic invariant is the ``isometry group’’ of a space, that is a set with a binary operation (we can think of it as ``multiplication’’); the combinatorial one is a graph (a set of vertices connected by edges) which decodes information about distance in a simpler way that is easier to implement. The main theorem shows how to compare these invariants. Professor Peter Clark (University of Georgia, USA) suggested on a mathematical forum, Math Overflow, that these invariants contain the same information about a given metric space. The paper shows that this in fact is not the case and studies the relation between them in detail.
Is knee joint degeneration predetermined by the shape of the patella?
I analyse data from 21,124 knee joint arthroscopies and find, that the most frequent location of articular cartilage degeneration (ACD) in the knee joint is a patella region. In the studied sample, there are 28 % cases of severe, 60 % – of mild, and 12 % – of normal type anatomic dysplasia of the patella shape. I show that the anatomic shape of the patella has strong influence on ACD in the knee joint. The patients with ACD in the patella region, possessing severe dysplasia of the patella shape, have larger areas of ACD and lower physical activity levels, compared to the patients with normal or mild dysplasia of the patelpa shape.
LOCUS OF THE FOCUS
Myopia (short-sightedness) occurs when the eye is too long for its focusing structures so that light rays converge in front of, rather than on the retina, producing a blurry image. Hannah investigated a recent theory that maintaining clear peripheral vision is important for slowing the progression of myopia. She studied rapidly growing chicks rather than human beings, fitting diffuser lenses over one eye of each to blur their central vision, their peripheral vision or both. The uncovered eyes of each chick were used as controls. After 3 days of these treatments the lenses were removed and she measured the focusing error, corneal curvature and the length of each eye in order to quantify the progression of myopia. Hannah found that the control eyes without the lenses were not myopic, but all eyes covered by a diffuser lens developed myopia. The degree of myopia was greatest in the eyes where both central and peripheral vision was blurred, followed by the eyes where the central vision was blurred. Myopia development was least, but still significant, where only the peripheral vision was blurred. These results indicate that form deprivation in any part of the retina contributes to myopia development, but unlike previous studies Hannah's suggests that the blurring in the central retina has a much greater effect than in the peripheral region. She goes on to suggest that it is the number of photoreceptors exposed to blurring rather than their location that matters in the development of myopia - an hypothesis now under test at the University of Auckland. Hannah's results have considerable relevance to the question of whether peripheral defocus caused by wearing glasses might accelerate myopia progression in children.
LSLLSLSLLSLLSLS – Modern Mathematics in Islamic Mosaics
“Islamic artists were 500 years ahead of Western scientists” –headlines like this one appeared all over the world when an acclaimed Science paper investigated Islamic mosaics in 2007. These so-called girih patterns were suggested to have been conceived as tilings, the jigsaw-like geometrical concept of putting shapes together to cover the plane. The Science authors argued that some of these Islamic tilings have a quasi-periodic structure – whereas in the Western world, such complex structures were not invented until the 1970s. However, my analysis of the Science patterns and six additional mosaics from Iran suggests that this sensational claim should be revised: The Science proof of the quasiperiodicity is not convincing. In addition, the mosaics were not constructed in the 15th but rather in the 18th or 19th century. My discovery of Ammann bars – a feature of quasi-periodic tilings – in girih patterns might serve as an alternative proof of their quasi-periodicity.
Lightning in the Rainy Window/A water curtain functioning as an air ionizer combining Kelvin dropper and Van-de-Graaf generator in a single device
This paper presents a water curtain which can be used for decorative purposes as a high-tech interior design feature. It differs from the known prototypes by an added function of an air ionizer. The device could contribute to health and safety of indoor public spaces and could be crucial for preventing epidemics. Indoor public spaces have a high concentration of germs and other harmful pollutants which are mainly charged positively and can be destroyed by negative ions. The presented device induces electrostatic charges on falling water drops and acts as a small scale model of lightning in the clouds. The water streams are electrified as in Lord Kelvin’s invention and the separated charges are transferred to spherical high voltage dischargers by a simple devise working on the principles invented by Van-de-Graaf. Electrostatic energy is released through coronal discharges generating ions of both signs. The relative concentration of positive and negative ions is adjustable.
Low cost equipment for the physics laboratory
Luca Maria Colombo Gomez, Daniele Maggioli, Gionata Pandini
This work was born by an original idea of Dr. Roberto Giudici, our lab engineer, with the aim of improve our physics laboratory. We bild two instrument, "The Magic Box" and "Panda". The Magic Box is a platform, composed of an electronic circuit and a pc software to which it is possible to connect some different sensors using common phone cables. The data acquired by the sensor are sent to a pc which elaborate them. We also built an alternative version of the platform, called Panda, cheaper in prize and easier to carry. Instead of a normal pc, we used a microcomputer called RaspberryPi, which can do all the essential operations needed during the experiments but costs much less than a traditional PC. This project was very useful for us because we had the opportunity to get into new scientific topics, such as programming and electronic, which are not studied in our school. All the equipment we produced will be used in our school laboratory to make several experiments. We also adopted an open-source mentality, in order to make all our work available to students of other schools, who will be able to change and improve the projects. With our instrumentation every school, or a single person too, could make his own laboratory with some cheap but fully functional workstations. It's really important to create a network of users, both school or private person, that will develop new equipments and help new users to join the project.
Map Coloring - Finding the Number of Colorings of Maps Colorable with Four Colors
Eero-Pekka Räty, Samuli Thomasson
The four-color theorem states that every planar map can be colored with four colors such that every adjacent region is colored with a different color. However nothing is said about the number of possible colorings. This research inspects the impact of the features of planar maps on the number of ways the regions can be colored through graph theory and computer-assisted simulation. We present a relation between the features and colorings applicable to e.g. time-complexity analysis and optimization of algorithms. We created a program which calculates the possible colorings and inspected a total of 633 maps used to prove the four-color theorem. We present a formula y = 3338754,0 ∙1, 7997866 ∙ z -17;705562, which ties together the colorings y, the vertices (regions) x and the vertices’ proportion to edges (adjacent regions) z. The formula’s coefficient of determination in our cases was 99.98%, which is enough for analysing computational complexity.
Measuring water waves
These slow-motion photographs never cease to amaze: A drop falls onto the surface of the water, which is smooth as glass. While part of the drop disappears into the water, the rest bounces out again. Invisible to the human eye, this spectacle repeats itself many times within fractions of a second. Usually, high-speed cameras are used to make this process visible. However, such cameras are expensive - which is why Daniel Pflueger thought of a more cost-effective method. His method does not photograph the drop hitting the water directly, but analyses the water waves generated in the process. By precisely measuring the height of the waves using a laser and digital camera, Daniel Pflueger was able to provide an initial approach to measuring the complex water play when the drop hits the water.
Molecular aspects of nitrogen excess on wheat plants in the vegetative stage
Increased content of inorganic nitrogen in soil is responsible not only for various ecological problems, but also induces stress to plants. In other hand, in some environmental or biotic stresses (pathogens) is it’s consumption increased over the optimal level of unstressed plants because of high energetic costs of defensive responses (synthesis of defensive proteins and other nitrogenous compounds). Unlike physiology of insufficient nitrogen nutrition or physiology of various stresses is physiology of excess nitrogen nutrition very poorly investigated. Objective of our work was to investigate impact of high nitrogen dosages in form of ammonium nitrate to plants of wheat in vegetative stage in ideal growth conditions. Our applied dosages of inorganic nitrogen in watering solutions (7 – 35 mM) induced inhibited growth and production of biomass, thinning of stems and decrease of count and total area of stomata in the epidermis of the leafs. Plants stressed by excess nitrogen in soil in our experiments in increased rate accumulated chlorophylls, free proline and soluble proteins, mostly small and large subunit of enzyme RuBisCO. Activity of isoforms of glucanase with lower molecular weights (30 kDa and 38 – 40 kDa) was impaired, rather than activity of heavier isoform (49 – 52 kDa) was stimulated. With our work we contributed to extension of still poor knowledge of physiology of plants in condition of excess nitrogen supply. Results can be used in further experiments leading to improvement of management of nitrogenous fertilization, especially in conditions of various stresses.
Music A' Clock
Music A’Clock is a musical innovation and a tool to studying music theory. It is a device and teaching method that looks like a round clock face with twelve keys on it. With pointers and stencils the student can learn names and use of notes, chords, intervals, scales and grades visually instead of having to memorize them. Music A’Clock revolutionizes the traditional approach to teaching music theory, because it is a sensible, logical and above all a visual tool to perceive music theory, which is often considered complicated and difficult. This is due to the fact that in the current system theoretical concepts are taught to too young children without connection to their playing skills and developmental stage. Music A’Clock fills this pedagogical gap, prevents students from frustration, helps to motivate and maintain interest – with visuality, practicality and playability. A’Clock enables the student to concentrate on music and musicality during the learning process. Patent for Music A‘ Clock is pending (Jan 2013) and marketing research is currently in process. Music A’Clock includes various applications. There are Piano A’Clock, Rhythm A’Clock, Guitar A’Clock and Xylo A’Clock.