pinecone/core-2020-05-10-deduplication huggingface dataset:
Bernstein modal basis: application to the spectral Petrov-Galerkin
  method for fractional partial differential equations

In the spectral Petrov-Galerkin methods, the trial and test functions are
required to satisfy particular boundary conditions. By a suitable linear
combination of orthogonal polynomials, a basis, that is called the modal basis,
is obtained. In this paper, we extend this idea to the non-orthogonal dual
Bernstein polynomials. A compact general formula is derived for the modal basis
functions based on dual Bernstein polynomials. Then, we present a
Bernstein-spectral Petrov-Galerkin method for a class of time fractional
partial differential equations with Caputo derivative. It is shown that the
method leads to banded sparse linear systems for problems with constant
coefficient. Some numerical examples are provided to show the efficiency and
the spectral accuracy of the method
Numerical Study of Nonlinear Dynamics of a Population System with Time
  Delay

Mathematical models of interacting populations are often constructed as
systems of differential equations, which describe how populations change with
time. Below we study one such model connected to the nonlinear dynamics of a
system of populations in presence of time delay. The consequence of the
presence of the time delay is that the nonlinear dynamics of the studied system
become more rich, e.g., new orbits in the phase space of the system arise which
are dependent on the time-delay parameters. In more detail we introduce a time
delay and generalize the model system of differential equations for the
interaction of three populations based on generalized Volterra equations in
which the growth rates and competition coefficients of populations depend on
the number of members of all populations
\cite{Dimitrova2001a},\cite{Dimitrova2001b} and then numerically solve the
system with and without time delay. We use a modification of the method of
Adams for the numerical solution of the system of model equations with time
delay. By appropriate selection of the parameters and initial conditions we
show the impact of the delay time on the dynamics of the studied population
system.Comment: 12 pages, no figure
On the mixed problem for the semilinear Darcy-Forchheimer-Brinkman PDE
  system in Besov spaces on creased Lipschitz domains

The purpose of this paper is to study the mixed Dirichlet-Neumann boundary
value problem for the semilinear Darcy-Forchheimer-Brinkman system in
$L_p$-based Besov spaces on a bounded Lipschitz domain in ${\mathbb R}^3$, with
$p$ in a neighborhood of $2$. This system is obtained by adding the semilinear
term $|{\bf u}|{\bf u}$ to the linear Brinkman equation. {First, we provide
some results about} equivalence between the Gagliardo and non-tangential
traces, as well as between the weak canonical conormal derivatives and the
non-tangential conormal derivatives. Various mapping and invertibility
properties of some integral operators of potential theory for the linear
Brinkman system, and well posedness results for the Dirichlet and Neumann
problems in $L_p$-based Besov spaces on bounded Lipschitz domains in ${\mathbb
R}^n$ ($n\geq 3$) are also presented. Then, employing integral potential
operators, we show the well-posedness in $L_2$-based Sobolev spaces for the
mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a
bounded Lipschitz domain in ${\mathbb R}^n$ $(n\geq 3)$. Further, by using some
stability results of Fredholm and invertibility properties and exploring
invertibility of the associated Neumann-to-Dirichlet operator, we extend the
well-posedness property to some $L_p$-based Sobolev spaces. Next we use the
well-posedness result in the linear case combined with a fixed point theorem in
order to show the existence and uniqueness for a mixed boundary value problem
of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman
system in $L_p$-based Besov spaces, with $p\in (2-\varepsilon ,2+\varepsilon)$
and some parameter $\varepsilon >0$.Comment: 51
Asymptotic approximation for the solution to a semi-linear parabolic
  problem in a thin star-shaped junction

A semi-linear parabolic problem is considered in a thin $3D$ star-shaped
junction that consists of several thin curvilinear cylinders that are joined
through a domain (node) of diameter $\mathcal{O}(\varepsilon).$
  The purpose is to study the asymptotic behavior of the solution
$u_\varepsilon$ as $\varepsilon \to 0,$ i.e. when the star-shaped junction is
transformed in a graph. In addition, the passage to the limit is accompanied by
special intensity factors $\{\varepsilon^{\alpha_i}\}$ and
$\{\varepsilon^{\beta_i}\}$ in nonlinear perturbed Robin boundary conditions.
  We establish qualitatively different cases in the asymptotic behaviour of the
solution depending on the value of the parameters $\{{\alpha_i}\}$ and
$\{\beta_i\}.$ Using the multi-scale analysis, the asymptotic approximation for
the solution is constructed and justified as the parameter $\varepsilon \to 0.$
Namely, in each case we derive the limit problem $(\varepsilon =0)$ on the
graph with the corresponding Kirchhoff transmission conditions (untypical in
some cases) at the vertex, define other terms of the asymptotic approximation
and prove appropriate asymptotic estimates that justify these coupling
conditions at the vertex and show the impact of the local geometric
heterogeneity of the node and physical processes in the node on some properties
of the solution.Comment: 37 pages, 4 figure
Boundedness and stabilization in a two-species chemotaxis-competition
  system of parabolic-parabolic-elliptic type

This paper deals with the two-species chemotaxis-competition system $u_t =
d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u(1 - u - a_1 v)$, $v_t
= d_2 \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v(1 - a_2 u - v)$, $0
= d_3 \Delta w + \alpha u + \beta v - \gamma w$, where $\Omega$ is a bounded
domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $\chi_i$ and $\mu_i$
are constants satisfying some conditions. The above system was studied in the
cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global
existence and asymptotic stability hold when $\frac{\chi_i}{\mu_i}$ are small.
However, the conditions in the above two cases strongly depend on $a_1,a_2$,
and have not been obtained in the case that $a_1,a_2\ge 1$. Moreover,
convergence rates in the cases that $a_1,a_2\in (0,1)$ and $a_1 > 1 > a_2$ have
not been studied. The purpose of this work is to construct conditions which
derive global existence of classical bounded solutions for all $a_1,a_2>0$
which covers the case that $a_1,a_2 \ge 1$, and lead to convergence rates for
solutions of the above system in the cases that $a_1,a_2\in (0,1)$ and $a_1\ge
1 >a_2$.Comment: 21 page
Concentration phenomena for a fractional Schr\"odinger-Kirchhoff type
  equation

In this paper we deal with the multiplicity and concentration of positive
solutions for the following fractional Schr\"odinger-Kirchhoff type equation
\begin{equation*} M\left(\frac{1}{\varepsilon^{3-2s}}
\iint_{\mathbb{R}^{6}}\frac{|u(x)- u(y)|^{2}}{|x-y|^{3+2s}} dxdy +
\frac{1}{\varepsilon^{3}} \int_{\mathbb{R}^{3}} V(x)u^{2}
dx\right)[\varepsilon^{2s} (-\Delta)^{s}u+ V(x)u]= f(u) \, \mbox{in}
\mathbb{R}^{3} \end{equation*} where $\varepsilon>0$ is a small parameter,
$s\in (\frac{3}{4}, 1)$, $(-\Delta)^{s}$ is the fractional Laplacian, $M$ is a
Kirchhoff function, $V$ is a continuous positive potential and $f$ is a
superlinear continuous function with subcritical growth. By using penalization
techniques and Ljusternik-Schnirelmann theory, we investigate the relation
between the number of positive solutions with the topology of the set where the
potential attains its minimum.Comment: Mathematical Methods in the Applied Sciences (2017
Existence and asymptotic behavior of the least energy solutions for
  fractional Choquard equations with potential well

In this paper, we are concerned with the existence and asymptotic behavior of
least energy solutions for following nonlinear Choquard equation driven by
fractional Laplacian $$(-\Delta)^{s} u+\lambda V(x)u=(I_{\alpha}\ast F(u))f(u)
\ \ in \ \ R^{N},$$ where $N> 2s$, $ (N-4s)^{+}<\alpha< N$, $\lambda$ is a
positive parameter and the nonnegative potential function $V(x)$ is continuous.
By variational methods, we prove the existence of least energy solution which
localize near the potential well $int (V^{-1}(0))$ as $\lambda$ large enough.Comment: 19 pages, text overlap with arXiv:1703.0802
IBVPs for Scalar Conservation Laws with Time Discontinuous Fluxes

The initial boundary value problem for a class of scalar non autonomous
conservation laws in one space dimension is proved to be well posed and stable
with respect to variations in the flux. Targeting applications to traffic, the
regularity assumptions on the flow are extended to a merely
$\mathbf{L}^{\infty}$ dependence on time. These results ensure, for instance,
the well posedness of a class of vehicular traffic models with time dependent
speed limits. A traffic management problem is then shown to admit an optimal
solution
IB2d Reloaded: a more powerful Python and MATLAB implementation of the
  immersed boundary method

The immersed boundary method (IB) is an elegant way to fully couple the
motion of a fluid and deformations of an immersed elastic structure. In that
vein, the IB2d software allows for expedited explorations of fluid-structure
interaction for beginners and veterans to the field of computational fluid
dynamics (CFD). While most open source CFD codes are written in low level
programming environments, IB2d was specifically written in high- level
programming environments to make its accessibility extend beyond scientists
with vast programming experience. Although introduced previously by Battista et
al. 2015, many improvements and additions have been made to the software to
allow for even more robust models of material properties for the elastic
structures, including a data analysis package for both the fluid and immersed
structure data, an improved time-stepping scheme for higher accuracy solutions,
and functionality for modeling slight fluid density variations as given by the
Boussinesq approximation.Comment: 41 pages, 23 figure
Nonlinear diffusion equations as asymptotic limits of Cahn--Hilliard
  systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give
existence and uniqueness of solutions to the initial-boundary problem (P) for
the nonlinear diffusion equation in an unbounded domain
$\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as
  \[
  \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u)
  = g \quad \mbox{in}\ \Omega\times(0, T),
  \] which represents the porous media, the fast diffusion equations, etc.,
where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and
$T>0$. Existence and uniqueness for (P) were directly proved under a growth
condition for $\beta$ even though the Stefan problem was excluded from examples
of (P). This paper completely removes the growth condition for $\beta$ by
confirming Cauchy's criterion for solutions of the following approximate
problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$:
  \[
  \frac{\partial u_{\varepsilon}}{\partial t}
  + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon})
+ \pi_{\varepsilon}(u_{\varepsilon}))
  = g \quad \mbox{in}\ \Omega\times(0, T),
  \] which is called the Cahn--Hilliard system, even if $\Omega \subset
\mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be
seen that the Stefan problem is covered in the framework of this paper
A Feynman integral in Lifshitz-point and Lorentz-violating theories in
  $R^D\oplus R^m$

We evaluate a one-loop, two-point, massless Feynman integral $I_{D,m}(p,q)$
relevant for perturbative field theoretic calculations in strongly anisotropic
$d=D+m$ dimensional spaces given by the direct sum $\mathbb R^D\oplus\mathbb
R^m$. Our results are valid in the whole convergence region of the integral for
generic (non-integer) co-dimensions $D$ and $m$. We obtain series expansions of
$I_{D,m}(p,q)$ in terms of powers of the variable $X:=4p^2/q^4$, where $p=|\bm
p|$, $q=|\bm q|$, $\bm p\in\mathbb R^D$, $\bm q\in\mathbb R^m$, and in terms of
generalised hypergeometric functions $_3F_2(-X)$, when $X<1$. These are
subsequently analytically continued to the complementary region $X\ge 1$. The
asymptotic expansion in inverse powers of $X^{1/2}$ is derived. The correctness
of the results is supported by agreement with previously known special cases
and extensive numerical calculations.Comment: 33 pages, 4 figure
Fast algorithm of adaptive Fourier series

Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was
originated for the goal of positive frequency representations of signals. It
achieved the goal and at the same time offered fast decompositions of signals.
There then arose several types of AFDs. AFD merged with the greedy algorithm
idea, and in particular, motivated the so-called pre-orthogonal greedy
algorithm (Pre-OGA) that was proven to be the most efficient greedy algorithm.
The cost of the advantages of the AFD type decompositions is, however, the high
computational complexity due to the involvement of maximal selections of the
dictionary parameters. The present paper offers one formulation of the 1-D AFD
algorithm by building the FFT algorithm into it. Accordingly, the algorithm
complexity is reduced, from the original $\mathcal{O}(M N^2)$ to $\mathcal{O}(M
N\log_2 N)$, where $N$ denotes the number of the discretization points on the
unit circle and $M$ denotes the number of points in $[0,1)$. This greatly
enhances the applicability of AFD. Experiments are carried out to show the high
efficiency of the proposed algorithm.Comment: 12 pages, 12 figure
Stability of two-component incommensurate fractional-order systems and
  applications to the investigation of a FitzHugh-Nagumo neuronal model

For two-dimensional autonomous linear incommensurate fractional-order
dynamical systems with Caputo derivatives of different orders, necessary and
sufficient conditions are obtained for the asymptotic stability and instability
of the null solution. These conditions are expressed in terms of the elements
of the system's matrix, as well as of the fractional orders of the Caputo
derivatives, leading to a generalization of the well known Routh-Hurwitz
conditions. These theoretical results are then used to investigate the
stability properties of a two-dimensional fractional-order FitzHugh-Nagumo
neuronal model. The occurrence of Hopf bifurcations is also discussed.
Numerical simulations are provided with the aim of exemplifying the theoretical
results, revealing rich spiking behavior, in comparison with the classical
integer-order FitzHugh-Nagumo model.Comment: 13 pages, 4 figures. arXiv admin note: text overlap with
  arXiv:1612.0538
Global existence and asymptotic behavior of classical solutions for a 3D
  two-species chemotaxis-Stokes system with competitive kinetics

This paper considers the two-species chemotaxis-Stokes system with
competitive kinetics under homogeneous Neumann boundary conditions in a
three-dimensional bounded domain with smooth boundary. Both chemotaxis-fluid
systems and two-species chemotaxis systems with competitive terms are studied
by many mathematicians. However, there has not been rich results on coupled
two-species-fluid systems. Recently, global existence and asymptotic stability
in this problem with convection term in the fluid equation of the above system
were established in the 2-dimensional case. The purpose of this paper is to
give results for global existence, boundedness and stabilization of solutions
to this system in the 3-dimensional case.Comment: 23 page
Generalised Magnetic Polarizability Tensors

We present a new complete asymptotic expansion for the low frequency
time-harmonic magnetic field perturbation caused by the presence of a
conducting (permeable) object as its size tends to zero for the eddy current
regime of Maxwell's equations. The new asymptotic expansion allows the
characterisation of the shape and material properties of such objects by a new
class of generalised magnetic polarizability tensors and we provide an explicit
formula for their calculation. Our result will have important implications for
metal detectors since it will improve small object discrimination and, for
situations where the background field varies over the inclusion, this
information will be useable, and indeed useful, in identifying their shape and
material properties. Thus, improving the ability of metal detectors to locate
landmines and unexploded ordnance, sort metals in recycling processes, ensure
food safety as well as enhancing security screening at airports and public
events
A computational algebraic geometry approach to classify partial Latin
  rectangles

This paper provides an in-depth analysis of how computational algebraic
geometry can be used to deal with the problem of counting and classifying
$r\times s$ partial Latin rectangles based on $n$ symbols of a given size,
shape, type or structure. The computation of Hilbert functions and triangular
systems of radical ideals enables us to solve this problem for all $r,s,n\leq
6$. As a by-product, explicit formulas are determined for the number of partial
Latin rectangles of size up to six. We focus then on the study of
non-compressible regular partial Latin squares and their equivalent incidence
structure called seminet, whose distribution into main classes is explicitly
determined for point rank up to eight. We prove in particular the existence of
two new configurations of point rank eight.Comment: 28 pages, 6 figures, 7 table
Early and late stage profiles for a new chemotaxis model with
  density-dependent jump probability and quorum-sensing mechanisms

In this paper, we derive a new chemotaxis model with degenerate diffusion and
density-dependent chemotactic sensitivity, and we provide a more realistic
description of cell migration process for its early and late stages. Different
from the existing studies focusing on the case of non-degenerate diffusion, the
new model with degenerate diffusion causes us some essential difficulty on the
boundedness estimates and the propagation behavior of its compact support. In
the presence of logistic damping, for the early stage before tumour cells
spread to the whole body, we first estimate the expanding speed of tumour
region as $O(t^{\beta})$ for $0<\beta<\frac{1}{2}$. Then, for the late stage of
cell migration, we further prove that the asymptotic profile of the original
system is just its corresponding steady state. The global convergence of the
original weak solution to the steady state with exponential rate $O(e^{-ct})$
for some $c>0$ is also obtained
A 3D individual-based model to investigate the spatially heterogeneous response of bacterial biofilms to antimicrobial agents

The response of bacterial biofilms to treatment with antimicrobial agents is often characterized by the emergence of recalcitrant cellular microcolonies. We present an individual-based model to investigate the biophysical mechanisms of the selective resistance that arises within the biofilm and leads to a spatially heterogeneous response upon treatment with antibiotics. The response occurs in 3 distinct phases. In the first phase, the subpopulation of metabolically active cells diminishes due to antibiotic-induced cell death. Subsequently, in the second phase, increased nutrient availability allows dormant cells in the lower layers of the biofilm to transform into metabolically active cells. In the third phase, survival of the biofilm is governed by the interplay between 2 contrasting factors: (1) rate of antibiotic-induced cell death and (2) rate of transformation of dormant cells into active ones. Metabolically active cells at the distal edge of the biofilm sacrifice themselves to protect the dormant cells in the interior by (1) reducing local antibiotic concentrations and (2) increasing nutrient availability. In the presence of quorum sensing, biofilms exhibit increased tolerance compared with the quorum sensing-negative strains. Extracellular polymeric substance (EPS) forms a protective layer at the top of the biofilm, thereby limiting antibiotic penetration. The surviving cells, in turn, produce EPS resulting in a feedback-like mechanism of resistance. Whereas resistance in QS- biofilms occurs because of transformation of dormant cells into metabolically active cells, this transformation is less pronounced in QS+ biofilms, and resistance is a consequence of the sequestration of the antibiotic by EPS
Stability and instability of the standing waves for the
  Klein-Gordon-Zakharov system in one space dimension

The orbital instability of standing waves for the Klein-Gordon-Zakharov
system has been established in two and three space dimensions under radially
symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one
space dimensional case, for the non-degenerate situation, we first check that
the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss'
assumptions on the stability and instability theorems for abstract Hamiltonian
systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the
degenerate case that the frequency $|\omega|=1/\sqrt{2}$, we follow Wu (ArXiv:
1705.04216, 2017) to describe the instability of the standing waves for the
Klein-Gordon-Zakharov system, by using the modulation argument combining with
the virial identity. For this purpose, we establish a modified virial identity
to overcome several troublesome terms left in the traditional virial identity
Stability and Hopf bifurcation analysis of a four-dimensional
  hypothalamic-pituitary-adrenal axis model with distributed delays

A four-dimensional mathematical model of the hypothalamus-pituitary-adrenal
(HPA) axis is investigated, incorporating the influence of the GR concentration
and general feedback functions. The inclusion of distributed time delays
provides a more realistic modeling approach, since the whole past history of
the variables is taken into account. The positivity of the solutions and the
existence of a positively invariant bounded region are proved. It is shown that
the considered four-dimensional system has at least one equilibrium state and a
detailed local stability and Hopf bifurcation analysis is given. Numerical
results reveal the fact that an appropriate choice of the system's parameters
leads to the coexistence of two asymptotically stable equilibria in the
non-delayed case. When the total average time delay of the system is large
enough, the coexistence of two stable limit cycles is revealed, which
successfully model the ultradian rhythm of the HPA axis both in a normal
disease-free situation and in a diseased hypocortisolim state, respectively.
Numerical simulations reflect the importance of the theoretical results.Comment: 14 pages, 6 figure
Position dependent mass Scarf Hamiltonians generated via the Riccati equation

Producción CientíficaThe construction of position dependent mass Scarf Hamiltonians of the trigonometric
as well as the hyperbolic types is addressed by means of the factorization
method and the Riccati equation. These Hamiltonians are shown to be independent of the ordering parameter of the kinetic term. Additionally, new families of Hamiltonians with the Scarf spectrum are also determined by supersymmetry. Some examples for masses with and without singularities are considered to illustrate our results