pinecone/core-2020-05-10-deduplication huggingface dataset:
On Some Sets of Dictionaries Whose omega-Powers Have a Given Complexity

To appear in Mathematical Logic QuarterlyInternational audienceA dictionary is a set of finite words over some finite alphabet X. The omega-power of a dictionary V is the set of infinite words obtained by infinite concatenation of words in V. Lecomte studied in [Omega-powers and descriptive set theory, JSL 2005] the complexity of the set of dictionaries whose associated omega-powers have a given complexity. In particular, he considered the sets $W({\bf\Si}^0_{k})$ (respectively, $W({\bf\Pi}^0_{k})$, $W({\bf\Delta}_1^1)$) of dictionaries $V \subseteq 2^\star$ whose omega-powers are ${\bf\Si}^0_{k}$-sets (respectively, ${\bf\Pi}^0_{k}$-sets, Borel sets). In this paper we first establish a new relation between the sets $W({\bf\Sigma}^0_{2})$ and $W({\bf\Delta}_1^1)$, showing that the set $W({\bf\Delta}_1^1)$ is ``more complex" than the set $W({\bf\Sigma}^0_{2})$. As an application we improve the lower bound on the complexity of $W({\bf\Delta}_1^1)$ given by Lecomte. Then we prove that, for every integer $k\geq 2$, (respectively, $k\geq 3$) the set of dictionaries $W({\bf\Pi}^0_{k+1})$ (respectively, $W({\bf\Si}^0_{k+1})$) is ``more complex" than the set of dictionaries $W({\bf\Pi}^0_{k})$ (respectively, $W({\bf\Si}^0_{k})$)
Bounds of some real (complex) solution of a finite system of polynomial
  equations with rational coefficients

We discuss two conjectures.
  (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such
that
  \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}}
  \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1)
  \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)
  \forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k)
  (II) Let G be an additive subgroup of C. Then for each x_1,...,x_n \in G
there exist y_1,...,y_n \in G \cap Q such that
  \forall i \in {1,...,n} |y_i| \leq 2^{n-1}
  \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1)
  \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)Comment: LaTeX2e, 28 pages, a shortened and revised version will appear in
  Mathematical Logic Quarterly 56 (2010), no.2, under the title ``Two
  conjectures on the arithmetic in R and C'
Uniform bounds on growth in o-minimal structures

We prove that a function definable with parameters in an o-minimal structure
is bounded away from infinity as its argument goes to infinity by a function
definable without parameters, and that this new function can be chosen
independently of the parameters in the original function. This generalizes a
result in a paper of Friedman and Miller. Moreover, this remains true if the
argument is taken to approach any element of the structure (or plus/minus
infinity), and the function has limit any element of the structure (or
plus/minus infinity).Comment: 3 pages. To appear in Mathematical Logic Quarterl
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
$\Sigma^0_1$-trees ($\Sigma^0_1$-WKL). This means that from every bounded
sequence of reals one can compute an infinite $\Sigma^0_1$-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
The Bergman-Shelah Preorder on Transformation Semigroups

This is the peer-reviewed version of the following article: Mesyan, Z., Mitchell, J. D., Morayne, M. and P??resse, Y. H. (2012), Mathematical Logic Quarterly, Vol. 58: 424???433, 'The Bergman-Shelah preorder on transformation semigroups', which has been published in final form at doi:10.1002/malq.201200002. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Copyright ?? 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. http://www.interscience.wiley.com/Let $\nat^\nat$ be the semigroup of all mappings on the natural numbers $\nat$, and let $U$ and $V$ be subsets of $\nat^\nat$. We write $U\preccurlyeq V$ if there exists a countable subset $C$ of $\nat^\nat$ such that $U$ is contained in the subsemigroup generated by $V$ and $C$. We give several results about the structure of the preorder $\preccurlyeq$. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder $\preccurlyeq$ is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on $\nat$. The results in this paper suggest that the preorder on subsemigroups of $\nat^\nat$ is much more complicated than that on subgroups of the symmetric group
Inconsistency lemmas in algebraic logic

In this paper, the inconsistency lemmas of intuitionistic and classical propositional logic are formulated abstractly.
We prove that, when a (finitary) deductive system   is algebraized by a variety K, then   has an inconsistency
lemma—in the abstract sense—iff every algebra in K has a dually pseudo-complemented join semilattice of
compact congruences. In this case, the following are shown to be equivalent: (1)   has a classical inconsistency
lemma; (2)   has a greatest compact theory and K is filtral, i.e., semisimple with EDPC; (3) the compact
congruences of any algebra in K form a Boolean lattice; (4) the compact congruences of any A ∈ K constitute
a Boolean sublattice of the full congruence lattice of A. These results extend to quasivarieties and relative
congruences. Except for (2), they extend even to protoalgebraic logics, with deductive filters in the role of
congruences. A protoalgebraic system with a classical inconsistency lemma always has a deduction-detachment
theorem (DDT), while a system with a DDT and a greatest compact theory has an inconsistency lemma. The
converses are false.http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1521-3870hb201
Linear extensions of partial orders and Reverse Mathematics

We introduce the notion of \tau-like partial order, where \tau is one of the
linear order types \omega, \omega*, \omega+\omega*, and \zeta. For example,
being \omega-like means that every element has finitely many predecessors,
while being \zeta-like means that every interval is finite. We consider
statements of the form "any \tau-like partial order has a \tau-like linear
extension" and "any \tau-like partial order is embeddable into \tau" (when
\tau\ is \zeta\ this result appears to be new). Working in the framework of
reverse mathematics, we show that these statements are equivalent either to
B\Sigma^0_2 or to ACA_0 over the usual base system RCA_0.Comment: 8 pages, minor changes suggested by referee. To appear in MLQ -
  Mathematical Logic Quarterl
From Bolzano-Weierstra{\ss} to Arzel\`a-Ascoli

We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using
suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we
can apply the results from \cite{aK} and obtain a classification of the
strength of instances of the Arzel\`a-Ascoli theorem and a variant of it.
  Let AA be the statement that each equicontinuous sequence of functions f_n:
[0,1] --> [0,1] contains a subsequence that converges uniformly with the rate
2^-k and let AA_weak be the statement that each such sequence contains a
subsequence which converges uniformly but possibly without any rate.
  We show that AA is instance-wise equivalent over RCA_0 to the
Bolzano-Weierstra{\ss} principle BW and that AA_weak is instance-wise
equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle
StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL
and StCOH + WKL are equivalent
Cardinal characteristics and countable Borel equivalence relations

Boykin and Jackson recently introduced a property of countable Borel
equivalence relations called Borel boundedness, which they showed is closely
related to the union problem for hyperfinite equivalence relations. In this
paper, we introduce a family of properties of countable Borel equivalence
relations which correspond to combinatorial cardinal characteristics of the
continuum in the same way that Borel boundedness corresponds to the bounding
number $\mathfrak b$. We analyze some of the basic behavior of these
properties, showing for instance that the property corresponding to the
splitting number $\mathfrak s$ coincides with smoothness. We then settle many
of the implication relationships between the properties; these relationships
turn out to be closely related to (but not the same as) the Borel Tukey
ordering on cardinal characteristics
Locally Finite ω-Languages and Effective Analytic Sets Have the Same Topological Complexity

International audienceLocal sentences and the formal languages they define were introduced by Ressayre in [Res88]. We prove that locally finite ω-languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω-languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non-null recursive ordinal α < ω_1^CK  there exist some Σ¨0_α-complete and some Π ^0_α-complete locally finite ω-languages, and the supremum of the set of Borel ranks of locally finite ω-languages is the ordinal γ^1_2 , which is strictly greater than the first non-recursive ordinal ω_1^ CK. This gives an answer to the question of the topological complexity of locally finite ω-languages, which was asked by Simonnet [Sim92] and also by Duparc, Finkel, and Ressayre in [DFR01]. Moreover we show that the topological complexity of a locally finite ω-language defined by a local sentence ϕ may depend on the models of the Zermelo-Fraenkel axiomatic system ZFC. Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω-languages are Π^1_2-complete, hence highly undecidable
A L\'opez-Escobar theorem for metric structures, and the topological
  Vaught conjecture

We show that a version of L\'opez-Escobar's theorem holds in the setting of
logic for metric structures. More precisely, let $\mathbb{U}$ denote the
Urysohn sphere and let $\mathrm{Mod}(\mathcal{L},\mathbb{U})$ be the space of
metric $\mathcal{L}$-structures supported on $\mathbb{U}$. Then for any
$\mathrm{Iso}(\mathbb{U})$-invariant Borel function $f\colon
\mathrm{Mod}(\mathcal{L}, \mathbb{U})\rightarrow \lbrack 0,1]$, there exists a
sentence $\phi $ of $\mathcal{L}_{\omega_{1}\omega}$ such that for all $M\in
\mathrm{Mod}(\mathcal{L},\mathbb{U})$ we have $f(M)=\phi ^{M}$. At the same
time we introduce a variant $\mathcal{L}_{\omega_1\omega}^\ast$ of
$\mathcal{L}_{\omega_1\omega}$ in which the usual quantifiers are replaced with
category quantifiers, and establish the analogous theorem for
$\mathcal{L}_{\omega_1\omega}^\ast$. This answers a question of Ivanov and
Majcher-Iwanow. We prove several consequences, for example every orbit
equivalence relation of a Polish group action is Borel isomorphic to the
isomorphism relation on the set of models of a given
$\mathcal{L}_{\omega_{1}\omega}$-sentence that are supported on the Urysohn
sphere. This in turn provides a model-theoretic reformulation of the
topological Vaught conjecture.Comment: 17 page
An infinite natural sum

As far as algebraic properties are concerned, the usual addition on the class
of ordinal numbers is not really well behaved; for example, it is not
commutative, nor left cancellative etc. In a few cases, the natural Hessemberg
sum is a better alternative, since it shares most of the usual properties of
the addition on the naturals.
  A countably infinite version of the natural sum has been used in a recent
paper by V\"a\"an\"anen and Wang, with applications to infinitary logics. We
provide an order theoretical characterization of this operation. We show that
this countable natural sum differs from the more usual infinite ordinal sum
only for an initial finite "head" and agrees on the remaining infinite "tail".
We show how to evaluate the countable natural sum just by computing a finite
natural sum. Various kinds of infinite mixed sums of ordinals are discussed.Comment: v3 added a remark connected with surreal number
Shelah's eventual categoricity conjecture in tame AECs with primes

A new case of Shelah's eventual categoricity conjecture is established:
  $\mathbf{Theorem}$
  Let $K$ be an AEC with amalgamation. Write $H_2 :=
\beth_{\left(2^{\beth_{\left(2^{\text{LS} (K)}\right)^+}}\right)^+}$. Assume
that $K$ is $H_2$-tame and $K_{\ge H_2}$ has primes over sets of the form $M
\cup \{a\}$. If $K$ is categorical in some $\lambda > H_2$, then $K$ is
categorical in all $\lambda' \ge H_2$.
  The result had previously been established when the stronger locality
assumptions of full tameness and shortness are also required.
  An application of the method of proof of the theorem is that Shelah's
categoricity conjecture holds in the context of homogeneous model theory (this
was known, but our proof gives new cases):
  $\mathbf{Theorem}$
  Let $D$ be a homogeneous diagram in a first-order theory $T$. If $D$ is
categorical in a $\lambda > |T|$, then $D$ is categorical in all $\lambda' \ge
\min (\lambda, \beth_{(2^{|T|})^+})$.Comment: 16 pages. Generalizes arXiv:1506.0702
Fermat's Last Theorem and Catalan's Conjecture in Weak Exponential
  Arithmetics

We study Fermat's Last Theorem and Catalan's conjecture in the context of
weak arithmetics with exponentiation. We deal with expansions (B,e) of models
of arithmetical theories (in the language L=(0,1,+,x,<)) by a binary (partial
or total) function e intended as an exponential. We provide a general
construction of such expansions and prove that it is universal for the class of
all exponentials e which satisfy a certain natural set of axioms Exp. We
construct a model (B,e) of Th(N) + Exp and a substructure (A,e) with e total
and A model of Pr (Presburger arithmetic) such that in both (B,e) and (A,e)
Fermat's Last Theorem for e is violated by cofinally many exponents n and (in
all coordinates) cofinally many pairwise linearly independent triples a,b,c. On
the other hand, under the assumption of ABC conjecture (in the standard model),
we show that Catalan's conjecture for e is provable in Th(N)+Exp (even in a
weaker theory) and thus holds in (B,e) and (A,e). Finally, we also show that
Fermat's Last Theorem for e is provable (again, under the assumption of ABC in
N) in Th(N)+Exp+"coprimality for e"
Cluster expansion and the boxdot conjecture

The boxdot conjecture asserts that every normal modal logic that faithfully
interprets T by the well-known boxdot translation is in fact included in T. We
confirm that the conjecture is true. More generally, we present a simple
semantic condition on modal logics $L_0$ which ensures that the largest logic
where $L_0$ embeds faithfully by the boxdot translation is $L_0$ itself. In
particular, this natural generalization of the boxdot conjecture holds for S4,
S5, and KTB in place of T.Comment: 9 page
Some transfinite natural sums

We study a transfinite iteration of the ordinal Hessenberg natural sum
obtained by taking suprema at limit stages and show that such an iterated
natural sum differs from the more usual transfinite ordinal sum only for a
finite number of iteration steps. The iterated natural sum of a sequence of
ordinals can be obtained as a "mixed sum" (in an order-theoretical sense) of
the ordinals in the sequence, in fact, it is the largest mixed sum which
satisfies a finiteness condition, relative to the ordering of the sequence. We
introduce other infinite natural sums which are invariant under permutations
and show that they all coincide in the countable case. Finally, in the last
section we use the above infinitary natural sums in order to provide a
definition of size for a well-founded tree, together with an order-theoretical
characterization in the countable case. The proof of this order-theoretical
characterization is mostly independent from the rest of this paper.Comment: v2 some improvement
Reverse mathematics of the finite downwards closed subsets of
  $\mathbb{N}^k$ ordered by inclusion and adjacent Ramsey for fixed dimension

We show that the well-partial orderedness of the finite downwards closed
subsets of $\mathbb{N}^k$ ,ordered by inclusion, is equivalent to the
well-foundedness of the ordinal $\omega^{\omega^\omega}$. This was conjectured
to be the case by Hatzikiriakou and Simpson. Since we use Friedman's adjacent
Ramsey theorem for fixed dimensions in the upper bound, we also give a
treatment of the reverse mathematical status of that theorem.Comment: This is a pre-peer-reviewed version of a paper which has been
  accepted for publication at Mathematical Logic Quarterl
Proteases to improve the mechanical characteristics of durable press finished cotton fabrics

N-Methylol reagents are conventional crosslinking agents that are still widely used in textile industry to produce crease-resistant cotton fabrics. In this work serine proteases were used to recover the strength of fabrics, crosslinked with N-hydroxymethylacrylamide. Nearly one half of the strength loss of  rosslinked cotton fabrics could be restored after protease treatment, while the wrinkle recovery angle (WRA) decreased only slightly. The enzymatic hydrolysis
of the amide cross-links in the durable pressed cellulose was confirmed by FT-IR analysis and dyeability with an acid
dye.
Effect of protease concentration on the tensile strength recovery, WRAand acid dye dyeability at 30 min reaction time
Laccases to improve the whiteness in a conventional bleaching of cotton

This study reports for the first time on the
enhancement of the bleaching effect achieved on cotton using
laccase enzyme. Laccases applied in short-time batchwise or
pad-dry processes prior to conventional peroxide bleaching,
improved the end fabric whiteness. The whiteness level
reached in the combined enzymatic/peroxide process was
comparable to the whiteness in two consecutive peroxide
bleaches.
Effect of 10 min laccase pre-treatment at 60 8C, pH 5 on
fabrics whiteness before and after a conventional hydrogen
peroxide bleaching
On toughness and stiffness of poly(butylene terephthalate) with epoxide-containing elastomer by reactive extrusion

To obtain a balance between toughness (as measured by notched impact strength) and elastic stiffness of poly(butylene terephthalate) (PBT), a small amount of tetrafunctional epoxy monomer was incorporated into PBT/ [ethylene/methyl acrylate/glycidyl methacrylate terpolymer (E-MA-GMA)] blends during the reactive extrusion process. The effectiveness of toughening by E-MA-GMA and the\ud
effect of the epoxy monomer were investigated. It was found that E-MA-GMA was finely dispersed in PBT matrix, whose toughness was significantly enhanced, but the stiffness decreased linearly, with increasing E-MA-GMA content. Addition of 0.2 phr epoxy monomer was noted to further improve the dispersion of E-MA-GMA particles by increasing the viscosity of the PBT matrix. While use of epoxy monomer had little influence on the notched impact strength of the blends, there was a distinct increase in the elastic stiffness. SEM micrographs of impact-fracture surfaces indicated that extensive\ud
matrix shear yielding was the main impact energy dissipation mechanism in both types of blends, with or without epoxy monomer, and containing 20 wt. or more\ud
elastomer
Thermal and thermomechanical behaviour of polycaprolactone and starch/polycaprolactone blends for biomedical applications

Polycaprolactone (PCL) and starch/PCL blends (SPCL) are shown to have the potential to be used in a range of biomedical applications and can be processed with conventional melting-based procedures. In this paper, the thermal and thermomechanical analyses of PCL and SPCL were performed, using DSC, optical microscopy and DMA. Starch effectively increased the non-isothermal crystallisation rate of PCL. Non-isothermal crystallisation kinetics was analyzed using Ozawa model, and a method, which combines the theories of Avrami and Ozawa. Starch effectively reinforced PCL and enhanced its damping properties, which indicated that SPCL could be more suitable than PCL in some biomedical applications, as it might help in the dissipation of the mechanical energy generated by the patient movements
Hydroxyapatite reinforced chitosan and polyester blends for biomedical applications

Hydroxyapatite, chitosan, and aliphatic polyester
were compounded using a twin-screw extruder. The polyesters
include poly(e-caprolactone) (PCL), poly(lactic acid) ,
poly(butylene succinate) (PBS), and poly(butylene terephthalate
adipate). The mass fraction of chitosan ranged from
17.5 to 45%, while that of HA ranged from 10 to 30%. These
blends were injection molded and evaluated for thermal,
morphological, and mechanical properties. The addition of
hydroxyapatite decreased the crystallinity in chitosan/PBS
blends, while in blends containing chitosan/PCL, the crystallinity
increased. Addition of hydroxyapatite significantly
decreased the tensile strength and elongation of polyester/
hydroxyapatite composites as well as chitosan/polyester/
hydroxyapatite composites with elongations undergoing decreases
over an order of magnitude. The tensile strength of the
composite was dictated by the adhesion of HA to the chitosan/
polyester matrix. The tensile strength of composites
containing hydroxyapatite could be predicted using the
Nicolai and Narkis equation for weak filler adhesion
(K 1.21). Tensile-fractured and cryogenically-fractured
surface indicates extensive debonding of hydroxyapatite
crystals from the matrix, indicating weak adhesion. The adhesion of hydroxyapatite was higher for pure polyester than
those containing chitosan and polyester. The modulus of the
composites registered modest increase. The two main diffraction
peaks observed using WAXS are unaffected by the
amount of chitosan or hydroxyapatite
The use of the indentation test for studying the solidification behaviour of different semicrystalline polymers during injection moulding

An in-line method for monitoring the solidification process during injection molding of semicrystalline polymers (demonstrated previously in J. Appl. Polym. Sci.2003, 89, 3713) is based on a simple device, where an additional ejector pin is pushed on the injection molded part at different times during the solidification phase. The ‘indentation depth profile’, i.e., residual deformation as a function of time, was obtained and allowed to determine the evolution of the solidification front in the mold as a function of the cooling time. The present work shows the reliability and the powerfulness of the aforementioned method for a large variety of different semicrystalline polymers (PET, PBT, polyamide-6 PA6, isotactic poly(propylene) iPP) characterized also by different molecular weight and/or nucleating agents. The results show that the indentation test may be considered as a ‘predictive’ tool to qualitatively and quantitatively compare the solidification process of different polymers and polymer grades during injection molding