arXiv subjects

Zhao Shen

Publications and source records attributed to Zhao Shen.

7 recordsLinked to original sources

arXiv2026

Intersections and Minkowski Sums of Four-Corner Cantor Dusts with the Unit Circle

For $0<\lambda<1/2$, let $K_{\lambda}$ be the attractor of the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$, and put $C_\lambda=K_{\lambda}\times K_{\lambda}$. We study the intersection \[ E_\lambda=C_\lambda\cap S^1 \] and the Minkowski sum \[ A_{\lambda}=C_\lambda+S^1, \] where $S^1$ is the unit circle. For the intersection problem, let $\lambda_{\infty}\approx 0.305854$ be the unique root in $(1/4,1/2)$ of $2x^3-3x^2+4x-1=0$. We prove that $E_\lambda$ is infinite for every $\lambda_{\infty}<\lambda<1/2$. In particular, $C_{1/3}\cap S^1$ is infinite, answering the first part of a question of Jiang, Kong, Li and Wang. Moreover, $E_\lambda$ has the cardinality of the continuum for $(\sqrt{3}-1)/2<\lambda<1/2$, and $\dim_{\rm H} E_\lambda>0$ for $\sqrt{2}-1<\lambda<1/2$. We also establish quantitative lower bounds for $\dim_{\rm H} E_\lambda$ for $\lambda$ near $1/2$; in particular, $\dim_{\rm H} E_\lambda$ approaches $1$ as $\lambda\uparrow1/2$. For the Minkowski sum problem, we prove that $A_{\lambda}$ has nonempty interior throughout the previously open range $1/4<\lambda<1/3$, answering a question of Simon and Taylor. Together with earlier results of Simon and Taylor, our theorem yields the complete classification: $A_{\lambda}$ has nonempty interior in $\mathbb{R}^2$ if and only if $1/4<\lambda<1/2$. More generally, we prove that $C_\lambda+\Gamma$ has nonempty interior for every regular $C^1$ closed curve $\Gamma$ whenever $1/4<\lambda<1/2$.

math.CA↗

arXiv2026

Reducing Every Set of 36 Consecutive Integers to Zero by Differences of Squares

For a finite multiset of integers, repeatedly choose two entries $a,b$ and replace them with $|a^{2}-b^{2}|$. Hickerson and Kleber asked whether, for every integer $n$, the set ${n,n+1,\ldots,n+35}$ can be reduced to zero. We give an explicit reduction. Together with their results for lengths 12 and 24, this gives, for every positive integer $L$, [ {n,n+1,\ldots,n+L-1}\text{ reduces to zero for every }n\in\mathbb Z \Longleftrightarrow 12\mid L\text{ and }L\ge 24. ]

math.NT↗

arXiv2026

2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function

Let $T(x)=\prod_{k=0}^{\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\sum_{n\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \[ \nu_2\bigl(t_5(4n+j)\bigr) =4\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,2,3\}, \] \[ \nu_2\bigl(t_9(8n+j)\bigr) =5\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-2\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,\ldots,7\}. \] These formulas confirm Conjecture~5.2 of Gawron--Miska--Ulas~\cite{ga} for $m=5$ and $m=9$, and imply that $t_5(n)\neq 0$ and $t_9(n)\neq 0$ for every $n\geq 0$. A key structural ingredient is a closed-form formula for the determinant of a family of matrices with binomial-coefficient entries.

math.CO↗

arXiv2026

Unboundedness of the Coefficients of Higher Powers of a Unimodular Power Series

Let $R(z)=\sum_{n=0}^{\infty} r_n z^n$ be a power series with $|r_n|=1$ for every $n\ge 0$. We show that for each integer $m\ge 2$, the coefficient sequence of $R(z)^m$ is unbounded. The proof combines Parseval's identity with Jensen's inequality. As a consequence, Conjecture~3.9 of Gawron, Miska, and Ulas \cite{gmu} is confirmed.

math.CO↗

arXiv2023

The effect of LPSO phase on the high-temperature oxidation of a stainless Mg-Y-Al alloy

In this study, we investigated the oxidation of the Mg-11Y-1Al alloy at 500{\deg}C in an Ar-20%O2 environment. Multiscale analysis showed the network-like long-period stacking ordered (LPSO) phase transformed into needle-like LPSO and polygonal Mg24Y5 phases, leading to the formation of a high-dense network of needle-like oxides at the oxidation front. These oxides grew laterally along the oxide/matrix interfaces, forming a thicker, continuous scale that effectively blocked elemental diffusion. Hence, the preferential oxidation along the needle-like LPSO is believed to accelerate the formation of a thicker and continuous oxide scale, further improving the oxidation resistance of the Mg-11Y-1Al alloy.

cond-mat.mtrl-sci↗

arXiv2022

On a conjecture of J. Shallit about Apéry-like numbers

Put $a(n)=\sum\limits_{k=0}^{n}\binom{n}{k}\binom{n+k}{k}$, and $b(n)=v_{3}(a(n))$, for all integers $n\geqslant 0$, where $v_{3}$ is the $3$-adic valuation. In this work, we shall confirm a formula about $b(n)$, conjectured by J. Shallit in 2000. As application, we show that the sequence $(b(n))_{n\geqslant 0}$ is $3$-regular.

math.NT↗

arXiv2018

Interpretable Credit Application Predictions With Counterfactual Explanations

We predict credit applications with off-the-shelf, interchangeable black-box classifiers and we explain single predictions with counterfactual explanations. Counterfactual explanations expose the minimal changes required on the input data to obtain a different result e.g., approved vs rejected application. Despite their effectiveness, counterfactuals are mainly designed for changing an undesired outcome of a prediction i.e. loan rejected. Counterfactuals, however, can be difficult to interpret, especially when a high number of features are involved in the explanation. Our contribution is two-fold: i) we propose positive counterfactuals, i.e. we adapt counterfactual explanations to also explain accepted loan applications, and ii) we propose two weighting strategies to generate more interpretable counterfactuals. Experiments on the HELOC loan applications dataset show that our contribution outperforms the baseline counterfactual generation strategy, by leading to smaller and hence more interpretable counterfactuals.

cs.AI↗