The Mouse Set Theorem Just Past Projective

We identify a particular mouse, $M^{\text{ld}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{\sharp}$ for all $n$, and we show that $\mathbb{R}\cap M^{\text{ld}} = Q_{\omega+1}$, the set of reals that are $\Delta^1_{\omega+1}$ in a countable ordinal. Thus $Q_{\omega+1}$ is a mouse set. This is analogous to the fact that $\mathbb{R}\cap M^{\sharp}_1 = Q_3$ where $M^{\sharp}_1$ is the the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are $\Delta^1_3$ in a countable ordinal. More generally $\mathbb{R}\cap M^{\sharp}_{2n+1} = Q_{2n+3}$. The mouse $M^{\text{ld}}$ and the set $Q_{\omega+1}$ compose the next natural pair to consider in this series of results. Thus we are proving the mouse set theorem just past projective. Some of this is not new. $\mathbb{R}\cap M^{\text{ld}} \subseteq Q_{\omega+1}$ was known in the 1990's. But $Q_{\omega+1} \subseteq M^{\text{ld}}$ was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin's proof.Comment: 30 page

A meta-analysis of the effect size of rheumatoid arthritis on left ventricular mass: comment on the article by Rudominer et al

We appreciate the work of Rudominer et al, who recently published a report describing the association of rheumatoid arthritis (RA) with increased left ventricular mass

Pharmacological And Genetic Reversal Of Age-Dependent Cognitive Deficits Attributable To Decreased Presenilin Function

Alzheimer\u27s disease (AD) is the leading cause of cognitive loss and neurodegeneration in the developed world. Although its genetic and environmental causes are not generally known, familial forms of the disease (FAD) are attributable to mutations in a single copy of the Presenilin (PS) and amyloid precursor protein genes. The dominant inheritance pattern of FAD indicates that it may be attributable to gain or change of function mutations. Studies of FAD-linked forms of presenilin (psn) in model organisms, however, indicate that they are loss of function, leading to the possibility that a reduction in PS activity might contribute to FAD and that proper psn levels are important for maintaining normal cognition throughout life. To explore this issue further, we have tested the effect of reducing psn activity during aging in Drosophila melanogaster males. We have found that flies in which the dosage of psn function is reduced by 50% display age-onset impairments in learning and memory. Treatment with metabotropic glutamate receptor (mGluR) antagonists or lithium during the aging process prevented the onset of these deficits, and treatment of aged flies reversed the age-dependent deficits. Genetic reduction of Drosophila metabotropic glutamate receptor (DmGluRA), the inositol trisphosphate receptor (InsP(3)R), or inositol polyphosphate 1-phosphatase also prevented these age-onset cognitive deficits. These findings suggest that reduced psn activity may contribute to the age-onset cognitive loss observed with FAD. They also indicate that enhanced mGluR signaling and calcium release regulated by InsP(3)R as underlying causes of the age-dependent cognitive phenotypes observed when psn activity is reduced

A Computer Game-Based Simulation for Teaching Photolithography

The CHIPS and Science Act in 2022 initiated a major push to boost semiconductor manufacturing infrastructure and workforce in the United States. There is a need for scalable and accessible training approaches to contribute to this goal. To address this need, we created an educational puzzle game using Unity 3D to teach broad-level microfabrication basics (high school through early graduate school). The game was designed to cover typical content of an introductory micromanufacturing course, and to help the students develop a practical understanding of microfabrication techniques. Development of this game necessitated the creation of a novel programming structure called a checkStructure to track the completion of gameplay objectives. The game was demonstrated at a graduate-level introductory microfabrication course at the University of California Davis. In addition to an in-class demonstration, the students got to play the game on their own and fill out a short survey about how the game helped the students learn microfabrication concepts. Out of the 24 respondents, 75% stated that the game helped them better understand the photolithography process, and 66% stated that the game had helped them learn how the tools of photolithography could be used to create devices. The game is expected to be broadly helpful in teaching microfabrication concepts, and could serve as part of an outreach campaign for the field of micromanufacturing as a whole. It is available to play on itch.io, and the source code can be found on GitHub

THE AREA AND CIRCUMFERENCE OF A

In grade school we were all given the formulas for the area and circumference of a circle: A = πr 2 and C =2πr where π ≈ 3.14159. Most likely these formulas were given with no justification, or even an intuitive explanation as to why they are true. In this article I will discuss the two formulas above, I will give elementary proofs of them, and I will discuss their history. To begin with, how should one interpret the formulas A = πr 2 and C =2πr? It is tempting to take the point of view that first there was this number π “out there, ” and subsequently it was discovered that π is useful for calculating the area and circumference of a circle. This of course would be misleading. The formulas A = πr 2 and C =2πr should be interpreted as both a theorem in geometry, and the definition of π. The theorem says that there is some constant k, such that for all circles, the area and circumference of the circle are given by: A = kr 2 and C =2kr. The definition says, let us agree to use the symbol π to refer to this constant. To be more precise, as I see it, there are at least seven important ideas associated with the area and circumference formulas given above. I describe these seven ideas below. (In order to understand what follows, it would be helpful to pretend for a moment that you have never heard of the number π. Then I will define π for you below.) Idea 1. The circumference of a circle is directly proportional to the radius of the circle. That is, there is some constant k such that for all circles, C = kr. This implies for instance that if you double the radius of a circle, then you double its circumference. Idea 2. The area of a circle is directly proportional to the square of the radius of the circle. That is, there is some constant h such that for all circles, A = hr 2. This implies for instance that if you double the radius of a circle, then you quadruple its area. Idea 3. The two constants of proportionality mentioned above are related to each other by the equation k =2h. Notation. Given Idea 3 above, let us agree to use the symbol π to refer to the constant h. Then Idea 1 says that C =2πr and Idea 2 says that A = πr 2. Idea 4. The number π is approximately equal to 3. To be more precise, 3