pinecone/core-2020-05-10-deduplication huggingface dataset:
The one-sided inverse along an element in semigroups and rings

The concept of the inverse along an element was introduced by Mary in 2011. Later, Zhu et al. introduced the one-sided inverse along an element. In this paper, we first give a new existence criterion for the one-sided inverse along a product and characterize the existence of Moore–Penrose inverse by means of one-sided invertibility of certain element in a ring. In addition, we show that a∈ S † ⋂ S # if and only if (a∗a)k is invertible along a if and only if (aa∗)k is invertible along a in a ∗ -monoid S, where k is an arbitrary given positive integer. Finally, we prove that the inverse of a along aa ∗ coincides with the core inverse of a under the condition a∈ S { 1 , 4 } in a ∗ -monoid S.FCT - Fuel Cell Technologies Program(CXLX13-072)This research was supported by the National Natural Science Foundation
of China (No. 11371089), the Specialized Research Fund for the Doctoral Program of
Higher Education (No. 20120092110020), the Natural Science Foundation of Jiangsu Province
(No. BK20141327) and the Foundation of Graduate Innovation Program of Jiangsu Province
(No. KYZZ15-0049).info:eu-repo/semantics/publishedVersio
New Techniques for Computing Geometric Index

We introduce \textcolor{red}{general} new techniques for computing the
geometric index of a link $L$ in the interior of a solid torus $T$. These
techniques simplify and unify previous ad hoc methods used to compute the
geometric index in specific examples \textcolor{red}{ and allow the simple
computation of geometric index for new examples where the index was not
previously known}. The geometric index measures the minimum number of times any
meridional disc of $T$ must intersect $L$. It is related to the algebraic index
in the sense that adding up signed intersections of an interior simple closed
curve $C$ in $T$ with a meridional disc gives $\pm$ the algebraic index of $C$
in $T$. One key idea is introducing the notion of geometric index for solid
chambers of the form $B^2\times I$ in $T$. After that we prove that if a solid
torus can be divided into solid chambers by meridional discs in a specific
\textcolor{red}{(and often easy to obtain)} way, then the geometric index can
be easily computed
On a Generalization for Tribonacci Quaternions

Let $V_{n}$ denote the third order linear recursive sequence defined by the
initial values $V_{0}$, $V_{1}$ and $V_{2}$ and the recursion
$V_{n}=rV_{n-1}+sV_{n-2}+tV_{n-3}$ if $n\geq 3$, where $r$, $s$, and $t$ are
real constants. The $\{V_{n}\}_{n\geq0}$ are generalized Tribonacci numbers and
reduce to the usual Tribonacci numbers when $r=s=t=1$ and to the $3$-bonacci
numbers when $r=s=1$ and $t=0$. In this study, we introduced a quaternion
sequence which has not been introduced before. We show that the new quaternion
sequence that we introduced includes the previously introduced Tribonacci,
Padovan, Narayana and Third order Jacobsthal quaternion sequences. We obtained
the Binet formula, summation formula and the norm value for this new quaternion
sequence
Nontrivial solutions of systems of Hammerstein integral equations with
  first derivative dependence

By means of classical fixed point index, we prove new results on the
existence, non-existence, localization and multiplicity of nontrivial solutions
for systems of Hammerstein integral equations where the nonlinearities are
allowed to depend on the first derivative. As a byproduct of our theory we
discuss the existence of positive solutions of a system of third order ODEs
subject to nonlocal boundary conditions. Some examples are provided in order to
illustrate the applicability of the theoretical results.Comment: 18 page
Characterization of curves that lie on a geodesic sphere or on a totally
  geodesic hypersurface in a hyperbolic space or in a sphere

The consideration of the so-called rotation minimizing frames allows for a
simple and elegant characterization of plane and spherical curves in Euclidean
space via a linear equation relating the coefficients that dictate the frame
motion. In this work, we extend these investigations to characterize curves
that lie on a geodesic sphere or totally geodesic hypersurface in a Riemannian
manifold of constant curvature. Using that geodesic spherical curves are normal
curves, i.e., they are the image of an Euclidean spherical curve under the
exponential map, we are able to characterize geodesic spherical curves in
hyperbolic spaces and spheres through a non-homogeneous linear equation.
Finally, we also show that curves on totally geodesic hypersurfaces, which play
the role of hyperplanes in Riemannian geometry, should be characterized by a
homogeneous linear equation. In short, our results give interesting and
significant similarities between hyperbolic, spherical, and Euclidean
geometries.Comment: 15 pages, 3 figures; comments are welcom
A comparison between numerical solutions to fractional differential
  equations: Adams-type predictor-corrector and multi-step generalized
  differential transform method

In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
A stronger form of the theorem constructing a rigid binary relation on
  any set

On every set A there is a rigid binary relation i.e. such a relation R
\subseteq A \times A that there is no homomorphism (A,R) \rightarrow (A,R)
except the identity (Vop{\v{e}}nka et al. [1965]). We prove that for each
infinite cardinal number \kappa if card A \leq 2^\kappa, then there exists a
relation R \subseteq A \times A with the following property:
  \forall (x \in A) \exists ({x} \subseteq A(x) \subseteq A, card A(x) \leq
\kappa) \forall (f: A(x) \rightarrow A, f \neq id_A(x))
  f is not a homomorphism of R.
  The above property implies that R is rigid. If a relation R \subseteq A
\times A has the above property, then card A \leq 2^\kappa.Comment: an enlarged version, 8 page
The Beckman-Quarles theorem for continuous mappings from C^n to C^n

Let varphi_n:C^n times C^n->C,
varphi_n((x_1,...,x_n),(y_1,...,y_n))=sum_{i=1}^n (x_i-y_i)^2. We say that
f:C^n->C^n preserves distance d>=0, if for each X,Y in C^n varphi_n(X,Y)=d^2
implies varphi_n(f(X),f(Y))=d^2. We prove: if n>=2 and a continuous f:C^n->C^n
preserves unit distance, then f has a form I circ (rho,...,rho), where
I:C^n->C^n is an affine mapping with orthogonal linear part and rho:C->C is the
identity or the complex conjugation. For n >=3 and bijective f the theorem
follows from Theorem 2 in [8].Comment: 10 pages, LaTeX2e, the version which appeared in Aequationes
  Mathematica
Antinorms and Radon curves

In this paper we consider two notions that have been discovered and
rediscovered by geometers and analysts since 1917 up to the present day: Radon
curves and antinorms.
  A Radon curve is a special kind of centrally symmetric closed convex curve in
the plane. A Radon plane is a normed plane obtained by using a Radon curve as
(the boundary of) the unit ball. Many known results in Euclidean geometry also
hold for Radon planes, for example the triangle and parallelogram area
formulas, certain theorems on angular bisectors, the area formula of a polygon
circumscribed about a circle, certain isoperimetric inequalities, and the
non-expansive property of certain non-linear projections.
  These results may be further generalized to arbitrary normed planes if we
formally change the statement of the result by referring in some places to the
antinorm instead of the norm. The antinorm is a norm dual to the norm of an
arbitrarily given normed plane, although it lives in the same plane as the
original norm.
  It is the purpose of this mainly expository paper to give a list of results
on antinorms that generalize results true for Radon norms, and in many cases
characterize Radon norms among all norms in the plane. Many of the results are
old, well-known, and have often been rediscovered. However, for most of the
results we give streamlined proofs. Also, some of the characterizations of
Radon curves given here seem not to have appeared previously in print.Comment: 24 pages, 7 figure
Hyers--Ulam stability of derivations and linear functions

In this paper the following implication is verified for certain basic
algebraic curves: if the additive real function $f$ approximately (i.e., with a
bounded error) satisfies the derivation rule along the graph of the algebraic
curve in consideration, then $f$ can be represented as the sum of a derivation
and a linear function. When, instead of the additivity of $f$, it is assumed
that, in addition, the Cauchy difference of $f$ is bounded, a stability theorem
is obtained for such characterizations of derivations.Comment: 9 pages; published in Aequationes Mathematicae in 201
Multi-variable translation equation which arises from homothety

In many regular cases, there exists a (properly defined) limit of iterations
of a function in several real variables, and this limit satisfies the
functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a
vector. This is a special case of a well-known translation equation. In this
paper we present a complete solution to this functional equation in case f is a
continuous function on a single point compactification of a 2-dimensional real
vector space. It appears that, up to conjugation by a homogeneous continuous
function, there are exactly four solutions. Further, in a 1-dimensional case we
present a solution with no regularity assumptions on f.Comment: 15 page
An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function U(a, b, z). By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the nth coefficient is expressed as the (n − 1)th moment of a measure, and also as the trace of the (n − 1)th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics
Axiomatizations of quasi-Lov\'asz extensions of pseudo-Boolean functions

We introduce the concept of quasi-Lov\'asz extension as being a mapping
$f\colon I^n\to\R$ defined on a nonempty real interval $I$ containing the
origin and which can be factorized as
$f(x_1,...,x_n)=L(\phi(x_1),...,\phi(x_n))$, where $L$ is the Lov\'asz
extension of a pseudo-Boolean function $\psi\colon\{0,1\}^n\to\R$ (i.e., the
function $L\colon\R^n\to\R$ whose restriction to each simplex of the standard
triangulation of $[0,1]^n$ is the unique affine function which agrees with
$\psi$ at the vertices of this simplex) and $\phi\colon I\to\R$ is a
nondecreasing function vanishing at the origin. These functions appear
naturally within the scope of decision making under uncertainty since they
subsume overall preference functionals associated with discrete Choquet
integrals whose variables are transformed by a given utility function. To
axiomatize the class of quasi-Lov\'asz extensions, we propose generalizations
of properties used to characterize the Lov\'asz extensions, including a
comonotonic version of modularity and a natural relaxation of homogeneity. A
variant of the latter property enables us to axiomatize also the class of
symmetric quasi-Lov\'asz extensions, which are compositions of symmetric
Lov\'asz extensions with 1-place nondecreasing odd functions
Metric trees of generalized roundness one

Every finite metric tree has generalized roundness strictly greater than one.
On the other hand, some countable metric trees have generalized roundness
precisely one. The purpose of this paper is to identify some large classes of
countable metric trees that have generalized roundness precisely one.
  At the outset we consider spherically symmetric trees endowed with the usual
combinatorial metric (SSTs). Using a simple geometric argument we show how to
determine decent upper bounds on the generalized roundness of finite SSTs that
depend only on the downward degree sequence of the tree in question. By
considering limits it follows that if the downward degree sequence $(d_{0},
d_{1}, d_{2}...)$ of a SST $(T,\rho)$ satisfies $|\{j \, | \, d_{j} > 1 \}| =
\aleph_{0}$, then $(T,\rho)$ has generalized roundness one. Included among the
trees that satisfy this condition are all complete $n$-ary trees of depth
$\infty$ ($n \geq 2$), all $k$-regular trees ($k \geq 3$) and inductive limits
of Cantor trees.
  The remainder of the paper deals with two classes of countable metric trees
of generalized roundness one whose members are not, in general, spherically
symmetric. The first such class of trees are merely required to spread out at a
sufficient rate (with a restriction on the number of leaves) and the second
such class of trees resemble infinite combs.Comment: 14 pages, 2 figures, 2 table
Functional definitions for $q$-analogues of eulerian functions and
  applications

We explore a number of functional properties of the $q$-gamma function and a
class of its quotients; including the $q$-beta function. We obtain formulas for
all higher logarithmic derivatives of these quotients and give precise
conditions on their sign. We prove how these and other functional properties,
such as the multiplication formula or the asymptotic expansion, together with
the fundamental functional equation of the $q$-gamma function uniquely define
those functions. We also study reciprocal "relatives" of the fundamental
$q$-gamma functional equation, and prove uniqueness of solution results for
them. In addition, we also use a reflection formula of Askey to derive
expressions relating the classical sine function and the number $\pi$ to the
$q$-gamma function. Throughout we highlight the similarities and differences
between the cases $0<q<1$ and $q>1$.Comment: 29 page
Ball and spindle convexity with respect to a convex body

Let C⊂R^n be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ∅ , or R^n . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces
Equalities and inequalities for Hermitian solutions and Hermitian
  definite solutions of the two matrix equations $AX = B$ and $AXA^* = B$

This paper studies algebraic properties of Hermitian solutions and Hermitian
definite solutions of the two types of matrix equation $AX = B$ and $AXA^* =
B$. We first establish a variety of rank and inertia formulas for calculating
the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian
definite solutions of the matrix equations $AX = B$ and $AXA^* = B$, and then
use them to characterize many qualities and inequalities for Hermitian
solutions and Hermitian definite solutions of the two matrix equations and
their variations.Comment: 22 page
The projective translation equation and rational plane flows. I

Let X=(x,y). A plane flow is a function F(X,t): R^2*R->R^2 such that
F(F(X,s),t)=F(X,s+t) for (almost) all real numbers x,y,s,t (the function F
might not be well-defined for certain x,y,t). In this paper we investigate
rational plane flows which are of the form F(X,t)=f(Xt)/t; here f is a pair of
rational functions in 2 real variables. These may be called projective flows,
and for a description of such flows only the knowledge of Cremona group in
dimension 1 is needed. Thus, the aim of this work is to completely describe
over R all rational solutions of the two dimensional translation equation
(1-z)f(X)=f(f(Xz)(1-z)/z). We show that, up to conjugation with a 1-homogenic
birational plane transformation (1-BIR), all solutions are as follows: a zero
flow, two singular flows, an identity flow, and one non-singular flow for each
non-negative integer N, called the level of the flow. The case N=0 stands
apart, while the case N=1 has special features as well. Conjugation of these
canonical solutions with 1-BIR produce a variety of flows with different
properties and invariants, depending on the level and on the conjugation
itself. We explore many more features of these flows; for example, there are 1,
4, and 2 essentially different symmetric flows in cases N=0, N=1, and N>=2,
respectively. Many more questions will be treated in the second part of this
work.Comment: 54 pages, 6 figures. Final version before proof
Beurling regular variation, Bloom dichotomy, and the Gołąb–Schinzel functional equation

The class of 'self-neglecting' functions at the heart of Beurling slow variation is expanded by permitting a positive asymptotic limit function λ(t), in place of the usual limit 1, necessarily satisfying the following 'self-neglect' condition:(Formula presented.)known as the Goła{ogonek}b-Schinzel functional equation, a relative of the Cauchy equation (which is itself also central to Karamata regular variation). This equation, due independently to Aczél and Goła{ogonek}b, occurring in the study of one-parameter subgroups, is here accessory to the λ -Uniform Convergence Theorem (λ-UCT) for the recent, flow-motivated, 'Beurling regular variation'. Positive solutions, when continuous, are known to be λ(t) = 1 + at (below a new, 'flow', proof is given); a = 0 recovers the usual limit 1 for self-neglecting functions. The λ-UCT allows the inclusion of Karamata multiplicative regular variation in the Beurling theory of regular variation, with λ (t) = 1 + t being the relevant case here, and generalizes Bloom's theorem concerning self-neglecting functions
Convexity with respect to families of means

In this paper we investigate continuity properties of functions
$f:\mathbb{R}_+\to\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity
inequality $$ f\big(H_p(x,y)\big)\leq H_q(f(x),f(y)) \qquad(x,y>0), $$ where
$H_p$ stands for the $p$th power (or H\"older) mean. One of the main results
shows that there exist discontinuous multiplicative functions that are
$(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of
this result states that if $f$ is $(p,p)$-Jensen convex for all $p\in
P\subseteq\mathbb{R}_+$, where $P$ is a set of positive Lebesgue measure, then
$f$ must be continuous
Identities for partial Bell polynomials derived from identities for
  weighted integer compositions

We discuss closed-form formulas for the (n; k)-th partial Bell polynomials
derived in Cvijovic. We show that partial Bell polynomials are special cases of
weighted integer compositions, and demonstrate how the identities for partial
Bell polynomials easily follow from more general identities for weighted
integer compositions. We also provide short and elegant probabilistic proofs of
the latter, in terms of sums of discrete integer-valued random variables.
Finally, we outline further identities for the partial Bell polynomials.Comment: Aequationes mathematicae, 2015, pp. 1-
Generalized Cauchy means

Given two means M and N, the operator  MM,NMM,N  assigning to a given mean μ the mean MM,N(μ)(x,y)=M(μ(x,N(x,y)),μ(N(x,y),y)) was defined in Berrone and Moro (Aequationes Math 60:1–14, 2000) in connection with Cauchy means: the Cauchy mean generated by the pair f, g of continuous and strictly monotonic functions is the unique solution μ to the fixed point equation MA(f),A(g)(μ)=μ, where A(f) and A(g) are the quasiarithmetic means respectively generated by f and g. In this article, the operator  MM,NMM,N  is studied under less restrictive conditions and a general fixed point theorem is derived from an explicit formula for the iterates  MnM,NMM,Nn . The concept of class of generalized Cauchy means associated to a given family of mixing pairs of means is introduced and some distinguished families of pairs are presented. The question of equality in these classes of means remains a challenging open problem.Fil: Berrone, Lucio Renato. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
Homomorphisms from functional equations: the Goldie equation

The theory of regular variation, in its Karamata and Bojani´c-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Gołab–Schinzel  functional equation (GS) and Goldie's equation (GBE) below, are prominent there. Here we unify their treatment by algebraicization: extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various (known) solutions into homomorphisms, in fact identifying them 'en passant', and show that (GS) is present everywhere, even if in a thick disguise
Basic inequalities for weighted entropies

The concept of weighted entropy takes into account values of different
outcomes, i.e., makes entropy context-dependent, through the weight function.
In this paper, we establish a number of simple inequalities for the weighted
entropies (general as well as specific), mirroring similar bounds on standard
(Shannon) entropies and related quantities. The required assumptions are
written in terms of various expectations of the weight functions. Examples are
weighted Ky Fan and weighted Hadamard inequalities involving determinants of
positive-definite matrices, and weighted Cram\'{e}r-Rao inequalities involving
the weighted Fisher information matrix.Comment: arXiv admin note: substantial text overlap with arXiv:1409.410