arXiv subjects

Boyuan Liu

Publications and source records attributed to Boyuan Liu.

89 records · Page 5Linked to original sources

arXiv2020

Exact minimum codegree thresholds for $K_4^-$-covering and $K_5^-$-covering

Given two $3$-graphs $F$ and $H$, an $F$-covering of $H$ is a collection of copies of $F$ in $H$ such that each vertex of $H$ is contained in at least one copy of them. Let {$c_2(n,F)$} be the maximum integer $t$ such that every 3-graph with minimum codegree greater than $t$ has an $F$-covering. In this note, we answer an open problem of Falgas-Ravry and Zhao (SIAM J. Discrete Math., 2016) by determining the exact value of {$c_2(n, K_4^-)$} and {$c_2(n, K_5^-)$}, where $K_t^-$ is the complete $3$-graph on $t$ vertices with one edge removed.

math.CO↗

arXiv2019

Extended range of dipole-dipole interactions in periodically structured photonic media

The interaction between quantum two-level systems is typically short-range in free space and most photonic environments. Here we show that diminishing momentum isosurfaces with equal frequencies can create a significantly extended range of interaction between distant quantum systems. The extended range is robust and does not rely on a specific location or orientation of the transition dipoles. A general relation between the interaction range and properties of the isosurface is described for structured photonic media. It provides a new way to mediate long-range quantum behavior.

physics.optics↗

arXiv2019

Neuromorphic Metasurface

Metasurfaces have been used to realize optical functions such as focusing and beam steering. They use sub-wavelength nanostructures to control the local amplitude and phase of light. Here we show that such control could also enable a new function of artificial neural inference. We demonstrate that metasurfaces can directly recognize objects by focusing light from an object to different spatial locations that correspond to the class of the object.

physics.app-ph↗

arXiv2019

Constraining the non-gravitational scattering of baryons and dark matter with early cosmic structure formation

We derive new constraints on the non-gravitational baryon-dark-matter scattering (BDMS) by evaluating the mass thresholds of dark matter (DM) haloes in which primordial gas can cool efficiently to form Population III (Pop III) stars, based on the timing of the observed 21-cm absorption signal. We focus on the BDMS model with interaction cross-section $σ=σ_{1}[v/(1\ \mathrm{km\ s^{-1}})]^{-4}$, where $v$ is the relative velocity of the encounter. Our results rule out the region in parameter space with $σ_{1}\gtrsim 10^{-19}\ \mathrm{cm^{2}}$ and DM particle mass $m_χc^{2}\lesssim 3\times 10^{-2}\ \mathrm{GeV}$, where the cosmic number density of Pop III hosts at redshift $z\sim 20$ is at least three orders of magnitude smaller than in the standard Lambda cold DM ($Λ$CDM) case. In these BDMS models, the formation of Pop III stars is significantly suppressed for $z\gtrsim 20$, inconsistent with the timing of the observed global 21-cm absorption signal. For the fiducial BDMS model with $m_χc^{2}=0.3$ GeV and $σ_{1}=8\times 10^{-20}\ \mathrm{cm^{2}}$, capable of accommodating the measured absorption depth, the number density of Pop III hosts is reduced by a factor of $3-10$ at $z\sim 15-20$, when the 21-cm signal is imprinted, compared with the $Λ$CDM model. The confluence of future detailed cosmological simulations with improved 21-cm observations promises to probe the particle-physics nature of DM at the small-scale frontier of early structure formation.

astro-ph.GA↗

arXiv2019

Constraining First Star Formation with 21cm-Cosmology

Within standard $Λ$CDM cosmology, Population III (Pop III) star formation in minihalos of mass $M_\mathrm{halo}\gtrsim 5\times10^5$ M$_\odot$ provides the first stellar sources of Lyman$α$ (Ly$α$) photons. The Experiment to Detect the Global Epoch of Reionization Signature (EDGES) has measured a strong absorption signal of the redshifted 21 cm radiation from neutral hydrogen at $z\approx 17$, requiring efficient formation of massive stars before then. In this paper, we investigate whether star formation in minihalos plays a significant role in establishing the early Ly$α$ background required to produce the EDGES absorption feature. We find that Pop III stars are important in providing the necessary Ly$α$-flux at high redshifts, and derive a best-fitting average Pop III stellar mass of $\sim$ 750M$_\odot{}$ per minihalo, corresponding to a star formation efficiency of 0.1%. Further, it is important to include baryon-dark matter streaming velocities in the calculation, to limit the efficiency of Pop~III star formation in minihalos. Without this effect, the cosmic dawn coupling between 21 cm spin temperature and that of the gas would occur at redshifts higher than what is implied by EDGES.

astro-ph.GA↗

arXiv2019

Global radiation signature from early structure formation

We use cosmological hydrodynamic zoom-in simulations to study early structure formation in two dark matter (DM) cosmologies, the standard CDM model, and a thermal warm DM (WDM) model with a particle mass of $m_χc^{2}=3\ \mathrm{keV}$. We focus on DM haloes with virial masses $M\sim 10^{10}\ M_{\odot}$. We find that the first star formation activity is delayed by $\sim 200\ \mathrm{Myr}$ in the WDM model, with similar delays for metal enrichment and the formation of the second generation of stars. However, the differences between the two models in globally-averaged properties, such as star formation rate density and mean metallicity, decrease towards lower redshifts ($z\lesssim 10$). Metal enrichment in the WDM cosmology is restricted to dense environments, while low-density gas can also be significantly enriched in the CDM case. The free-free contribution from early structure formation at redshifts $z>6$ to the cosmic radio background (CRB) is $3_{-1.5}^{+13}\%$ ($8_{-3.5}^{+33}\%$) of the total signal inferred from radio experiments such as ARCADE 2, in the WDM (CDM) model. The direct detection of the $\mathrm{H_{2}}$ emission from early structure formation ($z\gtrsim 7.2$), originating from the low-mass haloes explored here, will be challenging even with the next generation of far-infrared space telescopes, unless the signal is magnified by at least a factor of 10 via gravitational lensing or shocks. However, more massive haloes with $M\gtrsim 10^{12}\ M_{\odot}$ may be observable for $z\gtrsim 10$, even without magnification, provided that our extrapolation from the scale of our simulated haloes is valid.

astro-ph.GA↗

arXiv2018

The spectral radius of graphs without trees of diameter at most four

Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join of a complete graph of order $k$ and an empty graph of order $n-k$. In this paper, we show that the conjecture is true for trees of diameter at most four.

math.CO↗

arXiv2018

Codegree conditions for tilling balanced complete $3$-partite $3$-graphs and generalized 4-cycles

Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$ such that every $k$-graph $H$ on $n$ vertices with minimum codegree at least $t$ contains a perfect $F$-tiling. Mycroft (JCTA, 2016) determined the asymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$. Mycroft also conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replaced by a constant that depends only on $F$. In this paper, we improve the error term of Mycroft's result to a sub-linear term when $F=K^3(m)$, the complete $3$-partite $3$-graph with each part of size $m$. We also show that the sub-linear term is tight for $K^3(2)$, {the result also provides another family of counterexamples of Mycroft's conjecture (Gao, Han, Zhao (arXiv, 2016) gave a family of counterexamples when $H$ is a $k$-partite $k$-graph with some restrictions.)} Finally, we show that Mycroft's conjecture holds for generalized 4-cycle $C_4^3$ (the 3-graph on six vertices and four distinct edges $A, B, C, D$ with $A\cup B= C\cup D$ and $A\cap B=C\cap D=\emptyset$), i.e. we determine the exact value of $t_2(n, C_4^3)$.

math.CO↗

arXiv2018

Generalized Gilat-Raubenheimer method for density-of-states calculation in photonic crystals

Efficient numeric algorithm is the key for accurate evaluation of density of states (DOS) in band theory. Gilat-Raubenheimer (GR) method proposed in 1966 is an efficient linear extrapolation method which was limited in specific lattices. Here, using an affine transformation, we provide a new generalization of the original GR method to any Bravais lattices and show that it is superior to the tetrahedron method and the adaptive Gaussian broadening method. Finally, we apply our generalized GR (GGR) method to compute DOS of various gyroid photonic crystals of topological degeneracies.

physics.app-ph↗

arXiv2018

Effect of lithium hydride on the cooling of primordial gas

We complete the formulation of the standard model of first star formation by exploring the possible impact of $\mathrm{LiH}$ cooling, which has been neglected in previous simulations of non-linear collapse. Specifically, we find that at redshift $z\gtrsim 5$, the cooling by $\mathrm{LiH}$ has no effect on the thermal evolution of shocked primordial gas, and of collapsing primordial gas into minihaloes or relic HII regions, even if the primordial lithium abundance were enhanced by one order of magnitude. Adding the most important lithium species to a minimum network of primordial chemistry, we demonstrate that insufficient $\mathrm{LiH}$ is produced in all cases considered, about $[\mathrm{LiH/Li}]\sim 10^{-9}$ for $T\lesssim 100$ K. Indeed, $\mathrm{LiH}$ cooling would only be marginally significant in shocked primordial gas for the highly unlikely case that the $\mathrm{LiH}$ abundance were increased by nine orders of magnitude, implying that $all$ lithium would have to be converted into $\mathrm{LiH}$. In this study, photo-destruction processes are not considered, and the collisional disassociation rate of $\mathrm{LiH}$ is possibly underestimated, rendering our results an extreme upper limit. Therefore, the cooling by $\mathrm{LiH}$ can safely be neglected for the thermal evolution of Population~III star-forming gas.

astro-ph.GA↗

arXiv2017

$H$-Decomposition of $r$-graphs when $H$ is an $r$-graph with exactly $k$ independent edges

Let $ϕ_H^r(n)$ be the smallest integer such that, for all $r$-graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $ϕ_H^r(n)$ parts, of which every part either is a single edge or forms an $r$-graph isomorphic to $H$. The function $ϕ^2_H(n)$ has been well studied in literature, but for the case $r\ge 3$, the problem that determining the value of $ϕ_H^r(n)$ is widely open. Sousa (2010) gave an asymptotic value of $ϕ_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, and determined the exact value of $ϕ_H^r(n)$ in some special cases. In this paper, we first give the exact value of $ϕ_H^r(n)$ when $H$ is an $r$-graph with exactly 2 edges, which improves Sousa's result. Second we determine the exact value of $ϕ_H^r(n)$ when $H$ is an $r$-graph consisting of exactly $k$ independent edges.

math.CO↗

arXiv2017

The size of $3$-uniform hypergraphs with given matching number and codegree

Determine the size of $r$-graphs with given graph parameters is an interesting problem. Chvátal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014) studied the same problem for linear $3$-graphs with restricted matching number and maximum degree. In this paper, we give a tight upper bound of the size of $3$-graphs with bounded codegree and matching number.

math.CO↗

arXiv2017

Odd induced subgraphs in graphs with treewidth at most two

A long-standing conjecture asserts that there exists a constant $c>0$ such that every graph of order $n$ without isolated vertices contains an induced subgraph of order at least $cn$ with all degrees odd. Scott (1992) proved that every graph $G$ has an induced subgraph of order at least $|V(G)|/(2χ(G))$ with all degrees odd, where $χ(G)$ is the chromatic number of $G$, this implies the conjecture for graphs with { bounded} chromatic number. But the factor $1/(2χ(G))$ seems to be not best possible, for example, Radcliffe and Scott (1995) proved $c=\frac 23$ for trees, Berman, Wang and Wargo (1997) showed that $c=\frac 25$ for graphs with maximum degree $3$, so it is interesting to determine the exact value of $c$ for special family of graphs. In this paper, we further confirm the conjecture for graphs with treewidth at most 2 with $c=\frac{2}{5}$, and the bound is best possible.

math.CO↗

arXiv2017

Experimental discovery of nodal chains

Three-dimensional (3D) topological nodal points, such as Weyl and Dirac nodes have attracted wide-spread interest across multiple disciplines and diverse material systems. Unlike nodal points that contain little structural variations, nodal lines can have numerous topological configurations in the momentum space, forming nodal rings, nodal chains and potentially nodal links and nodal knots. However, nodal lines have much less development for the lack of ideal material platforms. In condensed matter for example, nodal lines are often fragile to spin-orbit-coupling, locating off the Fermi level, coexisting with energy-degenerate trivial bands and dispersing strongly in energy of the line degeneracy. Here, overcoming all above difficulties, we theoretically predict and experimentally observe nodal chains in a metallic-mesh photonic crystal having frequency-isolated linear bandtouching rings chained across the entire Brillouin zone (BZ). These nodal chains are protected by mirror symmetries and have a frequency variation less than 1%. We used angle-resolved transmission (ART) to probe the projected bulk dispersions and performed Fourier-transformed field scan (FTFS) to map out the surface dispersions, which is a quadratic touching between two drumhead surface bands. Our results established an ideal nodal-line material for further studies of topological line-degeneracies with nontrivial connectivities, as well as the consequent wave dynamics richer than 2D Dirac and 3D Weyl materials.

cond-mat.mtrl-sci↗

arXiv2016

Turán number and decomposition number of intersecting odd cycles

An extremal graph for a given graph $H$ is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erdős et al. (1995) determined the extremal graphs for $H_{s,0}$. Recently, Hou et al. (2016) determined the extremal graphs for $H_{0,t}$, where the $t$ cycles have the same odd length $q$ with $q\ge 5$. In this paper, we further determine the extremal graphs for $H_{s,t}$ with $s\ge 0$ and $t\ge 1$. Let $ϕ(n,H)$ be the largest integer such that, for all graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $ϕ(n, H)$ parts, of which every part either is a single edge or forms a graph isomorphic to $H$. Pikhurko and Sousa conjectured that $ϕ(n,H)=\ex(n,H)$ for $χ(H)\geqs3$ and all sufficiently large $n$. Liu and Sousa (2015) verified the conjecture for $H_{s,0}$. In this paper, we further verify Pikhurko and Sousa's conjecture for $H_{s,t}$ with $s\ge 0$ and $t\ge 1$.

math.CO↗

arXiv2015

Extremal graph for intersecting odd cycles

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

math.CO↗

arXiv2015

Decomposition of Graphs into $(k,r)$-Fans and Single Edges

Let $ϕ(n,H)$ be the largest integer such that, for all graphs $G$ on $n$ vertices, the edge set $E(G)$ can be partitioned into at most $ϕ(n, H)$ parts, of which every part either is a single edge or forms a graph isomorphic to $H$. Pikhurko and Sousa conjectured that $ϕ(n,H)=\ex(n,H)$ for $χ(H)\geqs3$ and all sufficiently large $n$, where $\ex(n,H)$ denotes the maximum number of edges of graphs on $n$ vertices that does not contain $H$ as a subgraph. A $(k,r)$-fan is a graph on $(r-1)k+1$ vertices consisting of $k$ cliques of order $r$ which intersect in exactly one common vertex. In this paper, we verify Pikhurko and Sousa's conjecture for $(k,r)$-fans. The result also generalizes a result of Liu and Sousa.

math.CO↗