Ge 2k Research Articles

The generalized quasilinear approximation of two-dimensional Rayleigh–Bénard convection

In the generalized quasilinear approximation (GQLA) (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 214501), a threshold wavenumber ( $k_0$ ) in the direction of translational symmetry segregates the total into large- ( $l$ ) and small-scale ( $h$ ) fields. While the governing equation for the large-scale field is fully nonlinear, that for the small scales is linearized with respect to the large-scale field. In addition, some nonlinear triad interactions are omitted in the GQLA. Herein, the GQLA is applied to two-dimensional planar Rayleigh–Bénard convection (RBC). A scale separation between the large-scale convection rolls and small-scale turbulent fluctuations is typical in RBC. The present work explores the efficacy of GQLA in capturing the scale-by-scale energy transfer processes in RBC. The initial condition for the GQLA simulations was either the statistically stationary state obtained in direct numerical simulation (DNS) or random fluctuations superimposed on the linear conductive temperature profile and $\boldsymbol {u}=0$ . The GQLA simulations can capture the convection rolls for $k_0$ larger than or equal to the dominant wavenumber for thermal driving of the flow ( $k_0 \ge k_{\hat {Q}}$ ). Additionally, the GQLA emulates the fully nonlinear dynamics for $k_0$ larger than or equal to the first harmonic of the convection-roll wavenumber ( $k_0 \ge 2k_{roll}$ ). In the intermediate regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$ , the dynamics captured in GQLA simulations is different from the DNS. In DNS, two primary energy transfer processes dominate: (i) the energy transfer to/from the convection rolls and (ii) the scale-by-scale inverse kinetic energy and forward thermal energy cascades mediated by the convection rolls. The fully nonlinear dynamics is emulated by GQLA when these energy transfer processes are faithfully reproduced. Utilizing the framework of altering the triad interactions in GQLA, an additional intrusive calculation, including target triad interactions, is performed here to study their influence. This intrusive calculation shows that the convection rolls are not captured in GQLA for $k_0 < k_{\hat {Q}}$ because of the exclusion of the $h \to l \to l$ and $l \to h \to h$ triad interactions in GQLA. The inclusion of these triad interactions in the intrusive calculation yields the convection rolls, and the reproduced dynamics is similar to that of the intermediate GQLA regime with $2k_{roll} > k_0 \ge k_{\hat {Q}}$ .

Realizations of Multiassociahedra via Rigidity

AbstractLet $$\Delta _{k}(n)$$ Δ k ( n ) denote the simplicial complex of $$(k+1)$$ ( k + 1 ) -crossing-free subsets of edges in $${\left( {\begin{array}{c}[n]\\ 2\end{array}}\right) }$$ [ n ] 2 . Here $$k,n\in \mathbb {N}$$ k , n ∈ N and $$n\ge 2k+1$$ n ≥ 2 k + 1 . Jonsson (2003) proved that [neglecting the short edges that cannot be part of any $$(k+1)$$ ( k + 1 ) -crossing], $$\Delta _{k}(n)$$ Δ k ( n ) is a shellable sphere of dimension $$k(n-2k-1)-1$$ k ( n - 2 k - 1 ) - 1 , and conjectured it to be polytopal. The same result and question arose in the work of Knutson and Miller (Adv Math 184(1):161-176, 2004) on subword complexes. Despite considerable effort, the only values of (k, n) for which the conjecture is known to hold are $$n\le 2k+3$$ n ≤ 2 k + 3 (Pilaud and Santos, Eur J Comb. 33(4):632–662, 2012. https://doi.org/10.1016/j.ejc.2011.12.003) and (2, 8) (Bokowski and Pilaud, On symmetric realizations of the simplicial complex of 3-crossing-free sets of diagonals of the octagon. In: Proceedings of the 21st annual Canadian conference on computational geometry, 2009). Using ideas from rigidity theory and choosing points along the moment curve we realize $$\Delta _{k}(n)$$ Δ k ( n ) as a polytope for $$(k,n)\in \{(2,9), (2,10) , (3,10)\}$$ ( k , n ) ∈ { ( 2 , 9 ) , ( 2 , 10 ) , ( 3 , 10 ) } . We also realize it as a simplicial fan for all $$n\le 13$$ n ≤ 13 and arbitrary k, except the pairs (3, 12) and (3, 13). Finally, we also show that for $$k\ge 3$$ k ≥ 3 and $$n\ge 2k+6$$ n ≥ 2 k + 6 no choice of points can realize $$\Delta _{k}(n)$$ Δ k ( n ) via bar-and-joint rigidity with points along the moment curve or, more generally, via cofactor rigidity with arbitrary points in convex position.

On the Maximum of the Sum of the Sizes of Non-trivial Cross-Intersecting Families

AbstractLet $$n \ge 2k \ge 4$$ n ≥ 2 k ≥ 4 be integers, $${[n]\atopwithdelims ()k}$$ [ n ] k the collection of k-subsets of $$[n] = \{1, \ldots , n\}$$ [ n ] = { 1 , … , n } . Two families $${\mathcal {F}}, {\mathcal {G}} \subset {[n]\atopwithdelims ()k}$$ F , G ⊂ [ n ] k are said to be cross-intersecting if $$F \cap G \ne \emptyset $$ F ∩ G ≠ ∅ for all $$F \in {\mathcal {F}}$$ F ∈ F and $$G \in {\mathcal {G}}$$ G ∈ G . A family is called non-trivial if the intersection of all its members is empty. The best possible bound $$|{\mathcal {F}}| + |{\mathcal {G}}| \le {n \atopwithdelims ()k} - 2 {n - k\atopwithdelims ()k} + {n - 2k \atopwithdelims ()k} + 2$$ | F | + | G | ≤ n k - 2 n - k k + n - 2 k k + 2 is established under the assumption that $${\mathcal {F}}$$ F and $${\mathcal {G}}$$ G are non-trivial and cross-intersecting. For the proof a strengthened version of the so-called shifting technique is introduced. The most general result is Theorem 4.1.

Sources of torsion in Poincarè gauge gravity

We study sources for torsion in Poincarè gauge theory in any dimension, signature, and spin. We find that symmetric kinetic terms for non-Yang–Mills bosonic fields of arbitrary rank drive torsion. Our detailed discussion of spin-3/2 Rarita–Schwinger fields shows that they source all independent parts of the torsion. We develop systematic notation for spin-(2k+1)/2 fields and find the spin tensor for arbitrary k in n ge 2k+1 dimensions. For k>0 there is a novel direct coupling between torsion and spinor fields. We also cast the well-known gauge relation between the canonical and Belinfante–Rosenfield energy tensors in terms of different choices of independent variables.

A Note on Exact Minimum Degree Threshold for Fractional Perfect Matchings

Rödl, Ruciński, and Szemerédi determined the minimum \((k-1)\)-degree threshold for the existence of fractional perfect matchings in k-uniform hypergrahs, and Kühn, Osthus, and Townsend extended this result by asymptotically determining the d-degree threshold for the range \(k-1>d\ge k/2\). In this note, we prove the following exact degree threshold: let k, d be positive integers with \(k\ge 4\) and \(k-1>d\ge k/2\), and let n be any integer with \(n\ge 2k(k-1)+1\). Then any n-vertex k-uniform hypergraph with minimum d-degree \(\delta _d(H)>{n-d\atopwithdelims ()k-d} -{n-d-(\lceil n/k\rceil -1)\atopwithdelims ()k-d}\) contains a fractional perfect matching. This lower bound on the minimum d-degree is best possible. We also determine the minimum d-degree threshold for the existence of fractional matchings of size s, where \(0<s\le n/k\) (when \(k/2\le d\le k-1\)), or with s large enough and \(s\le n/k\) (when \(2k/5<d<k/2\)).

Generating the Mapping Class Group by Two Torsion Elements

We prove that the mapping class group of a closed connected orientable surface of genus \(g\ge 6\) is generated by two elements of order g. Moreover, for \(g\ge 7\), we obtain a generating set of two elements, of order g and \(g'\), where \(g'\) is the least divisor of g greater than 2. We also prove that the mapping class group is generated by two elements of order \(g/\gcd (g,k)\) for \(g\ge 3k^2+4k+1\) and any positive integer k.

Cameron–Liebler k-sets in subspaces and non-existence conditions

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron–Liebler line classes in PG $$(n,q), n\ge 3$$ , to Cameron–Liebler sets of k-spaces in $${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$$ and $${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$$ . In his PhD thesis, Drudge proved that every Cameron–Liebler line class in $${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$$ intersects every 3-dimensional subspace in a Cameron–Liebler line class in that subspace. We are using the generalization of this result for sets of k-spaces in $${{\,\mathrm{\mathrm {PG}}\,}}(n,q)$$ and $${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$$ . Together with a basic counting argument this gives a very strong non-existence condition, $$n\ge 3k+3$$ . This condition can also be improved for k-sets in $${{\,\mathrm{\mathrm {AG}}\,}}(n,q)$$ , with $$n\ge 2k+2$$ .

Sharp Minimum Degree Conditions for the Existence of Disjoint Theta Graphs

In 1963, Corrádi and Hajnal showed that if $G$ is an $n$-vertex graph with $n \ge 3k$ and $\delta(G) \ge 2k$, then $G$ will contain $k$ disjoint cycles; furthermore, this result is best possible, both in terms of the number of vertices as well as the minimum degree. In this paper we focus on an analogue of this result for theta graphs. Results from Kawarabayashi and Chiba et al. showed that if $n = 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k \rceil$, or if $n$ is large with respect to $k$ and $\delta(G) \ge 2k+1$, respectively, then $G$ contains $k$ disjoint theta graphs. While the minimum degree condition in both results are sharp for the number of vertices considered, this leaves a gap in which no sufficient minimum degree condition is known. Our main result in this paper resolves this by showing if $n \ge 4k$ and $\delta(G) \ge \lceil \frac{5}{2}k\rceil$, then $G$ contains $k$ disjoint theta graphs. Furthermore, we show this minimum degree condition is sharp for more than just $n = 4k$, and we discuss how and when the sharp minimum degree condition may transition from $\lceil \frac{5}{2}k\rceil$ to $2k+1$.

On the $P_3$-Hull Number of Kneser Graphs

This paper considers an infection spreading in a graph; a vertex gets infected if at least two of its neighbors are infected. The $P_3$-hull number is the minimum size of a vertex set that eventually infects the whole graph.&#x0D; In the specific case of the Kneser graph $K(n,k)$, with $n\ge 2k+1$, an infection spreading on the family of $k$-sets of an $n$-set is considered. A set is infected whenever two sets disjoint from it are infected. We compute the exact value of the $P_3$-hull number of $K(n,k)$ for $n&gt;2k+1$. For $n = 2k+1$, using graph homomorphisms from the Knesser graph to the Hypercube, we give lower and upper bounds.

Exact reconstruction of sparse non-harmonic signals from their Fourier coefficients

In this paper, we derive a new reconstruction method for real non-harmonic Fourier sums, i.e., real signals which can be represented as sparse exponential sums of the form f(t) = sum _{j=1}^{K} gamma _{j} , cos (2pi a_{j} t + b_{j}), where the frequency parameters a_{j} in {mathbb {R}} (or a_{j} in {mathrm i} {mathbb {R}}) are pairwise different. Our method is based on the recently proposed numerically stable iterative rational approximation algorithm in Nakatsukasa et al. (SIAM J Sci Comput 40(3):A1494–A1522, 2018). For signal reconstruction we use a set of classical Fourier coefficients of f with regard to a fixed interval (0, P) with P>0. Even though all terms of f may be non-P-periodic, our reconstruction method requires at most 2K+2 Fourier coefficients c_{n}(f) to recover all parameters of f. We show that in the case of exact data, the proposed iterative algorithm terminates after at most K+1 steps. The algorithm can also detect the number K of terms of f, if K is a priori unknown and L ge 2K+2 Fourier coefficients are available. Therefore our method provides a new alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony’s method.

On the extension of bi-Lipschitz mappings

Let X be a closed semialgebraic set of dimension k. If \(n\ge 2k+1\), then there is a bi-Lipschitz and semialgebraic embedding of X into \(\mathbb {R}^n\). Moreover, if \(n \ge 2k+2\), then this embedding is unique (up to a bi-Lipschitz and semialgebraic homeomorphism of \(\mathbb {R}^n\)). We also give local and complex algebraic counterparts of these results.

Wiener Indices of Maximal k-Degenerate Graphs

A graph is maximal k-degenerate if each induced subgraph has a vertex of degree at most k and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal k-degenerate graphs of order $$n \ge k \ge 1$$ . A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal k-degenerate graphs of order $$n \ge k$$ are k-trees. For k-trees of order $$n \ge 2k+2$$ , we characterize all extremal graphs for the upper bound.

A sufficient condition for the unpaired k-disjoint path coverability of interval graphs

Given disjoint source and sink sets, $$S = \{ s_1,\ldots ,s_k\}$$ and $$T = \{t_1,\ldots , t_k\}$$ , in a graph G, an unpaired k-disjoint path cover joining S and T is a set of pairwise vertex-disjoint paths $$\{ P_1, \ldots , P_k\}$$ that altogether cover every vertex of the graph, in which $$P_i$$ is a path from source $$s_i$$ to some sink $$t_j$$ . In this paper, we prove that if the scattering number, $${{\,\mathrm{sc}\,}}(G)$$ , of an interval graph G of order $$n \ge 2k$$ is less than or equal to $$-k$$ , there exists an unpaired k-disjoint path cover joining S and T in G for any possible configurations of source and sink sets S and T of size k each. The bound $${{\,\mathrm{sc}\,}}(G) \le -k$$ is tight; moreover, the proof directly leads to a quadratic algorithm for building an unpaired k-disjoint path cover.

On Vertex-Disjoint Triangles in Tripartite Graphs and Multigraphs

Let G be a tripartite graph with tripartition $$(V_{1},V_{2},V_{3})$$ , where $$\mid V_{1}\mid =\mid V_{2}\mid =\mid V_{3}\mid =k>0$$ . It is proved that if $$d(x)+d(y)\ge 3k$$ for every pair of nonadjacent vertices $$x\in V_{i}, y\in V_{j}$$ with $$i\ne j(i,j\in \{1,2,3\})$$ , then G contains k vertex-disjoint triangles. As a corollary, if $$d(x)\ge \frac{3}{2}k$$ for each vertex $$x\in V(G)$$ , then G contains k vertex-disjoint triangles. Based on the above results, vertex-disjoint triangles in multigraphs are studied. Let M be a standard tripartite multigraph with tripartition $$(V_{1},V_{2},V_{3})$$ , where $$\mid V_{1} \mid =\mid V_{2}\mid =\mid V_{3} \mid =k>0$$ . If $$\delta (M)\ge 3k-1$$ for even k and $$\delta (M)\ge 3k$$ for odd k, then M contains k vertex-disjoint 4-triangles $$\varDelta _{4}$$ (a triangle with at least four edges). Furthermore, examples are given showing that the degree conditions of all our three results are best possible.

Reconfiguring 10-Colourings of Planar Graphs

Let $$k \ge 1$$ be an integer. The reconfiguration graph $$R_k(G)$$ of the k-colourings of a graph G has as vertex set the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. A conjecture of Cereceda from 2007 asserts that for every integer $$\ell \ge k + 2$$ and k-degenerate graph G on n vertices, $$R_{\ell }(G)$$ has diameter $$O(n^2)$$ . The conjecture has been verified only when $$\ell \ge 2k + 1$$ . We give a simple proof that if G is a planar graph on n vertices, then $$R_{10}(G)$$ has diameter at most $$n(n + 1)/ 2$$ . Since planar graphs are 5-degenerate, this affirms Cereceda’s conjecture for planar graphs in the case $$\ell = 2k$$ .

Structural Results for Conditionally Intersecting Families and some Applications

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$:&#x0D; (a) $(d,2k+d-3)$-conditionally intersecting families with $n\ge 3k^5$;&#x0D; (b) $(k,2k)$-conditionally intersecting families with $n\ge k^2/(k-1)$;&#x0D; (c) Nonintersecting $(3,2k)$-conditionally intersecting families with $n\ge 3k\binom{2k}{k}$.&#x0D; Our results for $(c)$ confirms a conjecture of Mammoliti and Britz for the case $d=3$.

Hamiltonian Cycle Properties in k-Extendable Non-bipartite Graphs with High Connectivity

Let G be a graph, $$\nu $$ the order of G and k a positive integer such that $$k\le (\nu -2)/2$$. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A graph G is Hamiltonian if it contains a Hamiltonian cycle. A graph G is Hamiltonian-connected if, for any two of its vertices, it contains a spanning path joining the two vertices. In this paper, we discuss k-extendable nonbipartite graphs with $$\kappa (G)\ge 2k+r$$ where $$k\ge 1$$ and $$r\ge 0$$. It is shown that if $$\nu \le 6k+2r$$, then G is Hamiltonian; and if $$\nu > 6k+2r$$, then G has a longest cycle C such that $$|V(C)|\ge 6k+2r$$; and if $$\nu <6k+2r$$, then G is Hamiltonian-connected; and if $$\nu \ge 6k+2r$$, then for each pair of vertices $$z_1$$ and $$z_2$$ of G, there is a path P of G joining $$z_1$$ and $$z_2$$ such that $$|V(P)|\ge 6k+2r-2$$. All the bounds are sharp and all results can be extended to 2k-factor-critical graphs.

On the sizes of bi-k-maximal graphs

Let $$k,n, s, t > 0$$ be integers and $$n = s+t \ge 2k+2$$. A simple bipartite graph G spanning $$K_{s,t}$$ is bi-k-maximal, if every subgraph of G has edge-connectivity at most k but any edge addition that does not break its bipartiteness creates a subgraph with connectivity at least $$k+1$$. We investigate the optimal size bounds of the bi-k-maximal simple graphs, and prove that if G is a bi-k-maximal graph with $$\min \{s, t \} \ge k$$ on n vertices, then each of the following holds.

$$C_{2k}$$-Saturated Graphs with No Short Odd Cycles

The saturation number of a graph F, written $$\text{ sat }(n,F)$$ , is the minimum number of edges in an n-vertex F-saturated graph. One of the earliest results on saturation numbers is due to Erdős et al. who determined $$\text{ sat }(n,K_r)$$ for all $$r \ge 3$$ . Since then, saturation numbers of various graphs and hypergraphs have been studied. Motivated by Alon and Shikhelman’s generalized Turan function, Kritschgau et al. defined $$\text{ sat }(n,H,F)$$ to be the minimum number of copies of H in an n-vertex F-saturated graph. They proved, among other things, that $$\text{ sat }(n,C_3,C_{2k}) = 0$$ for all $$k \ge 3$$ and $$n \ge 2k +2$$ . We extend this result to all odd cycles by proving that for any odd integer $$r \ge 5$$ , $$\text{ sat }(n , C_r ,C_{2k} ) = 0$$ for all $$2k \ge r+5$$ and $$n \ge 2kr$$ .

Power scaling for multi-pair massive MIMO two-way relaying system under Rician fading

In this work, we investigate a massive multi-input multi-output two-way relaying system, where K pairs single-antenna users exchange information through a relay station equipped with N ( $$N\ge 2K$$ ) antennas, employing the amplify-and-forward protocol and maximum ratio combining/maximum ratio transmission beamforming. Considering Rician fading channel with imperfect channel state information (CSI), we study the achievable sum rate and energy efficiency (EE) performance of the considered system. Firstly, we obtain the asymptotic results of the achievable sum rate and the EE, assuming a general power scaling scheme. It is shown that when $$N\rightarrow \infty $$ , the pilot sequences’ power, i.e., $$P_P$$ , each user’s transmit power, i.e., $$P_U$$ , and the relay station’s transmit power $$P_R$$ can be scaled down proportionally to $$1/N^a$$ , $$1/N^a$$ and $$1/N^b$$ , respectively, where $$a\ge 0$$ and $$b\ge 0$$ . Specifically, when perfect CSI is known and Rician factor $$\kappa =0$$ , our derived results reduce to the existing results in Rayleigh fading channel. Furthermore, the closed-form expressions of the achievable sum rate and EE are obtained approximately. Finally, simulation results are provided to confirm the validity of our analysis.