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 \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

Degree sum of a pair of independent edges and <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n ≥ 4 vertices such that G ∈ F if and only if d(e) + d(e’) ≥ 2n for every pair of independent edges e, e’ of G. We prove in this paper that for each G ∈ F, G is not Z ₃-connected if and only if G is one of K _2,n?2, K _3,n?3, K _2,n?2 ⁺ , K _3,n?3 ⁺ or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].     

Irreducible <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>1</Subscript><Superscript>(1)</Superscript></Stack>-modules from modules over two-dimensional non-abelian Lie algebra

For any module V over the two-dimensional non-abelian Lie algebra b and scalar α ∈ C, we define a class of weight modules F ^α (V) with zero central charge over the affine Lie algebra A ₁ ⁽¹⁾ . These weight modules have infinitedimensional weight spaces if and only if V is infinite dimensional. In this paper, we will determine necessary and sufficient conditions for these modules F ^α(V) to be irreducible. In this way, we obtain a lot of irreducible weight A ₁ ⁽¹⁾ -modules with infinite-dimensional weight spaces.     

Spectral invariant subalgebras of reduced groupoid <Emphasis Type="Italic">C</Emphasis>*-algebras

We introduce the notion of property (RD) for a locally compact, Hausdorff and r-discrete groupoid G, and show that the set S ₂ ^l (G) of rapidly decreasing functions on G with respect to a continuous length function l is a dense spectral invariant and Fréchet *-subalgebra of the reduced groupoid C*-algebra C _r ^* (G) of G when G has property (RD) with respect to l, so the K-theories of both algebras are isomorphic under inclusion. Each normalized cocycle c on G, together with an invariant probability measure on the unit space G ⁰ of G, gives rise to a canonical map τ _c on the algebra C _c (G) of complex continuous functions with compact support on G. We show that the map τ _c can be extended continuously to S ₂ ^l (G) and plays the same role as an n-trace on C _r ^* (G) when G has property (RD) and c is of polynomial growth with respect to l, so the Connes’ fundament paring between the K-theory and the cyclic cohomology gives us the K-theory invariants on C _r ^* (G).     

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.     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

On <Emphasis Type="Italic">r</Emphasis>-recognition by prime graph of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(3) where <Emphasis Type="Italic">p</Emphasis> is an odd prime

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. We denote by k(Γ(G)) the number of isomorphism classes of finite groups H satisfying Γ(G) = Γ(H). Given a natural number r, a finite group G is called r-recognizable by prime graph if k(Γ(G)) =  r. In Shen et al. (Sib. Math. J. 51(2):244–254, 2010), it is proved that if p is an odd prime, then B _p (3) is recognizable by element orders. In this paper as the main result, we show that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then \({G\cong B_p(3)}\) or C _p (3). Also if Γ(G) = Γ(B ₃(3)), then \({G\cong B_3(3), C_3(3), D_4(3)}\), or \({G/O_2(G)\cong {\rm Aut}(^2B_2(8))}\). As a corollary, the main result of the above paper is obtained.     

A REALIZATION OF CERTAIN MODULES FOR THE <Emphasis Type="Italic">N</Emphasis> = 4 SUPERCONFORMAL ALGEBRA AND THE AFFINE LIE ALGEBRA <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>2</Subscript><Superscript>(1)</Superscript></Stack>

We shall first present an explicit realization of the simple N = 4 superconformal vertex algebra L _c ^N?=?4 with central charge c = ?9. This vertex superalgebra is realized inside of the bcβγ system and contains a subalgebra isomorphic to the simple affine vertex algebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). Then we construct a functor from the category of L _c ^N?=?4 -modules with c = ?9 to the category of modules for the admissible affine vertex algebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). By using this construction we construct a family of weight and logarithmic modules for L _c ^N?=?4 and L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). We also show that a coset subalgebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \) is a logarithmic extension of the W(2; 3)-algebra with c = ?10. We discuss some generalizations of our construction based on the extension of affine vertex algebra L _A1 (kΛ ₀) such that k + 2 = 1/p and p is a positive integer.     

Equivalence of two definitions of a generalized <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> solution to the initial-boundary value problem for the wave equation

In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫ ₀ ^T (T ? t)|µ(t)| ^p dt exists. Here we prove that this solution is a unique solution to the problem in L _p that satisfies the corresponding integral identity.     

Improved bounds on the generalized acyclic chromatic number

An r-acyclic edge chromatic number of a graph G, denoted by a _r ^′ r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that a _r ^′ r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of a _r ^′ r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10.     

The SL(<Emphasis Type="Italic">V</Emphasis>)-ample cone of product of flag varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω = (\(\vec d\)(1), ..., \(\vec d\)(m)) of sequences of positive integers, denote by L _ω the ample line bundle corresponding to the polarization on the product X = Π _i=1 ^m Flag(V, \(\vec n\)(i)) of flag varieties of type \(\vec n\)(i) determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to L _ω. We give a sufficient and necessary condition on ω such that X ^ss (L _ω) ≠ \(\not 0\) (resp., X ^s (L _ω) ≠ \(\not 0\)). As a consequence, we characterize the SL(V)-ample cone (for the diagonal action of SL(V) on X), which turns out to be a polyhedral convex cone.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.     

Some new scales of weight characterizations of the class <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We present an equivalence theorem, which includes all known characterizations of the class B _p , i.e., the weight class of Ariño and Muckenhoupt, and also some new equivalent characterizations. We also give equivalent characterizations for the classes B _p ^* , B _∞ ^* and RB _p , and prove and apply a “gluing lemma” of independent interest.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Obstructions to the uniform stability of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let T _t : X → X be a C ₀-semigroup with generator A. We prove that if the abscissa of uniform boundedness of the resolvent s ₀(A) is greater than zero then for each nondecreasing function h(s): ?₊ → R ₊ there are x′ ∈ X′ and x ∈ X satisfying ∫ ₀ ^∞ h(|〈x′, T _x x〉|)dt = ∞. If i? ∩ Sp(A) ≠ Ø then such x may be taken in D(A ^∞).     

Characterizing projective general unitary groups PGU<Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis><Superscript>2</Superscript>) by their complex group algebras

Let G be a finite group. Let X ₁(G) be the first column of the ordinary character table of G. We will show that if X ₁(G) = X1(PGU₃(q ²)), then G ? PGU₃(q ²). As a consequence, we show that the projective general unitary groups PGU₃(q ²) are uniquely determined by the structure of their complex group algebras.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ(²G₂(q)), where q = 3²ⁿ⁺¹ for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to ² G ₂(q). We infer that if G is a finite group satisfying |G| = |² G ₂(q)| and Γ(G) = Γ (² G ₂(q)) then G ? = ² G ₂(q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-boundedness of a boundary integral operator on a contour with a peak

Results on the solvability of boundary integral equations on a plane contour with a peak obtained in collaboration with V.G. Maz’ya are developed. Earlier, it was proved that, on a contour Γ with an outward peak, the operator of the boundary equation of the Dirichlet boundary value problem maps the space ? _{p, β + 1} (Γ) continuously onto \(\mathcal{N}_{p,\beta } (\Gamma )\). The norm of a function in ? _{p, β} (Γ) is defined as

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, provided that the peak is at the origin. In this case, the norms on the spaces \(\mathcal{N}_{p,\beta }^ \mp (\Gamma )\) are defined by

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, where q _± are the intersection points of Γ with the circle {z: |z| = |q|} and δ > 0 is a fixed small number. On a contour with an inward peak, the operator of the boundary equation of the Dirichlet problem continuously maps ? _{p, β + 1} (Γ) onto ? _{p, β}(Γ), where ? _{p, β}(Γ) is the direct sum of \(\mathcal{N}_{p,\beta }^ + (\Gamma )\) (Γ) and the space

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(Γ) of functions on Γ of the form p(z) = Σ _{k = 0} ^m t ^(k)Rez ^k with the parameter m = [μ ? β ? p ^?1]. The operator I ? 2W of the boundary integral equation of plane elasticity theory, where W is the elastic double-layer potential, is considered. The main result is that the operator I ? 2W continuously maps the space ? _{p, β + 1} × ? _{p, β + 1}(Γ) to the space \(\mathcal{N}_{p,\beta }^ - \times \mathcal{N}_{p,\beta }^ - (\Gamma )\).

On a contour with an inward peak, the obtained representation of the operator I ? 2W and theorems on the boundedness of auxiliary integral operators imply that the images of vector-valued functions from ? _{p, β + 1} × ? _{p, β + 1}(Γ) have components representable as sums of functions from the spaces \(\mathcal{N}_{p,\beta }^ - (\Gamma )\)(Γ) and ? _{p, β}(Γ).