A model of Cummings and Foreman revisited

This paper concerns the model of Cummings and Foreman where from ω   supercompact cardinals they obtain the tree property at each _ℵn${\mathrm{\aleph }}_n$ for 2≤n<ω$2\le n<\omega$. We prove some structural facts about this model. We show that the combinatorics at _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ in this model depend strongly on the properties of _ω1${\omega }_1$ in the ground model. From different ground models for the Cummings–Foreman iteration we can obtain either _ℵω+1∈I[_ℵω+1]${\mathrm{\aleph }}_{\omega +1}\in I[{\mathrm{\aleph }}_{\omega +1}]$ and every stationary subset of _ℵω+1${\mathrm{\aleph }}_{\omega +1}$ reflects or there are a bad scale at _ℵω${\mathrm{\aleph }}_{\omega }$ and a non-reflecting stationary subset of _ℵω+1∩cof(_ω1)${\mathrm{\aleph }}_{\omega +1}\cap \mathrm{cof}({\omega }_1)$. We also prove that regardless of the ground model a strong generalization of the tree property holds at each _ℵn${\mathrm{\aleph }}_n$ for n≥2$n\ge 2$.     

Eigenvalue asymptotics of a modified Jaynes–Cummings model with periodic modulations

We analyze the influence of additive and multiplicative periodic modulations on the asymptotic behavior of eigenvalues of some Hermitian Jacobi Matrices related to the Jaynes–Cummings model using the so-called “successive diagonalization” method. This approach allows us to find the asymptotics of the nth eigenvalue λ_n as n→∞ stepwise with successively increasing precision. We bring to light the interplay of additive and multiplicative periodic modulations and their influence on the asymptotic behavior of eigenvalues. To cite this article: A. Boutet de Monvel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

具有相位阻尼的Milburn主方程控制的双光子JC模型

该文利用超算子技术求出了相位阻尼下非共振双光子JC模型主方程的解析解，研究了其相位阻尼对光子数分布振荡，原子数反转与恢复和亚泊松光子分布等非经典效应的影响。研究表明：相位阻尼能抑制原子反转与恢复和腔场的非经典效应。     

There Exists a Linear Problem with Infinite Combinatory Complexity

We present a linear problem whose information complexity is finite but whose combinatory complexity is infinite. Thus, this linear problem has infinite complexity due to infinite combinatory complexity and not due to information complexity. This holds in a real number model in which we only allow arithmetic operations, comparisons of real numbers as well as precomputation of finitely many arbitrary elements. The result is not true if we can also evaluate logarithms, exponentials, and ceilings.     

There exists a polyhedron with infinitely many left neighbors

We show that there exists a finite polyhedron homotopy dominating infinitely many finite polyhedra of different homotopy types such that there isn't any homotopy type between and . This answers negatively the question raised by K. Borsuk in 1975: Does every FANR have only finitely many left neighbors?

     

There are no graded domains with GK dimension strictly between 2 and 3

Let K be a field, and let R=⊕_n∈NR_n be a finitely generated, graded K-algebra which is a domain. It is shown that R cannot have Gelfand-Kirillov dimension strictly between 2 and 3. Mathematics Subject Classification (2000) 16D90, 16P40, 16S80, 16W50     

There is no plus-capping degree

Summary Answering a question of Per Lindström, we show that there is no plus-capping degree, i.e. that for any incomplete r.e. degreew, there is an incomplete r.e. degreea>w such that there is no r.e. degreev>w withav=w.Mathematics subject classification: 03D30The authors would like to thank Per Lindström for raising the question and Carl Jockusch for communicating it to them. The first author was partially supported by a U.S./New Zealand binational grant. The second author was partially supported by NSF grant DMS-9100114 and the U.S./New Zealand binational NSF grant INT-9020558     

There is no fat orbit

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.     

There exist graphs with super‐exponential Ramsey multiplicity constant

The Ramsey multiplicity M(G;n) of a graph G is the minimum number of monochromatic copies of G over all 2‐colorings of the edges of the complete graph K_n. For a graph G with a automorphisms, ν vertices, and E edges, it is natural to define the Ramsey multiplicity constant C(G) to be , which is the limit of the fraction of the total number of copies of G which must be monochromatic in a 2‐coloring of the edges of K_n. In 1980, Burr and Rosta showed that 0 ≥ C(G) ≤ 2^1?E for all graphs G, and conjectured that this upper bound is tight. Counterexamples of Burr and Rosta's conjecture were first found by Sidorenko and Thomason independently. Later, Clark proved that there are graphs G with E edges and 2^E?1C(G) arbitrarily small. We prove that for each positive integer E, there is a graph G with E edges and C(G) ≤ E^{?E/2 + o(E)}. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 89–98, 2008     

There are just four second-order quantifiers

Among the second-order quantifiers ranging over relations satisfying a first-order sentence, there are four for which any other one is bi-interpretable with one of them: the trivial, monadic, permutational, and full second order.     

There is no degree invariant half-jump

We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if is any definable operator on degrees such that on a cone then is low or high on a cone of degrees, i.e., there is a degree such that for every or for every .

     

There are many Kunen compact L-spaces

We prove that under CH there are non-homeomorphic Kunen compact L-spaces. Moreover there exist models of ZFC that have many non-homeomorphic Kunen spaces.

     

There are infinitely many Perrin pseudoprimes

This paper proves the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. The result uses ingredients of the proof of the infinitude of Carmichael numbers, along with zero-density estimates for Hecke L-functions.     

There is no bilinear Bishop-Phelps theorem

We answer a question posed by R. Aron, C. Finet and E. Werner, on the bilinear version of the Bishop-Phelps theorem, by exhibiting an example of a Banach spaceX such that the set of norm-attaining bilinear forms onX×X is not dense in the space of all continuous bilinear forms. Research partially supported by D.G.I.C.Y.T., project no. PB93-1142.     

There are Infinitely Many Williams Squares with Circular Structure and Order a Multiple of 4

Williams squares, also known as row quasi-complete Latin squares, that have circular structure are used in repeated measurements designs where error terms are correlated. They are known to exist for all orders that are not multiples of 4 and also order 8; there is no such square of order 4. We use a variation of the generating array method to construct a Williams square of order 12 with circular structure. We also give a product theorem that produces Williams squares of orders 8r and 12s, where the smallest prime divisor of r is at least 11 and the smallest prime divisor of s is at least 13, that have circular structure.     

准正交变换的三种等价定义及其判定

从正交变换的定义形式以及几何意义上推广了正交变换的定义,并证明了三种推广定义的等价性,将推广后的线性变换称为准正交变换,同时指出了参考文献[5]中推广正交变换时存在的一个重大问题.探讨了准正交变换的一系列性质和判定法.