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Full-Text Articles in Mathematics
Prouhet-Tarry-Escott Problem, Juan Manuel Gutierrez
Prouhet-Tarry-Escott Problem, Juan Manuel Gutierrez
Theses Digitization Project
The purpose of this research paper is to gain a deeper understanding of a famous unsolved mathematical problem known as the Prouhet-Terry-Escott Problem. The Prouhet-Terry-Escott Problem is a complex problem that still has much to be discovered. This fascinating problem shows up in many areas of mathematics such as the study of polynomials, graph theory, and the theory of integral quadratic forms.
The Reflection Property In Normed Linear Planes With Applications To Generalized Conics, Edward J. O'Neill, Mostafa Ghandehari
The Reflection Property In Normed Linear Planes With Applications To Generalized Conics, Edward J. O'Neill, Mostafa Ghandehari
Mathematics Technical Papers - Archive
The ancient Greek mathematician Heron was the first to solve the problem of finding the shortest path from point A to point B on one side of the line L, subject to the condition that the path goes from A to L and then to B (figure 1). Figure 1. His solution involved going from A to point R on L and then to B such that the line segments AR and BR make equal angles with L. This is exactly the path a light ray from A to B if L were a mirror. Heron included this proposition in …
The Least Square Values And The Shapley Value For Cooperative Tu Games, Irinel C. Dragan
The Least Square Values And The Shapley Value For Cooperative Tu Games, Irinel C. Dragan
Mathematics Technical Papers - Archive
The Least Square Values (briefly LS-values), represent a family of values for cooperative transferable utility games, introduced by L. Ruiz. F. Valenciano and J.Zarzuelo (1998). For a fixed set of players N, a set of weights [see pdf for notation], and any TU-game v , M.Keane (1968) has defined and solved a quadratic programming problem: minimize the sum of weighted squares of deviations of the excesses from their average, on the preimputation set. He has shown that this problem has a unique solution which depends on the weights and has also given the new computational formula. Further, Ruiz/Valenciano/Zarzuelo have called …
On The Inverse Problem For Semivalues Of Cooperative Tu Games, Irinel C. Dragan
On The Inverse Problem For Semivalues Of Cooperative Tu Games, Irinel C. Dragan
Mathematics Technical Papers - Archive
In the present paper, we define a basis of [see pdf for notation] relative to a Semivalue, we compute the potentials of the subgames of a given game, to show that the basis is a potential basis, from which we get the Semivalues of the basic vectors. In this way we discover a basis of the null space of a Semivalue and derive, as in the previous work, a solution of the inverse problem, this time for a Semivalue. As the Shapley value was considered in detail in the previous work, we give a complete description for the Banzhaf value. …
On A New Mathematical Model Of The Molecular Bond, Donald Greenspan
On A New Mathematical Model Of The Molecular Bond, Donald Greenspan
Mathematics Technical Papers - Archive
Using a new model of the molecular bond, we produce a set of initial conditions for the ground state H2 molecule which yield, over one complete period, bond length and vibrational frequency which are identical to the experimental results.
Minkowski's Inequality For Convex Curves, Mostafa Ghandehari
Minkowski's Inequality For Convex Curves, Mostafa Ghandehari
Mathematics Technical Papers - Archive
Minkowski's inequality is a relation between mixed areas of two curves and their respective areas. The concept of mixed area is defined. A variational technique is used to give a proof of Minkowski's inequality. As a special case the isoperimetric inequality is obtained.
Discrete Spring Solitons, Donald Greenspan
Discrete Spring Solitons, Donald Greenspan
Mathematics Technical Papers - Archive
A soliton is a local, traveling wave pulse. It is significant in a variety of unrelated areas, like studies of water waves, acoustic waves, electron transfer, gravity waves, plasmas, optics and condensed matter (Miura [4], Newell [5]). In this paper we will generate and study solitons in a new context, that is, from the point of view of discrete string motion in the XY plane. Our approach enables one to elucidate the complexities of soliton interactions easily using only a scientific personal computer and velocity profiles. Detailed examples are presented and displayed graphically.
Particle Simulation In Contact Mechanics Of A Bouncing Elastic Ball, Donald Greenspan
Particle Simulation In Contact Mechanics Of A Bouncing Elastic Ball, Donald Greenspan
Mathematics Technical Papers - Archive
An elastic ball is simulated by considering lumped mass molecular sets called particles. Particles are allowed to interact only locally in a fashion similar to that employed in molecular mechanics. Under local and gravity forces an n-body problem results when initial data are prescribed. By scaling the forces, the resulting n-body problem is solved numerically using only a scientific personal computer. A variety of examples of bouncing balls are described and discussed.
Molecular Mechanics Simulations Of The Three Dimensional Cavity Problem, Donald Greenspan
Molecular Mechanics Simulations Of The Three Dimensional Cavity Problem, Donald Greenspan
Mathematics Technical Papers - Archive
The rapid development of computer technology has resulted in broad interest in three dimensional fluid simulation, which is known to have more complex force interactions than occur in two dimensions (see, e.g., refs. [1-6], [9], [11-15], [17], [18] and the additional references therein). In this paper we develop and analyze some molecular mechanics simulations of the three dimensional cavity problem. The fluid considered is water at 15° C.
Tennis, Geometric Progression, Probability And Basketball, Mostafa Ghandehari
Tennis, Geometric Progression, Probability And Basketball, Mostafa Ghandehari
Mathematics Technical Papers - Archive
The following problem about a tennis match is well—known. See Halmos [1, 2]. Consider 2n tennis players playing a single elimination match. Ask the question: what are the number of games played? The answer can be obtained in two ways. First using the geometric progression 2n-1 + 2n-2 + • • • -2+1 we find that the answer is 2n — 1. We can also explain the answer as follows: for each game played there is a loser. Thus the total number of games played is equal to the number of losers. Since there is only one winner the total …
Molecular Cavity Flow, Donald Greenspan
Molecular Cavity Flow, Donald Greenspan
Mathematics Technical Papers - Archive
Using molecular mechanics, cavity flow is studied in a basin of 4235 water molecules at 15°C. Primary vortices are generated with wallspeeds [see pdf for notation]. The vortex motions agree with experimental results in the large. Fully turbulent flow is generated with wallspeed .V = 3000°A/ps. The mechanisms for primary vortex generation are clearly delineated, as are those for turbulent flow.
The Generation Of Turbulence In A Compressed Gas, Donald Greenspan
The Generation Of Turbulence In A Compressed Gas, Donald Greenspan
Mathematics Technical Papers - Archive
A molecular mechanics type formulation is applied to generate turbulence in a compressed gas. Turbulence is defined in terms of the rapid appearance and disappearance of many small vortices. The generating mechanism lies in the large repulsive forces between particles.
Ray Optics On Surfaces, Mostafa Ghandehari
Ray Optics On Surfaces, Mostafa Ghandehari
Mathematics Technical Papers - Archive
Variational techniques are used to find path of light rays on surfaces. Using Fermat's principle of least time the problem is treated as a constraint optimization to obtain a system of partial differential equations.
New Mathematical Properties Of The Least Square Value, Irinel Dragan
New Mathematical Properties Of The Least Square Value, Irinel Dragan
Mathematics Technical Papers - Archive
The Least Square Values for cooperative TU games (briefly, LS-values) represent a family of solutions of the following family of optimization problems associated with a cooperative TU game: minimize the sum of weighted squares of deviations of the excesses from their average, on the preimputation set; the weights were positive numbers associated to the coalitions, depending only on their size. For each system of weights a LS-value is obtained by the Lagrange multipliers method; the Shapley value belongs to the family for specified weights. L. Ruiz, F. Valenciano and J. Zarzuelo who introduced this class of values (see[9]), gave two …
Conservative Motion Of Discrete, Tetrahedral Tops And Gyroscopes, Donald Greenspan
Conservative Motion Of Discrete, Tetrahedral Tops And Gyroscopes, Donald Greenspan
Mathematics Technical Papers - Archive
Tetrahedral tops and gyroscopes are simulated as discrete, rigid bodies in rotation by introducing a molecular mechanics formulation. The conservative, dynamical differential equations are solved numerically in such a fashion that all the system invariants are preserved. Examples which include precession and nutation are described and discussed.
Self-Circumference Of Rotors, Edward J. O'Neill, Mostafa Ghandehari
Self-Circumference Of Rotors, Edward J. O'Neill, Mostafa Ghandehari
Mathematics Technical Papers - Archive
The law of cosines from trigonometry is used to obtain elliptic integrals of the second kind to calculate the "self-circumference" of a Reuleaux triangle and the self-circumference of a rotor in an equilateral triangle. The Euclidean lengths of the polar duals of these sets with respect to their centers are expressed in terms of elliptic integrals of the second kind. Geometric inequalities for the polar duals of rotors in the plane are discussed.
Dynamical Simulation Of The Simplest Hydrides, Donald Greenspan
Dynamical Simulation Of The Simplest Hydrides, Donald Greenspan
Mathematics Technical Papers - Archive
In agreement with recent results of Gell-Mann and Hartle, we approximate electron motions in ground state Li7H1 and Li7H2 using an energy conserving numerical method for the solution of Newton's equations and a novel assumption about the interaction of the bonding electrons. Initial calculations for the first excited state of Li7H1 are also discussed. I. Introduction. Quantum dynamics is usually perceived through the time dependent Schrödinger equation, for which related analytical and computational problems appear to be insurmountable at the present time. There is however an alternate approach which can be implemented readily when the dynamical behavior is periodic. This …
Two Remarks On Totally Balanced Games, Juan-Enrique Martínez-Legaz
Two Remarks On Totally Balanced Games, Juan-Enrique Martínez-Legaz
Mathematics Technical Papers - Archive
Two results on totally balanced TU games are presented. It is first shown that the core of any subgame of a non-negative totally balanced game can be easily obtained from the maximal average value function of the game. The second result is a characterization of convex games as those games all of whose marginal games are totally balanced.
Computer Experiments For Molecular Motions And Chemical Bonding, Donald Greenspan
Computer Experiments For Molecular Motions And Chemical Bonding, Donald Greenspan
Mathematics Technical Papers - Archive
It is shown how to simulate first the ground state molecule H. The correct bond length, period, and electron cloud result. The method is then extended to ground state H2 and it is shown that only a new model of the chemical bond can yield the correct bond length, period, and electron cloud.
New Mathematical Properties Of The Banzhaf Value, Irinel C. Dragan
New Mathematical Properties Of The Banzhaf Value, Irinel C. Dragan
Mathematics Technical Papers - Archive
In a paper by P. Dubey and L.S. Shapley an axiomatic definition of the Banzhaf value has been extracted from an axiomatic definition of the Banzhaf power index (see [6]). Briefly speaking, the Banzhaf value axioms can be obtained from the Shapley value axioms by replacing efficiency by a similar axiom. This fact suggest that other recently discovered properties of the Shapley value can lead to similar properties for the Banzhaf value and this is the motivation for the present work.
Heron's Problem In The Minkowski Plane, Mostafa Ghandehari
Heron's Problem In The Minkowski Plane, Mostafa Ghandehari
Mathematics Technical Papers - Archive
Heron's problem in the Euclidean plane is that of minimizing the length of a path that joins two points on one side of a line and intersects the line. A generalization in the Minkowski plane is given and used to discuss reflection properties of conics.
A Semiclassical, Dynamical Model Of The Water Molecule, Donald Greenspan
A Semiclassical, Dynamical Model Of The Water Molecule, Donald Greenspan
Mathematics Technical Papers - Archive
A model of the ground state water molecule is formulated dynamically and studied by computer simulations. Excellent approximations of NMR determined bond angles and bond lengths are achieved through electron pairing and simulation of ice I.
A New Method For Approximating Quantum Dynamical Phenomena Over Short (And Sometimes Long) Time Periods, Donald Greenspan
A New Method For Approximating Quantum Dynamical Phenomena Over Short (And Sometimes Long) Time Periods, Donald Greenspan
Mathematics Technical Papers - Archive
Approximate electron and proton motions are simulated for short time periods for using a special case of the quantum mechanical Ehrenfest equations. Attention is directed only to ground state. A numerical method is employed which conserves the ground state energy at all times. Large proton motions are found, with electron trajectories oscillating between the protons, but concentrating around them.
Computer Modelling Of A Dynamical Water Molecule, Donald Greenspan
Computer Modelling Of A Dynamical Water Molecule, Donald Greenspan
Mathematics Technical Papers - Archive
A model of a ground state water molecule is formulated dynamically and studied by computer simulations. The enigmatic 104.5° bond angle is approximated by introducing the effects of neighboring molecules in a crystaline state. The numerical methodology used conserves the system energy at all times. Illustrative examples are described and discussed.
Particle Simulation Of Large Carbon Dioxide Bubbles In Water, Donald Greenspan
Particle Simulation Of Large Carbon Dioxide Bubbles In Water, Donald Greenspan
Mathematics Technical Papers - Archive
A method which can be applied to simulate the motion of fluid drops within fluids is described through a detailed study of a prototype problem, that is, the motion of carbon dioxide bubbles in water. The mathematical formulation uses classical molecular type formulas and results in an n-body problem which is solved numerically. The rise of the bubbles is described, as is the motion of the water near the bubbles. For variety, both H2O water and D20 heavy water are considered. Only workstation computer capabilities are required.
A Semiclassical, Dynamical Model Of Methane, Donald Greenspan
A Semiclassical, Dynamical Model Of Methane, Donald Greenspan
Mathematics Technical Papers - Archive
In this paper we present a semiclassical, dynamical model of the ground state methane molecule. The resulting 13-body problem is solved numerically using energy conserving methodology. The bond length and bond angle results are in complete agreement with experimental results.
A Temperature-Adjusted Ozone Attainment Criterion Based On Extreme Value Regression Analysis, D. L. Hawkins
A Temperature-Adjusted Ozone Attainment Criterion Based On Extreme Value Regression Analysis, D. L. Hawkins
Mathematics Technical Papers - Archive
We propose a new statistical methodology to adjust observed peak ozone levels to "normal" meteorology (say, temperatures) across the year for a given site. The adjusted ozone levels can be used to establish a yearly ozone attainment criterion without undue influence of meteorology. They can also be used to estimate ozone trends resistant to meteorological influences. The adjustment process is based on an extreme value regression model of the relationship between peak ozone, temperature and various precursors of ozone. The proposed temperature-adjustment criterion is applied to data for seven sites in Los Angeles County for each of the years 1989-1993, …
Melting Points Of Atomic And Homogeneous, Diatomic Molecular Solids Via The Four-Body Problem, Donald Greenspan
Melting Points Of Atomic And Homogeneous, Diatomic Molecular Solids Via The Four-Body Problem, Donald Greenspan
Mathematics Technical Papers - Archive
For a regular tetrahedral arrangement of four identical atoms, the minimum velocity of one atom, required for that atom to pass through the plane of the other three, is used to define the melting point of any solid composed of such atoms. The formula which results is [see pdf for notation], in which h is Planck's constant. Computations and results are described for helium, neon, argon, krypton, xenon and copper. The methodology is then extended to homogeneous, diatomic molecular solids and results are described for H2, D2, N2, 02, and C12.
Inference About The Transition-Point In Nbue-Nwue Or Nwue-Nbue Models, Subhash Kochar, D. L. Hawkins
Inference About The Transition-Point In Nbue-Nwue Or Nwue-Nbue Models, Subhash Kochar, D. L. Hawkins
Mathematics Technical Papers - Archive
e(t) < e(0) for [see pdf for notation]. If the inequalities for e(t) are reversed on these time intervals, it is called NWUE-NBUE. Using a characterization of such distributions in terms of the scaled total-time-on-test transform (STTT), we first give tests of exponentiality versus NBUE-NWUE or NWUE-NBUE with to unknown. This extends the work of Klefsjo (1989), who devised tests assuming that [see pdf for notation] is known. Then, assuming that F is either NBUE-NWUE or NWUE-NBUE, we give point estimates and asymptotic confidence intervals for to and [see pdf for notation]. The point estimates are asymptotically normal. We rely heavily on the theory of the empirical STTT process discussed in Csorgo, Csorgo and Horvith (1986). Two examples of real-data applications are provided.
Classical, Computer Studies Of One-Electron And Two-Electron Atoms And Ions, Donald Greenspan
Classical, Computer Studies Of One-Electron And Two-Electron Atoms And Ions, Donald Greenspan
Mathematics Technical Papers - Archive
A geometrical, computer-oriented method is utilized to generate circular orbits of electrons in one-electron and two-electron atoms and ions. The radii of these orbits are defined to be the radii of the atoms or ions. Only energy conserving computer methodology is used. The results suggest a natural modification of the Bohr approach to hydrogen which is then applied heuristically to helium.