Abstract:
Several authors have conjectured that Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.

Abstract:
The class $\mathbf{No}$ of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers. Some work has also been done to develop analysis on $\mathbf{No}$. In this paper, we extend this work with a treatment of functions, limits, derivatives, power series, and integrals. We propose surreal definitions of the arctangent and logarithm functions using truncations of Maclaurin series. Using a new representation of surreals, we present a formula for the limit of a sequence, and we use this formula to provide a complete characterization of convergent sequences and to evaluate certain series and infinite Riemann sums via extrapolation. A similar formula allows us to evaluates limits (and hence derivatives) of functions. By defining a new topology on $\mathbf{No}$, we obtain the Intermediate Value Theorem even though $\mathbf{No}$ is not Cauchy complete, and we prove that the Fundamental Theorem of Calculus would hold for surreals if a consistent definition of integration exists. Extending our study to defining other analytic functions, evaluating power series in generality, finding a consistent definition of integration, proving Stokes' Theorem to generalize surreal integration, and studying differential equations remains open.

Abstract:
Let No be Conway's class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some "tameness" and uniformity condition, f must satisfy some interesting properties; in particular, the supremum of the class of element greater or equal to a fixed d in No is actually an element of No. For similar reasons, the concatenation function x:y cannot be defined recursively in a uniform way over polynomial functions.

Abstract:
We show that the natural embedding of the differential field of transseries into Conway's field of surreal numbers with the Berarducci-Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field of surreals with the Berarducci-Mantova derivation.

Abstract:
In his monograph, H. Gonshor showed that Conway's real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the classical Archimedean and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length of these exponential classes.

Abstract:
The relationship between the foundations of mathematics and physics is a topic of of much interest. This paper continues this exploration by examination of the effect of space and time dependent number scaling on theoretical descriptions of some physical and geometric quantities. Fiber bundles provide a good framework to introduce a space and time or space time dependent number scaling field. The effect of the scaling field on a few nonlocal physical and geometric quantities is described. The effect on gauge theories is to introduce a new complex scalar field into the derivatives appearing in Lagrangians. U(1) invariance of Lagrangian terms does not affect the real part of the scaling field. For this field, any mass is possible. The scaling field is also shown to affect quantum wave packets and path lengths, and geodesic equations even on flat space. Scalar fields described so far in physics, are possible candidates for the scaling field. The lack of direct evidence for the field in physics restricts the scaling field in that the gradient of the field must be close to zero in a local region of cosmological space and time. There are no restrictions outside the region. It is also seen that the scaling field does not affect comparisons of computation or measurements outputs with one another. However it does affect the assignment of numerical values to the outputs of computations or measurements. These are needed because theory predictions are in terms of numerical values.

Abstract:
All sciences need and many arts apply mathematics whereas mathematics seems to be independent of all of them, but only based upon logic. This conservative concept, however, needs to be revised because, contrary to Platonic idealism (frequently called "realism" by mathematicians), mathematical ideas, notions, and, in particular, numbers are not at all independent of physical laws and prerequisites.

Abstract:
Some aspects of the development of physics and the mathematics set one think about relation between complex numbers and reality around us. If number to spot as the relation of two quantities, from the fact of existence of complex numbers and accepted definition of number it is necessary necessity complex value to assign to all physical quantities. The basic property of quantity to be it is more or less, therefore field of complex quantities, if it exists, it is necessary is ranked. The hypothesis was proposed that lexicographic ordering may be applied to the complex physical quantities. A set of the ranked complex numbers is quite natural to arrange on a straight line that represents in this case a non-Archimedean complex numerical axis. All physical quantities are located on the relevant non-Archimedean complex numerical axes, forming a new reality - "complex-valued" world. Thus, we get the conclusion that the resulting non-Archimedean complex numerical axis may serve as an example of the ideal mathematical object - hyperreal numerical axis. So, differentiation and integration on the non-Archimedean complex numerical axis can be realized using methods of nonstandard analysis. Certain properties of a new "complex-valued" reality, its connection with our "real" world and possibility of experimental detection of complex physical quantities are discussed.

Abstract:
We dedicate this paper to investigate the most generalized form of Fibonacci Sequence, one of the most studied sections of the mathematical literature. One can notice that, we have discussed even a more general form of the conventional one. Although it seems the topic in the first section has already been covered before, but we present a different proof here. Later I found out that, the auxiliary theorem used in the first section was proven and even generalized further by F. T. Howard. Thanks to Curtis Cooper for pointing out the fact that this has already been studied and providing me with references. the At first, we prove that, only the common general Fibonacci Sequence can be a divisible sequence under some restrictions. In the latter part, we find some properties of the sequence, prove that there are infinite alternating bisquable Fibonacci sequence(defined later) and provide a lower bound on the number of divisors of Fibonacci numbers.

Abstract:
The first chapter of the classic book Principles of Mathematical Analysis (WALTER RUDIN, Third Edition) is the real and complex number systems. Theorem 1.20 of the chapter is extracted from the construction of real number R, and it provides a good illustration of what one can do with the least-upper-bound property. Besides, theorem 1.21 proves the existence of nth roots of positive real numbers. Remarks were given because of the importance of the two theorems. Particularly, the proof of “Hence there is an integer m (with?m2 ? nx ? m1 ) such that m ? 1 ? nx < m . ”which is not mentioned in theorem 1.20 was given in 2.2. A loophole “For every realx > 0 and every integern > 0 there is one and only one realy such that yn = x.” and a flaw of “If t = x 1+x then 0 < t < 1. Hence tn < t < x.” of theorem 1.21 were indicated in 3.1 and 3.2.