AZ-Identities and Strict 2-Part Sperner Properties of Product Posets

One of central issues in extremal set theory is Sperner’s theorem and its generalizations. Among such generalizations is the best-known LYM (also known as BLYM) inequality and the Ahlswede–Zhang (AZ) identity which surprisingly generalizes the BLYM into an identity. Sperner’s theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.     

If a polynomial identity guarantees that every partial order on a ring can be extended,then this identity is true only for a zero-ring

In his classical 1963 book on partially ordered algebraic systems, L. Fuchs formulated the following problem (No. 29). It is known that if an abelian groupG (i.e., a group that satisfies an identityxy=yx) can be linearly ordered, then every partial order onG can be extended to a linear order. Fuchs asked whether there exists a similar polynomial identityP=0 for (associative) rings. In other words, does there exist a polynomialP with a following property: if a ringR satisfies the identityP=0, andR can be linearly ordered, then every partial order onR can be extended to a linear order? We prove that no such non-trivial polynomial identity is possible. Namely, we prove that every ringR that satisfies such an identity is a zero-ring (i.e.,xy=0 for allx, y εR).     

An AZ-style identity and Bollobás deficiency

The powerful AZ identity is a sharpening of the famous LYM-inequality. More generally, Ahlswede and Zhang discovered a generalization in which the Bollobás inequality for two set families can be lifted to an identity.In this paper, we show another generalization of the AZ identity. The new identity implies an identity which characterizes the deficiency of the Bollobás inequality for an intersecting Sperner family. We also give some consequences relating to Helly families and LYM-style inequalities.     

A generalization of the AZ identity

The identity discovered in [1] can be viewed as a sharpening of the LYM inequality ([3], [4], [5]). It was extended in [2] so that it covers also Bollobás' inequality [6]. Here we present a further generalization and demonstrate that it shares with its predecessors the usefullness for uniqueness proofs in extremal set theory.     

Existence of extremal functions for a Hardy-Sobolev inequality

We present the best constant and the extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in RN.     

Weighted cross-intersecting families

In this paper we investigate weighted cross-intersecting families: if α,β>0 are given constants, we want to find the maximum of α|A|+β|B| for A,B uniform cross-intersecting families. We determine the maximum sum, even if we have restrictions of the size of A.As corollaries, we will obtain some new bounds on the shadows and the shades of uniform families. We give direct proofs for these bounds, as well, and show that the theorems for cross-intersecting families also follow from these results.Finally, we will generalize the LYM inequality not only for cross-intersecting families, but also for arbitrary Sperner families.     

Ornstein-Uhlenbeck processes on Lie groups

We consider Ornstein-Uhlenbeck processes (OU-processes) associated to hypo-elliptic diffusion processes on finite-dimensional Lie groups: let L be a hypo-elliptic, left-invariant “sum of the squares”-operator on a Lie group G with associated Markov process X, then we construct OU-processes by adding negative horizontal gradient drifts of functions U. In the natural case U(x)=−logp(1,x), where p(1,x) is the density of the law of X starting at identity e at time t=1 with respect to the right-invariant Haar measure on G, we show the Poincaré inequality by applying the Driver-Melcher inequality for “sum of the squares” operators on Lie groups. The resulting Markov process is called the natural OU-process associated to the hypo-elliptic diffusion on G. We prove the global strong existence of these OU-type processes on G under an integrability assumption on U. The Poincaré inequality for a large class of potentials U is then shown by a perturbation technique. These results are applied to obtain a hypo-elliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces M.     

Some remarks on normalized matching

A new class of LYM orders is obtained, and several results about general LYM orders are proved. (1) Let A₁ ? A₂ ? … ? A_r be a chain of subsets of [n] = {l,…,n}. Let 〈a_i〉 and 〈b_i〉 be two nondecreasing sequences with a_i ? b_i for l ? i ? r. Then {X ? [n]: a_i ? | ∩ A_i|? b_i}, ordered by inclusion, is a poset having the LYM property. (2) The smallest regular covering of an LYM order has M(P) chains, where M(P) is the least common multiple of the rank sizes. (3) Every LYM order has a smallest regular covering with at most || ? h(P) classes of distinct chains, where h(P) is the height of P. To obtain (3), we discuss “minimal sets” of covering relations between two adjacent levels of an LYM-order.     

Minimal differential identities in prime rings

It is known that a prime ring which satisfies a polynomial identity with derivations applied to the variables must satisfy a generalized polynomial identity, but not necessarily a polynomial identity. In this paper we determine the minimal identity with derivations which can be satisfied by a non-PI prime ringR. The main result shows, essentially, that this identity is the standard identityS ₃ withD applied to each variable, whereD = ad(y) fory inR, y ² = 0, andy of rank one in the central closure ofR.     

Self-orthogonal cyclicn-quasigroups

A quasigroup (Q,) satisfying the identityx(yx) =y (or the equivalent identity (xy)x =y) is called semisymmetric. Ann-quasigroup (Q, A) satisfying the identityA(A(x ₁, ...,x _n ),x ₁, ...,x _n–1) =x _n is called cyclic. So, cyclicn-quasigroups are a generalization of semisymmetric quasigroups. In this paper, self-orthogonal cyclicn-quasigroups (SOCnQs) are considered. Some constructions ofSOCnQs are described and the spectrum of suchn-quasigroups investigated.     

Local rearrangement invariant spaces and distribution of Rademacher series

We prove that a local version of Khintchine inequality holds for arbitrary rearrangement invariant (r.i.) spaces on an non-empty open set \(E\subset [0,1]\). For this, we give a definition of local r.i. space which is compatible with the notion of systems equivalent in distribution and prove that the Rademacher system \((r_{k+N})_{k=1}^\infty \) on an non-empty open set E is equivalent in distribution to \((r_k)_{k=1}^\infty \) on [0, 1], with N depending on E. The result can be generalized to a wider class of sets.     

Weak curvature conditions and functional inequalities

We give sufficient conditions for a measured length space (X,d,ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X,d,ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X,d,ν) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant N². The condition DM is preserved by measured Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally we derive a sharp global Poincaré inequality.     

Operator extensions of Hua’s inequality

We give an extension of Hua’s inequality in pre-Hilbert C^∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C^∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is equivalent to operator convexity of given continuous real function.     

Levitin–Polyak well-posedness by perturbations of inverse variational inequalities

The purpose of this paper is to investigate Levitin–Polyak type well-posedness for inverse variational inequalities. We establish some metric characterizations of Levitin–Polyak α-well-posedness by perturbations. Under suitable conditions, we prove that Levitin–Polyak well-posedness by perturbations of an inverse variational inequality is equivalent to the existence and uniqueness of its solution. Moreover, we show that Levitin–Polyak well-posedness by perturbations of an inverse variational inequality is equivalent to Levitin–Polyak well-posedness by perturbations of an enlarged classical variational inequality.     

On the construction of cyclic quasigroups

Let F(x,y) be the free groupoid on two generators x and y. Define an infinite class of words in F(x,y) by w₀(x,y) = x,w₁(x,y) = y and w_i+2(x,y) = w_i(x,y)w_i+1(x,y). An identity of the form w_3n(x,y) = x is called a cyclic identity and a quasigroup satisfying a cyclic identity is called a cyclic quasigroup. The most extensively studied cyclic quasigroups have been models of the identity y(xy) = x. The more general notion of cyclic quasigroups was introduced by N.S. Mendelsohn. In this paper a new construction for cyclic quasigroups is given. This construction is useful in constructing large numbers of nonisomorphic quasigroups satisfying a given cyclic identity or a consequence of a cyclic identity. The construction is based on a generalization of A. Sade's singular direct product of quasigroups.     

A connection between a convex programming problem and the LYM property on perfect graphs

We derive three equivalent conditions on a perfect graph concerning the optimal solution of a convex programming problem, the length-width inequality, and the simultaneous vertex covering by cliques and anticliques. By combining proof techniques including Lagrangian dual, Dilworth's Theorem, and Kuhn-Tucker Theorem, we establish a strong connection between the three topics. This provides new insights into the structure of perfect graphs. The famous Lubell-Yamamoto-Meschalkin (LYM) Property or Sperner Property for partially ordered sets is a specialization of our results to a subclass of perfect graphs.     

Hitchin–Thorpe-Type Inequalities for Pseudo-Riemannian 4-Manifolds of Metric Signature (++−−)

It is well known that the Euler characteristic χ and the Hirzebruch index τ of a compact Einstein 4-manifold satisfy the Hitchin–Thorpe inequality χ ≥ $\tfrac{3}{2}$ ; |τ|, and that these topological invariants of a compact pseudo-Riemannian Einstein 4-manifold of metric signature (++??), with certain curvature restrictions, also satisfy a similar inequality χ ≥ $\tfrac{3}{2}$ ; |τ|, which is called the Hitchin–Thorpe-type inequality. This shows that the Euler characteristic χ is nonpositive for the indefinite case. In this paper, it is shown that for a compact pseudo-Riemannian 4-manifold of signature (++??) the Hitchin–Thorpe-type inequality holds under a weaker condition, called the diagonal Einstein condition, than the Einstein condition. Our analysis is based on the fact that the existence of a metric of signature (++??) on a 4-manifold with the structure group SO _o (2,2) is equivalent to the existence of a pair of an almost complex structure and an opposite almost complex structure on the 4-manifold, which is also equivalent to the reduction of the structure group to the maximal compact subgroup SO(2) × SO(2) of SO _o (2,2).     

An equivalent characterization of BMO with Gauss measure

Let γ be the Gauss measure on ? ⁿ : We establish a Calderón-Zygmund type decomposition and a John-Nirenberg type inequality in terms of the local sharp maximal function and the median value of function over cubes. As an application, we obtain an equivalent characterization of known BMO space with Gauss measure.