On Local LYM Identities

The LYMinequality (Lubell, Yamamoto, Meshalkin) is a generalization of Sperner’s theorem for antichains. Kleitman and Harper independently proved that the LYM inequality and the normalized matching property (or local LYM inequality) are equivalent. Many contributions have been proposed to sharpen the LYM inequality. Noticeably, Ahlswede and Zhang lifted the LYM inequality to an identity, called the AZ identity. Thus, one expects that the same sharpening of the local LYM inequality is equivalent to the AZ identity. In this paper, we introduce a local LYM identity which sharpens the local LYM inequality and prove that it is equivalent to the AZ identity. The local LYM identity shows local relationships between components in the AZ identity.     

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.     

Pseudo-LYM Inequalities and AZ Identities

We give pseudo-LYM inequalities in some posets and we give a new restriction in this way for their antichains. Typically these posets fail the LYM inequality and some of them are known to not be Sperner.     

Reverse mathematics and order theoretic fixed point theorems

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within \(\mathsf {RCA_0}\) give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over \(\mathsf {RCA_0}\). Then we within \(\mathsf {RCA_0}\) give proofs of Knaster–Tarski fixed point theorem, Tarski–Kantorovitch fixed point theorem, Bourbaki–Witt fixed point theorem, and Abian–Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian–Brown least fixed point theorem, Davis’ converse for countable lattices, Markowski’s converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over \(\mathsf {RCA_0}\). Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.     

Unimodality and Lie Superalgebras

It is well-known how the representation theory of the Lie algebra sl(2, ?) can be used to prove that certain sequences of integers are unimodal and that certain posets have the Sperner property. Here an analogous theory is developed for the Lie superalgebra osp(1,2). We obtain new classes of unimodal sequences (described in terms of cycle index polynomials) and a new class of posets (the “super analogue” of the lattice L(m,n) of Young diagrams contained in an m × n rectangle) which have the Sperner property.     

Generalizations of Sherman’s Inequality Via Fink’s Identity and Green’s Function

Ukrainian Mathematical Journal - New generalizations of Sherman’s inequality for n-convex functions are obtained with the help of Fink’s identity and Green’s function. By using...     

Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems

In this paper, we establish some new generalizations of Darbo’s fixed point theorem for multivalued mappings. Moreover, we prove the existence of solutions for a class of integral equations by Darbo’s fixed point theorem and the existence of solutions for a class of differential inclusions using a generalization of Darbo’s fixed point theorem.     

Minuscule representations and Panyushev conjectures

Recently, Panyushev (2015) raised five conjectures concerning the structure of certain root posets arising from Z-gradings of simple Lie algebras. This paper aims to provide proofs for four of them. Our study also links these posets with Kostant-Macdonald identity, minuscule representations, Stembridge’s “t = ?1 phenomenon”, and the cyclic sieving phenomenon due to Reiner et al. (2004).     

On Weighted Kernels of Two Posets

A recent result of Aharoni Berger and Gorelik (Order 31(1), 35–43, 2014) is a weighted generalization of the well-known theorem of Sands Sauer and Woodrow (Theory Ser. B 33(3), 271–275, 1982) on monochromatic paths. The authors prove the existence of a so called weighted kernel for any pair of weighted posets on the same ground set. In this work, we point out that this result is closely related to the stable marriage theorem of Gale and Shapley (Amer. Math. Monthly 69(1), 9–15, 1962), and we generalize Blair’s theorem by showing that weighted kernels form a lattice under a certain natural order. To illustrate the applicability of our approach, we prove further weighted generalizations of the Sands Sauer Woodrow result.     

Characterizations of umbilic hypersurfaces in warped product manifolds

We consider the closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using an integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space.     

Hamiltonian Cycles and Symmetric Chains in Boolean Lattices

Let B(n) be the subset lattice of \({\{1,2,\dots, n\}.}\) Sperner’s theorem states that the width of B(n) is equal to the size of its biggest level. There have been several elegant proofs of this result, including an approach that shows that B(n) has a symmetric chain partition. Another famous result concerning B(n) is that its cover graph is hamiltonian. Motivated by these ideas and by the Middle Two Levels conjecture, we consider posets that have the Hamiltonian Cycle–Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. We show that the subset lattices have the HC-SCP property, and we obtain this result as a special case of a more general treatment.     

Schur’s theorem for a block Hadamard product

We define a generalized Kronecker product for block matrices, mention some of its properties, and apply it to the study of a block Hadamard product of positive semidefinite matrices, which was defined by Horn, Mathias, and Nakamura. Under strong commutation assumptions we obtain generalizations of Schur’s theorem and of Oppenheim’s inequality.     

Prime ideals in 0-distributive posets

In the first section of this paper, we prove an analogue of Stone’s Theorem for posets satisfying DCC by using semiprime ideals. We also prove the existence of prime ideals in atomic posets in which atoms are dually distributive. Further, it is proved that every maximal non-dense (non-principal) ideal of a 0-distributive poset (meet-semilattice) is prime. The second section focuses on the characterizations of (minimal) prime ideals in pseudocomplemented posets. The third section deals with the generalization of the classical theorem of Nachbin. In fact, we prove that a dually atomic pseudocomplemented, 1-distributive poset is complemented if and only if the poset of prime ideals is unordered. In the last section, we have characterized 0-distributive posets by means of prime ideals and minimal prime ideals.     

A generalization of a basic inequality of Telyakovski?

Two classes of numerical sequences are defined by means certain properties of the differences of terms of positive sequences, and their relationship to some newly defined classes and the well-known Sidon-Telyakovski? class is analyzed. It is also verified that if a sequence belongs to the newly defined wider class, then with this sequence in place of the sequence {1/k} an essential inequality established by Telyakovski? can be generalized notably. This new result and the previous generalizations of the original Telyakovski???s theorem are incomparable.     

Banaschewski’s theorem for S-posets: regular injectivity and completeness

In this paper we study the notion of injectivity in the category Pos-S of S-posets for a pomonoid S. First we see that, although there is no non-trivial injective S-poset with respect to monomorphisms, Pos-S has enough (regular) injectives with respect to regular monomorphisms (sub S-posets). Then, recalling Banaschewski’s theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for S-posets and get some homological classification of pomonoids and pogroups. Among other things, we also see that regular injective S-posets are exactly the retracts of cofree S-posets over complete posets.     

Ideal version of Ramsey’s theorem

We consider various forms of Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey’s theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems.     

Existence results for a class of hemivariational inequality problems on Hadamard manifolds

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.