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For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split into $f_1(p)+f_2(p),$ where $f_i(p)(i=1,\,2)$ counts those solutions with exactly $i$ of denominators$n_1,\,n_2,\,n_3$ divisible by $p.$ Recently Terence Tao proved that $$ \sum_{p< x}f_2(p)\ll x\log^2x\log\log x $$ with other results. But actually only the upper bound $x\log^2x\log\log^2x$ can be obtained in his discussion. In this note we shall use an elementary method to save a factor $\log\log x$ and recover the above estimate.
In this paper, we solve a problem of Terence Tao. We prove that for any $K\geq 2$ and sufficiently large $N$, the number of primes $p$ between $N$ and $(1+\frac{1}{K})N$ such that $\mid kp+ja^{i}+l\mid$ is composite for all $1\leq a, |j|, k\leq K$, $1\leq i \leq K\log N$ and $l$ in any set $L =L_{N}\subseteq\{-KN,\cdots, KN\}$ of cardinality $K$ with $ja^{i}+l eq0$ is at least $C_{K}\frac{N}{\log N}$, where $C_{K}>0$ depending only on $K$.
Many matching markets feature unknown, dynamic arrivals of agents that must match immediately. A caseworker must match an abused child to a foster home, a hospital must assign a patient in critical condition to a room, or a city must place a homeless individual into a shelter. We design an online matching algorithm -- the Sequential Equilibrium Mechanism (SEM) -- that approximates large market equilibria to match arriving agents to objects. SEM is asymptotically efficient, fair, and strategy-proof with probability one. Our application plans to deploy a lab-in-the-field experiment where real caseworkers match vulnerable children to host homes, and we provide simulation evidence that SEM can substantially improve welfare.
Recommendation systems have become essential in modern music streaming platforms, due to the vast amount of content available. A common approach in recommendation systems is collaborative filtering, which suggests content to users based on the preferences of others with similar patterns. However, this method performs poorly in domains where interactions are sparse, such as music. Content-based filtering is an alternative approach that examines the qualities of the items themselves. Prior work has explored a range of content-filtering techniques for music, including genre classification, instrument detection, and lyrics analysis. In the literature review component of this work, we examine these methods in detail. Music emotion recognition is a type of content-based filtering that is less explored but has significant potential. Since a user's emotional state influences their musical choices, incorporating user mood into recommendation systems is an alternative way to personalize the listening experience. In this study, we explore a mood-assisted recommendation system that suggests songs based on the desired mood using the energy-valence spectrum. Single-blind experiments are conducte
Serpentinization of ultramafic rocks is a naturally occurring mineralogical process that can generate molecular hydrogen through the oxidation of ferrous iron during water-rock reaction. Although the resource potential is large, the natural reaction is kinetically limited, and practical hydrogen recovery requires methods that can accelerate conversion without imposing an energy penalty that exceeds the value of the hydrogen produced. This short communication reports a preliminary atmospheric-pressure microwave serpentinization experiment using a water-saturated 2 g crushed olivine sample. Microwave irradiation produced a rapid increase in measured hydrogen concentration compared with conventional hot-plate heating under otherwise similar conditions. The preliminary experiment showed approximately a 12-fold increase in hydrogen concentration and an apparent rate increase from about 2 ppb s$^{-1}$ for conventional heating to about 10 ppb s$^{-1}$ during microwave exposure. These results suggest that electromagnetic stimulation can enhance serpentinization kinetics, likely through rapid volumetric heating, selective coupling to iron-bearing phases, and localized thermal gradients. The
The dimension of Kakeya sets can be bounded using sum-difference exponents $\SD(R;s)$ for various sets of rational slopes $R$ and output slope $s$; the arithmetic Kakeya conjecture, which implies the Kakeya conjecture in all dimensions, asserts that the infimum of such exponents is $1$. The best upper bound on this infimum currently is $1.67513\dots$. In this note, inspired by numerical explorations from the tool \texttt{AlphaEvolve}, we study the regime where the cardinality of the set of slopes $R$ is bounded. In this regime, we establish that these exponents converge to $2$ at a rate controlled by the \emph{rational complexity} of $s$ relative to $R$, which measures how efficiently $s$ can be expressed as a rational combination of slopes in $R$.
We study the problem of an organization that matches agents to objects where agents have preference rankings over objects and the organization uses algorithms to construct a ranking over objects on behalf of each agent. Our new framework carries the interpretation that the organization and its agents may be misaligned in pursuing some underlying matching goal. We design matching mechanisms that integrate agent decision-making and the algorithm by prioritizing matches that are unanimously agreeable between the two parties. Our mechanisms also satisfy restricted efficiency properties. Subsequently, we prove that no unanimous mechanism is strategy-proof but that ours can be non-obviously manipulable. We generalize our framework to allow for any preference aggregation rules and extend the famed Gibbard-Satterthwaite Theorem to our setting. We apply our framework to place foster children in foster homes to maximize welfare. Using a machine learning model that predicts child welfare in placements and a novel "elicited preferences" experiment that extracts real caseworkers' preferences, we empirically demonstrate that there are important match-specific welfare gains that our mechanisms ex
Caseworkers in foster care systems match waiting children to adoptive homes. We use dynamic matching market design to characterize a class of mechanisms that incentivize expedient matches that homes can accept or decline. We design mechanisms satisfying fairness and limited strategy-proofness. They also avoid costly patience. Our empirically-based simulations suggest the mechanisms could increase adoptions by at least 25% versus the status quo. A naive dynamic extension of Deferred Acceptance does not attain these benefits. Our mechanisms sidestep direct preference elicitation by predicting preferences, and they are robust to prediction error.
We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erdős. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns).
The dead internet theory is a conspiracy theory that states that all interactions and posts on social media are no longer being made by real people, but rather by autonomous bots. While the theory is obviously not true, an increasing amount of posts on social media have been made by bots optimised to gain followers and drive engagement on social media platforms. This paper looks at the recent phenomenon of these bots, analysing their behaviour through the lens of computational creativity to investigate the question: is computational creativity flourishing on the dead internet?
We obtain bounds on the distribution of normalized gaps of eigenvalues of $N \times N$ GUE matrix in the bulk, that do not lose logarithmic factors of $N$ in the limit $N \to \infty$. As an application, we obtain fixed index universality results for the GUE minor process, which in turn are useful for establishing limiting results for random hives with GUE boundary data.
A prototype Metaverse experience was created in which users could explore hierarchical cybersecurity data. A small group of participants were surveyed on their attitudes to the Metaverse. They then completed a short series of tasks in the environment. Questions were asked to assess if they were suffering from Cybersickness. After completing further tasks, their attitudes were surveyed regarding future uses of the metaverse in the organisation. A second cohort of participants attended an online seminar. They completed a survey about their attitudes to the Metaverse. They then watched a short video of the Metaverse experience. Afterwards, they answered questions related to their attitudes towards future uses of the metaverse in the organisation. The results of these questionnaires were assessed to see whether participants were receptive to the idea of working with data inside the Metaverse in the future.
A well known open problem of Meir and Moser asks if the squares of sidelength $1/n$ for $n \geq 2$ can be packed perfectly into a square of area $\sum_{n=2}^\infty \frac{1}{n^2} = \frac{π^2}{6}-1$. In this paper we show that for any $1/2 < t < 1$, and any $n_0$ that is sufficiently large depending on $t$, the squares of sidelength $n^{-t}$ for $n \geq n_0$ can be packed perfectly into a square of area $\sum_{n=n_0}^\infty \frac{1}{n^{2t}}$. This was previously known (if one packs a rectangle instead of a square) for $1/2 < t \leq 2/3$ (in which case one can take $n_0=1$).
Dying Sun-like stars may not fade away quietly。 As blobs of gas erupt unevenly from their swollen surfaces, each burst gives the star a tiny push in the opposite direction。 Thousands of these random kicks could eventually break apart distant stellar pairs or, in rare cases, drive two stars into a violent collision
“Halting this merger while our case proceeds is a critical victory," NY AG said
A giant planet had been hiding inside one of astronomy’s most closely studied planetary systems, concealed by a bright disk of cosmic dust。 Webb revealed Beta Pictoris d through atmospheric traces of carbon monoxide, water vapor, and methane, showcasing a promising new method for finding elusive exoplanets
There's no human safety or efficacy data, but that no longer seems to matter
Popular AI-powered food apps may make calorie counting easier, but they may also leave out a surprisingly large part of the meal。 Four apps underestimated calories and fat by about one-third when tested against carefully prepared meals。 High-fat ketogenic dishes appeared to cause the most trouble, while carbohydrates were measured more consistently