Fourier restriction in low fractal dimensions

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$.Comment: 31 pages. Minor revisio

Fourier restriction to convex surfaces of revolution in R3

If Γ is a C3 hypersurface in Rn and dσ is induced Lebesgue measure on Γ, then it is well known that a Tomas-Stein Fourier restriction estimate on Γ implies that Γ has a nowhere vanishing Gaussian curvature. In a recent paper, Carbery and Ziesler observed that if induced Lebesgue measure is replaced by affine surface area, then a Tomas-Stein restriction estimate on Γ implies that Γ satisfies the affine isoperimetric inequality. Since the only property needed for a hypersurface to satisfy the affine isoperimetric inequality is convexity, this raised the question of whether a TomasStein restriction estimate can be obtained for flat but convex hypersurfaces in Rn such as Γ(x) = (x, e−1/ $m ), m = 1, 2, . . . . We prove that this is indeed the case in dimension n = 3

Nonlinear Transient Field Problems with Phase Change using the Boundary Element Method

Peer reviewe

Kolonisationsresistenz: Eine endogene Barriere gegen Campylobacter jejuni

Colonization resistance (CR) represents a fundamental defense mechanism which the resident microbiota deploy to protect the host from invading enteric bacterial pathogens including Campylobacter jejuni a leading cause of food-borne enteritis worldwide. However, understanding of CR in campylobacteriosis remains incomplete due to the complexity of the intestinal microbiome and the variability across different model systems. To address this, we employed various murine models displaying varying susceptibilities to intestinal colonization by C. jejuni, encompassing a range of microbiota-specific compositions, and a varying degree of microbiome complexity. We investigated pathogen colonization dynamics, microbiota composition, and metabolome signatures that are associated with mice showing CR. Notably, specific pathogen-free (SPF) mice exhibited robust CR against C. jejuni due to their distinctive murine intestinal microbiota, effectively suppressing pathogen proliferation within the gut. Similarly, murine microbiota-associated mice (mma) which were depleted of their microbiota and replenished with murine-specific microbiota demonstrated effective CR. In contrast, secondary abiotic mice (SAB), and human microbiota-associated (hma), were highly susceptible to pathogen colonization, showing the role of murine-specific microbiota in orchestrating CR. Moreover, microbial community analysis revealed that in comparison to hma mice, SPF and mma mice had higher numbers of lactobacilli and Mouse Intestinal Bacteroides, and lower numbers of enterobacteria, enterococci, and clostridia. Metabolomic profiling identified distinct patterns of metabolites in mice with CR, particularly elevated levels of antimicrobial bile acids and fatty acids, and reduced concentrations of amino acids essential for C. jejuni growth. Expanding our research to a gnotobiotic animal model with a defined microbial community, we utilized the OligoMM12 mice. This model, designed to mimic the complex murine microbial community through a defined low-complexity microbiome, demonstrated its applicability beyond Salmonella infections. Unlike SPF mice, OligoMM12 mice were susceptible to C. jejuni infection. 16S rRNA sequencing revealed that C. jejuni colonization did not cause any major shifts in the microbiota composition of these mice. Moreover, targeted metabolome analysis validated our findings from the other models of the specific metabolomic signatures associated with CR against C. jejuni, highlighting the critical roles of nutrient competition and antimicrobial metabolites such as fatty acids and bile acids. In summary, this thesis sheds light on the intricate dynamics of host-microbiota-pathogen interactions, laying the foundation for future research aimed at deciphering the precise mechanisms of CR.Kolonisationsresistenz (CR) ist ein wesentlicher Abwehrmechanismus der ansässigen Mikrobiota, die den Wirt vor eindringenden bakteriellen Darmpathogenen wie Campylobacter jejuni schützen, einem der weltweit führenden Verursacher lebensmittelbedingter Enteritis. Aufgrund der Komplexität des intestinalen Mikrobioms und der Variabilität verschiedener Modellsysteme ist das Verständnis von CR bei Campylobacteriose jedoch noch lückenhaft. Zur Problemlösung verwendeten wir verschiedene Mausmodelle mit variabler Empfindlichkeit bei einer Besiedlung des Darms mit C. jejuni, einer Reihe von mikrobiota-spezifischen Zusammensetzungen sowie einem unterschiedlichen Grad an Komplexität des Mikrobioms. Wir untersuchten die Dynamik der Erregerbesiedlung, die Zusammensetzung der Mikrobiota und die Metabolom-Signaturen, die mit CR-Mäusen assoziiert sind. Insbesondere spezifisch pathogenfreie (SPF) Mäuse zeigten wegen ihrer ausgeprägten murinen Darmmikrobiota eine robuste CR gegen C. jejuni, die die Vermehrung des Pathogens im Darm effektiv unterdrückt. Ähnlich zeigten Mäuse mit mausassoziierter Mikrobiota (mma), deren Mikrobiota entfernt und mit mma aufgefüllt wurde, eine effektive CR. Im Gegensatz dazu waren sekundäre abiotische Mäuse (SAB) und humane mikrobiota-assoziierte Mäuse (hma) sehr anfällig für die Besiedlung mit Krankheitserregern. Dies betont die Rolle der mma bei der Orchestrierung von CR. Zusätzlich ergab die Analyse der mikrobiellen Gemeinschaft, dass SPF- und mma-Mäuse im Vergleich zu hma-Mäusen eine höhere Anzahl von Laktobazillen und Mouse Intestinal Bacteroides und eine geringere Anzahl von Enterobakterien, Enterokokken und Clostridien aufwiesen. Bei der Erstellung von Stoffwechselprofilen wurden bei CR-Mäusen unterschiedliche Muster von Metaboliten identifiziert, v.a. erhöhte Werte antimikrobieller Gallen- und Fettsäuren sowie geringere Konzentrationen von für das Wachstum von C. jejuni essenziellen Aminosäuren. Zur Erweiterung unserer Forschung auf ein gnotobiotisches Tiermodell mit einer definierten mikrobiellen Gemeinschaft, nutzten wir OligoMM12-Mäuse, ein Modell, das die komplexe mikrobielle Gemeinschaft von Mäusen durch ein definiertes, niedrig-komplexes Mikrobiom imitieren soll. Im Gegensatz zu SPF-Mäusen waren OligoMM12-Mäuse anfällig für eine Infektion mit C. jejuni. Die 16S rRNA-Sequenzierung ergab bei der Besiedlung mit C. jejuni keine größeren Veränderungen in der Zusammensetzung der Mikrobiota dieser Mäuse. Weiterhin bestätigte eine gezielte Metabolomanalyse unsere Erkenntnisse aus den anderen Modellen über die spezifischen metabolomischen Signaturen, die mit der CR gegen C. jejuni in Verbindung stehen und betonte die tragende Rolle der Nährstoffkonkurrenz und antimikrobieller Metaboliten wie Fett- und Gallensäuren. Zusammenfassend beleuchtet diese Arbeit die komplexe Dynamik der Interaktionen zwischen Wirt, Mikrobiota und Pathogen und legt damit den Grundstein für künftige Forschung zur Entschlüsselung der genauen Mechanismen von CR

Weighted restriction estimates using polynomial partitioning

We use the polynomial partitioning method of Guth to prove weighted Fourier restriction estimates in $\Bbb R^3$ with exponents $p$ that range between $3$ and $3.25$, depending on the weight. As a corollary to our main theorem, we obtain new (non-weighted) local and global restriction estimates for compact $C^\infty$ surfaces $S \subset \Bbb R^3$ with strictly positive second fundamental form. For example, we establish the global restriction estimate $\| Ef \|_{L^p(\Bbb R^3)} \leq C \, \| f \|_{L^q(S)}$ in the full conjectured range of exponents for $p > 3.25$ (up to the sharp line), and the global restriction estimate $\| Ef \|_{L^p(\Omega)} \leq C \, \| f \|_{L^2(S)}$ for $p>3$ and certain sets $\Omega \subset \Bbb R^3$ of infinite Lebesgue measure. As a corollary to our main theorem, we also obtain new results on the decay of spherical means of Fourier transforms of positive compactly supported measures on $\Bbb R^3$ with finite $\alpha$-dimensional energies.Comment: 55 pages. Revised following the suggestions of the referee. Accepted for publication in Proceedings of the London Mathematical Societ

Associations Among Motives for Cannabis Use, Expectancies of Cannabis Use and Chronic Pain in a Young Adult Sample

Nationally representative data indicate that cannabis use and chronic pain are both highly prevalent in young adults aged 18-24. Preliminary research suggests that young adults use cannabis for pain relief. Additional research regarding the motives and expectancies of cannabis use in this population are needed to better understand the associations among cannabis use motives, expectancies of cannabis use and chronic pain in young adults. The purpose of this study was to extend prior work on pain, cannabis use motives and expectancies in young adult cannabis users in order to inform efforts towards prevention and intervention for both cannabis use and chronic pain. Young adults aged 18-24 were recruited for this study using an online convenience sampling platform, Amazon’s MTurk, where participants were recruited in exchange for monetary compensation. Participants completed a series of validated psychological measures regarding pain (Graded Chronic Pain Scale), cannabis use motives (Marijuana Motives Questionnaire) and expectancies for cannabis use (Marijuana Effect Expectancy Questionnaire). Multiple linear regressions were run to test associations between cannabis variables and chronic pain. Gender by pain interactions were evaluated to test for gender differences within the multiple linear regression models. After controlling for relevant sociodemographics and hazardous cannabis use, pain was uniquely associated with coping, conformity, expansion, routine and pain motives (ps ≤ 0.002). Additionally, pain was associated with expectancies for global negative effects (p = 0.000). These findings suggest that although young adults who experience pain may expect greater negative effects of cannabis use (e.g., mood swings, carelessness, short-tempered) they may also hold unique pain-related motives for their cannabis use. Researchers and clinicians should consider assessing pain in the context of cannabis use studies and interventions.https://orb.binghamton.edu/research_days_posters_2021/1014/thumbnail.jp

Improved Weighted Restriction Estimates in R3

Suppose 0 < α≤ n , H: Rn→ [0 , 1] is a Lebesgue measurable function, and Aα(H) is the infimum of all numbers C for which the inequality ∫ BH(x) dx≤ CRα holds for all balls B⊂ Rn of radius R≥ 1 . After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form ‖Ef‖Lp(B,Hdx)≲RϵAα(H)1/p‖f‖Lq(σ) have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer’s distance set conjecture in geometric measure theory. This paper improves on the known estimates when E is the extension operator associated with the unit paraboloid P⊂ R3 , reaching the full possible range of p, q exponents (up to the sharp line) for p≥ 3 + (α- 2) / (α+ 1) and 2 < α≤ 3 . © 2023, Mathematica Josephina, Inc

Hazardous drinking is associated with expectancies for the simultaneous use of alcohol and e-cigarettes

Background: Alcohol and tobacco use are especially prevalent among college students, with co-use rates being as high as 59% (Weitzman, 2005). Related adverse health outcomes are further magnified during simultaneous use. Expectancies for substance use are consistently associated with heaviness of use and substance-related consequences. Therefore, the current study examined associations between expectancies for e-cigarette use and heaviness of alcohol use. Methods: College students (N = 362; Mage = 19.32, SD = 0.98, 72% Female) completed psychological measures (i.e. The Alcohol Use Disorder Identification Test (AUDIT) and the Nicotine and Other Substance Interaction Expectancies-E-cig Revised (NOSIE-ER). Results: Half (51%) of participants endorsed both lifetime e-cigarette and alcohol use. Of these, 30% use an e-cigarette at least once a month and 62% report hazardous drinking (AUDIT ≥ 8). AUDIT scores were associated with expectancies that drinking increases e-cigarette consumption (r=.29, p \u3c .001), with expectancies that e-cigarette consumption increases drinking (r=.17, p \u3c .05), and with overall expectancies for the simultaneous use of alcohol and e-cigarettes (r=.30, p \u3c.001). Conclusions: Results suggest that as college students engage in more hazardous drinking, they report greater expectancies for simultaneous use of alcohol and e-cigarette. Future studies should use experimental paradigms to test causal links between alcohol and e-cigarette use, especially across individuals with varying degrees of alcohol and nicotine consumption.https://orb.binghamton.edu/research_days_posters_spring2020/1092/thumbnail.jp