# Functions and Analysis

## 1812 Submissions

[3] **viXra:1812.0345 [pdf]**
*replaced on 2018-12-20 08:27:21*

### New Equations of the Resolution of The Navier-Stokes Equations

**Authors:** Abdelmajid Ben Hadj Salem

**Comments:** 14 Pages. Submitted to the journal Annals of PDE. Comments welcome.

This paper represents an attempt to give a solution of the Navier-Stokes equations under the assumptions (A) of the problem as described by the Clay Mathematics Institute. After elimination of the pressure, we obtain the fundamental equations function of the velocity vector u and vorticity vector \Omega=curl(u), then we deduce the new equations for the description of the motion of viscous incompressible fluids, derived from the Navier-Stokes equations, given by:
\nu \Delta \Omega -\frac{\partial \Omega}{\partial t}=0
\Delta p=-\sum^{i=3}_{i=1}\sum^{j=3}_{j=1}\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}
Then, we give a proof of the solution of the Navier-Stokes equations u and p that are smooth functions and u verifies the condition of bounded energy.

**Category:** Functions and Analysis

[2] **viXra:1812.0321 [pdf]**
*submitted on 2018-12-18 12:16:21*

### Positivity of the Fourier Transform of the Shortest Maximal Order Convolution Mask for Cardinal B-splines

**Authors:** Markus Sprecher

**Comments:** 6 Pages.

Positivity of the Fourier transform of a convolution mask can be used to define an inverse convolution and show that the spatial dependency decays exponentially. In this document, we consider, for an arbitrary order, the shortest possible convolution mask which transforms samples of a function to Cardinal B-spline coefficients and show that it is unique and has indeed a positive Fourier transform. We also describe how the convolution mask can be computed including some code.

**Category:** Functions and Analysis

[1] **viXra:1812.0178 [pdf]**
*replaced on 2018-12-12 11:29:13*

### The Zeta Induction Theorem: The Simplest Equivalent to the Riemann Hypothesis?

**Authors:** Terrence P. Murphy

**Comments:** 4 Pages.

This paper presents an uncommon variation of proof by induction. We call it deferred induction by recursion. To set up our proof, we state (but do not prove) the Zeta Induction Theorem. We then assume that theorem is true and provide an elementary proof of the Riemann Hypothesis (showing their equivalence).

**Category:** Functions and Analysis