New asymptotic expansions related to Somosʼ quadratic recurrence constant

We derive new asymptotic expansions related to Somos? quadratic recurrence constant, in terms of the ordered Bell numbers.     

A note on Linnik’s theorem on quadratic non-residues

Archiv der Mathematik - We present a short and purely combinatorial proof of Linnik’s theorem: for any $$varepsilon >0$$ there exists a constant $$C_varepsilon $$ such that for any...     

A note on Thomassen?s conjecture

In 1983 C. Thomassen [8] conjectured that for every k,g∈N there exists d such that any graph with average degree at least d contains a subgraph with average degree at least k and girth at least g. A result of Pyber, Szemerédi, and the second author implies that the conjecture is true for every graph G with average .We strengthen this and show that the conjecture holds for every graph G with average for some constants α, β depending on k and g.     

A note on Poincaré recurrence for semigroup actions via Furstenberg family

This paper presents an explanation of recurrence for semigroup actions from two points of view. The first one considers the concept of recurrence in the Poincaré sense as stated in Souza (Semigroup Forum 83:351–370, 2011) while the second one considers the notion of recurrence via a Fursternberg family as defined in Akin (Recurrence in Topological Dynamics. Furstenberg and Ellis Action. Plenum Press, New York, 1997). A relationship between these notions of recurrence is presented.     

A note on Haj?asz-Sobolev spaces on fractals

We introduce Haj?asz-Sobolev spaces involving walk dimensions on subsets of that will generally be fractals. The relationship between this kind of new space and the Besov space of Jonsson and Wallin is investigated. The (compact) embedding theorems for Haj?asz-Sobolev spaces are obtained.     

A note on Stevi?'s iteration method

In the present paper, motivated by Stevi?'s iteration method, we show the stability result of the following unified iterative scheme

xn+1=tnTvn+(1−tn)xn+un     

A note on Chudnovsky?s Fuchsian equations

We show that four exceptional Fuchsian equations, each determined by the four parabolic singularities, known as the Chudnovsky equations, are transformed into each other by algebraic transformations. We describe equivalence of these equations and their counterparts on tori. The latters are the Fuchsian equations on elliptic curves and their equivalence is characterized by transcendental transformations which are represented explicitly in terms of elliptic and theta functions.     

A note on θ-tightness

In questo lavoro chiariamo il ruolo che la θ-tightness (introdotta dagli autori in [2]) ha nel determinare alcune estensioni di sottoinsiemi di uno spazio topologico.     

A note on Marino-Vafa formula

Hodge integrals over moduli spaces of curves appear naturally during the localization procedure in computation of Gromov-Witten invariants. A remarkable formula of Marino-Vafa expresses a generation function of Hodge integrals via some combinatorial and algebraic data seemingly unrelated to these apriori algebraic geometric objects. We prove in this paper by directly expanding the formula and estimating the involved terms carefully that except a specific type all the other Hodge integrals involving up to three Hodge classes can be calculated from this formula. This implies that amazingly rich information about moduli spaces and Gromov-Witten invariants is encoded in this complicated formula. We also give some low genus examples which agree with the previous results in literature. Proofs and calculations are elementary as long as one accepts Mumford relations on the reductions of products of Hodge classes.     

A note on β-connectedness

An example is given to show that there exists a generalized topological space which is irreducible but not β-connected. This gives an answer to Question 2.8 in [6]. Besides, some characterizations of β-connectedness are obtained.     

A note on Weil index

Let F be a non-archimedean local field of characteristic 0 and(?)a nontrivial additive character.Weil first defined the Weil indexγ(a,(?))(a∈F~*)in his famous paper,from which we know thatγ(a,(?))γ(b,(?))=γ(ab,(?))γ(1,(?))(a,b)andγ(a,(?))~4 =(-1,-1),where(a,b)is the Hilbert symbol for F.The Weil index plays an important role in the theory of theta series and in the general representation theory.In this paper,we establish an identity relating the Weil indexγ(a,(?))and the Gauss sum.     

A note on K?hler–Ricci flow

Let g(t) with ${t\in [0,T)}$ be a complete solution to the K?hler–Ricci flow: ${\frac{d}{dt}g_{i\bar j}=-R_{i\bar j}}$ where T may be ∞. In this article, we show that the curvature of g(t) is uniformly bounded if the solution g(t) is uniformly equivalent. This result is stronger than the main result in ?e?um (Am J Math 127(6):1315–1324, 2005) within the category of K?hler–Ricci flow.     

A note on a problem of Erd?s on right angles

Erd?s asked the following question: is it true that a set of n² points in the plane always contain 2n−2 points which do not determine a right angle? We show that, apart from a log-factor, the answer is in the affirmative.     

A note on λ-compact spaces

We extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.     

A note on noncommutative moment problems

Noncommutative moment problems for C*-algebras are studied. We generalize a result of Hadwin on tracial states to nontracial case. Our results are applied to obtain simple solutions to moment problems on the square and the circle as well as extend the positive unital functionals from a (discrete) complex group algebra to states on the group C*-algebra.     

A note on ‘on antipodal graphs’

A lemma stated in our earlier paper has been corrected without affecting the major content of that paper. An independent proof for the succeeding lemma in that paper has also been produced. It is be noted that given a graph G, the antipodal graph A(G) is uniquely defined, but given A(G), there exist more than one graph H such that A(H) = A(G).