On cardinal sequences of scattered spaces

It was proved by Dow and Simon that there are 2^ω₁ (as many as possible) pairwise nonhomeomorphic compact, T₂, scattered spaces of height ω₁ and width ω. In this paper, we prove that if is an ordinal withω₁ < ω₂ and θ = κ_ξ: ξ < is a sequence of cardinals such that either κ_ξ = ω or κ_ξ = ω₁ for every ξ < , then there are 2^ω₁ pairwise nonhomeomorphic compact, T₂, scattered spaces whose cardinal sequence is θ.     

On the weak Freese-Nation property of complete Boolean algebras

The following results are proved:

(a) In a model obtained by adding ₂ Cohen reals, there is always a c.c.c. complete Boolean algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. complete Boolean algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □_μ and cof([μ]^₀,)=μ⁺ hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of is equivalent to the weak Freese-Nation property of any of or for uncountable κ. (d) Modulo the consistency of (_ω+1,_ω)(₁,₀), it is consistent with GCH that does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding _ω Cohen reals destroys the weak Freese-Nation property of .

These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.     

Changing cofinalities and collapsing cardinals in models of set theory

If a˜cardinal κ₁, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ₀ and its new cofinality, ρ, is less than κ₀, then, under some additional assumptions, each cardinal λ>κ₁ less than cc(P(κ₁)/[κ₁]^<κ₁) is collapsed to κ₀ as well. If in addition N=M[f], where f : ρ→κ₁ is an unbounded mapping, then N is a˜|λ|=κ₀-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.     

Error bounds on the power method for determining the largest eigenvalue of a symmetric, positive definite matrix

Let A be a positive definite, symmetric matrix. We wish to determine the largest eigenvalue, λ₁. We consider the power method, i.e. that of choosing a vector v₀ and setting v_k = A^kv₀; then the Rayleigh quotients R_k = (Av_k, v_k)/(v_k, v_k) usually converge to λ₁ as k → ∞ (here (u, v) denotes their inner product). In this paper we give two methods for determining how close R_k is to λ₁. They are both based on a bound on λ₁ − R_k involving the difference of two consecutive Rayleigh quotients and a quantity ω_k. While we do not know how to directly calculate ω_k, we can given an algorithm for giving a good upper bound on it, at least with high probability. This leads to an upper bound for λ₁ − R_k which is proportional to (λ₂/λ₁)^2k, which holds with a prescribed probability (the prescribed probability being an arbitrary δ > 0, with the upper bound depending on δ).     

On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs.Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques.This connection provided a new proof of Turán classical result on the Turán density of complete graphs.Since then,Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs.Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range.They showed that if G is a 3-uniform graph with m edges containing a clique of order t-1,then λ(G)=λ([t-1]~((3))) provided (t-13)≤m≤(t-13)+_(t-22).They also conjectured:If G is an r-uniform graph with m edges not containing a clique of order t-1,then λ(G)λ([t-1]~((r))) provided (t-1r)≤ m ≤(t-1r)+(t-2r-1).It has been shown that to verify this conjecture for 3-uniform graphs,it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m=t-13+t-22.Regarding this conjecture,we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t-1]~((3))\E(G)|=p,then λ(G)λ([t-1]~((3))) provided m=(t-13)+(t-22) and t≥17p/2+11.     

On the minor-minimal 2-connected graphs having a fixed minor

Let H be a graph with κ₁ components and κ₂ blocks, and let G be a minor-minimal 2-connected graph having H as a minor. This paper proves that |E(G)|−|E(H)|(κ₁−1)+β(κ₂−1) for all (,β) such that +β5,2+5β20, and β3. Moreover, if one of the last three inequalities fails, then there are graphs G and H for which the first inequality fails.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron

It is known that the volume function for hyperbolic manifolds of dimension 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4π²/3. This is “half” the set of possible values for volumes, which is the integral multiples of 4π²/3 due to the Gauss-Bonnet formula Vol(M) = 4π²/3 · χ(M).     

An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS(v)”. Our main idea is to use transitive resolvable idempotent symmetric quasigroups instead of TKTS. As an application, we can combine the known results on LKTS and LR-designs to obtain the existence of an LKTS(3ⁿm(2·13^n₁+1)(2·13^n_t+1)) for n1, m{1,5,11,17,25,35,43,67,91,123}{2^2r+125^s+1 : r0,s0}, t0 and n_i1 (i=1,…,t).     

Partition numbers

We continue [21] and study partition numbers of partial orderings which are related to (ω)/fin. In particular, we investigate P_f, be the suborder of ((ω)/fin)^ω containing only filtered elements, the Mathias partial order M, and (ω), (ω)^ω the lattice of (infinite) partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for P_f. We show that the partition number of (ω) is C. We also show that consistently the distributivity number of (ω)^ω is smaller than the distributivity number of (ω)/fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size ₁ does not imply that a Polish space can be partitioned into ₁ many closed sets.     

On index of total boundedness of (strictly) o-bounded groups

Let G be a metrizable topological group. Denote by itb(G) the smallest cardinality of a cover of G by totally bounded subsets of G. A group G is defined to be σ-bounded if itb(G)₀. The group G is called o-bounded if for every sequence (U_n)_nω of neighborhoods of the identity in G there exists a sequence (F_n)_nω of finite subsets in G such that G=_nωF_n·U_n; G is called strictly o-bounded (respectively OF-determined) if the second player (respectively one of the players) has a winning strategy in the following game OF: two players, I and II, choose at every step n an open neighborhood U_n of the identity in G and a finite subset F_n of G, respectively. The player II wins if G=_nωF_n·U_n.

For a second countable group G the following results are proven. . If G is strictly o-bounded, then itb(G)₁ and G is σ-bounded or meager. If the space G is analytic, then the group is OF-determined and satisfies . G is σ-bounded if it is strictly o-bounded and one of the following conditions holds: (i) G is analytic; (ii) ; (iii) (MA+¬CH) holds; (iv) analytic games are determined; (v) there exists a measurable cardinal. Also we show that under (MA) every non-locally compact Polish Abelian divisible group contains a Baire o-bounded OF-undetermined subgroup.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.     

Uniqueness theorems for 3D inverse problems with incomplete data

In Ramm, Phys. Lett. 99A, (1983), 258-260, it is proved that a compactly supported inhomogeneity in the velocity profile is uniquely determined by the values of the acoustic pressure collected for all positions of the source and receiver on the surface of the earth ( on the whole plane P) at low frequencies. Here it is proved that the data collected on ω₁ x ω₂ suffice for the uniqueness theorem to hold, where ω₁ and ω₂ are arbitrary open sets on the plane P. This result holds also for the data collected on ω₁ × ω₂ at a fixed frequency.     

Remarks on navier-stokes equations with measures as data

We show the existence of a solution to the Navier-Stokes equation taking the vorticity ω as the unknown: ω_t + Aω + Bω = μ, ω(0) = ω₀. Here, ω₀ and μ are bounded Radon measures. This study is motivated by a numerical approximation which will be given in a forthcoming work [1].     

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model,

, where ₀ > 0, _i 0, I = 1, 2, …, p, {_t} is an i.i.d. normal white noise with E_t = 0, E_t² = 1, and _t is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ₁ + ₂ + ··· + _p < 1. In this note, we assume that _t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under E_t² = 1. When _t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.     

Multiplicity results on a nonlinear biharmonic equation

Let ω be a bounded open set in Rn with smooth boundary ω We are concerned with a fourth order semilinear elliptic boundary value problem Δ²u + cΔu = bu⁺ + s inω under Dirichlet boundary condition. We investigate the existence of solutions of the fourth order nonlinear equation (0.1) when the nonlinearity bu⁺ crosses eigenvalues of Δ² + cΔ under Dirichlet boundary condition.     

Possible values for 2

From GCH and P^{m(κ)-hypermeasurable} (1 <m<gw), we construct a model satisfying 2_ⁿ = _a(n) and 2_^ω = _ω+m for a monotone a:ω→ω satisfying a(n)>n.     

The deviation, density, and depth of partially ordered sets

Let P be a poset, and let γ be a linear order type with |γ| ≥ 3. The γ-deviation of P, denoted by γ-dev P, is defined inductively as follows: (1) γ-dev P=0, if P contains no chain of order type γ; (2) γ-dev P = , if γ-dev P and each chain C of type γ in P contains elements a and b such that a<b and [a, b] as an interval of P has γ-deviation <. There may be no ordinal such that γ-dev P = ; i.e., γ-dev P does not exist. A chain is γ-dense if each of its intervals contains a chain of order type γ. If P contains a γ-dense chain, then γ-dev P fails to exist. If either (1) P is linearly ordered or (2) a chain of order type γ does not contain a dense interval, then the converse holds. For an ordinal ξ, a special set S(ξ) is used to study ω_ξ-deviation. The depth of P, denoted by δ(P) is the least ordinal β that does not embed in P^*. Then the following statements are equivalent: (1) ω_ξ-dev P does not exist; (2) S(ξ) embeds in P; and (3) P has a subset Q of cardinality _ξ such that δ(Q^*) = ω_{ξ + 1}. Also ω_ξ-dev P = <ω_{ξ + 1} if and only if |δ(P^*)|_ξ; if these equivalent conditions hold, then ω^β_ξ < δ(P^*) ≤ ω ^{+ 1}_ξ for all β < . Applications are made to the study of chains of submodules of a module over an associative ring.     

New constructions of divisible designs

A construction is given for a (p^2a(p+1),p²,p^2a+1(p+1),p^2a+1,p^2a(p+1)) (p a prime) divisible difference set in the group H×Z²_pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ₁≠0, and those are fairly rare. We also give a construction for a (p^a−1+p^a−2+…+p+2,p^a+2, p^a(p^a+p^a−1+…+p+1), p^a(p^a−1+…+p+1), p^a−1(p^a+…+p²+2)) divisible difference set in the group H×Z_p2×Z^a_p. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ₁=λ₂, and this corresponds to the parameters for the ordinary Menon difference sets.