Optimal Global Test for Functional Regression

This paper studies the optimal testing for the nullity of the slope function in the functional linear model using smoothing splines. We propose a generalized likelihood ratio test based on an easily implementable data-driven estimate. The quality of the test is measured by the minimal distance between the null and the alternative set that still allows a possible test. The lower bound of the minimax decay rate of this distance is derived, and test with a distance that decays faster than the lower bound would be impossible. We show that the minimax optimal rate is jointly determined by the smoothing spline kernel and the covariance kernel. It is shown that our test attains this optimal rate. Simulations are carried out to confirm the finite-sample performance of our test as well as to illustrate the theoretical results. Finally, we apply our test to study the effect of the trajectories of oxides of nitrogen ($\text{NO}_{\text{x}}$) on the level of ozone ($\text{O}_3$)

A Medical Literature Search System for Identifying Effective Treatments in Precision Medicine

The Precision Medicine Initiative states that treatments for a patient should take into account not only the patient's disease, but his/her specific genetic variation as well. The vast biomedical literature holds the potential for physicians to identify effective treatment options for a cancer patient. However, the complexity and ambiguity of medical terms can result in vocabulary mismatch between the physician's query and the literature. The physician's search intent (finding treatments instead of other types of studies) is difficult to explicitly formulate in a query. Therefore, simple ad hot retrieval approach will suffer from low recall and precision. In this paper, we propose a new retrieval system that helps physicians identify effective treatments in precision medicine. Given a cancer patient with a specific disease, genetic variation, and demographic information, the system aims to identify biomedical publications that report effective treatments. We approach this goal from two directions. First, we expand the original disease and gene terms using biomedical knowledge bases to improve recall of the initial retrieval. We then improve precision by promoting treatment-related publications to the top using a machine learning reranker trained on 2017 Text Retrieval Conference Precision Medicine (PM) track corpus. Batch evaluation results on 2018 PM track corpus show that the proposed approach effectively improves both recall and precision, achieving performance comparable to the top entries on the leaderboard of 2018 PM track.Comment: 32 page

On The Critical Number of Finite Groups (II)

Let G be a finite group and S a subset of G\{0}. We call S an additive basis of G if every element of G can be expressed as a sum over a nonempty subset in some order. Let cr(G) be the smallest integer t such that every subset of G\{0} of cardinality t is an additive basis of G. In this paper, we determine cr(G) for the following cases: (i) G is a finite nilpotent group; (ii) G is a group of even order which possesses a subgroup of index 2

Multiply Warped Products with a Quarter-symmetric Connection

In this paper, we study the Einstein warped products and multiply warped products with a quarter-symmetric connection. We also study warped products and multiply warped products with a quarter-symmetric connection with constant scalar curvature. Then apply our results to generalized Robertson-Walker spacetimes with a quarter-symmetric connection and generalized Kasner space-times with a quarter-symmetric connection.Comment: 41 pages. arXiv admin note: text overlap with arXiv:1207.509

Simultaneous Sparse Dictionary Learning and Pruning

Dictionary learning is a cutting-edge area in imaging processing, that has recently led to state-of-the-art results in many signal processing tasks. The idea is to conduct a linear decomposition of a signal using a few atoms of a learned and usually over-completed dictionary instead of a pre-defined basis. Determining a proper size of the to-be-learned dictionary is crucial for both precision and efficiency of the process, while most of the existing dictionary learning algorithms choose the size quite arbitrarily. In this paper, a novel regularization method called the Grouped Smoothly Clipped Absolute Deviation (GSCAD) is employed for learning the dictionary. The proposed method can simultaneously learn a sparse dictionary and select the appropriate dictionary size. Efficient algorithm is designed based on the alternative direction method of multipliers (ADMM) which decomposes the joint non-convex problem with the non-convex penalty into two convex optimization problems. Several examples are presented for image denoising and the experimental results are compared with other state-of-the-art approaches

Global classical solutions to partially dissipative hyperbolic systems violating the Kawashima condition

This paper considers the Cauchy problem for the quasilinear hyperbolic system of balance laws in $\mathbb{R}^d$, $d\ge 2$. The system is partially dissipative in the sense that there is an eigen-family violating the Kawashima condition. By imposing certain supplementary degeneracy conditions with respect to the non-dissipative eigen-family, global unique smooth solutions near constant equilibria are constructed. The proof is based on the introduction of the partially normalized coordinates, a delicate structural analysis, a family of scaled energy estimates with minimum fractional derivative counts and a refined decay estimates of the dissipative components of the solution.Comment: 44pp. It is revised so that the results can be applied to Euler equation

Optimal Estimation for the Functional Cox Model

Functional covariates are common in many medical, biodemographic, and neuroimaging studies. The aim of this paper is to study functional Cox models with right-censored data in the presence of both functional and scalar covariates. We study the asymptotic properties of the maximum partial likelihood estimator and establish the asymptotic normality and efficiency of the estimator of the finite-dimensional estimator. Under the framework of reproducing kernel Hilbert space, the estimator of the coefficient function for a functional covariate achieves the minimax optimal rate of convergence under a weighted $L_2$-risk. This optimal rate is determined jointly by the censoring scheme, the reproducing kernel and the covariance kernel of the functional covariates. Implementation of the estimation approach and the selection of the smoothing parameter are discussed in detail. The finite sample performance is illustrated by simulated examples and a real application

Davenport constant of the multiplicative semigroup of the quotient ring \frac{\F_p[x]}{\langle f(x)\rangle}

Let $\mathcal{S}$ be a finite commutative semigroup. The Davenport constant of $\mathcal{S}$, denoted $D(\mathcal{S})$, is defined to be the least positive integer $d$ such that every sequence $T$ of elements in $\mathcal{S}$ of length at least $d$ contains a subsequence $T'$ with the sum of all terms from $T'$ equaling the sum of all terms from $T$. Let \F_p[x] be a polynomial ring in one variable over the prime field \F_p, and let f(x)\in \F_p[x]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring \frac{\F_p[x]}{\langle f(x)\rangle}. Among other results, we mainly prove that, for any prime $p>2$ and any polynomial f(x)\in \F_p[x] which can be factorized into several pairwise non-associted irreducible polynomials in \F_p[x], then $$D(\mathcal{S}_{f(x)}^p)=D(U(\mathcal{S}_{f(x)}^p)),$$ where $\mathcal{S}_{f(x)}^p$ denotes the multiplicative semigroup of the quotient ring \frac{\F_p[x]}{\langle f(x)\rangle} and $U(\mathcal{S}_{f(x)}^p)$ denotes the group of units of the semigroup $\mathcal{S}_{f(x)}^p$.Comment: 9 page

A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of \F_2[x]

Let \F_q[x] be the ring of polynomials over the finite field \F_q, and let $f$ be a polynomial of \F_q[x]. Let R=\frac{\F_q[x]}{(f)} be a quotient ring of \F_q[x] with 0\neq R\neq \F_q[x]. Let $\mathcal{S}_R$ be the multiplicative semigroup of the ring $R$, and let ${\rm U}(\mathcal{S}_R)$ be the group of units of $\mathcal{S}_R$. The Davenport constant ${\rm D}(\mathcal{S}_R)$ of the multiplicative semigroup $\mathcal{S}_R$ is the least positive integer $\ell$ such that for any $\ell$ polynomials g_1,g_2,\ldots,g_{\ell}\in \F_q[x], there exists a subset $I\subsetneq [1,\ell]$ with $$\prod\limits_{i\in I} g_i \equiv \prod\limits_{i=1}^{\ell} g_i\pmod f.$$ In this manuscript, we proved that for the case of $q=2$, $${\rm D}({\rm U}(\mathcal{S}_R))\leq {\rm D}(\mathcal{S}_R)\leq {\rm D}({\rm U}(\mathcal{S}_R))+\delta_f,$$ where \begin{displaymath} \delta_f=\left\{\begin{array}{ll} 0 & \textrm{if $\gcd(x*(x+1_{\mathbb{F}_2}),\ f)=1_{\F_{2}}}\\ 1 & \textrm{if \gcd(x*(x+1_{\mathbb{F}_2}),\ f)\in \{x, \ x+1_{\mathbb{F}_2}\}}\\ 2 & \textrm{if gcd(x*(x+1_{\mathbb{F}_2}),f)=x*(x+1_{\mathbb{F}_2}) }\\ \end{array} \right. \end{displaymath} which partially answered an open problem of Wang on Davenport constant for the multiplicative semigroup of \frac{\F_q[x]}{(f)}$ (G.Q. Wang, \emph{Davenport constant for semigroups II,} Journal of Number Theory, 155 (2015) 124--134).Comment: 12 page

New Constructions of Permutation Polynomials of the Form xrh(xq−1)x^rh\left(x^{q-1}\right) over Fq2\mathbb{F}_{q^2}

Permutation polynomials over finite fields have been studied extensively recently due to their wide applications in cryptography, coding theory, communication theory, among others. Recently, several authors have studied permutation trinomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$, where $q=2^k$, $h(x)=1+x^s+x^t$ and $r, s, t, k>0$ are integers. Their methods are essentially usage of a multiplicative version of AGW Criterion because they all transformed the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing the corresponding fractional polynomials permute a smaller set $\mu_{q+1}$, where $\mu_{q+1}:=\{x\in\mathbb{F}_{q^2} : x^{q+1}=1\}$. Motivated by these results, we characterize the permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$ such that $h(x)\in\mathbb{F}_q[x]$ is arbitrary and $q$ is also an arbitrary prime power. Using AGW Criterion twice, one is multiplicative and the other is additive, we reduce the problem of proving permutation polynomials over $\mathbb{F}_{q^2}$ into that of showing permutations over a small subset $S$ of a proper subfield $\mathbb{F}_{q}$, which is significantly different from previously known methods. In particular, we demonstrate our method by constructing many new explicit classes of permutation polynomials of the form $x^rh\left(x^{q-1}\right)$ over $\mathbb{F}_{q^2}$. Moreover, we can explain most of the known permutation trinomials, which are in [6, 13, 14, 16, 20, 29], over finite field with even characteristic.Comment: 29 pages. An early version of this paper was presented at Fq13 in Naples, Ital