On the Coates–Sinnott Conjecture

Let E signify a totally real Abelian number field with a prime power conductor and ring of pintegers R _E for a prime p. Let G denote the Galois group of E over the rationals, and let be a padic character of G of order prime to p. Theorem A calculates, under a minor restriction on , the Fitting ideals of H ² _ét(R _E;Z _p (n/2+1))() over Z _p [G](). Here we require that n2 mod 4. These Fitting ideals are principal and generated by a Stickelberger element. This gives a partial verification and also a strong indication of the Coates–Sinnott conjecture.     

On Landis’ Conjecture in the Plane

In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W(z) be a measurable real vector-valued function and V(z) ≥0 be a real measurable scalar function, satisfying ‖W‖ _{L ^∞(R ²)} ≤ 1 and ‖V‖ _{L ^∞(R ²)} ≤ 1. Let u be a real solution of Δu ? ?(Wu) ? Vu = 0 in R ². Assume that u(0) = 1 and |u(z)| ≤exp (C ₀|z|). Then u satisfies inf _{|z 0| =R}  sup _{|z?z 0| <1}|u(z)| ≥exp (?CRlog R), where C depends on C ₀. In addition to the case of the whole plane, we also establish a quantitative form of Landis’ conjecture defined in an exterior domain.     

On the Superlinear Lazer–McKenna Conjecture: Part II

ABSTRACT

We prove that certain weakly nonlinear elliptic equations have many solutions when a paragraph is large. The nonlinearity grows superlinearly for y positive but grows linearly for y negative.     

On the 1/3–2/3 Conjecture

Let (P, ≤) be a finite poset (partially ordered set), where P has cardinality n. Consider linear extensions of P as permutations x₁x₂?x_n in one-line notation. For distinct elements x, y ∈ P, we define ?(x ? y) to be the proportion of linear extensions of P in which x comes before y. For \(0\leq \alpha \leq \frac {1}{2}\), we say (x, y) is an α-balanced pair if α ≤ ?(x ? y) ≤?1 ? α. The 1/3–2/3 Conjecture states that every finite partially ordered set which is not a chain has a 1/3-balanced pair. We make progress on this conjecture by showing that it holds for certain families of posets. These include lattices such as the Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. We also consider various posets which satisfy the stronger condition of having a 1/2-balanced pair. For example, this happens when the poset has an automorphism with a cycle of length 2. Various questions for future research are posed.     

On Levine–O'Sullivan's Sequence and Its Conjecture

Let q_1=1,and q_n be the largest value of(k+1)(n-q_k) for all integers 1≤k≤n-1with n≥2.The sequence Q={q1,q2,q3,...} is called Levine–O'Sullivan sequence.In this paper,we use the combinational and analysis skill and the mathematical induction to study the asymptotic properties of q_n,and give an interesting asymptotic formula for it.This solved a conjecture proposed by Professor Chen Yonggao.     

On the Classification of Finite R-Trivial P-Semigroups

In this paper we discuss the classification of finite R-triVial P-semigroups (congruence permutable semigroup).It is clear that the monogenic semigroup C_(n,1)and c_(n,1)~1 are R-triVial P-semigYoups.F01"the Proper ideal R={a~k,a~(k ),…a~k of C_(n,1)~1,the branehextension[4]C_(n,1)~1×I/R is a P-sereigroup,denoted by C_(x,1,k)~1 It is a R-triVial semigroulp. Let N be the set of nattlral nulnbers,l≠s∈N,e_l,e_2,a are transfotmations of     

On the Syskin Problem of Finite Groups

In 1980, at the Seventh All-Union Symposium on Group Theory, ProfessorS.A .Syskin raised the following well-known problem: In a finite group G, supposeany two elemeuts of the same order are conjugate. Is |G|≤6? It is quite easy to     

On the Truth of a Conjecture Concerning the q-Derangement Numbers

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On Π-Permutable Subgroups in Finite Groups

Ukrainian Mathematical Journal - Let σ = {σi | i ∈ I} be a partition of the set of all primes ? and let Π be a nonempty subset of the set σ. A set ? of...     

On the Characterization of the Full p-Defective Finite Group

A finite group G is called a full p-defective group if every p-block of G isof the highest defect.In this paper, we shall mainly study the full p-defectivefinite groups and characterize them.     

On a Theorem of Barker–Puig that Refines Alperin’s Weight Conjecture for Blocks in p-Solvable Finite Groups via Clifford Theory

We present a much shorter Clifford Theoretic Proof of an important Theorem of Barker and Puig on a refinement of Alperin’s Weight Conjecture for Blocks in p-solvable finite groups. Our proof employs a standard Clifford Theoretic approach. We also demonstrate a relationship between the Green correspondence and Sibley’s concept of a vertex pair. Consequently the main theorem can be stated in terms of Sibley’s vertex pairs.     

On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross–Prasad Conjecture

In this paper, we would like to formulate a conjecture on a relation between a certain period of automorphic forms on special orthogonal groups and some L-value. Our conjecture can be considered as a refinement of the global Gross–Prasad conjecture.     

On an Empty Convex Partition of a Finite Set in the Plane

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On the Number of Conjugacy Classes in Semi-Products of Certain Finite Groups

Let V be a vector space of n-dimension over the field GF(p) of p elements,where p is a prime. V is also an elementary abelian p-group.Let G be a p'-group oflinear transformations on V.Theorem 1 Let π_v(a_1,a_2) be the number of the common fixed points of a_1and a_2 on V, a_1, a_2 ∈G. Let k(GV) be the number of conjugacy classes in thesemi-product GV (We also denote it by GV) of G and V. Then     

On the Scope of Averaging for Frankl’s Conjecture

Let be a union-closed family of subsets of an m-element set A. Let . For b ∈ A let w(b) denote the number of sets in containing b minus the number of sets in not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2 ^m  − 2 ^m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2 ^m/2 with the upper integer part of 2 ^m /3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and then the averaged Frankl’s property holds (i.e., 2 ^m/2 can be replaced with the lower integer part of 2 ^m /3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148.     

On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.