The Existence of LHMTS(<Emphasis Type="Italic">m</Emphasis><Superscript><Emphasis Type="Italic">v</Emphasis></Superscript>) and LHDTS(<Emphasis Type="Italic">m</Emphasis><Superscript><Emphasis Type="Italic">v</Emphasis></Superscript>)

In this article, we establish the existence of an LHMTS(m^v) for v ≡ 2 (mod 6) and m≡ 3 (mod 6). Thus there exists an LHMTS(m^v) if and only if v(v-1)m² ≡ 0 (mod 3) except possibly for v=6, m≡ 1, 5 (mod 6) and m≠1. In the similar way, the existence of LHDTS(m^v) is completely determined, i.e., there exists an LHDTS(m^v) if and only if v(v-1)m² ≡ 0 (mod 3).     

Biased partitions and judicious <Emphasis Type="Italic">k</Emphasis>-partitions of graphs

Let G = (V,E) be a graph with m edges. For reals p ∈ [0, 1] and q = 1- p, let m_p(G) be the minimum of qe(V₁) +pe(V₂) over partitions V = V₁ ∪ V₂, where e(V_i) denotes the number of edges spanned by V_i. We show that if mp(G) = pqm-δ, then there exists a bipartition V₁, V₂ of G such that e(V₁) ≤ p²m - δ + p√m/2 + o(√m) and e(V₂) ≤ q²m - δ + q √m/2 + o(√m) for δ = o(m^2/3). This is sharp for complete graphs up to the error term o(√m). For an integer k ≥ 2, let fk(G) denote the maximum number of edges in a k-partite subgraph of G. We prove that if fk(G) = (1 - 1/k)m + α, then G admits a k-partition such that each vertex class spans at most m/k² - Ω(m/k^7.5) edges for α = Ω(m/k⁶). Both of the above improve the results of Bollobás and Scott.     

Some Ore-type Results for Matching and Perfect Matching in <Emphasis Type="Italic">k</Emphasis>-uniform Hypergraphs

Let S₁ and S₂ be two (k-1)-subsets in a k-uniform hypergraph H. We call S₁ and S₂ strongly or middle or weakly independent if H does not contain an edge e ∈ E(H) such that S₁ ∩ e ≠∅ and S₂ ∩ e ≠∅ or e ⊆ S₁ ∪ S₂ or e ⊇ S₁ ∪ S₂, respectively. In this paper, we obtain the following results concerning these three independence. (1) For any n ≥ 2k²-k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent (k-1)-sets equal to 2n-4(k-1), contains no perfect matching; (2) Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd+(k-2)k. If the degree sum of any two middle independent (k-1)-subsets is larger than 2(d-1), then H contains a d-matching; (3) For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent (k-1)-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.     

Solution to an extremal problem on bigraphic pairs with a <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connected realization

Let S=(a₁,...,a_m; b₁,...,b_n), where a₁,...,a_m and b₁,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S=(a₁,...,a_m; b₁,...,b_n) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a₁,...,a_m and b₁,...,b_n are the degrees of the vertices in X and Y, respectively. Let Z₃ be the cyclic group of order 3. Define σ(Z₃, m, n) to be the minimum integer k such that every bigraphic pair S=(a₁,...,a_m; b₁,...,b_n) with a_m, b_n ≥ 2 and σ(S)=a₁ +... + a_m ≥ k has a Z₃-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z₃, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z₃, m, n) for m ≥ n ≥ 4.     

Banach upper density recurrent points of <Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>-flows

Let X denote a compact metric space with distance d and F:X×R→X or F_t:X→X denote a C⁰-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C⁰-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates of rough maximal functions along surfaces with applications

In this paper, we study the L^p mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the L^p boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(S^n-1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.     

Solvable <Emphasis Type="Italic">D</Emphasis><Subscript>2</Subscript>-groups

Let G be a finite group, Irr¹(G) be the set of nonlinear irreducible characters of G and cd¹(G) the set of degrees of the characters in Irr¹(G). A group G is said to be a D₂-group if|cd¹(G)|=|Irr¹(G)|-2. In this paper, we give a complete classification of solvable D₂-groups.     

<Emphasis Type="Italic">n</Emphasis>-transitivity of bisection groups of a Lie groupoid

The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈ N by an arbitrary group of C^r-bisections of a Lie groupoid Γ of class C^r, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if Γ is source connected for any arrow γ ∈ Γ, there is a bisection passing through γ.     

Equivalent characterization of centralizers on <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">H</Emphasis>)

Let H be a Hilbert space with dim H≥2 and Z∈ß(H) be an arbitrary but fixed operator. In this paper we show that an additive map Φ:ß(H) → ß(H) satisfies Φ(AB)=Φ(A)B=AΦ(B) for any A,B∈ß(H) with AB=Z if and only if Φ(AB)=Φ(A)B=AΦ(B), ∀A, B∈ß(H), that is, Φ is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Φ(A²)=AΦ(A)=Φ(A)A for any A∈ ß(H) with AA²=0 if and only if Φ(A)=AΦ(I)=Φ(I)A, ∀A∈ß(H), and generalize main results in Linear Algebra and its Application, 450, 243-249 (2014) to infinite dimensional case. New equivalent characterization of centralizers on ß(H) is obtained.     

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.     

On a dendrite generated by a zero-dimensional weak self-similar set

Let S be a zero-dimensional, perfect, compact weak self-similar set generated in dendrite X by a family {f_j} of weak contractions from X to itself. Decomposition space D_f of S due to a continuous mapping f from S onto X is also a dendrite. In the dendrite D_f, there exists a zero-dimensional, perfect, compact weak self-similar set S¹ based on a family each of which is topologically conjugate to f_j. Decomposition space D_f¹ of S¹ due to a continuous mapping f¹ from S¹ onto D_f is again a dendrite. In this manner, through the successive formation of weak self-similar set, we can obtain a sequence X,D_f,D_f¹,… of dendrite any pair in which are mutually homeomorphic.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for bi-parameter and bilinear Fourier integral operators

Fourier integral operators play an important role in Fourier analysis and partial differential equations. In this paper, we deal with the boundedness of the bilinear and bi-parameter Fourier integral operators, which are motivated by the study of one-parameter FIOs and bilinear and bi-parameter Fourier multipliers and pseudo-differential operators. We consider such FIOs when they have compact support in spatial variables. If they contain a real-valued phase φ(x, ξ, η) which is jointly homogeneous in the frequency variables ξ, η, and amplitudes of order zero supported away from the axes and the antidiagonal, we can show that the boundedness holds in the local-L² case. Some stronger boundedness results are also obtained under more restricted conditions on the phase functions. Thus our results extend the boundedness results for bilinear and one-parameter FIOs and bilinear and bi-parameter pseudo-differential operators to the case of bilinear and bi-parameter FIOs.     

Pontryagin–Thom construction for approximation of mappings by embeddings

Let n3 and be positive integers, f :Sⁿ→Sⁿ be a C⁰-mapping, and denote the standard embedding. As an application of the Pontryagin–Thom construction in the special case of the two-point configuration space, we construct complete algebraic obstructions O(f) and to discrete and isotopic realizability (realizability as an embedding) of the mapping Jf. The obstructions are described in terms of stable (equivariant) homotopy groups of neighborhoods of the singular set Σ(f)={(x,y)Sⁿ×Sⁿf(x)=f(y), x≠y}.

A standard method of solving problems in differential topology is to translate them into homotopy theory by means of bordism theory and Pontryagin–Thom construction. By this method we give a generalization of the van-Kampen–Skopenkov obstruction to discrete realizability of f and the van-Kampen–Melikhov obstruction to isotopic realizability of f. The latter are complete only in the case d=0 and are the images of our obstructions under a Hurewicz homomorphism.

We consider several examples of computation of the obstructions.     

(<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-bi-Koszul algebras

The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.     

MEROMORPHIC FUNCTIONS SHARING TWO FINITE SETS

Let S1 ＝ {∞} and S2 ＝ {w: Ps(w)＝ 0}, Ps(w) being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f-1(Si) ＝ g-1(Si)(i ＝ 1,2), where f-1(Si) and g-1(Si) denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.     

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

The <Emphasis Type="Italic">s</Emphasis>-energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S ². A spherical n-design is a point set on S ² that gives rise to an equal weight cubature rule which is exact for all spherical polynomials of degree ≤n. The s-energy E _s (X) of a point set of m distinct points is the sum of the potential for all pairs of distinct points . A sequence Ξ = {X _m } of point sets X _m ⊂S ², where X _m has the cardinality card(X _m )=m, is well separated if for each pair of distinct points , where the constant λ is independent of m and X _m . For all s>0, we derive upper bounds in terms of orders of n and m(n) of the s-energy E _s (X _m(n)) for well separated sequences Ξ = {X _m(n)} of spherical n-designs X _m(n) with card(X _m(n))=m(n).      

<Emphasis Type="Italic">K</Emphasis>-homology of <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-categories and symmetric spectra representing <Emphasis Type="Italic">K</Emphasis>-homology

We define K-homology groups K _* () for small C ^* -categories in terms of Hilbert modules over the C ^* -category . We also define a functor A ^f from the category of small C ^* -categories into the category of C ^* -algebras and show that there is a natural isomorphism . In addition, we give an easy construction of a functor from the category of C ^* -algebras into the category of symmetric spectra which represents K-homology, i.e. we show that the functor comes with a natural isomorphism for C ^* -algebras A. It then follows that the composition A ^f provides a functor that can be used in the Davis-Lück approach for constructing the Baum-Connes map.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake