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Guozhen Shen

Publications and source records attributed to Guozhen Shen.

15 recordsLinked to original sources
arXiv2026

A choice-free proof of the Erd\H{o}s--Dushnik--Miller theorem

The Erd\H{o}s--Dushnik--Miller theorem states that for every aleph $\kappa$, \[ \kappa\to(\kappa,\omega); \] that is, every coloring $c:[\kappa]^2\to2$ has either a $0$-homogeneous set of cardinality $\kappa$ or a $1$-homogeneous set of cardinality $\omega$. In this article, we present a purely combinatorial proof of this theorem in $\mathsf{ZF}$ (i.e., Zermelo--Fraenkel set theory without the axiom of choice), avoiding any metamathematical considerations.

math.LO↗
arXiv2025

Amorphous sets and dual Dedekind finiteness

A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists an amorphous set $A$ whose power set $\mathscr{P}(A)$ is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss [J. Truss, Fund. Math. 84, 187--208 (1974)]. Nevertheless, we prove in $\mathsf{ZF}$ that, for all strictly amorphous sets $A$ and all natural numbers $n$, $\mathscr{P}(A)^n$ is dually Dedekind finite, which generalizes a result of Goldstern.

math.LO↗
arXiv2025

On ordering of surjective cardinals

Let $\mathrm{Card}$ denote the class of cardinals. For all cardinals $\mathfrak{a}$ and $\mathfrak{b}$, $\mathfrak{a}\leqslant\mathfrak{b}$ means that there is an injection from a set of cardinality $\mathfrak{a}$ into a set of cardinality $\mathfrak{b}$, and $\mathfrak{a}\leqslant^\ast\mathfrak{b}$ means that there is a partial surjection from a set of cardinality $\mathfrak{b}$ onto a set of cardinality $\mathfrak{a}$. A doubly ordered set is a triple $\langle P,\preccurlyeq,\preccurlyeq^\ast\rangle$ such that $\preccurlyeq$ is a partial ordering on $P$, $\preccurlyeq^\ast$ is a preordering on $P$, and ${\preccurlyeq}\subseteq{\preccurlyeq^\ast}$. In 1966, Jech proved that for every partially ordered set $\langle P,\preccurlyeq\rangle$, there exists a model of $\mathsf{ZF}$ in which $\langle P,\preccurlyeq\rangle$ can be embedded into $\langle\mathrm{Card},\leqslant\rangle$. We generalize this result by showing that for every doubly ordered set $\langle P,\preccurlyeq,\preccurlyeq^\ast\rangle$, there exists a model of $\mathsf{ZF}$ in which $\langle P,\preccurlyeq,\preccurlyeq^\ast\rangle$ can be embedded into $\langle\mathrm{Card},\leqslant,\leqslant^\ast\rangle$.

math.LO↗
arXiv2025

A note on surjective cardinals

For cardinals $\mathfrak{a}$ and $\mathfrak{b}$, we write $\mathfrak{a}=^\ast\mathfrak{b}$ if there are sets $A$ and $B$ of cardinalities $\mathfrak{a}$ and $\mathfrak{b}$, respectively, such that there are partial surjections from $A$ onto $B$ and from $B$ onto $A$. $=^\ast$-equivalence classes are called surjective cardinals. In this article, we show that $\mathsf{ZF}+\mathsf{DC}_κ$, where $κ$ is a fixed aleph, cannot prove that surjective cardinals form a cardinal algebra, which gives a negative solution to a question proposed by Truss [J. Truss, Ann. Pure Appl. Logic 27, 165--207 (1984)]. Nevertheless, we show that surjective cardinals form a ``surjective cardinal algebra'', whose postulates are almost the same as those of a cardinal algebra, except that the refinement postulate is replaced by the finite refinement postulate. This yields a smoother proof of the cancellation law for surjective cardinals, which states that $m\cdot\mathfrak{a}=^\ast m\cdot\mathfrak{b}$ implies $\mathfrak{a}=^\ast\mathfrak{b}$ for all cardinals $\mathfrak{a},\mathfrak{b}$ and all nonzero natural numbers $m$.

math.LO↗
arXiv2025

The 2D Materials Roadmap

Over the past two decades, 2D materials have rapidly evolved into a diverse and expanding family of material platforms. Many members of this materials class have demonstrated their potential to deliver transformative impact on fundamental research and technological applications across different fields. In this roadmap, we provide an overview of the key aspects of 2D material research and development, spanning synthesis, properties and commercial applications. We specifically present roadmaps for high impact 2D materials, including graphene and its derivatives, transition metal dichalcogenides, MXenes as well as their heterostructures and moiré systems. The discussions are organized into thematic sections covering emerging research areas (e.g., twisted electronics, moiré nano-optoelectronics, polaritronics, quantum photonics, and neuromorphic computing), breakthrough applications in key technologies (e.g., 2D transistors, energy storage, electrocatalysis, filtration and separation, thermal management, flexible electronics, sensing, electromagnetic interference shielding, and composites) and other important topics (computational discovery of novel materials, commercialization and standardization). This roadmap focuses on the current research landscape, future challenges and scientific and technological advances required to address, with the intent to provide useful references for promoting the development of 2D materials.

cond-mat.mtrl-sci↗
arXiv2025

A note on dual Dedekind finiteness

A set $A$ is dually Dedekind finite if every surjection from $A$ onto $A$ is injective; otherwise, $A$ is dually Dedekind infinite. It is proved consistent with $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) that there exists a family $\langle A_n\rangle_{n\inω}$ of sets such that, for all $n\inω$, $A_n^n$ is dually Dedekind finite whereas $A_n^{n+1}$ is dually Dedekind infinite. This resolves a question that was left open in [J. Truss, Fund. Math. 84, 187--208 (1974)].

math.LO↗
arXiv2025

Boundedly finite-to-one functions

A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\mathsf{ZF}$ (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) several results concerning this notion, among which are the following: (1) For each infinite set $A$ and natural number $n$, there is no boundedly finite-to-one function from $\mathcal{S}(A)$ to $\mathcal{S}_{\leq n}(A)$, where $\mathcal{S}(A)$ is the set of all permutations of $A$ and $\mathcal{S}_{\leq n}(A)$ is the set of all permutations of $A$ moving at most $n$ points. (2) For each infinite set $A$, there is no boundedly finite-to-one function from $\mathcal{B}(A)$ to $\mathrm{fin}(A)$, where $\mathcal{B}(A)$ is the set of all partitions of $A$ such that every block is finite and $\mathrm{fin}(A)$ is the set of all finite subsets of $A$.

math.LO↗
arXiv2025

A choice-free proof of Mal'cev's theorem on quasivarieties

In 1966, Mal'cev proved that a class $\mathcal{K}$ of first-order structures with a specified signature is a quasivariety if and only if $\mathcal{K}$ contains a unit and is closed under isomorphisms, substructures, and reduced products. In this article, we present a proof of this theorem in $\mathsf{ZF}$ (the Zermelo--Fraenkel set theory without the axiom of choice).

math.LO↗
arXiv2024

The finitary partitions with $n$ non-singleton blocks of a set

A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{fin}(\mathfrak{a})$ and $\mathscr{B}_{n}(\mathfrak{a})$ be the cardinalities of the set of finite subsets and the set of finitary partitions with exactly $n$ non-singleton blocks of a set which is of cardinality $\mathfrak{a}$, respectively. In this paper, we prove in $\mathsf{ZF}$ (without the axiom of choice) that for all infinite cardinals $\mathfrak{a}$ and all non-zero natural numbers $n$, \[ (2^{\mathscr{B}_{n}(\mathfrak{a})})^{\aleph_0}=2^{\mathscr{B}_{n}(\mathfrak{a})} \] and \[ 2^{\mathrm{fin}(\mathfrak{a})^n}=2^{\mathscr{B}_{2^n-1}(\mathfrak{a})}. \] It is also proved consistent with $\mathsf{ZF}$ that there exists an infinite cardinal $\mathfrak{a}$ such that \[ 2^{\mathscr{B}_{1}(\mathfrak{a})}<2^{\mathscr{B}_{2}(\mathfrak{a})}<2^{\mathscr{B}_{3}(\mathfrak{a})}<\cdots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}. \]

math.LO↗
arXiv2023

Cantor's theorem may fail for finitary partitions

A partition is finitary if all its members are finite. For a set $A$, $\mathscr{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{ZF}$ (without the axiom of choice) that there exist an infinite set $A$ and a surjection from $A$ onto $\mathscr{B}(A)$. On the other hand, we prove in $\mathsf{ZF}$ some theorems concerning $\mathscr{B}(A)$ for infinite sets $A$, among which are the following: (1) If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathscr{B}(A)$ and no finite-to-one functions from $\mathscr{B}(A)$ to $A$. (2) For all $n\inω$, $|A^n|<|\mathscr{B}(A)|$. (3) $|\mathscr{B}(A)|\neq|\mathrm{seq}(A)|$, where $\mathrm{seq}(A)$ is the set of all finite sequences of elements of $A$.

math.LO↗
arXiv2021

A generalized Cantor theorem in ZF

It is proved in $\mathsf{ZF}$ (without the axiom of choice) that, for all infinite sets $M$, there are no surjections from $ω\times M$ onto $\mathscr{P}(M)$.

math.LO↗
arXiv2019

A choice-free cardinal equality

For a cardinal $\mathfrak{a}$, let $\mathrm{fin}(\mathfrak{a})$ be the cardinality of the set of all finite subsets of a set which is of cardinality $\mathfrak{a}$. It is proved without the aid of the axiom of choice that for all infinite cardinals $\mathfrak{a}$ and all natural numbers $n$, \[ 2^{\mathrm{fin}(\mathfrak{a})^n}=2^{[\mathrm{fin}(\mathfrak{a})]^n}. \] On the other hand, it is proved that the following statement is consistent with $\mathsf{ZF}$: there exists an infinite cardinal $\mathfrak{a}$ such that \[ 2^{\mathrm{fin}(\mathfrak{a})}<2^{\mathrm{fin}(\mathfrak{a})^2}<2^{\mathrm{fin}(\mathfrak{a})^3}<\dots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}. \]

math.LO↗
arXiv2018

Factorials of infinite cardinals in ZF

For a set $x$, let $\mathcal{S}(x)$ be the set of all permutations of $x$. We study several aspects of this notion in $\mathsf{ZF}$. The main results are as follows: (1) $\mathsf{ZF}$ proves that for all sets $x$, if $\mathcal{S}(x)$ is Dedekind infinite, then there are no finite-to-one maps from $\mathcal{S}(x)$ into $\mathcal{S}_{\mathrm{fin}}(x)$, where $\mathcal{S}_{\mathrm{fin}}(x)$ is the set of all permutations of $x$ which move only finitely many elements. (2) $\mathsf{ZF}$ proves that for all sets $x$, the cardinality of $\mathcal{S}(x)$ is strictly greater than that of $[x]^2$. (3) It is consistent with $\mathsf{ZF}$ that there exists an infinite set $x$ such that the cardinality of $\mathcal{S}(x)$ is strictly less than that of $[x]^3$. (4) It is consistent with $\mathsf{ZF}$ that there exists an infinite set $x$ such that there is a finite-to-one map from $\mathcal{S}(x)$ into $x$.

math.LO↗
arXiv2013

Laterally emitted surface second harmonic generation in a single ZnTe nanowire

We report a direct observation on the unique laterally emitted surface second harmonic generation (SHG) in a single ZnTe nanowire. The highly directional surface SHG signal that polarizes along the direction vertical to the nanowire growth axis, is significantly larger than the bulk SHG contribution, indicating a high efficiency of surface SHG. Two strong SHG peaks are observed on both sides of the nanowire surfaces in the far-field scanning images, which is further supported by FDTD simulations, demonstrating that the unique laterally emitted signal is ascribed to surface SHG in the ZnTe nanowire. The surface SHG in a single ZnTe nanowire with unique lateral emission and high conversion efficiency shows great potential applications in short-wavelength nanolasers, nonlinear microscopy and polarization dependent photonic integrating.

physics.optics↗