Geometry, Analysis and Mechanics

The following sections are included:

• PREFACE

• CONTENTS

ARCHIMEDES was born in the Greek city Syracuse on the island of Sicily around 287 B.C. and was killed by a Roman soldier in Syracuse around 212 B.C. when he told him “Do not disturb my circles”. Little is known of the life of Archimedes —the greatest mathematician, physicist and engineer. His father Pheidias (Phidias) was an astronomer who studied the distances and the sizes of the moon and the sun. Archimedes was intimate with the ruler of Syracuse, King Hieron II and his son Gelo(n), and was probably a relative to them (according to Plutarch and Polybius). He visited the great intellectual center of Alexandria where he worked with Conon of Samos, Dositheus of Pelusium, and Eratosthenes of Cyrene, who were pupils of Euclid. The siege and capture of Syracuse by Marcellus provided the occasion for the appearance of Archimedes as a personage in history. His burning mirrors, catapults, cranes, and many other machines were used with great effect against the Romans in the siege of Syracuse. Archimedes took all the mathematical sciences for his province: Arithmetic, Geometry, Astronomy, Mechanics, and Hydrostatics. Of his writings, although some have been lost the most important have survived. Archimedes’ entire preoccupation by his abstract studies is illustrated by a number of stories. He would forget all about his food and neglect.his person and all necessities of life. He used to trace geometrical figures in the ashes of the fire or in the oil of his body. The most famous of these stories is when he discovered in a bath the solution of the question referred to him by Hieron, as to whether a certain newly made crown supposed to have been made of gold did not in fact contain a certain proportion of silver, he ran naked through the streets to his home shouting “Eureka, Eureka!” (I have found it, I have found it!). This story is associated with his discovery of the basic law of Hydrostatics which states that a floating body displaces its own weight of liquid…

Efficient schemes for numerical integration in multi-dimensions are presented. We convert the computation of an integral in n-dimensions to the solution of n first order ordinary differential equations. Angelopoulos’ method is then applied to solve these initial value problems. Numerical experiments are presented to demonstrate the efficiency of the proposed schemes.

In recent years considerable interest has been focused on nonlinear diffusion problems, a typical equation for these being: . Here, the solution u = u(x,t) is defined over some space-time domain of the form Ω ×[0,T], and f(u) is a given real function of u. These applications have become more widespread since many problems have been shown to lead to an equation of this type or to its time-independent counterpart, elliptic equation of equilibrium:. Particular cases arise, for example, in population genetics, the physics of nuclear stability, the Lend-Emden equation of astrophysics, various combustion models, and in determining metrics which realize given scalar or Gaussian curvatures. In this paper we propose finite difference schemes to solve numerically these equations. Our idea is to introduce a pseudo-time variable and then solve numerically the so obtained partial differential equation using techniques developed by the authors in some of their previous published work. Numerical experiments are conducted in order to illustrate the power of these methods.

Model equations describing velocity and stress in a one-dimensional elastic body are discussed. Solutions satisfying a proportionality assumption are characterized by a single hyperbolic PDE. Conditions for the existence of classical solutions to this equation are given, weak solutions are studied as well. For the numerical solution a semidicretization method using a moving grid is proposed.

We provide an existence and uniqueness theorem for strong solutions of a general class of integral equations involving operators acting on the Finite-Difference Fock space. We also examine the dependence of solutions on initial conditions and coefficients, and provide a necessary and sufficient condition in order for the solutions to be unitary.

The Euler formula for convex polyhedra is characterized by means of several sets of functional equations which arise when some perturbations are performed on polyhedra. Some single perturbations characterize the formula by itself, but in other cases two perturbations are required.

In the context of algebraic varieties over a field k relative homotopy groups based on parabolic and spherical cylinders (among other types) are shown to exist. A sequence of relative homotopy groups analagous to that found in ordinary homotopy theory, but more suitable for specialized studies in algebraic geometry and alternating between the parabolic and spherical cases, is proven to be of order two.

The functional equation method is used to show that the uniqueness and existence of solutions of a mixed problem for a hyperbolic equation depend on both the parameter and the solution space. A new kind of phenomenon is then revealed.

Archimedes computed π very accurately. Much later, Ramanujan discovered several infinite series for 1/π that enables one to compute π even more accurately. The most impressive one is^[1] ((a)_k denotes, as usual, a(a+1)…(a+k−1).)

The following sections are included:

• INTRODUCTION

• PROBLEMS

• References

The following sections are included:

• INTRODUCTION

• PRELIMINARIES

• THE WELL KNOWN CONSTRUCTIONS OF Z

• OUR CONSTRUCTION OF Z AND OUR PROOF OF (P)

• References

It was proved by Euler in 1754 that a prime of the form p ≡ 1 (mod 4) can be uniquely represented as a sum of two squares. But other integers of the form n ≡ 1 (mod 4) also enjoy this property. For example, 45 = 3² + 6² and 637 = 14² + 21² are unique as such representations. It is clear, therefore, that not all composite integers of the above form are representable as a sum of two squares in at least two ways. This, however, is inevitable for composite integers, all the prime factors of whose are of the form p ≡ 1 (mod 4).

It is rather easy to construct odd composite integers free from primes of the form q ≡ 3 (mod 4), as it is enough to keep a and b prime to each other, in the constructive formula n = a² + b². The proof for that is already known as an application of the Fermat's Little Theorem, and will be repeated herein for easy reference.

The Two Squares Theorem, which is the main task in this piece of work, states by induction that an integer, all the prime factors of whose are of the form, p ≡ 1 (mod 4), is composite, if and. only if, it can be expressed as a sum of two squares in at least two ways.

The steady free surface flow induced by a submerged source in a fluid of finite or infinite depth is examined analytically and numerically.

This note analyzes the main ideas in a little known paper by Georges Louis Leclerc, Comte de Buffon (1707-88) on Archimedes and the invention of longdistance burning mirrors as an informative document on Buffon himself as well as on eighteenth century physics and its roots.

We prove that the fundamental matrix function c₂(A) is characterized up to a constant by the following properties: (i) c₂(AB) = c₂(BA), (ii) c₂ (A) is a quadratic function, and (iii) c₂(J) = 0 where J is the matrix will all entries equal to 1.

Most work to date on computational complexity has been for combinatorial complexity. Over the last thirty years, however, also a complexity theory for numerical (or continuous) problems was developed. There is much recent research on the complexity (worst case complexity, average case complexity, Monte Carlo complexity) of problems such as numerical integration, optimization of continuous functions, zero finding, solution of differential and integral equations, and ill-posed problems.

This survey is based on recent work of different authors and on several previous surveys listed at the end of the paper. I also used unpublished notes for several lectures, given at Normal University Beijing in 1990.

F. G. Tricomi (1923) originated the theory of boundary value problems for mixed type equations by imposing the Tricomi equation: yu_xx + u_yy = 0 which is hyperbolic for y < 0, elliptic for y > 0 and parabolic for y = 0. Then A. V. Bitsadze and M. A. Lavrentjev (1950) considered the Bitsadze-Lavrentjev equation: sgn (y)u_xx + u_yy = 0. S. Nocilla (1957), C. Ferrari (1959) have applied this theory in fluid mechanics . In 1990, we established well-posedness for the Tricomi-Bitsadze- Lavrentjev Problem for equation: sgn (y)u_xx +u_yy +r(x,y)u = f(x,y) in the sense that there is at most one quasi-regular solution and a weak solution exists. In this paper we consider the Tricomi Problem for equation: k(y)u_xx + u_yy + r(x,y)u = f(x,y) and establish well-posedness in the above sense. This problem is interesting in transonic aerodynamics.

Applying the Cayley-Hamilton theorem and standard trace, and introducing tracelike forms, we establish a new formula for the computation of the inverse of an invertible n × n matrix A via a polynomial where with t₁(A) = tr (A): standard trace, and t2(A),…t_n−1(A): trace-like forms of A. This formula is very efficient on a computer.

Whereas this problem is traditionally solved using the fundamentals of integral calculus, the solution given in this manuscript will feature the same technique used by Archimedes when he found the volume of the sphere using a balancing scheme involving three solids.

The famous Archimedes’ Law claims:

A body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body…

In this paper we consider the Dirichlet problem to a degenerate elliptic equation in a domain whose interior contains a degenerate surface. By means of the method of expansion of Poisson kernel we obtain a twice continuously differentiable solution of the Dirichlet problem.

The oblique derivative problem can be reduced either to the singular integral equation [1] or to the pseudodifferential equation [2]. If the normality condition of these equations is violated, then one does not succeed to essentially advance in the investigation of the solvability character of such equations. For pseudodifferential equations, appearing in the investigation of the oblique derivative problem, one can develop a procedure analogous to the regularization of one-dimensional singular integral equations. We shall present here this procedure by an example of the oblique derivative problem with the boundary condition [3]

for regular in the halfspace H : {x_n,> 0} harmonic functions u(X), X = (x₁…,x_n)…

R. S. Bucy and G. Maltese [1] in 1966, and M. S. Espelie [2] in 1980 proved that in commutative Banach *-algebras (E,p) the Hermitian characters of E coincide with the extreme points of the unit sphere

W. R. Hamilton (1844) found a way to represent certain objects in the plane and in space that led to the discovery of “quaternions”. His definition of the quaternion included what he called a “real part” and an “imaginary part”. Hamilton defined in his paper (1844) the imaginary part of his “quaternion” as the vector part. This paper marked the beginning of modem vector analysis. Then A. Cayley (1857) developed the algebra of matrices, that is, the rules by which matrices can be added and multiplied. Hamilton, Cayley, and all the research workers in the field of Linear Algebra have restricted their attention to the rectangular system of coordinates and not considered the oblique system of coordinates because of the difficulties that arise. In this paper we establish definitions, operations’ formulas and theorems on vectors’ and matrices in the oblique system of coordinates.

The paper deals with the initial global problem for the class of quasilinear hyperbolic equations u_xx +f₁(u_x)f₂(u_y)u_xy =0. Theorems on the unique global solvability of regular solutions are proved and solutions as well as domains of their definition are constructed in the explicit form…

In 1940 S .M. Ulam (“Problems in Modern Mathematics”, Wiley, N.Y., 1964) imposed the following problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist” [J. Approx. Th. 57, (1989), 268-273; Discus. Math., to appear in Vol 12 (1992 )]. In this paper we establish stability results for an analogous problem for approximately multi-dimensional nonlinear Euler-Lagrange wave mappings.

Let X be a complex Banach space, and let t → T(t) (l‖ T(t)≤1) be a strongly contraction semigroup (for t ≥ 0), or a group (for t ɛ R), or a cosine function (for t ≥ o) on X, with infinitesimal generator A such that Aⁿ × ≠ 0, n = 2,3,4,5,… Then, we prove that inequality

Problem 1.(due to John M. Rassias): Assume m_o = 1, m₄ > m₃ > m₂ > m_l > m_o…

In 1940 S. M. Ulam (Intersci : Publ., Inc., New York, 1960) imposed before the Mathematics Club of the University of Wisconsin the following problem:

“Give conditions in order for a linear mapping near an approximately linear mapping to exist.”

Then D. H. Dyers (Proc. Nat. Acad. Sci., 27 (1941), 411 - 416) established this stability problem with Cauchy inequality involving a non - negative constant. Then J. M. Rassias (J. Approx. Th.,57,No.3(1989), 268 - 273) solved Ulam problem with Cauchy functional inequality, involving a product of powers of norms. Recently J. M. Rassias (Discuss. Math. 12 (1992), 95 - 103) established the general version of this problem with multi dimensional Cauchy inequalities involving a non - negative real - valued function K : K(0) = 0. In this paper J. M. Rassias introduces the 2 - dimensional Cauchy type functional inequality :

for all x₁, x₂ ɛ X (: = normed linear space), c (: = const. ) ≥ 0 with

and assumes mapping f : X → Y (: = complete normed linear space) with f (tx) continued in t for each fixed x. Then he establishes the stability Problem for above inequality and for the corresponding p - dimensional inequality with p = 2, 3, 4,….