arXiv subjects

Colin Tan

Publications and source records attributed to Colin Tan.

12 recordsLinked to original sources

arXiv2026

Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set can be represented by some polynomial with only positive coefficients. This result generalizes a result of P\'olya which corresponds to the case when the semi-algebraic set is the standard simplex. Our proof uses the Archimedean Representation Theorem from real algebra.

math.AG↗

arXiv2025

Cayley graphs on elementary abelian groups of extreme degree have complete cores

Ne\v{s}et\v{r}il and \v{S}\'{a}mal asked whether every cubelike graph has a cubelike core. Man\v{c}inska, Pivotto, Roberson and Royle answered this question in the affirmative for cubelike graphs whose core has at most $32$ vertices. When the core of a cubelike graph has at most $16$ vertices, they gave a list of these cores, from which it follows that every cubelike graph with degree strictly less than $5$ has a complete core. We prove the following extension: if the degree of a cubelike graph is either strictly less than $5$ or at least $5$ less than the number of its vertices, then its core is complete and induced by a $\mathbb{F}_2$-vector subspace of its vertices. Thus we also answer Ne\v{s}et\v{r}il and \v{S}\'{a}mal's question in the affirmative for cubelike graphs with degree at least $5$ less than the number of vertices. Our result is sharp as the $5$-regular folded $5$-cube and its graph complement are both non-complete cubelike graph cores. We also prove analogous results for Cayley graphs on elementary abelian $p$-groups for odd primes $p$.

math.CO↗

arXiv2022

Probability bounds for $n$ random events under $(n-1)$-wise independence

A collection of $n$ random events is said to be $(n - 1)$-wise independent if any $n - 1$ events among them are mutually independent. We characterise all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent. We provide sharp upper and lower bounds on the probability that at least $k$ out of $n$ events with given marginal probabilities occur over these probability measures. The bounds are shown to be computable in polynomial time.

math.PR↗

arXiv2019

Characterization of polynomials whose large powers have fully positive coefficients

We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of the authors, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.

math.AG↗

arXiv2018

Improved effective estimates of P\'olya's Theorem for quadratic forms

Following de Loera and Santos, the P\'olya exponent of a $n$-ary real form (i.e. a homogeneous polynomial in $n$ variables with real coefficients) $f$ is the infimum of the upward closed set of nonnegative integers $m$ such that $(x_1 + \cdots + x_n)^m f$ strictly has positive coefficients. By a theorem of P\'olya, a form assumes only positive values over the standard $(n - 1)$-simplex in Euclidean $n$-space if and only if its P\'olya exponent is finite. In this note, we compute an upper bound of the P\'olya exponent of a quadratic form $f$ that assumes only positive values over the standard simplex. Our bound improves a previous upper bound due to de Klerk, Laurent and Parrilo. For example, for the binary quadratic form $f_\kappa = \lambda^2 x_1^2 - 2 \kappa \lambda x_1 x_2 + x_2^2$, which assumes only positive values over the standard $1$-simplex whenever $0 \le \kappa < 1 < \lambda$, our upper bound of its P\'olya's exponent is $O(1/\lambda)$ times that of de Klerk, Laurent and Parrilo's as $\lambda$ tends to infinity.

math.AG↗

arXiv2017

Archimedean Representation Theorem for modules over a commutative ring

P\'olya's Positivstellensatz and Handelman's Positivstellensatz are known to be concrete instances of the abstract Archimedean Representation Theorem for (commutative unital) rings. We generalise the Archimedean Representation Theorem to modules over rings. For example, consider the module of all symmetric matrices with entries in a polynomial ring, also known as matrix polynomials. P\'olya's Positivstellensatz and Handelman's Positivstellensatz had been generalised by Scherer and Hol, and L\^{e} and Du' respectively to matrix polynomials, using the method of effective estimates from analysis. We show that these two Positivstellens\"atze for matrix polynomials are concrete instances of our Archimedean Representation Theorem in the case of the module of symmetric matrix polynomials over the polynomial ring.

math.AG↗

arXiv2017

A Novel Multi-task Deep Learning Model for Skin Lesion Segmentation and Classification

In this study, a multi-task deep neural network is proposed for skin lesion analysis. The proposed multi-task learning model solves different tasks (e.g., lesion segmentation and two independent binary lesion classifications) at the same time by exploiting commonalities and differences across tasks. This results in improved learning efficiency and potential prediction accuracy for the task-specific models, when compared to training the individual models separately. The proposed multi-task deep learning model is trained and evaluated on the dermoscopic image sets from the International Skin Imaging Collaboration (ISIC) 2017 Challenge - Skin Lesion Analysis towards Melanoma Detection, which consists of 2000 training samples and 150 evaluation samples. The experimental results show that the proposed multi-task deep learning model achieves promising performances on skin lesion segmentation and classification. The average value of Jaccard index for lesion segmentation is 0.724, while the average values of area under the receiver operating characteristic curve (AUC) on two individual lesion classifications are 0.880 and 0.972, respectively.

cs.CV↗

arXiv2017

Characterization of polynomials whose large powers have all positive coefficients

We give a criterion which characterizes a homogeneous real multi-variate polynomial to have the property that all sufficiently large powers of the polynomial (as well as their products with any given positive homogeneous polynomial) have positive coefficients. Our result generalizes a result of De Angelis, which corresponds to the case of homogeneous bi-variate polynomials, as well as a classical result of P\'olya, which corresponds to the case of a specific linear polynomial. As an application, we also give a characterization of certain polynomial beta functions, which are the spectral radius functions of the defining matrix functions of Markov chains.

math.CV↗

arXiv2017

A Positivstellensatz for forms on the positive orthant

Let $p$ be a nonconstant form in $\mathbb{R}[x_1,\dots,x_n]$ with $p(1,\dots,1)>0$. If $p^m$ has strictly positive coefficients for some integer $m\ge1$, we show that $p^m$ has strictly positive coefficients for all sufficiently large $m$. More generally, for any such $p$, and any form $q$ that is strictly positive on $(\mathbb{R}_+)^n\setminus\{0\}$, we show that the form $p^mq$ has strictly positive coefficients for all sufficiently large $m$. This result can be considered as a strict Positivstellensatz for forms relative to $(\mathbb{R}_+)^n\setminus\{0\}$. We give two proofs, one based on results of Handelman, the other on techniques from real algebra.

math.AG↗

arXiv2016

Eventual positivity of Hermitian algebraic functions and associated integral operators

Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert's seventeenth problem in the nondegenerate case. Later Catlin-D'Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator. The arguments of Catlin-D'Angelo, as well as that of a subsequent refinement by Varolin, involve subtle asymptotic estimates of the Bergman kernel. In this article, we give an elementary and geometric proof of the eventual positivity of this integral operator, thereby yielding another proof of the corresponding positivstellensatz.

math.DG↗

arXiv2015

Loop homological invariants associated to real projective spaces

Let A be a based subspace of Y. Under the assumptions that Y is path-connected and that the reduced diagonal map of A induces the zero map in all mod 2 reduced homology groups, we compute a formula for the mod 2 reduced Poincar\'{e} series of the loop space $\Omega ((A \wedge \mathbb{RP}^\infty) \cup_{A \wedge \mathbb{RP}^1} (Y \wedge \mathbb{RP}^1))$. Here $\mathbb{RP}^\infty$ and $\mathbb{RP}^1$ denote the infinite real projective space and the real projective line respectively.

math.AT↗

arXiv2014

Eventual positivity of Hermitian polynomials and integral operators

Quillen proved that, if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.

math.DG↗