For a simple digraph G without directed triangles or digons, let β(G) be the size of the smallest subset X ⊆ E(G) such that G \ X has no directed cycles, and let γ(G) be the number of unordered pairs of nonadjacent vertices in G. In 2008, Chudnovsky, Seymour, and Sullivan showed that β(G) ≤ γ(G), and conjectured that β(G) ≤ γ(G)/2. Recently, Dunkum, Hamburger, and Pór proved that β(G) ≤ 0.88γ(G...

We extend Crapo’s β invariant from matroids to greedoids, concentrating especially on antimatroids. Several familiar expansions for β(G) have greedoid analogs. We give combinatorial interpretations for β(G) for simplicial shelling antimatroids associated with chordal graphs. When G is this antimatroid and b(G) is the number of blocks of the chordal graph G, we prove β(G) = 1− b(G).

For α, β > 0 and for a locally integrable function (or, more generally , a distribution) ϕ on (0, ∞), we study integral ooperators G α,β ϕ on L 2 (R +) defined by G α,β ϕ f (x) = R+ ϕ x α + y β f (y)dy. We describe the bounded and compact operators G α,β ϕ and operators G α,β ϕ of Schatten–von Neumann class S p. We also study continuity properties of the averaging projection Q α,β onto the oper...

We discuss the analytic properties of the Callan-Symanzik β-function β(g) associated with the zero-momentum four-point coupling g in the two-dimensional φ4 model with O(N) symmetry. Using renormalization-group arguments, we derive the asymptotic behavior of β(g) at the fixed point g∗. We find that β(g) is not analytic at g = g∗: in the Ising model, β(g) has an expansion for g → g∗ with noninteg...

Natural kaolinite was subjected to a successful exfoliation process into separated nanosheets (KNs), followed by hybridization with β-cyclodextrin biopolymer (β-CD), forming an advanced bio-nanocomposite (β-CD/KNs). The synthetic products were evaluated as enhanced delivery structures for oxaliplatin chemotherapy (OXAPN). of KNs β-CD polymer notably the loading capacity 355.3 mg/g (β-CD/KNs) co...

A bipartite graph G is called (α, β)-biregular if all vertices in one part of G have the degree α and all vertices in the other part have the degree β. An edge coloring of a graph G with colors 1, 2, 3, . . . , t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored ...

β-thalassemia, a monogenic autosomal recessive disorder, is prevalent in middle east, particularly in iran. in iran, near to 20 mutations in the β-globin gene are introduced as common mutations with varying incidence frequencies in each city. therefore, detection and screening for couples at high risk can help to solve the problems of this disease. in this study, optimized genotyping of two com...

We present an elegant exact formula for the gaugino β-function in a softly-broken supersymmetric gauge theory, of the form β M = O(β g /g), where β g is the gauge β function and O is a simple differential operator acting on the gauge coupling g and the Yukawa

introduction: β-globin gene cluster haplotypes are useful in diagnosis of particular molecular defects in β-thalassemia, prenatal diagnosis of β-thalassemia, and elucidating population affinities. methods: β-globin gene cluster haplotypes were studied in 150 β-thalassemia minor and 52 healthy in-dividuals from the fars province of iran. dna was extracted from leukocytes of whole blood by phe-no...

and Applied Analysis 3 called cofinite if the recession function f0 of f satisfies f0 y ∞, for all y / 0 see 15, page 116 . Then f is cofinite if and only if dom f∗ R by means of 15, Corollary 13.3.1 . The terminology “cofinite” is renewed as “coercive” in 16, 3.26 Theorem . Now we take a look at Atteia and Raı̈ssouli 11, Proposition 4.4 with a refined proof. Proposition 2.1 See Atteia and Raı̈ss...