Mathematics

In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness ("n-awareness"). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the mathematical formalism of higher category theory. The beauty of higher category theory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and (∞,1) -topoi which will give rise to our meta-language. We then posit a "grammar" ("n-declension") which could express n-awareness, accompanied by a new temporal ontology ("n-time"). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of "personal identity over time" is thus introduced.

We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the points where the cubic polynomial attains zero. We then reformulate a necessary and sufficient condition for a quartic polynomial to be nonnegative for all positive reals. From this, we derive a necessary and sufficient condition for a quartic polynomial to be nonnegative and positive for all reals. Our condition explicitly exhibits the scope and role of some coefficients, and has strong geometrical meaning. In the interior of the nonnegativity region for all reals, there is an appendix curve. The discriminant is zero at the appendix, and positive in the other part of the interior of the nonnegativity region. By using the Sturm sequences, we present a necessary and sufficient condition for a quintic polynomial to be positive and nonnegative for all positive reals. We show that for polynomials of a fixed even degree higher than or equal to four, if they have no real roots, then their discriminants take the same sign, which depends upon that degree only, except on an appendix set of dimension lower by two, where the discriminants attain zero.

This note investigates the prime values of the polynomial f(t)=q t 2 +a for any fixed pair of relatively prime integers a≥1 and q≥1 of opposite parity. For a large number x≥1 , an asymptotic result of the form ∑ n≤ x 1/2 ,n odd Λ(q n 2 +a)≫q x 1/2 /2φ(q) is achieved for q≪(logx ) b , where b≥0 is a constant.

Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and estimation of the number of primes on the twin primes rows are obtained. A new method of twin prime infinity is proposed.

Two prominent methods for integer factorization are those based on general integer sieve and elliptic curve. The general integer sieve method can be specialized to quadratic integer sieve method. In this paper, a probability analysis for the success of these methods is described, under some reasonable conditions. The estimates presented are specialized for the elliptic curve factorization. These methods are compared through heuristic estimates. It is shown that the elliptic curve method is a probabilistic polynomial time algorithm under the assumption of uniform probability distribution for the arising group orders and clearly more likely to succeed, faster asymptotically.

Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann's exterior algebra and the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that the measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics.

In this article, we will show that Collatz is theorem and we proof it by method that we made in section 2 and 3. In section 1, first we introduction Collatz problem and idea of mathematician about this problem then we change the function of this problem and we make a new definition of Collatz set and generalize Collatz problem in the set theory. In section 2 we decrease all of natural numbers to Z 10 and make a model with lemma that we said. Then in section 3 we say 3 properties of numbers that are in our models and then we make a new definition of coloring of graph to complete our model and make a new model to explain Collatz system with 3 numbers. Finally in section 4 we begin proof some part of first model and we use properties that we proved in section 3, to proof our model completely.

We present a new proof of Cramer's rule by interpreting a system of linear equations as a transformation of n -dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to our best knowledge. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system.

In this paper proof of the twin prime conjecture is going to be presented. Originally very difficult problem (in observational space) has been transformed into a sampler one (in generative space) that can be solved. It will be shown that twin primes could be obtained through two stage sieve process, and that will be used to obtain a reasonable estimation of the number of twin primes. The same approach is used to prove the Polignac's conjecture for cousin primes.

Lattice polytope representation of natural numbers is introduced based on the fundamental theorem of arithmetic. The combinatorial and geometric properties of the polytopes are studied using Polymake and Qhull software. The volume of the polytope representing a natural number and the sum of the volumes of polytopes representing a subset of natural numbers are further examined.