Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

On judicious bipartitions of graphs

For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G)=V ₁?V ₂ such that G has at most s edges with both incident vertices in V _i . Alon proved that the limsup of \(f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)\) tends to infinity as m tends to infinity, establishing a conjecture of Erd?s. Bollobás and Scott proposed the following judicious version of Erd?s' conjecture: the limsup of \(m/4 + \left( {\sqrt {m/32} - g(m)} \right)\) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon's result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to k-partitions for odd integers k.     

Acceleration of convergence of general linear sequences by the Shanks transformation

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A _n }. It generates a two-dimensional array of approximations \({A^{(j)}_n}\) to the limit or anti-limit of {A _n } defined as solutions of the linear systems

$A_l=A^{(j)}_n +\sum^{n}_{k=1}\bar{\beta}_k(\Delta A_{l+k-1}),\ \ j\leq l\leq j+n,$

where \({\bar{\beta}_{k}}\) are additional unknowns. In this work, we study the convergence and stability properties of \({A^{(j)}_n}\) , as j → ∞ with n fixed, derived from general linear sequences {A _n }, where \({{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}\) as n → ∞, where ζ _k  ≠ 1 are distinct and |ζ ₁| = ... = |ζ _m | = θ, and γ _k  ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A _n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems

$\begin{array}{ll}{\quad\quad\quad\quad\max\limits_{s_1,\ldots,s_m}\sum\limits_{k=1}^{m}\left[(\Re\gamma_k)s_k-s_k(s_k-1)\right],}\\ {{\rm subject\,to}\,\, s_1\geq0,\ldots,s_m\geq0\quad{\rm and}\quad \sum\limits_{k=1}^{m} s_k = n,}\end{array}$

have unique (integer) solutions for s ₁, . . . , s _m . A special case of our convergence result concerns the situation in which \({{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}\) and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads \({A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}\) as j → ∞. When compared with A _j ? A = O(θ ^j j ^α ) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.

     

Elementary Graphs with Respect to <Emphasis Type="Italic">f</Emphasis>-Parity Factors

This note concerns the f-parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function \({f : V(G) \rightarrow \mathbb{N}}\), and our goal is to find a spanning subgraph F of G with deg _F ≤ f and minimizing the number of vertices x with \({\deg_F(x) \not\equiv f(x) \, {\rm mod} \, {2}}\) . First we prove a Gallai–Edmonds type structure theorem and some other known results on the f-parity subgraph problem, using an easy reduction to the matching problem. Then we use this reduction to investigate barriers and elementary graphs with respect to f-parity factors, where an elementary graph is a graph such that the union of f-parity factors form a connected spanning subgraph.     

On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.

     

A threshold phenomenon for embeddings of $${H^m_0}$$ into Orlicz spaces

Given an open bounded domain \({\Omega\subset\mathbb {R}^{2m}}\) with smooth boundary, we consider a sequence \({(u_k)_{k\in\mathbb{N}}}\) of positive smooth solutions to

$\left\{\begin{array}{ll} (-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2} \quad\quad\quad\quad\quad {\rm in}\,\Omega\\ u_k=\partial_\nu u_k=\cdots =\partial_\nu^{m-1} u_k=0 \quad {\rm on }\, \partial \Omega, \end{array}\right.$

where λ _k → 0⁺. Assuming that the sequence is bounded in \({H^m_0(\Omega)}\) , we study its blow-up behavior. We show that if the sequence is not precompact, then

$\liminf_{k\to\infty}\|u_k\|^2_{H^m_0}:=\liminf_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx\geq \Lambda_1,$

where Λ₁ = (2m ? 1)!vol(S ^2m ) is the total Q-curvature of S ^2m .

     

Solutions with many mixed positive and negative interior spikes for a semilinear Neumann problem

We study the existence of sign-changing multiple interior spike solutions for the following Neumann problem $$\varepsilon^2\Delta v-v+f(v) = 0 \,\, {\rm in} \,\, \Omega, \quad \frac{\partial v}{\partial \nu} = 0 \,\, {\rm on} \,\, \partial \Omega,$$ where ?? is a smooth bounded domain of ${\mathbb {R}^N}$ , ?? is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. No symmetry on ?? is assumed. To our knowledge, only positive interior peak solutions have been obtained for this problem and it remains a question whether or not multiple interior peak solutions with mixed positive and negative peaks exist. In this paper we assume that ?? is a two-dimensional strictly convex domain and, provided that k is sufficiently large, we construct a (k?+?1)-peak solutions with k positive interior peaks aligned on a closed curve near ??? and 1 negative interior peak located in a more centered part of ??.     

A Neumann problem of Ambrosetti–Prodi type

The aim of this paper is to establish an Ambrosetti–Proditype result for the problem

$$\left\{ \begin{array}{ll}-\Delta{u} = g(x, u,\nabla{u}) + t\varphi \quad {\rm in}\, \Omega,\\ \frac{\partial{u}}{\partial\eta} = 0 \qquad\qquad\qquad\quad {\rm on}\, \partial\Omega ;\end{array} \right.$$

i.e., under appropriate conditions, we will show that there exists a constant t ₀ such that the problem above has no solution if t > t ₀, at least a solution if t =  t ₀ and at least two solutions if t < t ₀. The proof is based on a combination of upper and lower solutions method and the Leray–Schauder degree.

     

Neighbor sum distinguishing chromatic index of sparse graphs via the combinatorial Nullstellensatz

Let ?: E(G) → {1, 2, · · ·, k} be an edge coloring of a graph G. A proper edge-k-coloring of G is called neighbor sum distinguishing if \(\sum\limits_{e \mathrel\backepsilon u} {\phi \left( e \right)} \ne \sum\limits_{e \mathrel\backepsilon v} {\phi \left( e \right)} \) for each edge uv ∈ E(G). The smallest value k for which G has such a coloring is denoted by χ′_Σ(G), which makes sense for graphs containing no isolated edge (we call such graphs normal). It was conjectured by Flandrin et al. that χ′_Σ(G) ≤ Δ(G) + 2 for all normal graphs, except for C₅. Let mad(G) = \(\max \left\{ {\frac{{2\left| {E\left( h \right)} \right|}}{{\left| {V\left( H \right)} \right|}}|H \subseteq G} \right\}\) be the maximum average degree of G. In this paper, we prove that if G is a normal graph with Δ(G) ≥ 5 and mad(G) < 3 ? \(\frac{2}{{\Delta \left( G \right)}}\), then χ′_Σ(G) ≤ Δ(G) + 1. This improves the previous results and the bound Δ(G) + 1 is sharp.     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

Graphs with small total rainbow connection number

A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we investigate graphs with small total rainbow connection number. First, for a connected graph G, we prove that \({\text{trc(G) = 3 if}}\left( {\begin{array}{*{20}{c}}{n - 1} \\2\end{array}} \right) + 1 \leqslant \left| {{\text{E(G)}}} \right| \leqslant \left( {\begin{array}{*{20}{c}}n \\2\end{array}} \right) - 1\), and \({\text{trc(G)}} \leqslant {\text{6 if }}\left| {{\text{E(G)}}} \right| \geqslant \left( {\begin{array}{*{20}{c}}{n - 2} \\2\end{array}} \right) + 2\). Next, we investigate the total rainbow connection numbers of graphs G with |V(G)| = n, diam(G) ≥ 2, and clique number ω(G) = n ? s for 1 ≤ s ≤ 3. In this paper, we find Theorem 3 of [Discuss. Math. Graph Theory, 2011, 31(2): 313–320] is not completely correct, and we provide a complete result for this theorem.     

*-regularity of certain generalized group algebras

Let B be a *-semisimple Banach algebra with a bounded approximate identity and \({\alpha: G \longrightarrow {\rm Aut}_{*}(B)}\) (isometric *-automorphisms group of B) an action of a locally group G on B. Let (D, G, γ) be the associated dynamical system, where D = C ₀(G, B) is the Banach *-algebra of all continuous B-valued functions on G vanishing at infinity and the action γ : G → Aut D is given by γ _s (y)(t) = α _s (y(s ^?1 t)) for \({y \in D}\) and \({s, t \in G}\) . Recall that B is said to be *-regular if the natural mapping \({I\in {\rm Prim} \, C^{*}(B) \mapsto I\cap B\in {\rm Prim}_{*}(B)}\) is a homeomorphism under the hull-kernel topology. When G is amenable, we show that if B is *-regular, then the generalized group algebra L ¹(G, D; γ) is *-regular. The converse is also true if we further assume that G is countable discrete. Finally the case of compact groups is studied.     

Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

Total Rainbow Connection Number and Complementary Graph

A vertex-colored graph G is rainbow vertex connected if any two distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we prove that for a connected graph G, if \({{\rm diam}(\overline{G}) \geq 3}\), then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. Next, we obtain that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 2}\), if G is connected, then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we prove that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 3}\), if G is connected, then trc\({(G) \leq 5}\), and this bound is tight. Next, a Nordhaus–Gaddum-type result for the total rainbow connection number is provided. We show that if G and \({\overline{G}}\) are both connected, then \({6 \leq {\rm trc} (G) + {\rm trc}(\overline{G}) \leq 4n - 6.}\) Examples are given to show that the lower bound is tight for \({n \geq 7}\) and n = 5. Tight lower bounds are also given for n = 4, 6.     

On <Emphasis Type="Italic">ω</Emphasis>-strongly quasiconvex and <Emphasis Type="Italic">ω</Emphasis>-strongly quasiconcave sequences

Let ω ≥ 0 be a given number and let I be a subinterval of \({{\mathbb Z}}\). We say that a sequence \({(f_k)_{k \in I}}\) is ω -strongly quasiconvex, ω-strongly quasiconcave, ω-strongly quasiaffine if

$\begin{array}{lll}f_k \leq \max(f_{k-1},f_{k+1})-\omega\quad\quad{\rm for}\,\,\,k:k-1, k, k+1 \in I;\\ f_k \geq \max(f_{k-1},f_{k+1})-\omega\quad\quad{\rm for}\,\,\,k:k-1, k, k+1 \in I;\\ f_k = \max(f_{k-1},f_{k+1})-\omega\quad\quad{\rm for}\,\,\,k:k-1, k, k+1 \in I.\end{array}$

We characterize ω-strongly quasiconvex, ω-strongly quasiconcave and ω-strongly quasiaffine sequences. We also show that these notions lead naturally to analogous notions for functions defined on subintervals of \({{\mathbb R}}\).

     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms \({\phi, \psi: C\to A}\) are approximately unitarily equivalent if and only if

$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$

where T(A) is the tracial state space of A. In this paper we prove the following: Given \({\kappa\in KL(C,A)}\) with \({\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) and with κ([1 _C ]) = [1 _A ] and a continuous affine map \({\lambda: T(A)\to T_{\mathfrak f}(C)}\) which is compatible with κ, where \({T_{\mathfrak f}(C)}\) is the convex set of all faithful tracial states, there exists a unital monomorphism \({\phi: C\to A}\) such that

$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$

for all \({c\in C_{s.a.}}\) and \({\tau\in T(A).}\) Denote by \({{\rm Mon}_{au}^e(C,A)}\) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map

$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$

where KLT(C, A)⁺⁺ is the set of compatible pairs of elements in KL(C, A)⁺⁺ and continuous affine maps from T(A) to \({T_{\mathfrak f}(C).}\) Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and \({\kappa\in KL(C(X), A)}\) with \({\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) for which there is no homomorphism h: C(X) → A so that [h] = κ.

     

On a vectorized version of a generalized Richardson extrapolation process

Let {x _m } be a vector sequence that satisfies

$$\boldsymbol{x}_{m}\sim \boldsymbol{s}+\sum\limits^{\infty}_{i=1}\alpha_{i} \boldsymbol{g}_{i}(m)\quad\text{as \(m\to\infty\)}, $$

s being the limit or antilimit of {x _m } and \(\{\boldsymbol {g}_{i}(m)\}^{\infty }_{i=1}\) being an asymptotic scale as m → ∞, in the sense that

$$\lim\limits_{m\to\infty}\frac{\|\boldsymbol{g}_{i+1}(m)\|}{\|\boldsymbol{g}_{i}(m)\|}=0,\quad i=1,2,\ldots. $$

The vector sequences \(\{\boldsymbol {g}_{i}(m)\}^{\infty }_{m=0}\), i = 1, 2,…, are known, as well as {x _m }. In this work, we analyze the convergence and convergence acceleration properties of a vectorized version of the generalized Richardson extrapolation process that is defined via the equations

$$\sum\limits^{k}_{i=1}\langle\boldsymbol{y},{\Delta}\boldsymbol{g}_{i}(m)\rangle\widetilde{\alpha}_{i}=\langle\boldsymbol{y},{\Delta}\boldsymbol{x}_{m}\rangle,\quad n\leq m\leq n+k-1;\quad \boldsymbol{s}_{n,k}=\boldsymbol{x}_{n}+\sum\limits^{k}_{i=1}\widetilde{\alpha}_{i}\boldsymbol{g}_{i}(n), $$

s _{n, k} being the approximation to s. Here, y is some nonzero vector, 〈? ,?〉 is an inner product, such that \(\langle \alpha \boldsymbol {a},\beta \boldsymbol {b}\rangle =\overline {\alpha }\beta \langle \boldsymbol {a},\boldsymbol {b}\rangle \), and Δx _m = x _{m + 1}? x _m and Δg _i (m) = g _i (m + 1)?g _i (m). By imposing a minimal number of reasonable additional conditions on the g _i (m), we show that the error s _{n, k} ? s has a full asymptotic expansion as n→∞. We also show that actual convergence acceleration takes place, and we provide a complete classification of it.

     

Asymptotically radial solutions in expanding annular domains

In this paper we consider the problem

$\left\{\begin{array}{ll}-\Delta u=u^{p}\quad {\rm in}\, \Omega_R,\\ u=0 \quad \quad \quad {\rm on}\, \partial\Omega_R,\quad\quad\quad (0.1)\end{array}\right.$

where p > 1 and Ω _R is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as \({R \longrightarrow \infty}\). The main goal of the paper is to prove, for large R, the existence of a positive solution to (0.1) which is close to the positive radial solution in the corresponding diffeomorphic annulus. The proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions.

     

The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups

The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm).