Judiciously 3‐partitioning 3‐uniform hypergraphs

Bollobás, Reed, and Thomason proved every 3‐uniform hypergraph ? with m edges has a vertex‐partition V()=V₁?V₂?V₃ such that each part meets at least edges, later improved to 0.6m by Halsegrave and improved asymptotically to 0.65m+o(m) by Ma and Yu. We improve this asymptotic bound to , which is best possible up to the error term, resolving a special case of a conjecture of Bollobás and Scott.     

Uniform Mixed Hypergraphs: The Possible Numbers of Colors

For a mixed hypergraph , where and are set systems over the vertex set X, a coloring is a partition of X into ‘color classes’ such that every meets some class in more than one vertex, and every has a nonempty intersection with at least two classes. The feasible set of , denoted , is the set of integers k such that admits a coloring with precisely k nonempty color classes. It was proved by Jiang et al. [Graphs and Combinatorics 18 (2002), 309–318] that a set S of natural numbers is the feasible set of some mixed hypergraph if and only if either or S is an ‘interval’ for some integer k ≥ 1. In this note we consider r-uniform mixed hypergraphs, i.e. those with |C| = |D| = r for all and all , r ≥ 3. We prove that S is the feasible set of some r-uniform mixed hypergraph with at least one edge if and only if either for some natural number k ≥ r − 1, or S is of the form where S′′ is any (possibly empty) subset of and S′ is either the empty set or {r − 1} or an ‘interval’ {k, k + 1, ..., r − 1} for some k, 2 ≤ k ≤ r − 2. We also prove that all these feasible sets can be obtained under the restriction , i.e. within the class of ‘bi-hypergraphs’. Research supported in part by the Hungarian Scientific Research Fund, OTKA grant T-049613.     

A noncommutative symmetric system over the Grossman-Larson Hopf algebra of labeled rooted trees

In this paper, we construct explicitly a noncommutative symmetric ( CS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the CS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a graded Hopf algebra homomorphism from the Connes-Kreimer Hopf algebra of labeled rooted forests to the Hopf algebra of quasi-symmetric functions. A connection of the coefficients of the third generating function of the constructed CS system with the order polynomials of rooted trees is also given and proved.     

Upper chromatic number of finite projective planes

For a finite projective plane , let denote the maximum number of classes in a partition of the point set, such that each line has at least two points in the same partition class. We prove that the best possible general estimate in terms of the order of projective planes is , which is tight apart from a multiplicative constant in the third term :

• (1) As holds for every projective plane of order q.
• (2) If q is a square, then the Galois plane of order q satisfies .

Our results asymptotically solve a ten‐year‐old open problem in the coloring theory of mixed hypergraphs, where is termed the upper chromatic number of . Further improvements on the upper bound (1) are presented for Galois planes and their subclasses. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 221–230, 2008     

Judicious bisection of hypergraphs

Judicious bisection of hypergraphs asks for a balanced bipartition of the vertex set that optimizes several quantities simultaneously.In this paper,we prove that if G is a hypergraph with n vertices and m_i edges of size i for i=1,2,...,k,then G admits a bisection in which each vertex class spans at most(m_1)/2+1/4m_2+…+(1/(2~k)+m_k+o(m_1+…+m_k) edges,where G is dense enough or △(G)= o(n) but has no isolated vertex,which turns out to be a bisection version of a conjecture proposed by Bollobas and Scott.     

On induced Ramsey numbers for k‐uniform hypergraphs

Let denote the complete k‐uniform k‐partite hypergraph with classes of size t and the complete k‐uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = s^o(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k‐partite k‐uniform hypergraph with classes of size t and an arbitrary k‐graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016     

An introduction to the perplex number system

The perplex number system is a generalization of the abstract logical relationships among electrical particles. The inferential logic of the new number system is homologous to the inferential logic of the progression of the atomic numbers. An electrical progression is defined categorically as a sequence of objects with teridentities. Each identity infers corresponding values of an integer, units and a correspondence relation between each unit and its integer. Thus, in this logical system, each perplex numeral contains an exact internal representational structure; it carries an internal message. This structure is a labeled bipartite graph that is homologous to the internal electrical structure of a chemical atom. The formal logical operations are conjunctions and disjunctions. Combinations of conjunctions and disjunctions compose the spatiality of objects. Conjunctions may include the middle term of pairs of propositions with a common term, thereby creating new information. The perplex numerals are used as a universal source of diagrams.The perplex number system, as an abstract generalization of concrete objects and processes, constitutes a new exact notation for chemistry without invoking alchemical symbols. Practical applications of the number system to concrete objects (chemical elements, simple ions and molecules, and the perplex isomers, ethanol and dimethyl ether) are given. In conjunction with the real number system, the relationships between the perplex number system and scientific theories of concrete systems (thermodynamics, intra-molecular dynamics, molecular biology and individual medicine) are described.     

非均齐超网络中标度律的涌现—–富者愈富导致幂律分布吗?

建立非线性择优连接非均齐超网络演化模型,研究非均齐超网络演化机制和拓扑性质.使用Poisson过程理论和连续化方法对模型进行分析,给出超网络超度的特征方程.利用超度特征方程不仅证明网络稳态平均超度分布存在,而且获得超度分布的解析表达式.分析表明这个网络具有"富者愈富"现象.仿真实验和理论分析相符合.随着网络规模的增大,这个动态演化的非均齐超网络的超度分布表现出拉直指数分布的特征,而不一定是幂律分布.结果表明"富者愈富"不一定导致幂律分布.     

Partitioning 3-uniform hypergraphs

Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges can be partitioned into r sets so that each set meets at least rm/(2r−1) edges. For r=3, Bollobás, Reed and Thomason proved the lower bound (1−1/e)m/3≈0.21m, which was improved to (5/9)m by Bollobás and Scott and to 0.6m by Haslegrave. In this paper, we show that any 3-uniform hypergraph with m edges can be partitioned into 3 sets, each of which meets at least 0.65m−o(m) edges.     

Determination of [13C-15N] Labeled Methylxanthines by Gas Chromatography-Atomic Emission Detection (GC-AED) and Compound-Independent Calibration

ABSTRACT

Compound independent calibration (CIC) can be used to measure stable isotopically labeled molecules and their metabolites. This study was designed to demonstrate that gas chromatography coupled to atomic emission detection and CIC are useful analytical tools for the specific and quantitative determination of ¹³C and ¹⁵N labeled caffeine and its metabolites. These labeled compounds can be measured by monitoring either ¹³C or ¹⁵N atomic emission lines and using only one calibration function from one of the analytes. This method can be helpful when it is difficult or impossible to obtain an authentic reference material, which is often the case in metabolic studies.