Polynomial Roth Theorems on Sets of Fractional Dimensions

Let $E\subset{\mathbb{R}}$ be a closed set of Hausdorff dimension $\alpha \in (0, 1)$. Let $P: {\mathbb{R}}\to{\mathbb{R}}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Łaba and Pramanik [ 11] to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot et al. [ 7].

Progress on Proving the Mass gap for Yang Mills and Gravity (Maybe it’s already proven…)

Proving and constructing viable Yang Mills Gauge is a key concern for the Standard Model and an open problem. It has only be solved on lattices. Yet, gravity is not modeled in the Standard Model. We discuss that in a multi-fold universe where gravity emerges from entanglement effects, the spacetime is discrete (fractal with fractional dimensions, noncommutative and still Lorentz invariant). For any Lorentz invariant discrete spacetime, the lattice proofs and their lattice cell size independence completes the proof of the mass gap for Yang Mills Gauge theories. Continuous spacetime may or may not have a mass gap; but it does not matter if the real universe is discrete and Lorentz invariant.

Exercise your imagination, with fractional dimensions

“Exercise your imagination, with fractional dimensions” offers a basic introduction to fractional dimensional objects. Unlike a one-dimensional line, a two-dimensional piece of paper, or a three-dimensional box, the dimension of a fractional dimensional object may not be represented by a whole number. The fractional dimensional Koch curve, for example, has approximately 1.26185 dimensions. The discussion is enhanced with numerous hand-drawn sketches providing instruction on how to construct and compute the dimension of the Koch curve. Mathematics students and enthusiasts are encouraged to exercise their imaginations in mathematical and life pursuits as a way of opening themselves up to more of life’s unusual possibilities. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.

Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

In this study, a dimensionally-consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multi-dimensional unconfined aquifer, a previously-developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, a numerical application to an unconfined aquifer groundwater flow problem is presented to show the skills of the proposed fractional governing equation.