Heat kernel estimates and their stabilities for symmetric jump processes with general mixed polynomial growths on metric measure spaces

In this paper, we consider a symmetric pure jump Markov process $X$ on a general metric measure space that satisfies volume doubling conditions. We study estimates of the transition density $p(t,x,y)$ of $X$ and their stabilities when the jumping kernel for $X$ has general mixed polynomial growths. Unlike [24], in our setting, the rate function which gives growth of jumps of $X$ may not be comparable to the scale function which provides the borderline for $p(t,x,y)$ to have either near-diagonal estimates or off-diagonal estimates. Under the assumption that the lower scaling index of scale function is strictly bigger than $1$, we establish stabilities of heat kernel estimates. If underlying metric measure space admits a conservative diffusion process which has a transition density satisfying a general sub-Gaussian bounds, we obtain heat kernel estimates which generalize [2, Theorems 1.2 and 1.4]. In this case, scale function is explicitly given by the rate function and the function $F$ related to walk dimension of underlying space. As an application, we proved that the finite moment condition in terms of $F$ on such symmetric Markov process is equivalent to a generalized version of Khintchine-type law of iterated logarithm at the infinity

An Lq(Lp)L_{q}(L_{p})-regularity theory for parabolic equations with integro-differential operators having low intensity kernels

In this article, we present the existence, uniqueness and regularity of solutions to parabolic equations with non-local operator $$\partial_{t}u(t,x) = \mathcal{L}u(t,x) + f(t,x), \quad t>0$$ in $L_{q}(L_{p})$ space. Our spatial operator $\mathcal{L}$ is an integro-differential operator of the form $$\int_{\mathbb{R}^{d}} \left( u(x+y)-u(x) -\nabla u(x) \cdot y \mathrm{1}_{|y|\leq 1} \right) j_{d}(|y|)dy.$$ We investigate conditions on $j_{d}(r)$ which yield $L_{q}(L_{p})$-regularity of solution. Our assumptions on $j_d$ are general so that $j_d(r)$ may be comparable to $r^{-d}\ell(r^{-1})$ for a function $\ell$ which is slowly varying at infinity. For example, we can take $\ell(r)=\log{(1+r^{\alpha})}$ or $\ell(r) = \min{\{r^{\alpha},1\}}$ ($\alpha\in(0,2)$). Indeed, our result covers the operators $\mathcal{L}$, whose Fourier multiplier $\psi(\xi)$ does not have any scaling condition for $|\xi|\geq 1$. Furthermore, we give some examples of $\mathcal{L}$, which cannot be covered by previous results where smoothness or scaling conditions on $\psi$ are considered

A relative of Hadwiger's conjecture

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.Comment: 6 page

Developing parametric design fashion products using 3D printing technology

This study created wearable fashion products with parametric design characteristics, using 3D printing technology. The goal of the study was to understand what parametric design features can be simulated with 3D modeling and printing technology, as well as to demonstrate what techniques can be used to produce fashion products using 3D printing technology. This study created two different parametric motifs using an FDM-type 3D printer with TPU and ABS as the printing materials. With those motifs, we produced three garments and two accessories. The limitations found during the process were modeling the exact measurement of the motifs that will merge with the apparel design seamlessly while maintaining the parametric features, as well as attaching the printed motifs to fabric without ruining the integrity of the textile. A significant implication of this study is that it recreates parametric designs on the human body and utilizes 3D printing technology for fashion products. This paper cast a light on a discussion about the technique can be applied on fashion design with full-sized body and encouraged designers to explore further with technological advancements in the future