[4] **viXra:1606.0344 [pdf]**
*replaced on 2016-09-07 12:56:55*

**Authors:** Valdir Monteiro dos Santos Godoi

**Comments:** 7 Pages.

§ 1: remembering the need of imposed the boundary condition u(x,t)=0 at infinity to ensure uniqueness solutions to the Navier-Stokes equations. This section is historical only. § 2: verifying that for potential and incompressible flows there is no uniqueness solutions when the velocity is equal to zero at infinity. More than this, when the velocity is equal to zero at infinity for all t≥0 there is no uniqueness solutions, in general case. Exceptions when u^0=0. § 3: non-uniqueness in time for incompressible and potential flows, if u≠0. § 4: a more general solution of Euler and Navier-Stokes equations for incompressible and irrotational (potential) flows, given the initial velocity. § 5: Solution for Euler and Navier-Stokes equations using Taylor’s series of powers of t around t=0.

**Category:** Functions and Analysis

[3] **viXra:1606.0324 [pdf]**
*replaced on 2017-01-20 16:47:57*

**Authors:** Furkan Semih Dundar

**Comments:** 6 Pages. v3. Clarifications have been made.

We provide a theory of $n$-scales previously called as $n$ dimensional time scales. In previous approaches to the theory of time scales, multi-dimensional scales were taken as product space of two time scales \cite{bohner2005multiple,bohner2010surface}. $n$-scales make the mathematical structure more flexible and appropriate to real world applications in physics and related fields. Here we define an $n$-scale as an arbitrary closed subset of $\mathbb R^n$. Modified forward and backward jump operators, $\Delta$-derivatives and multiple integrals on $n$-scales are defined.

**Category:** Functions and Analysis

[2] **viXra:1606.0180 [pdf]**
*submitted on 2016-06-17 22:42:03*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 18 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Relative Metric That Generates A Field Super-Set To The Fields Generated By
Various Distinct Relative Metrics’.

**Category:** Functions and Analysis

[1] **viXra:1606.0141 [pdf]**
*replaced on 2016-06-21 11:42:55*

**Authors:** Richard J. Mathar

**Comments:** 24 Pages. Added Section 4 in version 2.

The definite integrals int_0^oo x^j I_0^s(x) I_1^t(x) K_0^u(x)K_1^v(x) dx
are considered for non-negative integer j and four integer exponents s+t+u+v=4, where I and K are
Modified Bessel Functions. There are essentially 15 types of the 4-fold product.
Partial integration of each of these types leads correlations between these integrals. The main result are
(forward) recurrences of the integrals with respect to the exponent j
of the power.

**Category:** Functions and Analysis