Let X be a non empty normed structure and let s1 be a sequence of X. The functor ( ∑ κ α=0(s1)(α))κ∈N yielding a sequence of X is defined as follows: (Def. 1) ( ∑ κ α=0(s1)(α))κ∈N(0) = s1(0) and for every natural number n holds ( ∑ κ α=0(s1)(α))κ∈N(n + 1) = ( ∑ κ α=0(s1)(α))κ∈N(n) + s1(n + 1). One can prove the following proposition (1) Let X be an add-associative right zeroed right complementa...

We study a one-parameter family of discrete dynamical systems called the κ-color firefly cellular automata, proposed by the author in [13] as discrete models for finite-state pulse-coupled inhibitory oscillators. At each discrete time t , each vertex in a graph has a state (color) in {0, · · · ,κ−1}, and a special color b(κ) = bκ−1 2 c is designated as the ‘blinking’ color. During each transiti...

During B cell development, the precursor B cell receptor (pre-BCR) checkpoint is thought to increase immunoglobulin κ light chain (Igκ) locus accessibility to the V(D)J recombinase. Accordingly, pre-B cells lacking the pre-BCR signaling molecules Btk or Slp65 showed reduced germline V(κ) transcription. To investigate whether pre-BCR signaling modulates V(κ) accessibility through enhancer-mediat...

The following pcf results are proved: 1. Assume that κ > א0 is a weakly compact cardinal. Let μ > 2κ be a singular cardinal of cofinality κ. Then for every regular λ < pp+Γ(κ)(μ) there is an increasing sequence ⟨λi | i < κ⟩ of regular cardinals converging to μ such that λ = tcf( ∏ i<κ λi, <Jbd κ ). 2. Let μ be a strong limit cardinal and θ a cardinal above μ. Suppose that at least one of them h...

The weakly compact reflection principle Reflwcpκq states that κ is a weakly compact cardinal and every weakly compact subset of κ has a weakly compact proper initial segment. The weakly compact reflection principle at κ implies that κ is an ω-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that κ is pω ` 1qweakly compact. Moreover, we...

It is shown that if κ > 2א0 and κ is less than the first inaccessible cardinal then every automorphism of P(κ)/[κ]<א0 is trivial outside of a set of cardinality 2א0 .

We show that for any infinite cardinal κ, every (κ+1)-strategically closed poset is κ+-strategically closed if and only if κ holds. This extends previous results of Velleman, et.al.

Lemma 1. For any > 0 and κ ∈ [, 1], we have V(κ) − V(κ −) ≤ Cα log 2 (1//). Proof. By monotonicity, it suffices to prove that result for κ = 1. Notice that for K ≥ 2α,

structures Jia-Liang Le, Zdeněk P. Bažant, and Martin Z. Bazant Department of Civil and Environmental Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA Department of Materials Science, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA Department of Chemical Engineering and Department of Mathematics, Massachusetts Institute of Techno...

Mediate Disparate Responses to β B κ and I α B κ B Inhibitory Proteins I κ The NF