Trivial automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?₁ and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.     

On strong chains of uncountable functions

For functionsf,g:ω₁ → ω₁, where ω₁ is the first uncountable cardinal, we write thatf?g if and only if {ξ ∈ ω₁ :f(ξ)≥g(ξ)} is finite. We prove the consistency of the existence of a well-ordered increasing ?-chain of length ω₁₂, solving a problem of A. Hajnal. The methods previously developed by us involveforcing with side conditions in morasses which is a variation on Todorcevic'sforcing with models as side conditions. The paper is self-contained and requires from the reader knowledge of Kunen's textbook and some basic experience with proper forcing and elementary submodels.     

A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ ⁺). Then with further forcing we show that it is consistent that GCH fails at ? _ω , ? _ω strong limit, while there is a lightface definable wellorder of H(? _ω+1) (“definable failure” of the singular cardinal hypothesis at ? _ω ). The large cardinal hypothesis used is the existence of a κ ⁺⁺-strong cardinal, where κ is κ ⁺⁺-strong if there is an embedding j: V → M with critical point κ such that H(κ ⁺⁺) ? M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ ⁺⁺-strong cardinal is used throughout.     

Quotient polytopes of cyclic polytopes part II: Stability of thef-vector and of thek-skeleton

This paper is the continuation of an earlier paper on quotient polytopesC(v, 2m)/F of cyclic polytopes and the associated quotient complexesC(V, 2m)/J. Here, we study mainly what changes in the faceJ do not affect thef-vector of the quotientC(V, 2m)/J. In the last section we examine the corresponding question fork-skeleta, i.e., what changes inJ do not affect the isomorphism type of skel _k C(V, 2m)/J.     

Projecting precipitousness

This paper is about “strong” ideals on small cardinals. It is shown that a typical property of large cardinal measures does not transfer to these ideals. More specifically, that precipitous ideals onP _ω1λ spaces may not project down to precipitous ideals on “smaller”P _ω1λ′ spaces. Also, that the existence of a presaturated ideal on the bigger space does not imply the existence of a presaturated ideal on the smaller space.     

On almost precipitous ideals

With less than 0^# two generic extensions ofL are identified: one in which ${\aleph_1}With less than 0^# two generic extensions ofL are identified: one in which à₁{\aleph_1}, and the other à₂{\aleph_2}, is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in Gitik M, Magidor M (On partialy wellfounded generic ultrapowers, in Pillars of Computer Science, 2010), and answers some questions raised there. Also, main results of Gitik (On normal precipitous ideals, 2010), are generalized—assumptions on precipitousness are replaced by those on ∞-semi precipitousness. As an application it is shown that if δ is a Woodin cardinal and there is an f:w₁ ? w₁{f:\omega_1 \to \omega_1} with ||f||=w₂{\|f\|=\omega_2}, then after Col(à₂,d){Col(\aleph_2,\delta)} there is a normal precipitous ideal over à₁{\aleph_1}. The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal.     

Destroying stationary sets

We present a forcing poset for destroying the stationarity of certain subsets ofP _kk⁺. Using this poset along with Prikry forcing techniques we establish some consistency results concerning saturated ideals andS(k, k ⁺). This paper forms a part of the author’s Ph.D. dissertation written under the supervision of Professor Cummings at Carnegie Mellon University.     

The Strengths of Some Violations of Covering

We consider two models V₁, V₂ of ZFC such that V₁ ⊆ V₂, the cofinality functions of V₁ and of V₂ coincide, V₁ and V₂ have that same hereditarily countable sets, and there is some uncountable set in V₂ that is not covered by any set in V₁ of the same cardinality. We show that under these assumptions there is an inner model of V₂ with a measurable cardinal κ of Mitchell order κ⁺⁺. This technical result allows us to show that changing cardinal characteristics without changing cofinalities or ω‐sequences (which was done for some characteristics in [13]) has consistency strength at least Mitchell order κ⁺⁺. From this we get that the changing of cardinal characteristics without changing cardinals or ω‐sequences has consistency strength Mitchell order ω₁, even in the case of characteristics that do not stem from a transitive relation. Hence the known forcing constructions for such a change have lowest possible consistency strength. We consider some stronger violations of covering which have appeared as intermediate steps in forcing constructions.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

Topologies on products of partially ordered sets I: Interval topologies

Given a certain construction principle assigning to each partially ordered setP some topology θ(P) onP, one may ask under what circumstances the topology θ(P) of a productP = ?_j∈J P _j of partially ordered setsP _i agrees with the product topology ?_j∈Jθ(P _i) onP. We shall discuss this question for several types ofinterval topologies (Part I), forideal topologies (Part II), and fororder topologies (Part III). Some of the results contained in this first part are listed below:

1. Let θ_i(P) denote thesegment topology. For any family of posetsP _j ?_j∈Jθ_s(P_j)=θ_s(?_j∈JP_i) iff at most a finite number of theP _j has more than one element (1.1).
2. Let θ_cs(P) denote theco-segment topology (lower topology). For any family of lower directed posetsP _j ?_j∈Jθ_cs(P_i)=θ_cs(?_j∈JP_i) iff eachP _j has a least element (1.5).
3. Let θ_i(P) denote theinterval topology. For a finite family of chainsP _j,P _j ?_j∈Jθ_i(P_i)=θ_i(?_j∈JP_i) iff for allj∈k, P _j has a greatest element orP _k has a least element (2.11).
4. Let θ_ni(P) denote thenew interval topology. For any family of posetsP _j,P _j ?_j∈Jθ_ni(P_j)=θ_ni(?_j∈JP_j) whenever the product space is ab-space (i.e. a space where the closure of any subsetY is the union of all closures of bounded subsets ofY) (3.13).

In the case oflattices, some of the results presented in this paper are well-known and have been shown earlier in the literature. However, the case of arbitraryposets often proved to be more difficult.     

The near-ring of congruence-preserving functions on an expanded group

Let V be a finite expanded group, e.g., a ring or a group. We investigate the near-ring 〈C₀(V);+,°〉 of zero-preserving congruence-preserving functions on V. We obtain some information on the structure of 〈C₀(V);+,°〉 from the lattice of ideals of V: for example, the number of maximal ideals of 〈C₀(V);+,°〉 is completely determined by the isomorphism class of the ideal lattice of V.     

Ideals in tensor powers of the enveloping algebra u(sl 2)

Let U = U(_sl₂)^?n be the tensor power of n copies of the enveloping algebra U(sl ₂) over an arbitrary field K of characteristic zero. In this paper we list the prime ideals of U by generators and classify them by height. If Z is the center of U and J is a prime ideal of Z, there are exactly 2₅ prime ideals I of U with I ∩ Z = J, where 0 ≤ s = s(J) ≤ n is an integer. Indeed, with respect to inclusion, they form a lattice isornorphic to the lattice of subsets of a set. When J is a maximal ideal of Z, there are only finitely many two-sided ideals of U containing J, They are presented by generators and their lattice is described, In particular, for each such J there exists a unique maximal ideal of U containing J and a unique ideal of U minimal with respect to the property that it properly contains JU. Similar results are given in the case when U is the tensor product of infinitely many copies of U(sl ₂).     

An ℵ1-dense ideal on ℵ2

This paper establishes the consistency of a countably complete, uniform, ℵ₁-dense ideal on ℵ₂. As a corollary, it is consistent that there exists a uniform ultrafilterD on ω₂ such that |ω ₁ ^ω2 D|=ω₁. A general “transfer” result establishes the consistency of countably complete uniform ideal K on ω₂ such thatP(ω₂)/K≅P(ω₁)/ {countable sets}. Partially supported by an NSF grant.     

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J:Z→R be a lower semicontinuous function bounded from below and p?1. This paper is concerned with the perturbed optimization problem of finding z₀∈Z such that ‖x−z₀p^‖+J(z₀)=infz∈Z{‖x−zp^‖+J(z)}, which is denoted by minJ(x,Z). The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x∈X for which every minimizing sequence of the problem minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z₀, where Z₀ is the set of all points z∈Z such that z is a solution of the problem minJ(z,Z). If additionally p>1 and X is J-strictly convex with respect to Z, then the set of all x∈X for which the problem minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z₀.     

Equivariant embeddings of principal Z-bundles into complex vector bundles

Let π: E→X be a principal $\text{Z}$_n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and $\text{zε}\text{Z}{}_{\mathrm{n}}\text{?S}{}^1\text{?}\text{Z}$. We compute the primary obstruction $\text{cεH}{}^{\mathrm{2m}}\text{(X;}\text{Z}\text{)}$ to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle ε^m. We use these criteria for a discussion of the classification of the equivalence classes of principal $\text{Z}$-bundles that admit equivariant embeddings into ε^m. Finally, we offer an example of a principal $\text{Z}$-bundle that admit an ordinary but not an equivariant embedding into ε¹.     

Comparing Zagreb indices for connected graphs

It was conjectured that for each simple graph G=(V,E) with n=|V(G)| vertices and m=|E(G)| edges, it holds M₂(G)/m≥M₁(G)/n, where M₁ and M₂ are the first and second Zagreb indices. Hansen and Vuki?evi? proved that it is true for all chemical graphs and does not hold in general. Also the conjecture was proved for all trees, unicyclic graphs, and all bicyclic graphs except one class. In this paper, we show that for every positive integer k, there exists a connected graph such that m−n=k and the conjecture does not hold. Moreover, by introducing some transformations, we show that M₂/(m−1)>M₁/n for all bicyclic graphs and it does not hold for general graphs. Using these transformations we give new and shorter proofs of some known results.     

Porosity of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem infz∈Z{J(z)+‖x−z‖}, denoted by (x,J)-inf for x∈X. In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist z₀∈Z such that J(z₀)+‖x−z₀‖=infz∈Z{J(z)+‖x−z‖} is a σ-porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points x∈X?Z⁰ such that the problem (x,J)-inf fails to be approximately compact, is a σ-porous set in X?Z⁰, where Z⁰ denotes the set of all z∈Z such that z∈PZ(z). Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417-424] fail is provided.     

Generalized Prikry forcing and iteration of generic ultrapowers

It is known that there is a close relation between Prikry forcing and the iteration of ultrapowers: If U is a normal ultrafilter on a measurable cardinal κ and 〈M_n, j_m,n | m ≤ n ≤ ω〉 is the iteration of ultrapowers of V by U, then the sequence of critical points 〈j_0,n(κ) | n ∈ ω〉 is a Prikry generic sequence over M_ω. In this paper we generalize this for normal precipitous filters. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On leastp-th power methods in multiple regressions and location estimations

The solution of the matricial systemZ=HX+V is derived. The observations are given by the matrixZ(n×m) and arem-components vectors. The parameter matrix isH(n×g), whileX(g×m) is the unknown andV(n×m) the residual matrix. The solutionX is estimated in order to minimize anL _p -norm ofV defined with the help of anL _r -norm on theV rows. The minimization algorithm is a steepest descent method with a specially chosen step size to “avoid” the singularities. The general problem shrinks in simple regression or location problems whilem org is reduced to the unit. Possibly more important is the recursive solution of this same general problem; a solutionX _(n) being obtained, we require the solutionX _(n+1) of the same system plus one row. In this context the productHX can be seen as a filtered version of the originalZ. The inversion of a system ofg linear equations is required at each step of the recursive solution.