2,322 research outputs found
The cohesive principle and the Bolzano-Weierstra{\ss} principle
The aim of this paper is to determine the logical and computational strength
of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of
it.
We show that BW is instance-wise equivalent to the weak K\"onig's lemma for
-trees (-WKL). This means that from every bounded
sequence of reals one can compute an infinite -0/1-tree, such that
each infinite branch of it yields an accumulation point and vice versa.
Especially, this shows that the degrees d >> 0' are exactly those containing an
accumulation point for all bounded computable sequences.
Let BW_weak be the principle stating that every bounded sequence of real
numbers contains a Cauchy subsequence (a sequence converging but not
necessarily fast). We show that BW_weak is instance-wise equivalent to the
(strong) cohesive principle (StCOH) and - using this - obtain a classification
of the computational and logical strength of BW_weak. Especially we show that
BW_weak does not solve the halting problem and does not lead to more than
primitive recursive growth. Therefore it is strictly weaker than BW. We also
discuss possible uses of BW_weak.Comment: corrected typos, slightly improved presentatio
Generalized cohesiveness
We study some generalized notions of cohesiveness which arise naturally in
connection with effective versions of Ramsey's Theorem. An infinite set of
natural numbers is --cohesive (respectively, --r--cohesive) if is
almost homogeneous for every computably enumerable (respectively, computable)
--coloring of the --element sets of natural numbers. (Thus the
--cohesive and --r--cohesive sets coincide with the cohesive and
r--cohesive sets, respectively.) We consider the degrees of unsolvability and
arithmetical definability levels of --cohesive and --r--cohesive sets.
For example, we show that for all , there exists a
--cohesive set. We improve this result for by showing that there is
a --cohesive set. We show that the --cohesive and
--r--cohesive degrees together form a linear, non--collapsing hierarchy of
degrees for . In addition, for we characterize the jumps
of --cohesive degrees as exactly the degrees {\bf \geq \jump{0}{(n+1)}}
and show that each --r--cohesive degree has jump {\bf > \jump{0}{(n)}}
Iterative forcing and hyperimmunity in reverse mathematics
The separation between two theorems in reverse mathematics is usually done by
constructing a Turing ideal satisfying a theorem P and avoiding the solutions
to a fixed instance of a theorem Q. Lerman, Solomon and Towsner introduced a
forcing technique for iterating a computable non-reducibility in order to
separate theorems over omega-models. In this paper, we present a modularized
version of their framework in terms of preservation of hyperimmunity and show
that it is powerful enough to obtain the same separations results as Wang did
with his notion of preservation of definitions.Comment: 15 page
-Generic Computability, Turing Reducibility and Asymptotic Density
Generic computability has been studied in group theory and we now study it in
the context of classical computability theory. A set A of natural numbers is
generically computable if there is a partial computable function f whose domain
has density 1 and which agrees with the characteristic function of A on its
domain. A set A is coarsely computable if there is a computable set C such that
the symmetric difference of A and C has density 0. We prove that there is a
c.e. set which is generically computable but not coarsely computable and vice
versa. We show that every nonzero Turing degree contains a set which is not
coarsely computable. We prove that there is a c.e. set of density 1 which has
no computable subset of density 1. As a corollary, there is a generically
computable set A such that no generic algorithm for A has computable domain. We
define a general notion of generic reducibility in the spirt of Turing
reducibility and show that there is a natural order-preserving embedding of the
Turing degrees into the generic degrees which is not surjective
Freely forming groups: trying to be rare
A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components
EVIDENCE OF COMMUNAL OVIPOSITION AND NEST ABANDONMENT IN THE NORTHERN TWO-LINED SALAMANDER (EURYCEA BISLINEATA, (GREEN, 1818)) IN NORTHEASTERN CONNECTICUT
Most plethodontid salamanders oviposit their eggs in an individual nest and attend the clutch until hatching. Here, we describe aspects of the reproduction of Eurycea bislineata (Northern Two-lined Salamander) from three field sites in northeastern Connecticut that contrast with the typical plethodontid reproductive behavior. Rocks used as oviposition sites contained up to 296 eggs, with an average of more than 100. These numbers exceed the maximum ovarian egg counts for this species, indicating that communal oviposition is common. The lack of correlation between rock size and number of eggs, as well as the lack of discrete clutches when eggs are laid in large clusters, suggests that communal oviposition may be caused by something other than nest site limitation. Additionally, the rate of maternal attendance at nests was low. Thus, communal oviposition with high rates of nest abandonment is the dominant reproductive strategy in E. bislineata at these sites
Asymptotic density and the Ershov hierarchy
We classify the asymptotic densities of the sets according to
their level in the Ershov hierarchy. In particular, it is shown that for , a real is the density of an -c.e.\ set if and only if
it is a difference of left- reals. Further, we show that the densities
of the -c.e.\ sets coincide with the densities of the
sets, and there are -c.e.\ sets whose density is not the density of an
-c.e. set for any .Comment: To appear in Mathematical Logic Quarterl
Asymptotic density and the coarse computability bound
For we say that a set is \emph{coarsely
computable at density} if there is a computable set such that has lower density at least . Let . We study the interactions of
these concepts with Turing reducibility. For example, we show that if there are sets such that
where is coarsely computable at density while is not coarsely
computable at density . We show that a real is equal to
for some c.e.\ set if and only if is left-. A
surprising result is that if is a -generic set, and with , then is coarsely computable at density
Use of a novel collagen matrix with oriented pore structure for muscle cell differentiation in cell culture and in grafts
Tissue engineering of skeletal muscle from cultured cells has been attempted using a variety of synthetic and natural macromolecular scaffolds. Our study describes the application of artificial scaffolds (collagen sponges, CS) consisting of collagen-I with parallel pores (width 20–50 μm) using the permanent myogenic cell line C2C12. CS were infiltrated with a high-density cell suspension, incubated in medium for proliferation of myoblasts prior to further culture in fusion medium to induce differentiation and formation of multinucleated myotubes. This resulted in a parallel arrangement of myotubes within the pore structures. CS with either proliferating cells or with myotubes were grafted into the beds of excised anterior tibial muscles of immunodeficient host mice. The recipient mice were transgenic for enhanced green fluorescent protein (eGFP) to determine a host contribution to the regenerated muscle tissue. Histological analysis 14–50 days after surgery showed that donor muscle fibres had formed in situ with host contributions in the outer portions of the regenerates. The function of the regenerates was assessed by direct electrical stimulation which resulted in the generation of mechanical force. Our study demonstrated that biodegradable CS with parallel pores support the formation of oriented muscle fibres and are compatible with force generation in regenerated muscle
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