Mathematics

We show that the generalised Stieltjes constants may be represented by infinite series involving logarithmic terms. Some relations involving the derivatives of the Hurwitz zeta function are also investigated

We show that there are infinitely many square numbers , which are constrocted by putting two square numbers together , that non of them are divisible by 10 . We also investigate the interesting properties of some square numbers.

In this paper we introduce a new kind of topological space, called 'structured space', which locally resembles various kinds of algebraic structures. This can be useful, for instance, to locally study a space that cannot be globally endowed with an algebraic structure using tools from algebra. The definition of these spaces will be made more precise via one of our main result, which involves the 'structure map'. This will also lead us to a rigorous and unambiguous definition of algebraic structure. After showing some examples which naturally arise in this context, we study various properties and develop some theory for these new spaces; in particular, we consider partitions (with respect to some measure μ ). We then prove one of the most important Theorem of this paper (Theorem 4.1), which states that every structured space, under some assumptions, induces a lattice, and conversely every lattice induces a structured space satisfing such hypothesis. We conclude with some relations with connected spaces.

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra C ℓ 3,3 of the quadratic space R 3,3 . We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspective projection can be written in this Clifford algebra as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation. An important point is that the expression for the operations of reflection and rotation in C ℓ 3,3 preserve the subspaces that can be associated with the algebras C ℓ 3,0 and C ℓ 0,3 , so that reflection and rotation can be expressed in terms of C ℓ 3,0 or C ℓ 0,3 , as well-known. However, all other operations mix those subspaces in such a way that they need to be expressed in terms of the full Clifford algebra C ℓ 3,3 . An essential aspect of our formulation is the representation of points in terms of objects called paravectors. Paravectors have been used previously to represents points in terms of an algebra closely related to the Clifford algebra C ℓ 3,3 . We compare these different approaches.

In this work the generalized Collatz problem qn+1 ( q odd) is studied. As a natural generalization of the original 3n+1 problem, it consists of a discrete dynamical system of an arithmetical kind. Using standard methods of number theory and dynamical systems, general properties are established, such as the existence of finitely many periodic sequences for each q . In particular, when q is a Mersenne number, q= 2 p −1 , there only exists one such cycle, known as the trivial one. Further analysis based on a probabilistic model shows that for q=3 the asymptotic behavior of all sequences is always convergent, whereas for q≥5 the asymptotic behavior of the sequences is divergent for almost all numbers (for a set of natural density one). This leads to the conclusion that the so called Collatz Conjecture is true, and that q=3 is a very special case among the others (Crandall conjecture). Indeed, it is conjectured that the general problem qn+1 is undecidable.

It will be shown a different way to find infinite series for Pi involving complex conjugates.

This paper is a numerical evaluation of some trajectories of the Collatz function. Specifically, I assess the coalescence points of each integer n≡0(mod2) and n≡2(mod3) through a sophisticated algorithm that has been developed to test on any different modulus classes. The data discovered illustrate that the distribution of the first point of coalescence is closely related to the solutions of some exponential diophantine equation. Afterwards, I show that the first point of coalescence of the integers n and 3n+2 appear to tend to an expected value of 4/5n . When the algorithm was pushed to its peak estimation it has been discovered that the expected value begins to deviate from the initial estimation of 4/5n . The first point of coalescence of the integers n and 3n+2 appear eradicate from a "step by step" point of view but from a topological point of view seem to be localized around the diophantine solution of some particular functions.

In this paper we formulate and prove several variants of the Erdős-Turán additive bases conjecture.

This paper constructs a t r -norm and a t r -conorm on the set of all normal and convex functions from [0,1] to [0,1] , which are not obtained by using the following two formulas on binary operations ⋏ and ⋎ : (f⋏g)(x)=sup{f(y)∗g(z)∣y△z=x}, (f⋎g)(x)=sup{f(y)∗g(z)∣y ▽ z=x}, where f,g∈Map([0,1],[0,1]) , △ and ▽ are respectively a t -norm and a t -conorm on [0,1] , and ∗ is a binary operation on [0,1] . {\color{blue}This result answers affirmatively an open problem posed in \cite{HCT2015}. Moreover, the duality between t r -norms and t r -conorms is obtained by the introduction of operations dual to binary operations on Map([0,1],[0,1]) .}

In this paper, we study quadric surfaces in the 3-dimensional Euclidean space whose Gauss map n is of coordinate finite I-type, i.e., the position vector n satisfies the relation {\Delta}In = {\Lambda}n, where {\Delta}I is the Laplace operator with respect to the first fundamental form I of the surface and {\Lambda} is a square matrix of order 3. We show that helicoids and spheres are the only quadric surfaces of coordinate finite I-type Gauss map.