A Tur\'an-type problem on degree sequence

Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular type. Denote by $ex_p(n,H)$ the maximum value of $e_p(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Tur\'an number, i.e., the maximum number of edges among all $H$-free graphs with $n$ vertices. Pikhurko and Taraz generalize this Tur\'an-type problem: let $f$ be a non-negative increasing real function and $e_f(G)=\sum_{i=1}^n f(d_i)$, and then define $ex_f(n,H)$ as the maximum value of $e_f(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Observe that $ex_f(n,H)=ex(n,H)$ if $f(x)=x/2$, $ex_f(n,H)=ex_p(n,H)$ if $f(x)=x^p$. Bollob\'as and Nikiforov mentioned that it is important to study concrete functions. They gave an example $f(x)=\phi(k)={x\choose k}$, since $\sum_{i=1}^n{d_i\choose k}$ counts the $(k+1)$-vertex subgraphs of $G$ with a dominating vertex. Denote by $T_r(n)$ the $r$-partite Tur\'an graph of order $n$. In this paper, using the Bollob\'as--Nikiforov's methods, we give some results on $ex_{\phi}(n,K_{r+1})$ $(r\geq 2)$ as follows: for $k=1,2$, $ex_\phi(n,K_{r+1})=e_\phi(T_r(n))$; for each $k$, there exists a constant $c=c(k)$ such that for every $r\geq c(k)$ and sufficiently large $n$, $ex_\phi(n,K_{r+1})=e_\phi(T_r(n))$; for a fixed $(r+1)$-chromatic graph $H$ and every $k$, when $n$ is sufficiently large, we have $ex_\phi(n,H)=e_\phi(n,K_{r+1})+o(n^{k+1})$.Comment: 9 page

Two-Particle Correlations and Meson-Antimeson Mixing Effects

We discuss 2-particle correlations which arise in the time evolution of C-odd and C-even meson--antimeson states of flavoured neutral mesons. In order to keep our discussion general, we do not use the Weisskopf -- Wigner approximation. Possible deviations from quantum-mechanical coherence effects are parameterized by a so-called decoherence parameter $\zeta$. In particular, we study the $\zeta$-dependence of the asymmetry of unlike and like-flavoured events which was recently observed experimentally in the $K^0 \bar{K}^0$ system. In this $\zeta$-dependence, we point out some important general features which do not rely on the Weisskopf -- Wigner approximation. Some other related results are derived more generally than in the literature.Comment: 12 pages, revtex, no figure

Making Triangles Colorful

We prove that for any point set P in the plane, a triangle T, and a positive integer k, there exists a coloring of P with k colors such that any homothetic copy of T containing at least ck^8 points of P, for some constant c, contains at least one of each color. This is the first polynomial bound for range spaces induced by homothetic polygons. The only previously known bound for this problem applies to the more general case of octants in R^3, but is doubly exponential.Comment: 6 page

The Spend-It-All Region and Small Time Results for the Continuous Bomber Problem

A problem of optimally allocating partially effective ammunition $x$ to be used on randomly arriving enemies in order to maximize an aircraft's probability of surviving for time~$t$, known as the Bomber Problem, was first posed by \citet{Klinger68}. They conjectured a set of apparently obvious monotonicity properties of the optimal allocation function $K(x,t)$. Although some of these conjectures, and versions thereof, have been proved or disproved by other authors since then, the remaining central question, that $K(x,t)$ is nondecreasing in~$x$, remains unsettled. After reviewing the problem and summarizing the state of these conjectures, in the setting where $x$ is continuous we prove the existence of a ``spend-it-all'' region in which $K(x,t)=x$ and find its boundary, inside of which the long-standing, unproven conjecture of monotonicity of~$K(\cdot,t)$ holds. A new approach is then taken of directly estimating~$K(x,t)$ for small~$t$, providing a complete small-$t$ asymptotic description of~$K(x,t)$ and the optimal probability of survival

RED: Reinforced Encoder-Decoder Networks for Action Anticipation

Action anticipation aims to detect an action before it happens. Many real world applications in robotics and surveillance are related to this predictive capability. Current methods address this problem by first anticipating visual representations of future frames and then categorizing the anticipated representations to actions. However, anticipation is based on a single past frame's representation, which ignores the history trend. Besides, it can only anticipate a fixed future time. We propose a Reinforced Encoder-Decoder (RED) network for action anticipation. RED takes multiple history representations as input and learns to anticipate a sequence of future representations. One salient aspect of RED is that a reinforcement module is adopted to provide sequence-level supervision; the reward function is designed to encourage the system to make correct predictions as early as possible. We test RED on TVSeries, THUMOS-14 and TV-Human-Interaction datasets for action anticipation and achieve state-of-the-art performance on all datasets

Multiple Instance Curriculum Learning for Weakly Supervised Object Detection

When supervising an object detector with weakly labeled data, most existing approaches are prone to trapping in the discriminative object parts, e.g., finding the face of a cat instead of the full body, due to lacking the supervision on the extent of full objects. To address this challenge, we incorporate object segmentation into the detector training, which guides the model to correctly localize the full objects. We propose the multiple instance curriculum learning (MICL) method, which injects curriculum learning (CL) into the multiple instance learning (MIL) framework. The MICL method starts by automatically picking the easy training examples, where the extent of the segmentation masks agree with detection bounding boxes. The training set is gradually expanded to include harder examples to train strong detectors that handle complex images. The proposed MICL method with segmentation in the loop outperforms the state-of-the-art weakly supervised object detectors by a substantial margin on the PASCAL VOC datasets.Comment: Published in BMVC 201