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Tao Jiang

Publications and source records attributed to Tao Jiang.

At least 361 records · Page 20Linked to original sources
arXiv2015

Turán numbers of hypergraph trees

An $r$-graph is an $r$-uniform hypergraph tree (or $r$-tree) if its edges can be ordered as $E_1,\ldots, E_m$ such that $\forall i>1 \, \exists α(i)<i$ such that $E_i\cap (\bigcup_{j=1}^{i-1} E_j)\subseteq E_{α(i)}$. The Turán number $ex(n,{\cal H})$ of an $r$-graph ${\cal H}$ is the largest size of an $n$-vertex $r$-graph that does not contain ${\cal H}$. A cross-cut of ${\cal H}$ is a set of vertices in ${\cal H}$ that contains exactly one vertex of each edge of ${\cal H}$. The cross-cut number $σ({\cal H})$ of ${\cal H}$ is the minimum size of a cross-cut of ${\cal H}$. We show that for a large family of $r$-graphs (largest within a certain scope) that are embeddable in $r$-trees, $ex(n,{\cal H})=(σ-1)\binom{n}{r-1}+o(n^{r-1})$ holds, and we establish structural stability of near extremal graphs. From stability, we establish exact results for some subfamilies.

math.CO↗
arXiv2015

Demonstration of nonlinear-energy-spread compensation in relativistic electron bunches with corrugated structures

High quality electron beams with flat distributions in both energy and current are critical for many accelerator-based scientific facilities such as free-electron lasers and MeV ultrafast electron diffraction and microscopes. In this Letter we report on using corrugated structures to compensate for the beam nonlinear energy chirp imprinted by the curvature of the radio-frequency field, leading to a significant reduction in beam energy spread. By using a pair of corrugated structures with orthogonal orientations, we show that the quadrupole wake fields which otherwise increase beam emittance can be effectively canceled. This work also extends the applications of corrugated structures to the low beam charge (a few pC) and low beam energy (a few MeV) regime and may have a strong impact in many accelerator-based facilities.

physics.acc-ph↗
arXiv2015

Small dense subgraphs of a graph

Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Turán number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order of a graph is the number of vertices in it. In this paper, we study the Turán number of the family of graphs with bounded order and high average degree. For every real $d\geq 2$ and positive integer $m\geq 2$, let ${\cal F}_{d,m}$ denote the family of graphs on at most $m$ vertices that have average degree at least $d$. It follows from the Erdős-Rényi bound that $ex(n,{\cal F}_{d,m})=Ω(n^{2-\frac{2}{d}+\frac{c}{dm}})$, for some positive constant $c$. Verstraëte asked if it is true that for each fixed $d$ there exists a function $ε_d(m)$ that tends to $0$ as $m\to \infty$ such that $ex(n,{\cal F}_{d,m})=O(n^{2-\frac{2}{d}+ε_d(m)})$. We answer Verstraëte's question in the affirmative whenever $d$ is an integer. We also prove an extension of the cube theorem on the Turán number of the cube $Q_3$, which partially answers a question of Pinchasi and Sharir.

math.CO↗
arXiv2015

A Lower Bound on the Average-Case Complexity of Shellsort

We prove a general lower bound on the average-case complexity of Shellsort: the average number of data-movements (and comparisons) made by a $p$-pass Shellsort for any incremental sequence is $Ω(pn^{1 + 1/p})$ for every $p$. The proof method is an incompressibility argument based on Kolmogorov complexity. Using similar techniques, the average-case complexity of several other sorting algorithms is analyzed.

cs.CC↗
arXiv2014

Linear Turan numbers of r-uniform linear cycles and related Ramsey numbers

An $r$-uniform hypergraph is called an $r$-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear $r$-graph $H$ and a positive integer $n$, the linear Turán number $ex_L(n,H)$ is the maximum number of edges in a linear $r$-graph $G$ that does not contain $H$ as a subgraph. For each $\ell\geq 3$, let $C^r_\ell$ denote the $r$-uniform linear cycle of length $\ell$, which is an $r$-graph with edges $e_1,\ldots, e_\ell$ such that $\forall i\in [\ell-1]$, $|e_i\cap e_{i+1}|=1$, $|e_\ell\cap e_1|=1$ and $e_i\cap e_j=\emptyset$ for all other pairs $\{i,j\}, i\neq j$. For all $r\geq 3$ and $\ell\geq 3$, we show that there exist positive constants $c_{m,r}$ and $c'_{m,r}$, depending only $m$ and $r$, such that $ex_L(n,C^r_{2m})\leq c_{m,r} n^{1+\frac{1}{m}}$ and $ex_L(n,C^r_{2m+1})\leq c'_{m,r} n^{1+\frac{1}{m}}$. This answers a question of Kostochka, Mubayi, and Verstraëte. For even cycles, our result extends the result of Bondy and Simonovits on the Turán numbers of even cycles to linear hypergraphs. Using our results on linear Turán numbers we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constants $a_{m,r}$ and $b_{m,r}$, depending only on $m$ and $r$, such that $R(C^r_{2m}, K^r_t)\leq a_{m,r} (\frac{t}{\ln t})^\frac{m}{m-1}$ and $R(C^r_{2m+1}, K^r_t)\leq b_{m,r} t^\frac{m}{m-1}$.

math.CO↗
arXiv2013

Sensing and Recognition When Primary User Has Multiple Power Levels

In this paper, we present a new cognitive radio (CR) scenario when the primary user (PU) operates under more than one transmit power levels. Different from the existing studies where PU is assumed to have only one constant transmit power, the new consideration well matches the practical standards, i.e., IEEE 802.11 Series, GSM, LTE, LTE-A, etc., as well as the adaptive power concept that has been studied over the past decades. The primary target in this new CR scenario is, of course, still to detect the presence of PU. However, there appears a secondary target as to identify the PU's transmit power level. Compared to the existing works where the secondary user (SU) only senses the ``on-off'' status of PU, recognizing the power level of PU achieves more ``cognition", and could be utilized to protect different powered PU with different interference levels. We derived quite many closed-form results for either the threshold expressions or the performance analysis, from which many interesting points and discussions are raised. We then further study the cooperative sensing strategy in this new cognitive scenario and show its significant difference from traditional algorithms. Numerical examples are provided to corroborate the proposed studies.

cs.IT↗
arXiv2013

Feasibility Study on SRF Cavity Improvement by Doping Method

In this article a new idea is proposed by PKU group to improve the RF superconductive property of niobium cavity. Rare earth elements like scandium and yttrium are doped into ingot niobium during the smelting processes. A series of experiments have been done since 2010, and the preliminary testing results show that the magnetic property of niobium materials has been changed with different doping elements and proportions while the superconductive transition temperature almost does not change. Then a large ingot niobium doped with scandium was melted. RRR and mechanical properties were measured. This method may increase the superheating magnetic field and decrease surface resistance of niobium so as to improve the performance of niobium cavity which is a key component of SRF accelerators. A single-cell cavity made of Sc-doped niobium has been fabricated.

physics.acc-ph↗
arXiv2013

On the co-degree threshold for the Fano plane

Given a 3-graph H, let \ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. Let F denote the Fano plane, which is the 3-graph \{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\}. Mubayi proved that \ex_2(n,F)=(1/2+o(1))n and conjectured that \ex_2(n, F)=\floor{n/2} for sufficiently large n. Using a very sophisticated quasi-randomness argument, Keevash proved Mubayi's conjecture. Here we give a simple proof of Mubayi's conjecture by using a class of 3-graphs that we call rings. We also determine the Turán density of the family of rings.

math.CO↗
arXiv2013

Hypergraph Turan numbers of linear cycles

A k-uniform linear cycle of length s is a cyclic list of k-sets A_1,..., A_s such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k at least 5 and s at least 3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length s. For odd s=2t+1 the unique extremal family F_S consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even s=2t+2, the unique extremal family consists of F_S plus all the k-sets outside S containing some fixed two elements. For k at least 4 and large n we also establish an exact result for so-called minimal cycles. For all k at least 4 our results substantially extend Erdos' result on largest k-uniform families without t+1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraete. Our main method is the delta system method.

math.CO↗
arXiv2012

Galaxy growth by merging in the nearby universe

We measure the mass growth rate by merging for a wide range of galaxy types. We present the small-scale (0.014 < r < 11 h70^{-1} Mpc) projected cross-correlation functions w(rp) of galaxy subsamples from the spectroscopic sample of the NYU VAGC (5 \times 10^5 galaxies of redshifts 0.03 < z < 0.15) with galaxy subsamples from the SDSS imaging (4 \times 10^7 galaxies). We use smooth fits to de-project the two-dimensional functions w(rp) to obtain smooth three-dimensional real-space cross-correlation functions ξ(r) for each of several spectroscopic subsamples with each of several imaging subsamples. Because close pairs are expected to merge, the three-space functions and dynamical evolution time estimates provide galaxy accretion rates. We find that the accretion onto massive blue galaxies and onto red galaxies is dominated by red companions, and that onto small-mass blue galaxies, red and blue galaxies make comparable contributions. We integrate over all types of companions and find that at fixed stellar mass, the total fractional accretion rates onto red galaxies (\sim 1.5 h70 percent per Gyr) is greater than that onto blue galaxies (\sim 0.5 h70 percent per Gyr). Although these rates are very low, they are almost certainly over-estimates because we have assumed that all close pairs merge as quickly as dynamical friction permits.

astro-ph.GA↗
arXiv2012

The Extreme Small Scales: Do Satellite Galaxies Trace Dark Matter?

We investigate the radial distribution of galaxies within their host dark matter halos by modeling their small-scale clustering, as measured in the Sloan Digital Sky Survey. Specifically, we model the Jiang et al. (2011) measurements of the galaxy two-point correlation function down to very small projected separations (10 < r < 400 kpc/h), in a wide range of luminosity threshold samples (absolute r-band magnitudes of -18 up to -23). We use a halo occupation distribution (HOD) framework with free parameters that specify both the number and spatial distribution of galaxies within their host dark matter halos. We assume that the first galaxy in each halo lives at the halo center and that additional satellite galaxies follow a radial density profile similar to the dark matter Navarro-Frenk-White (NFW) profile, except that the concentration and inner slope are allowed to vary. We find that in low luminosity samples, satellite galaxies have radial profiles that are consistent with NFW. M_r < -20 and brighter satellite galaxies have radial profiles with significantly steeper inner slopes than NFW (we find inner logarithmic slopes ranging from -1.6 to -2.1, as opposed to -1 for NFW). We define a useful metric of concentration, M_(1/10), which is the fraction of satellite galaxies (or mass) that are enclosed within one tenth of the virial radius of a halo. We find that M_(1/10) for low luminosity satellite galaxies agrees with NFW, whereas for luminous galaxies it is 2.5-4 times higher, demonstrating that these galaxies are substantially more centrally concentrated within their dark matter halos than the dark matter itself. Our results therefore suggest that the processes that govern the spatial distribution of galaxies, once they have merged into larger halos, must be luminosity dependent, such that luminous galaxies become poor tracers of the underlying dark matter.

astro-ph.CO↗
arXiv2011

Exact solution of the hypergraph Turán problem for $k$-uniform linear paths

A $k$-uniform linear path of length $\ell$, denoted by $P^{(k)}_\ell$, is a family of $k$-sets $\{F_1,..., F_\ell\}$ such that $|F_i\cap F_{i+1}|=1$ for each $i$ and $F_i\cap F_j=\emptyset$ whenever $|i-j|>1$. Given a $k$-uniform hypergraph $H$ and a positive integer $n$, the {\it $k$-uniform hypergraph Turán number} of $H$, denoted by $\ex_k(n,H)$, is the maximum number of edges in a $k$-uniform hypergraph $\cF$ on $n$ vertices that does not contain $H$ as a subhypergraph. With an intensive use of the delta-system method, we determine $\ex_k(n,P^{(k)}_\ell)$ exactly for all fixed $\ell\geq 1, k\geq 4$, and sufficiently large $n$. We show that $$\ex_k(n,P^{(k)}_{2t+1})={n-1\choose k-1}+{n-2\choose k-1}+...+{n-t\choose k-1}.$$ The only extremal family consists of all the $k$-sets in $[n]$ that meet some fixed set of $t$ vertices. We also show that $$\ex(n, P^{(k)}_{2t+2})={n-1\choose k-1}+{n-2\choose k-1}+...+{n-t\choose k-1}+{n-t-2\choose k-2},$$ and describe the unique extremal family. Stability results on these bounds and some related results are also established.

math.CO↗
arXiv2011

Set families with a forbidden induced subposet

For each poset $H$ whose Hasse diagram is a tree of height $k$, we show that the largest size of a family $\cF$ of subsets of $[n]=\{1,..., n\}$ not containing $H$ as an induced subposet is asymptotic to $(k-1){n\choose \fl{n/2}}$. This extends the result of Bukh \cite{bukh}, which in turn generalizes several known results including Sperner's theorem.

math.CO↗
arXiv2009

Sub-diffraction-limit Observation Realized by Nonlinear Metamaterial Lens

In this paper, we show by experiment that by covering a thin flat nonlinear lens on the sources, the sub-diffraction-limit observation can be achieved by measuring either the near-field distribution or the far-field radiation of the sources at the harmonic frequencies and calculating the inverse Fourier transformation to obtain the sub-wavelength imaging. Especially, the sub-wavelength image calculated from measured far-field data demonstrates very clear resolution. Since metamaterials included with active elements can easily behave strong nonlinearity under very weak incident electromagnetic powers, the application of the nonlinear lens proposed in this paper would have important potential in improving the sub-wavelength resolution in the near future.

physics.optics↗
arXiv2009

Second-Harmonic Generation and Spectrum Modulation by Active Nonlinear Metamaterial

The nonlinear properties of a metamaterial sample composed of double-layer metallic patterns and voltage controllable diodes are experimentally investigated. Second harmonics and spectrum modulations are clearly observed in a wide band of microwave frequencies, showing that this kind of metamaterial is not only tunable by low DC bias voltage, but also behaves strong nonlinear property under a small power incidence. These properties are difficult to be found in normal, naturally occurring materials.

physics.optics↗
arXiv2004

Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Binary Codes

Given positive integers $n$ and $d$, let $A_2(n,d)$ denote the maximum size of a binary code of length $n$ and minimum distance $d$. The well-known Gilbert-Varshamov bound asserts that $A_2(n,d) \geq 2^n/V(n,d-1)$, where $V(n,d) = \sum_{i=0}^{d} {n \choose i}$ is the volume of a Hamming sphere of radius $d$. We show that, in fact, there exists a positive constant $c$ such that $$ A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1) $$ whenever $d/n \le 0.499$. The result follows by recasting the Gilbert- Varshamov bound into a graph-theoretic framework and using the fact that the corresponding graph is locally sparse. Generalizations and extensions of this result are briefly discussed.

math.CO↗
arXiv2003

The Average-Case Area of Heilbronn-Type Triangles

From among $ {n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < μ_n < C/n^3$ for all large enough values of $n$, where $μ_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''

math.CO↗
arXiv2001

Two heads are better than two tapes

We show that a Turing machine with two single-head one-dimensional tapes cannot recognize the set {x2x'| x \in {0,1}^* and x' is a prefix of x} in real time, although it can do so with three tapes, two two-dimensional tapes, or one two-head tape, or in linear time with just one tape. In particular, this settles the longstanding conjecture that a two-head Turing machine can recognize more languages in real time if its heads are on the same one-dimensional tape than if they are on separate one-dimensional tapes.

cs.CC↗