Additive completition of thin sets

Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A=\{a_1<a_2<\cdots\}$ be a set of positive integers. Denote $A(x)$ by the counting function of $A$ and $a^*(x)$ by the largest element in $A\bigcap [1,x]$. Following the work of Ruzsa and Chen-Fang, we prove that, for exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$, we have $A(x)B(x)-x\ge \frac{a^*(x)}{A(x)}+o\left(\frac{a^*(x)}{A(x)^2}\right)$ as $x\rightarrow +\infty$. On the other hand, we also construct exact additive complements $A,B$ with $\frac{a_{n+1}}{na_n}\rightarrow\infty$ such that $A(x)B(x)-x\le \frac{a^*(x)}{A(x)}+(1+o(1))\left(\frac{a^*(x)}{A(x)^2}\right)$ holds for infinitely many positive integers $x$.Comment: 7 page

Lagrange-like spectrum of perfect additive complements

Two infinite sets $A$ and $B$ of non-negative integers are called \emph{perfect additive complements of non-negative integers}, if every non-negative integer can be uniquely expressed as the sum of elements from $A$ and $B$. In this paper, we define a Lagrange-like spectrum of the perfect additive complements ($\mathfrak{L}$ for short). As a main result, we obtain the smallest accumulation point of the set $\mathfrak{L}$ and prove that the set $\mathfrak{L}$ is closed. Other related results and problems are also contained

A description of the transverse momentum distributions of charged particles produced in heavy ion collisions at RHIC and LHC energies

By assuming the existing of memory effects and long-range interactions in the hot and dense matter produced in high energy heavy ion collisions, the nonextensive statistics together with the relativistic hydrodynamics including phase transition is used to discuss the transverse momentum distributions of charged particles produced in heavy ion collisions. It is shown that the combined contributions from nonextensive statistics and hydrodynamics can give a good description to the experimental data in Au+Au collisions at sqrt(s_NN )= 200 GeV and in Pb+Pb collisions at sqrt(s_NN) )= 2.76 TeV for pi^(+ -) , K^(+ -) in the whole measured transverse momentum region, and for p(p-bar) in the region of p_T<= 2.0 GeV/c. This is different from our previous work, where, by using the conventional statistics plus hydrodynamics, the describable region is only limited in p_T<= 1.1 GeV/c.Comment: 14 pages, 3 figures, 2 table

Efficient IoT Inference via Context-Awareness

While existing strategies to execute deep learning-based classification on low-power platforms assume the models are trained on all classes of interest, this paper posits that adopting context-awareness i.e. narrowing down a classification task to the current deployment context consisting of only recent inference queries can substantially enhance performance in resource-constrained environments. We propose a new paradigm, CACTUS, for scalable and efficient context-aware classification where a micro-classifier recognizes a small set of classes relevant to the current context and, when context change happens (e.g., a new class comes into the scene), rapidly switches to another suitable micro-classifier. CACTUS features several innovations, including optimizing the training cost of context-aware classifiers, enabling on-the-fly context-aware switching between classifiers, and balancing context switching costs and performance gains via simple yet effective switching policies. We show that CACTUS achieves significant benefits in accuracy, latency, and compute budget across a range of datasets and IoT platforms.Comment: 12 pages, 8 figure