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Title: Almost Everywhere Behavior of Functions According to Partition Measures

Abstract

This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.

The following summarizes the main results proved under suitable partition hypotheses.

If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.

If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.

If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.

Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.

Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).

Griesmer, John T.; Le, Anh N.; Lê, Thái Hoàng(
, Forum of Mathematics, Sigma)

Abstract

We prove three results concerning the existence of Bohr sets in threefold sumsets. More precisely, lettingGbe a countable discrete abelian group and$\phi _1, \phi _2, \phi _3: G \to G$be commuting endomorphisms whose images have finite indices, we show that

If$A \subset G$has positive upper Banach density and$\phi _1 + \phi _2 + \phi _3 = 0$, then$\phi _1(A) + \phi _2(A) + \phi _3(A)$contains a Bohr set. This generalizes a theorem of Bergelson and Ruzsa in$\mathbb {Z}$and a recent result of the first author.

For any partition$G = \bigcup _{i=1}^r A_i$, there exists an$i \in \{1, \ldots , r\}$such that$\phi _1(A_i) + \phi _2(A_i) - \phi _2(A_i)$contains a Bohr set. This generalizes a result of the second and third authors from$\mathbb {Z}$to countable abelian groups.

If$B, C \subset G$have positive upper Banach density and$G = \bigcup _{i=1}^r A_i$is a partition,$B + C + A_i$contains a Bohr set for some$i \in \{1, \ldots , r\}$. This is a strengthening of a theorem of Bergelson, Furstenberg and Weiss.

All results are quantitative in the sense that the radius and rank of the Bohr set obtained depends only on the indices$[G:\phi _j(G)]$, the upper Banach density ofA(in (1)), or the number of sets in the given partition (in (2) and (3)).

Joshi, P.; Anderson, W.(
, Journal of Fluid Mechanics)

Large-eddy simulation was used to model turbulent atmospheric surface layer (ASL) flow over canopies composed of streamwise-aligned rows of synthetic trees of height,$h$, and systematically arranged to quantify the response to variable streamwise spacing,$\delta _1$, and spanwise spacing,$\delta _2$, between adjacent trees. The response to spanwise and streamwise heterogeneity has, indeed, been the topic of a sustained research effort: the former resulting in formation of Reynolds-averaged counter-rotating secondary cells, the latter associated with the$k$- and$d$-type response. No study has addressed the confluence of both, and results herein show secondary flow polarity reversal across ‘critical’ values of$\delta _1$and$\delta _2$. For$\delta _2/\delta \lesssim 1$and$\gtrsim 2$, where$\delta$is the flow depth, the counter-rotating secondary cells are aligned such that upwelling and downwelling, respectively, occurs above the elements. The streamwise spacing$\delta _1$regulates this transition, with secondary cell reversal occurring first for the largest$k$-type cases, as elevated turbulence production within the canopy necessitates entrainment of fluid from aloft. The results are interpreted through the lens of a benchmark prognostic closure for effective aerodynamic roughness,$z_{0,{Eff.}} = \alpha \sigma _h$, where$\alpha$is a proportionality constant and$\sigma _h$is height root mean square. We report$\alpha \approx 10^{-1}$, the value reported over many decades for a broad range of rough surfaces, for$k$-type cases at small$\delta _2$, whereas the transition to$d$-type arrangements necessitates larger$\delta _2$. Though preliminary, results highlight the non-trivial response to variation of streamwise and spanwise spacing.

Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni(
, Forum of Mathematics, Pi)

Abstract

We study higher uniformity properties of the Möbius function$\mu $, the von Mangoldt function$\Lambda $, and the divisor functions$d_k$on short intervals$(X,X+H]$with$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$for a fixed constant$0 \leq \theta < 1$and any$\varepsilon>0$.

More precisely, letting$\Lambda ^\sharp $and$d_k^\sharp $be suitable approximants of$\Lambda $and$d_k$and$\mu ^\sharp = 0$, we show for instance that, for any nilsequence$F(g(n)\Gamma )$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$

As a consequence, we show that the short interval Gowers norms$\|f-f^\sharp \|_{U^s(X,X+H]}$are also asymptotically small for any fixedsfor these choices of$f,\theta $. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$L^2$.

Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$II$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$I_2$sums.

Jiang, Tao; Longbrake, Sean; Ma, Jie(
, Combinatorics, Probability and Computing)

Abstract

Given a family$\mathcal{F}$of bipartite graphs, theZarankiewicz number$z(m,n,\mathcal{F})$is the maximum number of edges in an$m$by$n$bipartite graph$G$that does not contain any member of$\mathcal{F}$as a subgraph (such$G$is called$\mathcal{F}$-free). For$1\leq \beta \lt \alpha \lt 2$, a family$\mathcal{F}$of bipartite graphs is$(\alpha,\beta )$-smoothif for some$\rho \gt 0$and every$m\leq n$,$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$(\alpha,\beta )$-smooth family$\mathcal{F}$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$is bipartite. In this paper, we strengthen their result by showing that for every real$\delta \gt 0$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\delta n^{\alpha -1}$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$\mathcal{F}$consisting of the single graph$K_{s,t}$when$t\gg s$. We also prove an analogous result for$C_{2\ell }$-free graphs for every$\ell \geq 2$, which complements a result of Keevash, Sudakov and Verstraëte.

Cobos, Richard; Khair, Aditya S.(
, Journal of Fluid Mechanics)

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength$\beta =a^*e^*E_\infty ^*/k_B^*T^*$, defined as the ratio of the product of the applied electric field magnitude$E_\infty ^*$and particle radius$a^*$, to the thermal voltage$k_B^*T^*/e^*$, where$k_B^*$is Boltzmann's constant,$T^*$is the absolute temperature, and$e^*$is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density$\sigma$over a wide range of$\beta$. Here,$\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$, where$\sigma ^*$is the dimensional surface charge density, and$\epsilon ^*$is the permittivity of the electrolyte. For moderately charged particles ($\sigma ={O}(1)$), the electrophoretic velocity is linear in$\beta$when$\beta \ll 1$, and its dependence on the ratio of the Debye length ($1/\kappa ^*$) to particle radius (denoted by$\delta =1/(\kappa ^*a^*)$) agrees with Henry's formula. As$\beta$increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is$\delta$-dependent. For$\beta \gg 1$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all$\delta$. For highly charged particles ($\sigma \gg 1$) in the thin-Debye-layer limit ($\delta \ll 1$), our computations are in good agreement with recent experimental and asymptotic results.

Chan, William, Jackson, Stephen, and Trang, Nam. Almost Everywhere Behavior of Functions According to Partition Measures. Retrieved from https://par.nsf.gov/biblio/10501532. Forum of Mathematics, Sigma 12. Web. doi:10.1017/fms.2023.130.

Chan, William, Jackson, Stephen, & Trang, Nam. Almost Everywhere Behavior of Functions According to Partition Measures. Forum of Mathematics, Sigma, 12 (). Retrieved from https://par.nsf.gov/biblio/10501532. https://doi.org/10.1017/fms.2023.130

Chan, William, Jackson, Stephen, and Trang, Nam.
"Almost Everywhere Behavior of Functions According to Partition Measures". Forum of Mathematics, Sigma 12 (). Country unknown/Code not available: Cambridge University Press. https://doi.org/10.1017/fms.2023.130.https://par.nsf.gov/biblio/10501532.

@article{osti_10501532,
place = {Country unknown/Code not available},
title = {Almost Everywhere Behavior of Functions According to Partition Measures},
url = {https://par.nsf.gov/biblio/10501532},
DOI = {10.1017/fms.2023.130},
abstractNote = {Abstract This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals. The following summarizes the main results proved under suitable partition hypotheses.•If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.•If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.•If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.•Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.•Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).},
journal = {Forum of Mathematics, Sigma},
volume = {12},
publisher = {Cambridge University Press},
author = {Chan, William and Jackson, Stephen and Trang, Nam},
}

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