In this note, a simple description of zone theorem in three dimensions is given.Arrangements in three dimensions are useful for constructing higher-order Voronoi diagrams in plane. An elementary and very intuitive treatment of this result is also given.

<p><em>Dark matter is one of the most mysterious physics observations today and has puzzled the scientific community for decades. We can detect its gravitational effect but not the matter itself. After years of study and billions of dollars in spending by some of the top researchers in the world, we are only uncovering more questions. By looking at the problem from a different angle, the fundamentals of gravitation, I believe the following new gravity force equations in this paper can solve dark matter puzzle we face. </em></p> <p>F=Y_1 (f_1 f_2)/r^2</p> <p>F=Y_2 fm/r^2</p> <p><em>With every month comes new papers and ideas on this topic. One theory attempts to explain dark matter as photons with very small amounts mass and that the numbers of photons in the universe could cover the observed 85% of missing matter. This theory has not progressed far, as with every other </em>hypothetical tiny particle theory for dark matter; however, I do believe that photons create the dark matter observations we see.</p> <p>My study believes EM waves, or photons, are the reason for the observations of dark matter but not because they have mass. If my new theory of gravity is correct, EM waves or photons are massless but would still have a gravitational effect. While most theories try to explain the source of dark matter, I used dark matter observations as the best evidence for this new theory of gravity.</p>

In standard quantum theory, symmetry is defined in the spirit of Klein's Erlangen Program: the background space has a symmetry group, and the basic operators should commute according to the Lie algebra of that group. We argue that the definition should be the opposite: background space has a direct physical meaning only on classical level while on quantum level symmetry should be defined by a Lie algebra of basic operators. Then the fact that de Sitter symmetry is more general than Poincare one can be proved mathematically. The problem of explaining cosmological acceleration is very difficult but, as follows from our results, there exists a scenario that the phenomenon of cosmological acceleration can be explained proceeding from basic principles of quantum theory. The explanation has nothing to do with existence or nonexistence of dark energy and therefore the cosmological constant problem and the dark energy problem do not arise. We consider finite quantum theory (FQT) where states are elements of a space over a finite ring or field with characteristic $p$ and operators of physical quantities act in this space. We prove that, with the same approach to symmetry, FQT and finite mathematics are more general than standard quantum theory and classical mathematics, respectively: the latter theories are special degenerated cases of the former ones in the formal limit $p\to\infty$.

This paper solves the problem of division by zero. Starting with a deep analysis of multiplication and division, it is presented that they are one in the same, as an operation. Together with the operation of selection, they are different forms of the transformation operation, that changes one pair of numbers into another pair of numbers. It is presented that the numbers are always in reference to some other numbers. Therefore, the only correct form of the number, is when it is a ratio between the certain value and the base measure, which this value is related to. It is clearly proven that the problem of division by zero is the result of unauthorized simplification, which was done by bringing all the rational numbers (which are represented by the ratio of the value to the measure) to the fractions, with a denominator of 1. This paper shows why we should not make fractions with different numerators and denominators equal, even so they seem to be in the same proportion. Finally, there are two graphs with the functions presented 1/x and tg(x) with their discontinuity points, which are visible only on the real numbers graph. This result is from projecting points of the rational numbers graph onto the real numbers graph with a denominator of 1. These discontinuity points disappear when we present our graphs in rational numbers. This paper fixes one of the foundations of mathematics and shows us how to divide by zero. This is only the first issue that was solved, but based on the proper understanding of numbers in their natural form, still many other things need to be fixed. It is still difficult to predict all the consequences, which this change can bring, to everything that was created based on this wrong foundation.

<p>A distributed brute force algorithm to solve the problem of diminishing returns when more processors are added to a password hash cracking system. Using GBFP, the brute force password cracking algorithm can be greatly reduced in computation time and greatly optimized. This algorithm also helps with the issue of cracking a password hash with many processors with differing clock speeds as well.</p>

In this book authors for the first time introduce a new method of building algebraic structures on the interval [0, n). This study is interesting and innovative. However, [0, n) is a semigroup under product, × modulo n and a semigroup under min or max operation. Further [0, n) is a group under addition modulo n.

In this book the authors introduce the notion of finite complex modulo integer intervals. Finite complex modulo integers were introduced by the authors in 2011. Now, using this finite complex modulo integer intervals several algebraic structures are built. Open problems are suggested.

<p>This paper is devoted to the concepts of neutrosophic upper and neutrosophic lower contra-continuous multifunctions and also some of their characterizations are considered.</p>

<p>In this paper we introduce the concepts of neutrosophic upper and neutrosophic lower semi-continuous multifunctions and study some of their basic properties.</p>

Application of force accompany with energy consumption is the basic common sense. Based on this common sense, Newtonian mechanics system is studied according to energy transformation and conservation law. A concept of dissipation work is proposed and a calculation formula of dissipation work is derived. Meanwhile, thermal effect of force is obtained through inference, kinetic energy theorem is amended, and supplements of angular kinetic energy theorem and the law of conservation of angular kinetic energy are made. Through these researches, celestial body movement is found to comply with conservation of angular kinetic energy, rather than the conservation of angular momentum or the so-called conservation of mechanical energy. Also, the potential energy, which is superfluous in Newtonian mechanics, is found not to be energy. Therefore, the unity of conservative forces and non-conservative forces is realized. The revised Newtonian mechanics will have a profound influence on subjects and relative research fields of physics, astronomy and astronautics.