pinecone/core-2020-05-10-deduplication huggingface dataset:
Algebras of multiplace functions for signatures containing antidomain

We define antidomain operations for algebras of multiplace partial functions.
For all signatures containing composition, the antidomain operations and any
subset of intersection, preferential union and fixset, we give finite
equational or quasiequational axiomatisations for the representation class. We
do the same for the question of representability by injective multiplace
partial functions. For all our representation theorems, it is an immediate
corollary of our proof that the finite representation property holds for the
representation class. We show that for a large set of signatures, the
representation classes have equational theories that are coNP-complete.Comment: 33 pages. Added brief discussion of square algebra
A uniform Birkhoff theorem

Garret Birkhoff's HSP theorem characterizes the classes of models of
algebraic theories as those being closed with respect to homomorphic images,
subalgebras, and products. In particular, it implies that an algebra
$\mathbf{B}$ satisfies all equations that hold in an algebra $\mathbf{A}$ of
the same type if and only if $\mathbf{B}$ is a homomorphic image of a
subalgebra of a (possibly infinite) direct power of $\mathbf{A}$. The former
statement is equivalent to the existence of a natural map sending term
functions of the algebra $\mathbf{A}$ to those of $\mathbf{B}$, and it is
natural to wonder about continuity properties of this mapping. We show that
this map is uniformly continuous if and only if every finitely generated
subalgebra of $\mathbf{B}$ is a homomorphic image of a subalgebra of a finite
power of $\mathbf{A}$ -- without any additional assumptions concerning the
algebras $\mathbf{A}$ and $\mathbf{B}$. Moreover, provided that $\mathbf{A}$ is
almost locally finite (for instance if $\mathbf{A}$ is locally oligomorphic or
locally finite), the considered map is uniformly continuous if and only if it
is Cauchy-continuous. In particular, our results extend a recent theorem by
Bodirsky and Pinsker beyond the countable $\omega$-categorical setting.Comment: 15 page
On the J\'onsson distributivity spectrum

Suppose throughout that $\mathcal V$ is a congruence distributive variety. If
$m \geq 1$, let $ J _{ \mathcal V} (m) $ be the smallest natural number $k$
such that the congruence identity $\alpha ( \beta \circ \gamma \circ \beta
\dots ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ
\dots $ holds in $\mathcal V$, with $m$ occurrences of $ \circ$ on the left and
$k$ occurrences of $\circ$ on the right. We show that if $ J _{ \mathcal V} (m)
=k$, then $ J _{ \mathcal V} (m \ell ) \leq k \ell $, for every natural number
$\ell$. The key to the proof is an identity which, through a variety, is
equivalent to the above congruence identity, but involves also reflexive and
admissible relations. If $ J _{ \mathcal V} (1)=2 $, that is, $\mathcal V$ is
$3$-distributive, then $ J _{ \mathcal V} (m) \leq m $, for every $m \geq 3$
(actually, a more general result is presented which holds even in
nondistributive varieties). If $\mathcal V$ is $m$-modular, that is, congruence
modularity of $\mathcal V$ is witnessed by $m+1$ Day terms, then $ J _{
\mathcal V} (2) \leq J _{ \mathcal V} (1) + 2m^2-2m -1 $. Various problems are
stated at various places.Comment: v. 4, added somethin
Unique inclusions of maximal C-clones in maximal clones

$\mathit{C}$-clones are polymorphism sets of so-called clausal relations, a
special type of relations on a finite domain, which first appeared in
connection with constraint satisfaction problems in [Creignou et al. 2008]. We
completely describe the relationship w.r.t. set inclusion between maximal
$\mathit{C}$-clones and maximal clones. As a main result we obtain that for
every maximal $\mathit{C}$-clone there exists exactly one maximal clone in
which it is contained. A precise description of this unique maximal clone, as
well as a corresponding completeness criterion for $\mathit{C}$-clones is
given.Comment: 20 page
Galois connection for multiple-output operations

It is a classical result from universal algebra that the notions of
polymorphisms and invariants provide a Galois connection between suitably
closed classes (clones) of finitary operations $f\colon B^n\to B$, and classes
(coclones) of relations $r\subseteq B^k$. We will present a generalization of
this duality to classes of (multi-valued, partial) functions $f\colon B^n\to
B^m$, employing invariants valued in partially ordered monoids instead of
relations. In particular, our set-up encompasses the case of permutations
$f\colon B^n\to B^n$, motivated by problems in reversible computing.Comment: 36 pages; to appear in Algebra Universali
Canonical extensions and ultraproducts of polarities

J{\'o}nsson and Tarski's notion of the perfect extension of a Boolean algebra
with operators has evolved into an extensive theory of canonical extensions of
lattice-based algebras. After reviewing this evolution we make two
contributions. First it is shown that the failure of a variety of algebras to
be closed under canonical extensions is witnessed by a particular one of its
free algebras. The size of the set of generators of this algebra can be made a
function of a collection of varieties and is a kind of Hanf number for
canonical closure. Secondly we study the complete lattice of stable subsets of
a polarity structure, and show that if a class of polarities is closed under
ultraproducts, then its stable set lattices generate a variety that is closed
under canonical extensions. This generalises an earlier result of the author
about generation of canonically closed varieties of Boolean algebras with
operators, which was in turn an abstraction of the result that a first-order
definable class of Kripke frames determines a modal logic that is valid in its
so-called canonical frames
A problem of Kusner on equilateral sets

R. B. Kusner [R. Guy, Amer. Math. Monthly 90 (1983), 196--199] asked whether
a set of vectors in a d-dimensional real vector space such that the l-p
distance between any pair is 1, has cardinality at most d+1. We show that this
is true for p=4 and any d >= 1, and false for all 1<p<2 with d sufficiently
large, depending on p.
  More generally we show that the maximum cardinality is at most $(2\lceil
p/4\rceil-1)d+1$ if p is an even integer, and at least $(1+\epsilon_p)d$ if
1<p<2, where $\epsilon_p>0$ depends on p.Comment: 6 pages. Small correction to Proposition
On quasi-contractivity of C 0-semigroups on Banach spaces

A basic result in semigroup theory states that every C-0-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula (e(t/n) (A) p)(n) (where P denotes a bounded projection), we prove that whenever the generator A is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is not quasi-contractive
A Construction of Metabelian Groups

In 1934, Garrett Birkhoff has shown that the number of isomorphism classes of
finite metabelian groups of order $p^{22}$ tends to infinity with $p$. More
precisely, for each prime number $p$ there is a family
$(M_\lambda)_{\lambda=0,...,p-1}$ of indecomposable and pairwise nonisomorphic
metabelian $p$-groups of the given order. In this manuscript we use recent
results on the classification of possible embeddings of a subgroup in a finite
abelian $p$-group to construct families of indecomposable metabelian groups,
indexed by several parameters, which have upper bounds on the exponents of the
center and the commutator subgroup.Comment: 5 pages; to appear in Archiv der Mathemati
Resolutions over Koszul algebras

In this paper we show that if $\Lambda=\amalg_{i\geq 0}\Lambda_i$ is a Koszul
algebra with $\Lambda_0$ isomorphic to a product of copies of a field, then the
minimal projective resolution of $\Lambda_0$ as a right $\Lambda$-module
provides all the information necessary to construct both a minimal projective
resolution of $\Lambda_0$ as a left $\Lambda$-module and a minimal projective
resolution of $\Lambda$ as a right module over the enveloping algebra of
$\Lambda$. The main tool for this is showing that there is a comultiplicative
structure on a minimal projective resolution of $\Lambda_0$ as a right
$\Lambda$-module
Improvement of an estimate of H. Mueller involving the order of 2(mod u)
  II

Let m>=1 be an arbitrary fixed integer and let N_m(x) count the number of odd
integers u<=x such that the order of 2 modulo u is not divisible by m. In case
m is prime estimates for N_m(x) were given by H. Mueller that were subsequently
sharpened into an asymptotic estimate by the present author. Mueller on his
turn extended the author's result to the case where m is a prime power and gave
bounds in the case m is not a prime power. Here an asymptotic for N_m(x) is
derived that is valid for all integers m. This asymptotic would easily have
followed from Mueller's approach were it not for the fact that a certain
Diophantine equation has non-trivial solutions. All solutions of this equation
are determined. We also generalize to other base numbers than 2. For a very
sparse set of these numbers Mueller's approach does work.Comment: 11 pages, 2 Tables; Proposition 3 has now been corrected along with a
  few typo
Permutable subnormal subgroups of finite groups

The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT -groups. In particular, it is shown that the finite solvable PT -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugate-permutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not PT -groups but all subnormal subgroups of defect two are permutable
Three cubes in arithmetic progression over quadratic fields

We study the problem of the existence of arithmetic progressions of three
cubes over quadratic number fields Q(sqrt(D)), where D is a squarefree integer.
For this purpose, we give a characterization in terms of Q(sqrt(D))-rational
points on the elliptic curve E:y^2=x^3-27. We compute the torsion subgroup of
the Mordell-Weil group of this elliptic curve over Q(sqrt(D)) and we give
partial answers to the finiteness of the free part of E(Q(sqrt(D))). This last
task will be translated to compute if the rank of the quadratic D-twist of the
modular curve X_0(36) is zero or not
The affine preservers of non-singular matrices

When K is an arbitrary field, we study the affine automorphisms of M_n(K)
that stabilize GL_n(K). Using a theorem of Dieudonn\'e on maximal affine
subspaces of singular matrices, this is easily reduced to the known case of
linear preservers when n>2 or #K>2. We include a short new proof of the more
general Flanders' theorem for affine subspaces of M_{p,q}(K) with bounded rank.
We also find that the group of affine transformations of M_2(F_2) that
stabilize GL_2(F_2) does not consist solely of linear maps. Using the theory of
quadratic forms over F_2, we construct explicit isomorphisms between it, the
symplectic group Sp_4(F_2) and the symmetric group S_6.Comment: 13 pages, very minor corrections from the first versio
Automorphisms of prime order of smooth cubic n-folds

In this paper we give an effective criterion as to when a prime number p is
the order of an automorphism of a smooth cubic hypersurface of P^{n+1}, for a
fixed n > 1. We also provide a computational method to classify all such
hypersurfaces that admit an automorphism of prime order p. In particular, we
show that p<2^{n+1} and that any such hypersurface admitting an automorphism of
order p>2^n is isomorphic to the Klein n-fold. We apply our method to compute
exhaustive lists of automorphism of prime order of smooth cubic threefolds and
fourfolds. Finally, we provide an application to the moduli space of
principally polarized abelian varieties.Comment: 10 page
PLB-spaces of holomorphic functions with logarithmic growth conditions

Countable projective limits of countable inductive limits, called PLB-spaces,
of weighted Banach spaces of continuous functions have recently been
investigated by Agethen, Bierstedt and Bonet. In a previous article, the author
extended their investigation to the case of holomorphic functions and
characterized when spaces over the unit disc w.r.t. weights whose decay,
roughly speaking, is neither faster nor slower than that of a polynomial are
ultrabornological or barrelled. In this note, we prove a similar
characterization for the case of weights which tend to zero logarithmically.Comment: Version of November 23, 2011. 9 page
The Moment Problem for Continuous Positive Semidefinite Linear
  functionals

Let $\tau$ be a locally convex topology on the countable dimensional
polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed
subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated
quadratic module in $\rx$. We investigate the following question: When is the
cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of
$M$? We give an interpretation of this inclusion with respect to representing
continuous linear functionals by measures. We discuss several examples; we
compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies.
We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a
certain convex compact polyhedron.Comment: 14 page
Determination of Conjugacy Class Sizes from Products of Characters

Robinson showed that the character degrees are determined by knowing, for all
$n$, the number of ways that the identity can be expressed as a product of $n$
commutators. Earlier, Strunkov showed that the existence of characters of
$p$-defect 0 can be determined by counting solutions to certain equations
involving commutators and conjugates. In this paper, we prove analogs to
Robinson's and Strunkov's theorems by switching conjugacy classes and
characters. We show that counting the multiplicity of the trivial character in
certain products of characters determines the conjugacy class sizes and
existence of conjugacy classes with $p$-defect 0.Comment: 7 page
Generators for Cubic Surfaces with two Skew Lines over Finite Fields

Let S be a smooth cubic surface defined over a field K. As observed by Segre
and Manin, there is a secant and tangent process on S that generates new
K-rational points from old. It is natural to ask for the size of a minimal
generating set for S(K). In a recent paper, for fields K with at least 13
elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can
be generated from one point. In this paper we prove the corresponding version
of this result for fields K having at least 4 elements, and slightly milder
results for #K=2 or 3.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1012.1838 by
  other author
A Note on Schanuel's Conjectures for Exponential Logarithmic Power
  Series Fields

In [1], J. Ax proved a transcendency theorem for certain differential fields
of characteristic zero: the differential counterpart of the still open
Schanuel's conjecture about the exponential function over the field of complex
numbers [11, page 30]. In this article, we derive from Ax's theorem
transcendency results in the context of differential valued exponential fields.
In particular, we obtain results for exponential Hardy fields,
Logarithmic-Exponential power series fields and Exponential-Logarithmic power
series fields.Comment: 6 pages, to appear in Archiv der Mathemati
A few remarks on the Generalized Vanishing Conjecture

We show that the Generalized Vanishing Conjecture $$\forall_{m \ge 1} [\Lam^m
f^m = 0] \Longrightarrow \forall_{m \gg 0} [\Lam^m (g f^m) = 0]$$ for a fixed
differential operator $\Lam \in k[\partial]$ follows from a special case of it,
namely that the additional factor $g$ is a power of the radical polynomial $f$.
Next we show that in order to prove the Generalized Vanishing Conjecture (up to
some bound on the degree of $\Lam$), we may assume that $\Lam$ is a linear
combination of powers of distinct partial derivatives. At last, we show that
the Generalized Vanishing Conjecture holds for products of linear forms in
$\partial$, in particular homogeneous differential operators $\Lambda \in
k[\partial_1,\partial_2]$.Comment: 7 page
Strings of special primes in arithmetic progressions

The Green-Tao Theorem, one of the most celebrated theorems in modern number
theory, states that there exist arbitrarily long arithmetic progressions of
prime numbers. In a related but different direction, a recent theorem of Shiu
proves that there exist arbitrarily long strings of consecutive primes that lie
in any arithmetic progression that contains infinitely many primes. Using the
techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain
subsets of the primes such as primes of the form $\lfloor \pi n\rfloor$ and
some of arithmetic density zero such as primes of the form $\lfloor n\log\log
n\rfloor$.Comment: 14 pages; preprint of article published in Archiv der Mathemati
Equidistribution of Signs for Modular Eigenforms of Half Integral Weight

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype
character whose Fourier coefficients a(n) are all real. We study an
equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We
prove this conjecture for certain subfamilies of coefficients that are
accessible via the Shimura lift by using the Sato-Tate equidistribution theorem
for integral weight modular forms. Firstly, an unconditional proof is given for
the family {a(tp^2)}_p where t is a squarefree number and p runs through the
primes. In this case, the result is in terms of natural density. To prove it
for the family {a(tn^2)}_n where t is a squarefree number and n runs through
all natural numbers, we assume the existence of a suitable error term for the
convergence of the Sato-Tate distribution, which is weaker than one conjectured
by Akiyama and Tanigawa. In this case, the results are in terms of
Dedekind-Dirichlet density.Comment: 8 pages; typos corrected, final version, accepted for publication in
  Archiv der Mathemati
Some exact values of the Harborth constant and its plus-minus weighted analogue

minimal correctionInternational audienceThe Harborth constant of a finite abelian group is the smallest integer $\ell$ such that each subset of $G$ of cardinality $\ell$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this constant is defined in the same way except that instead of considering the sum of all elements of the subset one can choose to add either the element or its inverse. We determine these constants for certain groups, mainly groups that are the direct sum of a cyclic group and a group of order $2$. Moreover, we contrast these results with existing results and conjectures on these problems
A criterion for I-adic completeness

Let $I$ denote an ideal in a commutative Noetherian ring $R$. Let $M$ be an
$R$-module. The $I$-adic completion is defined by $\hat{M}^I =
\varprojlim{}_{\alpha} M/I^{\alpha}M$. Then $M$ is called $I$-adic complete
whenever the natural homomorphism $M \to \hat{M}^I$ is an isomorphism. Let $M$
be $I$-separated, i.e. $\cap_{\alpha} I^{\alpha}M = 0$. In the main result of
the paper it is shown that $M$ is $I$-adic complete if and only if
$\Ext_R^1(F,M) = 0$ for the flat test module $F = \oplus_{i = 1}^r R_{x_i}$
where $\{x_1,\ldots,x_r\}$ is a system of elements such that $\Rad I = \Rad \xx
R$. This result extends several known statements starting with C. U. Jensen's
result (see \cite[Proposition 3]{J}) that a finitely generated $R$-module $M$
over a local ring $R$ is complete if and only if $\Ext^1_R(F,M) = 0$ for any
flat $R$-module $F$.Comment: to appear in Archiv der Mat
Gaps in the space of skeletal signatures

Skeletal signatures were introduced in [J W Anderson and A Wootton, A Lower
Bound for the Number of Group Actions on a Compact Riemann Surface, Algebr.
Geom. Topol. 12 (2012) 19--35.] as a tool to describe the space of all
signatures with which a group can act on a surface of genus $\sigma \geq 2$. In
the present paper we provide a complete description of the gaps that appear in
the space of skeletal signatures, together with proofs of the conjectures posed
in our earlier work.Comment: 10 pages, 1 figur
Thin sequences and the Gram matrix

We provide a new proof of Volberg's Theorem characterizing thin interpolating
sequences as those for which the Gram matrix associated to the normalized
reproducing kernels is a compact perturbation of the identity. In the same
paper, Volberg characterized sequences for which the Gram matrix is a compact
perturbation of a unitary as well as those for which the Gram matrix is a
Schatten-$2$ class perturbation of a unitary operator. We extend this
characterization from $2$ to $p$, where $2 \le p \le \infty$.Comment: v1: 6 pages. v2: 6 pages, referee comments incorporate
An Elementary Proof That Rationally Isometric Quadratic Forms Are
  Isometric

Let $R$ be a valuation ring with fraction field $K$ and $2\in R^\times$. We
give an elementary proof of the following known result: Two unimodular
quadratic forms over $R$ are isometric over $K$ if and only if they are
isometric over $R$. Our proof does not use Witt's Cancelation Theorem and
yields an explicit algorithm to construct an isometry over $R$ from a given
isometry over $K$. The statement actually holds for hermitian forms over
valuated involutary division rings, provided mild assumptions.
  A python implementation of the algorithm derived from the proof can be found
on the author's home page.Comment: 5 page
H\"older continuity for support measures of convex bodies

The support measures of a convex body are a common generalization of the
curvature measures and the area measures. With respect to the Hausdorff metric
on the space of convex bodies, they are weakly continuous. We provide a
quantitative improvement of this result, by establishing a H\"older estimate
for the support measures in terms of the bounded Lipschitz metric, which
metrizes the weak convergence. Specializing the result to area measures yields
a reverse counterpart to earlier stability estimates, concerning Minkowski's
existence theorem for convex bodies with given area measure.Comment: The manuscript is an extended and improved version of the second part
  of the manuscript number arxiv:1310.151
Resolutions of General Canonical Curves on Rational Normal Scrolls

Let $C\subset \mathbb{P}^{g-1}$ be a general curve of genus $g$ and let $k$
be a positive integer such that the Brill-Noether number $\rho(g,k,1)\geq 0$
and $g > k+1$. The aim of this short note is to study the relative canonical
resolution of $C$ on a rational normal scroll swept out by a $g^1_k=|L|$ with
$L\in W^1_k(C)$ general. We show that the bundle of quadrics appearing in the
relative canonical resolution is unbalanced if and only if $\rho>0$ and
$(k-\rho-\frac{7}{2})^2-2k+\frac{23}{4}>0$.Comment: 11 pages, 1 figur