text:
\section{Introduction} \label{intro}

According to the Standard Model of particle physics and to some of its extensions, spontaneous symmetry breaking and phase transitions constitute a crucial aspect for the early evolution of the universe.

When the temperature starts decreasing because of the expansion of the universe, spontaneous symmetry breaking can be triggered so that the interactions among elementary particles undergo (dis)continuous jumps from one phase to another. New phases with a broken symmetry will form in many regions at the same time, and in each of them only one single vacuum state will be spontaneously chosen. Sufficiently separated spatial regions may not be in causal contact, so that it is quite natural to assume that the early universe is divided into many causally disconnected patches whose size is roughly given by the Hubble radius\footnote{The Hubble radius is given by $R_{\rm H}\sim H^{-1}=a(t)/\dot{a}(t)$ where $a(t)$ is the scale factor depending on the cosmic time $t$. More precisely speaking, the vacuum is chosen over the region with the correlation length of the fields at that time.}, and in each of which the vacuum is independently determined.
As the universe expands, it can eventually happen that patches with different vacua collide in such a way that boundaries begin to form between adjacent regions with a different vacuum state.
Since the field associated with the spontaneous breaking has to vary in a continuous way between different vacua, it must interpolate smoothly from one vacuum to another via the hill of the potential. This implies that finite-energy field configurations must form at the boundaries separating patches with different vacua, and must persist even after the phase transition is completed. These objects are called \textit{topological defects}~\cite{Vilenkin:2000jqa}, and their formation mechanism (in a cosmological context) is known as \textit{Kibble mechanism}~\cite{Kibble:1976sj,Kibble:1980mv}.

Topological defects can be of several type and different spatial dimensions, and their existence is in one-to-one correspondence with the topology of the vacuum manifold~\cite{Vilenkin:2000jqa}. Domain walls are two-dimensional objects that form when a discrete symmetry is broken, so that the associated vacuum manifold is disconnected. Strings are one-dimensional objects associated to a symmetry breaking whose corresponding vacuum manifold is not simply-connected, and their formation could be predicted both by some extensions of the Standard Model of particle physics and by some classes of Grand Unified Theory (GUT). Monopoles are zero-dimensional objects whose existence is ensured when the vacuum manifold is characterized by non-contractible two-spheres, and they constitute an inevitable prediction of GUT. Moreover, there exist other topological objects called textures that can form when larger groups are broken and whose vacuum-manifold topology is more complicated.

Since the existence of topological defects is intrinsically related to the particular topology of the vacuum manifold, they can naturally appear in several theories beyond the Standard Model that predict a spontaneous symmetry breaking at some high-energy scale.
For instance, the spontaneous breaking of $SU(5)$ symmetry in GUT leads to the formation of various topological defects.
Therefore, observations and phenomenology of topological defects are very important, and should be considered as unique test-benches to test and constrain theories of particle physics and of the early universe. This also means that for any alternative theory - e.g. that aims at giving a complete ultraviolet (UV) description of the fundamental interactions - it is worth studying the existence of topological defects, investigating how their properties differ with the respect to other models/theories, and putting them to the test with current and future experiments.

In this paper we discuss for the first time topological defects in the context of \textit{nonlocal field theories} in which the Lagrangians contain infinite-order differential operators. In particular, we will make a very detailed analysis of domain wall solutions.
The type of differential operator that we will consider do not lead to ghost degrees of freedom in the particle spectrum despite the presence of higher-order time derivatives in the Lagrangian.

The work is organized as follows:
\begin{enumerate}

\item[\textbf{Sec.~\ref{sec:nonlocal-review}:}] we introduce nonlocal field theories by discussing the underlying motivations and their main properties.

\item[\textbf{Sec.~\ref{sec-review}:}] we briefly review the domain wall solution in the context of standard (local) two-derivative theories by highlighting various features whose mathematical and physical meanings will be important for the subsequent sections.

\item[\textbf{Sec.~\ref{sec-nft-dw}:}] we analyze for the first time domain wall solutions in the context of ghost-free nonlocal field theories by focusing on the simplest choice for the infinite-order differential operator in the Lagrangian. Despite the high complexity of non-linear and infinite-order differential equations, we will be able to find an approximate analytic solution by relying on the fact that the topological structure of the vacuum manifold ensures the existence of an exact domain wall configuration.
Firstly, we analytically study the asymptotic behavior of the solution close to the two symmetric vacua.
Secondly, we find a linearized nonlocal solution by perturbing around the local domain wall configuration. We show that the linearized treatment agrees with the asymptotic analysis, and make remarks on the peculiar behavior close to the origin. We perform an order-of-magnitude estimation of width and energy per unit area of the domain wall. Furthermore, we derive a theoretical lower bound on the scale of nonlocality for the specific domain wall configuration under investigation.

\item[\textbf{Sec.~\ref{sec-other}:}] we briefly comment on other topological defects like string and monopole.

\item[\textbf{Sec.~\ref{sec-dis}:}] we summarize our results, and discuss both theoretical and phenomenological future tasks.

\item[\textbf{App.~\ref{sec-corr}:}] we develop a formalism to confirm the validity of the linearized solution close to the origin.

\item[\textbf{App.~\ref{app-em}:}] we find a compact expression for the energy-momentum tensor in a generic nonlocal (infinite-derivative) field theory, and apply it to the specific nonlocal scalar model analyzed in the main text.

\end{enumerate}

We adopt the mostly positive convention for the metric signature, $\eta=\diag(-,+,+,+),$ and work with the natural units system, $c=\hbar=1.$

\section{Nonlocal field theories}\label{sec:nonlocal-review}

The wording `nonlocal theories' is quite generic and, in principle, can refer to very different theories due to the fact that the nonlocal nature of fields can manifest in various ways. In this work with `nonlocality' we specifically mean that Lagrangians are made up of certain non-polynomial differential operators containing infinite-order derivatives.

A generic nonlocal Lagrangian contains both polynomial and non-polynomial differential operators, i.e. given a field $\phi(x)$ one can have
\begin{equation}
\mathcal{L}\equiv\mathcal{L}\left(\phi,\partial\phi,\partial^2\phi,\dots,\partial^n\phi,\frac{1}{\Box}\phi,{\rm ln}\left(- \Box/M_s^2\right)\phi,e^{\Box/M_s^2}\phi,\dots\right),\label{nonlocal-lagr}
\end{equation}
where $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ is the flat d'Alambertian and $M_s$ is the energy scale at which nonlocal effects are expected to become important.
Non-analytic differential operators like $1/\Box$ and ${\rm log}(-\Box)$ are usually important at infrared (IR) scales, e.g. they can appear as contributions in the finite part of the quantum effective action in standard perturbative quantum field theories~\cite{Barvinsky:2014lja,Woodard:2018gfj}. Whereas analytic operators like $e^{\Box/M_s^2}$ are usually responsible for UV modifications and do not affect the IR physics. Such a transcendental differential operator typically appears in the context of string field theory~\cite{Witten:1985cc,Gross:1987kza,Eliezer:1989cr,Tseytlin:1995uq,Siegel:2003vt,Pius:2016jsl,Erler:2020beb} and p-adic string~\cite{Freund:1987kt,Brekke:1988dg,Freund:1987ck,Frampton:1988kr,Dragovich:2020uwa}.

We are interested in alternative theories that extend the Standard Model in the UV regime, therefore we will focus on analytic differential operators. In general, we can consider a scalar Lagrangian of the following type\footnote{To keep the formula simpler we do not write the scale of nonlocality in the argument of $F(-\Box)$ which, to be more precise, should read $F(-\Box/M_s^2).$}:
\begin{equation}
\mathcal{L}=-\frac{1}{2}\phi F(-\Box)\phi - V(\phi)\,,\label{analytic-Lag}
\end{equation}
where $V(\phi)$ is a potential term, and the the kinetic operator can be defined through its Taylor expansion
\begin{equation}
F(-\Box)=\sum\limits_{n=0}^\infty f_n(-\Box)^n\,,\label{nl-oper}
\end{equation}
where $f_n$ are constant coefficients. It should now be clear that the type of nonlocality under investigation manifests through the presence of infinte-order derivatives.

To recover the correct low-energy limit and avoid IR modifications, it is sufficient to require that the function $F(z),$ with $z\in \mathbb{C},$ does not contain any poles in the complex plane. Thus, we choose $F(-\Box)$ to be an \textit{entire function} of the d'Alembertian $\Box$.

By making use of the Weierstrass factorization theorem for entire functions we can write
\begin{equation}
F(-\Box)=e^{\gamma(-\Box)}\prod\limits_{i=1}^{N}(-\Box+m_i^2)^{r_i}\,,
\end{equation}
where $\gamma(-\Box)$ is another entire function, $m_i^2$ are the zeroes of the kinetic operator $F(-\Box),$ and $r_i$ is the multiplicity of the $i$-th zero. The integer $N\geq 0$ counts the number of zeroes and, in general, can be either finite or infinite.

To prevent the appearance of ghost degrees of freedom, it is sufficient to exclude the possibility to have extra zeroes besides the standard two-derivative one\footnote{It is worth mentioning that this is \textit{not} the unique possibility for ghost-free higher derivative theories. In fact, we can allow additional pairs of complex conjugate poles and still avoid ghost degrees of freedom and respect unitarity, in both local~\cite{Modesto:2015ozb,Modesto:2016ofr,Anselmi:2017yux,Anselmi:2017lia} and nonlocal theories~\cite{Buoninfante:2018lnh,Buoninfante:2020ctr}. Moreover, tree-level unitarity was shown to be satisfied also if one admits branch cuts in the bare propagator~\cite{Abel:2019ufz,Abel:2019zou}.}. We impose that the kinetic operator does not contain any additional zeroes, so that the effects induced by new physics are entirely captured by the differential operator $e^{\gamma(-\Box)}.$ Therefore, we consider the following Lagrangian:
\begin{equation}
\mathcal{L}=\frac{1}{2}\phi\, e^{\gamma(-\Box)}(\Box-m^2)\,\phi -V(\phi)\,,\label{exp-nonlocal}
\end{equation}
whose propagator reads
\begin{equation}
\Pi(p^2)=-i\frac{e^{-\gamma(p^2)}}{p^2+m^2}\,.\label{propag-nonlocal}
\end{equation}
From the last equation it is evident that no additional pole appears other than $p^2=-m^2,$ because $e^{-\gamma(p^2)}$ is an exponential of an entire function and as such does not have poles in the complex plane.

More generally, under the assumption that the transcendental function $e^{-\gamma(p^2)}$ is convergent in the limits $p^0\rightarrow\pm i\infty,$ it was shown that an S-matrix can be well defined for the Lagrangian~\eqref{exp-nonlocal}, and it can be proven to satisfy perturbative unitarity at any order in loop~\cite{Pius:2016jsl,Briscese:2018oyx,Chin:2018puw,Koshelev:2021orf}. Moreover, the presence of the exponential function can make loop-integrals convergent so that the scalar theory in Eq.~\eqref{exp-nonlocal} turns out to be finite in the high-energy regime~\cite{Krasnikov:1987yj,Moffat:1990jj,Tomboulis:2015gfa,Buoninfante:2018mre}. Very interestingly for such nonlocal theories, despite the presence of infinite-order time derivatives, a consistent initial value problem can be formulated in terms of a finite number of initial conditions~\cite{Barnaby:2007ve,Calcagni:2018lyd,Erbin:2021hkf}.

This type of transcendental operators with some entire function $\gamma(-\Box)$ have been intensely studied in the past years not only in the context of quantum field theories in flat space~\cite{Krasnikov:1987yj,Moffat:1990jj,Tomboulis:2015gfa,Buoninfante:2018mre,Boos:2018kir,Boos:2019fbu}, but also to formulate ghost-free infinite-derivative theories of gravity~\cite{Krasnikov:1987yj,Kuzmin:1989sp,Tomboulis:1997gg,Biswas:2005qr,Modesto:2011kw,Biswas:2011ar,Frolov:2015bia,Frolov:2015bta,Buoninfante:2018xiw,Buoninfante:2018stt,Buoninfante:2018xif,Koshelev:2016xqb,Koshelev:2017tvv,Koshelev:2020foq,Kolar:2020max}.

In this work we assume that fundamental interactions are intrinsically nonlocal, and that nonlocality becomes relevant in the UV regime. Thus, we consider nonlocal quantum field theories as possible candidates for UV-complete theories beyond the Standard Model. With this in mind, we will analyze topological defects in infinite-derivative field theories, and investigate the physical implications induced by nonlocality in comparison to standard (local) two-derivative theories.

In what follows, we work with the simplest ghost-free nonlocal model for which the entire function is given by
\begin{equation}
\gamma(-\Box)=-\frac{\Box}{M_s^2}\,.\label{entire-box^1}
\end{equation}

\section{Standard domain wall: a brief review} \label{sec-review}

Before discussing domain walls in the context of nonlocal quantum field theories, it is worth reminding some of their basic properties in standard two-derivative field theories, which will then be useful for the main part of this work.

In presence of a domain wall one has to deal with a static scalar field that only depends on one spatial coordinate, e.g. $x,$ and whose Lagrangian reads~\cite{Vilenkin:2000jqa,Saikawa:2017hiv}
\begin{align}
	{\cal L} = \frac{1}{2}\qty(\partial_x\phi)^2 -U(\phi)\,,\qquad U(\phi)=\frac{\lambda}{4}(\phi^2-v^2)^2\,,\label{local-Lag-wall}
\end{align}
which is $\mathbb{Z}_2$-symmetric as it is invariant under the transformation $\phi\rightarrow-\phi;$ $\lambda>0$ is a dimensionless coupling constant and $v>0$ is related to the symmetry-breaking energy scale. The quartic potential has two degenerate minima at $\phi=\pm v$ ($U(\pm v)=0$).

As mentioned in the Introduction, the discrete symmetry $\mathbb{Z}_2$ can be spontaneously  broken, for instance, in the early universe because of thermal effects. As a consequence, causally disconnected regions of the universe can be characterized by a different choice of the vacuum (i.e. $\phi=+v$ or $\phi=-v$), and when two regions with different vacua collide a continuous two-dimensional object -- called domain wall -- must form at the boundary of these two regions.

Let us now determine explicitly such a finite-energy configuration interpolating $\pm v.$
First of all, we impose the asymptotic boundary conditions
\begin{align}
	\phi(-\infty) = -v\,,
	\qquad
	\phi(\infty) = v\,.\label{boundary conditions}
\end{align}
The field configuration must be non-singular and of finite energy, therefore $\phi(x)$ must interpolate smoothly between the two vacua, this implies that there exists a point $x_0\in \mathbb{R}$ such that $\phi(x_0)=0.$ Without any loss of generality, we can choose the reference frame such that the centre of the wall is at the origin $x_0=0,$ i.e. $\phi(0)=0.$

The energy density can be computed as
\begin{align}
	\mathcal{E}(x) \equiv T_{0}^0(x)= \frac{1}{2}(\del_x\phi)^2 + \frac{\lambda}{4}(\phi^2-v^2)^2\,,\label{local-e-dens}
\end{align}
from which it follows  $\mathcal{E}(x)\geq U(0)=\lambda v^2/4,$ and this implies that  there exists a solution that does not dissipate at infinity. Hence, the topological structure of the vacuum manifold -- which is disconnected in the case of $\mathbb{Z}_2$ symmetry -- ensures the existence of a non-trivial field configuration of finite energy.

We can determine qualitatively the behavior of this field configuration by making an order-of-magnitude estimation of the width $R$ (along the $x$-direction), and of the energy per unit area $E$ of the wall.

In fact, we can define the width of the wall in three ways. The first one is to use the energy density in Eq.~\eqref{local-e-dens}. The lowest energy configuration interpolating the two vacua can be found by balancing the kinetic and potential term in the energy density $\mathcal{E}(x)$.

By approximating the gradient with the inverse of the width, $\partial_x\sim 1/R,$ and the field value with $\phi\sim v,$ Eq.~\eqref{local-e-dens} gives
\begin{align}
	\frac{1}{2} \frac{1}{R^2} v^2 \sim \frac{\lambda}{4} v^4\quad
		\Rightarrow\quad
	R \sim \sqrt{\frac{2}{\lambda}} \frac{1}{v}\,,\label{local-radius-estim}
\end{align}
from which it follows that the width of the wall is of the same order of the Compton wavelength $R\sim(\sqrt{\lambda}v)^{-1}\sim m^{-1}$.
Whereas, the energy per unit area can be estimated as
\begin{eqnarray}
	E&=& \int_\mathbb{R}{{\rm d}x}\qty[\frac{1}{2}(\del_x\phi)^2+\frac{\lambda}{4}(\phi^2-v^2)^2] \nonumber\\[2mm]
	 &\sim& (\text{width of the wall}) \times (\text{energy density}) \nonumber\\[2mm]
		& \sim& R \times \lambda v^4 \nonumber\\[2mm]
		&\sim&\sqrt{\lambda}v^3\,.\label{local-energ-estim}
\end{eqnarray}
The other two ways are to use the exact configuration (solution) of the domain wall.
The field equation
\begin{align}
	\partial_x^2 \phi(x) = \lambda \phi(\phi^2-v^2)\label{local-field-eq}
\end{align}
can be solved by quadrature, and an exact analytic solution can be found, and it satisfies all the qualitative properties discussed above.
The exact solution is sometime called `kink', and it reads~\cite{Vilenkin:2000jqa}
\begin{align}
	\phi(x) = v \tanh \left(\sqrt{\frac{\lambda}{2}}vx\right)\,.\label{local-dom-wal-sol}
\end{align}
Its asymptotic behavior is given by
\begin{align}
|x|\rightarrow \infty\quad \Rightarrow \quad \phi(x)\sim \pm v\qty(1-2e^{-\sqrt{2\lambda}vx})\,.
		\label{eq:asymp sol in local case}
\end{align}
Through the exact solution~\eqref{local-dom-wal-sol}, we can define the width of the wall in two ways. One way is to identify it with the typical length scale over which $\phi(x)$ changes in proximity of the origin, that is, the length scale $\ell$ defined as the inverse of the gradient at the origin, i.e. $\ell\sim v/(\partial_x\phi|_{x=0}),$ where the scale $v$ is introduced for dimensional reasons. From Eq.~\eqref{local-dom-wal-sol} we have
\begin{equation}
\del_x\phi(x)|_{x=0}=v^2\sqrt{\frac{\lambda}{2}},
\end{equation}
which yields
\begin{equation}
\ell\sim \sqrt{\frac{2}{\lambda}}\frac{1}{v} = R\,.\label{ell-linearized-local}
\end{equation}
The last way is to use the asymptotic behavior given in Eq.~\eqref{eq:asymp sol in local case}. The width of the wall, $\widetilde{R}$, can be defined as
\begin{align}
|x|\rightarrow \infty\quad \Rightarrow \quad \phi(x)\sim \pm v\qty(1-2e^{-\frac{2x}{\widetilde{R}}})\,,
		\label{eq:asymp radius}
\end{align}
which yields $\widetilde{R} \sim \sqrt{\frac{2}{\lambda}} \frac{1}{v} = R = \ell\,.$ In the local case, all of the three definitions give the same expressions and we need not discriminate them. But, as we will show, in the nonlocal case, all of the definitions would give different expressions in the sub-leading order, though two of them ($R$ and $\widetilde{R}$) have the similar feature. In fact, $R$ and/or $\widetilde{R}$ might be more appropriate as the definition of the width (or the radius) of a domain wall because $\ell$ is related to the behavior of the solution close to the origin and far from the vacuum.

We can obtain the energy per unit area as
\begin{eqnarray}
       E = \int_{\mathbb{R}} {\rm d}x\mathcal{E}(x) =\int_{\mathbb{R}} {\rm d}x\qty(\frac{{\rm d}\phi}{{\rm d}x})^2
		= \frac{4}{3} \sqrt{\frac{\lambda}{2}}v^3\,,
\end{eqnarray}
which is consistent with the estimation in Eq.~\eqref{local-energ-estim} up to an order-one numerical factor.

The discussion in this Section was performed for a local two-derivative theory in one spatial dimension. However, the essential concepts and methods, like the condition for the existence of a solution related to the non-trivial topology of the vacuum manifold, and the order-of-magnitude estimations, can be applied to nonlocal field theories and to higher dimensional cases (e.g. string and monopole).

\section{Domain wall in nonlocal field theories} \label{sec-nft-dw}

In this Section we analyze the domain wall solution for the nonlocal field theory~\eqref{exp-nonlocal} with the simplest choice of entire function given in~\eqref{entire-box^1}. Hence, we consider a nonlocal generalization of the Lagrangian~\eqref{eq:Lagrangian for DW} given by
\begin{align}
	{\cal L}
		= \frac{1}{2}\phi e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi
				-\frac{\lambda}{4}\qty(\phi^4+v^4)\,,
	\label{eq:Lagrangian for DW}
\end{align}
whose field equation reads
\begin{align}
	e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi = \lambda\phi^3\,.
		\label{eq:the equation of the motion for DW}
\end{align}
We can easily verify that in the limit $\partial_x^2/M_s^2\rightarrow 0$ we consistently recover the local two-derivative case, i.e. Eqs.~\eqref{local-Lag-wall} and~\eqref{local-field-eq}.

In this case the differential equation is non-linear and highly nonlocal, and finding a solution seems to be very difficult not only analytically but even numerically. However, despite the complexity of the scalar field equation, we can still find a domain wall configuration. First of all, the existence of such a solution is still guaranteed by the non-trivial topological structure of the vacuum manifold which is disconnected in the case of $\mathbb{Z}_2$ symmetry. Indeed, also in this case a finite-energy field configuration that smoothly interpolates between the two vacua $\phi=\pm v$ must exist. Furthermore, by relying on the fact that the presence of the exponential operator $e^{-\partial_x^2/M_s^2}$ should not change the number of degrees of freedom and of initial conditions~\cite{Barnaby:2007ve}, we can impose the same boundary conditions as done in the two-derivative case in Eq.~\eqref{boundary conditions}, i.e. $\phi(\pm \infty)=\pm v.$ We can set $x_0=0$ to be the centre of the wall such that $\phi(0)=0,$ and expect $\phi(x)$ to be a smooth odd function.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.6]{schematic-illustration.jpg}
	\caption{Schematic illustration of the analysis made in this Section to study the domain wall configuration in nonlocal field theory, whose qualitative behavior is drawn consistently with the boundary conditions $\phi(\pm\infty)=\pm v$, and with the choice of the origin $\phi(0)=0.$ We analyze the behavior of the domain wall configuration in several regimes, and use different methods to find approximate analytic solutions. (i) We study the asymptotic behavior of the domain wall solution in the limit $|x|\rightarrow \infty.$ (ii) We find a linearized nonlocal solution by perturbing around the local domain wall configuration treated as a background, and analyse its behavior not only at infinity but also close to the origin. (iii) We make an order-of-magnitude estimation for the width and the energy per unit area of the wall, and verify the consistency with the analytic approximate solutions.}
	\label{fig:conceptual figure for analysis of nonlocal domain wall}
\end{figure}

Very interestingly, knowing that a domain wall solution must exist, and equipped with a set of boundary conditions, we can still find an approximate analytic solution by working perturbatively in some regime. We will proceed as follows. In Sec.~\ref{subsec-asym} we will determine the solution asymptotically close to the vacua $\pm v$ (i.e. for $|x|\rightarrow \infty$). In Sec.~\ref{subsec-pert} we will find a linearized nonlocal solution by perturbing around the known local `kink' configuration treated as a background. Finally, in Sec.~\ref{subsec-order} we will make an order-of-magnitude estimation for the width and the energy of the nonlocal domain wall, and check the consistency with the approximate analytic solutions. See Fig.~\ref{fig:conceptual figure for analysis of nonlocal domain wall} for a schematic illustration of our analysis.

\subsection{Asymptotic solution for $|x|\rightarrow \infty$} \label{subsec-asym}

As a first step we analyze the asymptotic behavior of the solution close to the two vacua $\phi=\pm v$, i.e. in the regime $|x|\rightarrow \infty.$ Let us first consider the perturbation around $\phi=+v,$ i.e. we write
\begin{equation}
    \phi(x)=v+\delta\phi(x)\,,\qquad \frac{|\delta\phi|}{v}\ll 1\,,
\end{equation}
so that the linearized field equation reads
\begin{align}
	e^{-\del_x^2/M_s^2} (\del_x^2+\lambda v^2) \delta\phi = 3\lambda v^2 \delta\phi\,.
\end{align}
Taking inspiration from the asymptotic behavior of the domain wall in the local case (see Eq.~\eqref{eq:asymp sol in local case}), as an ansatz we assume that $\phi(x)$ approaches the vacuum exponentially, i.e. we take
\begin{align}
	\delta\phi = \phi - v = A e^{-B x} \label{eq:def for asymp sol}
\end{align}
where $A,$ and $B>0$ are two constants.

Since the exponential is an eigenfunction of the kinetic operator, we can easily obtain an equation for $B,$
\begin{align}
	e^{-B^2/M_s^2} (B^2+\lambda v^2)  = 3\lambda v^2\,.\label{eq-for-B}
\end{align}
By using the principal branch $W_0(x)$ of the Lambert-W function (defined as the inverse function of $f(x)=xe^x$) we can solve Eq.~\eqref{eq-for-B} as follows
\begin{align}
	B^2 = - M_s^2\, W_0\qty(-\frac{3\lambda v^2}{M_s^2}e^{-\lambda v^2/M_s^2})-\lambda v^2\,.
		\label{eq:coeff B in asymp sol}
\end{align}
Before continuing let us make some remarks on the Lambert-W function. It is a multivalued function that has an infinite number of branches $W_n$ with $n\in \mathbb{Z}.$ The only real solutions are given by the branch $W_0(x)$ for $x\geq -1/e,$ and an additional real solution comes from the branch $W_{-1}(x)$ for $-1/e\leq x<0.$ In the equations above we have taken the so-called principal branch $W_0,$ and we will do the same in the rest of the paper. However, we will also comment on the branch $W_{-1}$ and the physical implications associated to it. Regarding higher order branches $n>0,$ they will generate non-physical complex values, and in some cases they do not even recover the local limit; therefore, we discard such solutions.

Note that, by means of the asymptotic analysis above the coefficient $A$ cannot be determined as it factors out from the field equation but, as we will explain in Sec.~\ref{subsec-pert}, we will be able to determine it up to order $\mathcal{O}(1/M_s^2)$.

\subsubsection{A theoretical constraint on the scale of nonlocality} \label{subsubsec-cond}

The use of the Lambert-W function to obtain the solution for $B$ in Eq.~\eqref{eq:coeff B in asymp sol} relied on the fact that  Eq.~\eqref{eq-for-B} could be inverted. As explained above this inversion is valid if and only if $c \in \{xe^x|x\in\mathbb{R}\}$, i.e. if $c\geq -1/e$.

Applying this condition to~\eqref{eq:coeff B in asymp sol}, we get
\begin{align}
	-\frac{3\lambda v^2}{M_s^2}e^{-\lambda v^2/M_s^2}\geq -\frac{1}{e}\,,\label{condition}
\end{align}
and inverting in terms of the (principal branch) Lambert-W function we obtain the following \textit{theoretical constraint}:
\begin{align}
M_s^2\geq  -\frac{\lambda v^2}{W_0\qty(-1/3e)}\,,	\label{eq:neccessary cond for asymp sol}
\end{align}
where we have used the fact that $W_0(x)$ is a monotonically increasing function. The inequality~\eqref{eq:neccessary cond for asymp sol} means that
the energy scale of nonlocality must be greater than the symmetry-breaking scale $\sqrt{\lambda} v.$ We can evaluate $W_0\qty(-1/3e)\simeq -0.14,$  so that the lower bound reads $M_s^2\gtrsim 7.14 \lambda v^2.$
One usually obtains constraints on the free parameters of a theory by using experimental data. In the present work, instead, we found a purely theoretical constraint. See Sec.~\ref{sec-dis} for further discussions on this feature.

Given the fact that $\lambda v^2/M_s^2< 0.14,$ we can expand~\eqref{eq:coeff B in asymp sol} and obtain
\begin{align}
	B^2
		= 2 \lambda v^2 \qty(1+\frac{3\lambda v^2}{M_s^2})
			+ {\cal O}\qty(\qty(\frac{\lambda v^2}{M_s^2})^2)\,,
	\label{eq:series expansion for coeff B}
\end{align}
or by taking the square root,
\begin{align}
	B
		= \sqrt{2 \lambda} v \qty(1+\frac{3}{2}\frac{\lambda v^2}{M_s^2})
			+ {\cal O}\qty(\qty(\frac{\lambda v^2}{M_s^2})^2)\,\,.
	\label{eq:series expansion for coeff B-sqrt}
\end{align}
From this expression, we can obtain the width of wall, $\widetilde{R}$, defined in Eq.~\eqref{eq:asymp radius} as
\begin{equation}
\widetilde{R} \sim \frac{2}{B}\sim \sqrt{\frac{2}{\lambda}}\frac{1}{v}\left(1-\frac32\frac{\lambda v^2}{M_s^2}\right)\,.\label{eq:tildeR}
\end{equation}
Also, from Eq.~\eqref{eq:series expansion for coeff B-sqrt} we can check that in the local limit $M_s\rightarrow \infty$ we recover the two-derivative case in Eq.~\eqref{eq:asymp sol in local case}:
\begin{align}
	\lim_{M_s\to\infty} B^2 = 2 \lambda v^2 = \qty(\sqrt{2\lambda}v)^2\equiv B_{\rm L}^2\,.
\end{align}
Furthermore, from~\eqref{eq:series expansion for coeff B} we can notice that $B\geq B_{\rm L}=\qty(\sqrt{2\lambda}v).$ This physically means that the nonlocal domain wall solution approaches the vacuum $\phi=+v$ \textit{faster} as compared to the local two-derivative case. This feature, which is manifest in Eq.~\eqref{eq:tildeR}, may also suggest that the width of the nonlocal domain wall is \textit{smaller} as compared to the local case; indeed this fact will also be observed with the expression of $R$ in the next Subsections.

So far we have only focused on the asymptotic solution for $x\rightarrow +\infty$ ($\phi(+\infty)=+v$) but the same analysis can be applied to the other asymptotic $x\rightarrow -\infty$ ($\phi(-\infty)=-v$), and the same results hold because of the $\mathbb{Z}_2$ symmetry.

\paragraph{Remark 1.} Before concluding this Subsection it is worth commenting on the validity of the asymptotic solution we determined. On one hand, we know that the existence of the domain wall is ensured by the topological structure of the vacuum manifold, and this should not depend on the value of $M_s.$ On the other hand, it appears that the asymptotic solution we found is only valid for some values of $M_s$ satisfying the inequality in Eq.~\eqref{eq:neccessary cond for asymp sol}, which seems to imply that the domain wall solution does not exist for other values of $M_s$. Is this a contradiction? The answer is no, all is consistent, and in fact the domain wall solution exists for any value of $M_s.$

First of all, we should note that to solve Eq.~\eqref{condition} in terms of $M_s$ we have used the principal branch $W_0(x)$ which is a monotonic increasing function, but an additional real solution can be found by using the branch $W_{-1}(x)$ which is, instead, a monotonically decreasing function. Thus, given the opposite monotonicity behavior of $W_{-1}$ as compared to $W_{0},$ if we solve Eq.~\eqref{condition} by means of $W_{-1}$ we get $M_s\leq -\lambda v^2/W_{-1}(-1/3e)\simeq 0.30\lambda v^2.$
Moreover, in the range of values $0.30 \lambda v^2 \lesssim M_s^2\lesssim 7.21 \lambda v^2$ the functional form in Eq.~\eqref{eq:def for asymp sol} does not represent a valid asymptotic behavior for the domain wall. In this case the domain wall configuration may be characterized by a completely different profile, but its existence is still guaranteed by the non-trivial topology.
Anyway, as already mentioned above, in this paper we only work with $W_0,$ therefore with values of $M_s$ satisfying the inequality~\eqref{eq:neccessary cond for asymp sol}. See also Sec.~\ref{sec-dis} for more discussion on this in relation to physical implications.

\subsection{Perturbation around the local solution} \label{subsec-pert}

Let us now implement an alternative method to determine the behavior of the nonlocal domain wall not only at infinity but also close to the origin.

We consider a linear perturbation around the standard two-derivative domain wall configuration $\phi_{\rm L}(x)=v\tanh(\sqrt{\lambda/2} vx)$. Let us define the deviation from the local solution as
\begin{align}
	\delta\phi (x) = \phi(x) - \phi_{\rm L}(x)\,, \qquad  \left|\frac{\delta\phi}{\phi_{\rm L}}\right|\ll 1\,, \label{pert-loc-nonloc}
\end{align}
in terms of which we can linearize the field equation~\eqref{eq:the equation of the motion for DW}:
\begin{eqnarray}
\left[e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)-3\lambda\phi_{\rm L}^2\right]\delta\phi=\lambda (1-e^{-\partial_x^2/M_s^2})\phi_{\rm L}^3\,.\label{linear-lnl}
\end{eqnarray}
Since the nonlocal scale appears squared in~\eqref{eq:the equation of the motion for DW}, we would expect that $\delta\phi\sim \mathcal{O}(1/M_s^2)$ such that the local limit is consistently recovered, i.e. $\delta\phi\to0$ when $M_s^2\to\infty$. We now write Eq.~\eqref{pert-loc-nonloc} up to order $\mathcal{O}(1/M_s^2)$ in order to extract the leading nonlocal correction to $\phi_{\rm L}.$ By expanding the nonlocal terms as follows
\begin{eqnarray}
	e^{-\del_x^2/M_s^2} \delta\phi&=&
	\delta\phi
	+ {\cal O}\qty(\frac{\del_x^2}{M_s^2}) \delta\phi\,,\\[2mm]
	e^{-\del_x^2/M_s^2} \phi_{\rm L}^3&=& \phi_{\rm L}^3-\frac{\partial_x^2}{M_s^2}\phi_{\rm L}^3
	+{\cal O}\qty(\qty(\frac{\del_x^2}{M_s^2})^2)\phi_{\rm L}^3\,,
\end{eqnarray}
we can write~\eqref{linear-lnl} up to order $\mathcal{O}(1/M_s^2):$
\begin{align}
	\qty[\partial_x^2+\lambda v^2-3\lambda\phi_{\rm L}^2(x)] \delta\phi
		= \frac{\lambda}{M_s^2}\del_x^2(\phi_{\rm L}(x)^3)\,.
		\label{lin-eq-nln-2}
\end{align}
This expansion is valid as long as the following inequality holds:
\begin{equation}
\frac{1}{|\delta\phi|}\left|\frac{\del_x^2}{M_s^2}\delta\phi\right| \ll 1\,.\label{ineq-2}
\end{equation}

We now introduce the dimensionless variable $s=\sqrt{\lambda/2}\,vx$ and the function $f(s)=\delta\phi(x)/v,$ so that we can recast Eq.~\eqref{lin-eq-nln-2} as
\begin{align}
	f^{\prime\prime}(s)+2(1-3\tanh^2 s)f(s) = \frac{\lambda v^2}{M_s^2}(\tanh^3 s)^{\prime\prime}\label{diff-eq-s}
\end{align}
where the prime $'$ denotes the derivative with respect to $s$. The above differential equation can be solved analytically, and its solution reads
\begin{eqnarray}
\!\!f(s)&=&
	\frac{C_1}{\cosh^2s}
	+ \frac{3C_2/2+27\lambda v^2/32M_s^2}{\cosh^2s} \log\frac{1+\tanh s}{1-\tanh s}
\nonumber\\[2mm]
	&&+\qty[
		\qty(2C_2+\frac{\lambda v^2}{8M_s^2})\cosh^2s
			+ \qty(3C_2-\frac{61\lambda v^2}{16M_s^2})
			+ \frac{2\lambda v^2}{M_s^2}(1+\tanh^2 s)
	] \tanh s\,,\,\,\,
\end{eqnarray}
where $C_1$ and $C_2$ are two integration constants to be determined.

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.48]{perturb-sol.pdf}
	\caption{
			%
			In this figure we show the linearized nonlocal domain wall solution $\phi(x)=\phi_{\rm L}(x)+\delta\phi(x)$ (solid blue line) in comparison with the local domain wall (orange dashed line). The nonlocal configuration approaches the asymptotic vacua at $x\rightarrow \pm \infty$ faster as compared to the local case.
			In the smaller plot we showed the behavior of the two solutions over a smaller interval in order to make more evident the differences between local and nonlocal cases. We can notice that when going from $x=0$ to $x\rightarrow \pm \infty$ the nonlocal curve slightly oscillates around the local one.

			We set $\lambda=2$, $v=1$ and $M_s^2=14.3,$ which are consistent with the theoretical constraint $M_s^2\geq -\lambda v^2/W_0(-1/3e)$ in Eq.~\eqref{eq:neccessary cond for asymp sol}.
			}
	\label{fig:illustration for the perturbative solution}
\end{figure}

The boundary conditions $\phi(\pm \infty)=\pm v$ in terms of the linearized deviation read $\delta\phi(\pm \infty)=0,$ or equivalently $f(\pm \infty)=0$. These are satisfied if and only if the algebraic relation $2C_2+\lambda v^2/8M_s^2=0$ holds true, which means that $C_2=-\lambda v^2/16M_s^2$.
Moreover, the constant $C_1$ must be zero because of the $\mathbb{Z}_2$-symmetry.
Thus, the solution for $\delta \phi$ is given by
\begin{align}
	\delta\phi
		= vf(s)
		= \frac{\lambda v^3}{M_s^2}\frac{1}{\cosh^2\sqrt{\frac{\lambda}{2}}vx}
			\qty[\frac{3}{4}\log\frac{1+\tanh\sqrt{\frac{\lambda}{2}}vx}{1-\tanh\sqrt{\frac{\lambda}{2}}vx}
					-2\tanh\sqrt{\frac{\lambda}{2}}vx]\,.
	\label{eq:exact sol for delta phi}
\end{align}
In Fig.~\ref{fig:illustration for the perturbative solution} we showed the behavior of the nonlocal domain wall solution $\phi=\phi_{\rm L}+\delta\phi$ in comparison with the local two-derivative one $\phi_{\rm L};$ we have set values for $v,$ $\lambda$ and $M_s$ consistently with the theoretical lower bound in Eq.~\eqref{eq:neccessary cond for asymp sol}. From the plot we can notice that the nonlocal solution approaches the vacua $\pm v$ faster as compared to the local case, which is in agreement with the asymptotic analysis in the previous Subsection. Indeed, we can expand the solution $\phi=\phi_{\rm L}+\delta\phi$ in the regime $|x|\rightarrow \infty$, and obtain
\begin{eqnarray}
\phi(x)&\simeq& \pm v\left[1-2\left(\frac{4v^2\lambda}{M^2_s}\right)e^{-\sqrt{2\lambda}vx}-2e^{-\sqrt{2\lambda}vx}\left(1-\frac{3}{2}\sqrt{2\lambda}\frac{\lambda v^3}{M_s^2}x\right)\right]\nonumber\\[2mm]
&\simeq& \pm v\left[1-2\left(1+\frac{4v^2\lambda}{M^2_s}\right)e^{-\sqrt{2\lambda}v(1+3\lambda v^2/2M_s^2)x}\right]\nonumber\\[2mm]
&\simeq& \pm v\left[1-2\left(1+\frac{4v^2\lambda}{M^2_s}\right)e^{-B x}\right]\,,
\end{eqnarray}
where to go from the first to the second line we have used the freedom to add negligible terms of order higher than $\mathcal{O}(1/M_s^2),$ i.e. $4v^2\lambda/M_s^2\simeq 4v^2\lambda/M_s^2(1-3\sqrt{2\lambda}\lambda v^3x/2M_s^2)$ and $1-3\sqrt{2\lambda}\lambda v^3/2M_s^2x\simeq e^{-(3\sqrt{2\lambda}\lambda v^3/2M_s^2)x}.$ Remarkably, the asymptotic behavior of the linearized solution perfectly matches the result obtained in Eq.~\eqref{eq:series expansion for coeff B-sqrt}, indeed the coefficient $B$ in the exponent turns out to be exactly the same in both approaches. Moreover, from the linearized solution we can also determine the coefficient $A$ up to order $\mathcal{O}(1/M_s^2)$, i.e. $A=-2v-8v^3\lambda/M_s^2,$ which could not be determined through the asymptotic analysis in Sec.~\ref{subsec-asym} (see Eq.~\eqref{eq:def for asymp sol}).

Furthermore, the behavior of the linearized solution close to the origin is quite peculiar as the nonlocal domain wall profile slightly oscillates around the local one. In other words, when going from $x=0$ to $x\rightarrow \infty$ the perturbation $\delta\phi$ is initially negative and then becomes positive; whereas the opposite happens when going from $x=0$ to $x\rightarrow-\infty.$
This property may suggest that the typical length scale $\ell$ over which $\phi(x)$ changes in proximity of the origin is larger as compared to the local case. As done for the local domain wall in Sec.~\ref{sec-review}, we can estimate such a length scale as the inverse of the gradient at the origin times the energy scale $v$, i.e.  $\ell\sim v/(\partial_x\phi|_{x=0}).$
By doing so, up to order $\mathcal{O}(1/M_s^2)$ we get
\begin{equation}
\del_x\phi(x)|_{x=0}=v^2\sqrt{\frac{\lambda}{2}}\left(1-\frac{\lambda v^2}{2M_s^2}\right)+\mathcal{O}\left(\frac{1}{M_s^4}\right)\,,\label{grad-zero}
\end{equation}
which yields
\begin{equation}
\ell\sim \sqrt{\frac{2}{\lambda}}\frac{1}{v}\left(1+\frac{\lambda v^2}{2M_s^2}\right)\,.\label{ell-linearized}
\end{equation}
Note that such a length scale does \textit{not} coincide with the width of the wall because it is related to the behavior of the solution close to the origin and far from the vacuum. In standard two-derivative theories the above computation would give a result for $\ell$ that coincides with the size of the wall, but this is just a coincidence. We will comment more on this in Sec.~\ref{subsec-order}.

\subsubsection{Validity of the linearized solution}

The above linearized solution was found perturbatively, and it is valid as long as the two inequalities in Eq.~\eqref{pert-loc-nonloc} and~\eqref{ineq-2} are satisfied. We now check when these conditions are verified.

By working with the variable $s=\sqrt{\lambda/2} vx$ and the field redefinition $\delta\phi=vf(s),$ the  inequality~\eqref{pert-loc-nonloc} reads:
\begin{equation}
  |H(s)|= \left|\frac{f(s)}{\tanh s}\right|\ll 1\,,
\end{equation}
where $H(s):=f(s)/\tanh s\,.$ By analyzing the behavior of $|H(s)|$ we can notice that it is always less than unity, thus supporting the validity of the linearized solution in Eq.~\eqref{eq:exact sol for delta phi}; see the left panel in Fig.~\ref{fig:graph for verifivation of perturbative condition}.

Let us now focus on the inequality~\eqref{ineq-2}. By introducing also in this case the variable $s=\sqrt{\lambda/2}vx$ we can write

\begin{eqnarray}
	\frac{\partial_x^2}{M_s^2}\delta\phi
	&=&\frac{\lambda v^2/2}{M_s^2} v f^{\prime\prime}(s) \nonumber\\[2mm]
	&=&\frac{\lambda v^3/2}{M_s^2} \frac{\lambda v^2}{M_s^2} \frac{{\rm d}^2}{{\rm d}s^2}
		\qty{
			\frac{1}{\cosh^2 s}
			\left(
				\frac{3}{4} \log\frac{1+\tanh s}{1-\tanh s} - 2\tanh s
			\right)
		 } \nonumber\\[2mm]
	&=&v \times
		\underbrace{\frac{1}{2}\qty(\frac{\lambda v^2}{M_s^2})^2\frac{{\rm d}^2}{{\rm d}s^2}
			\qty{
				\frac{1}{\cosh^2 s}
				\left(
					\frac{3}{4} \log\frac{1+\tanh s}{1-\tanh s} - 2\tanh s
				\right)
			}
		}_{=:\,g(s)}\,,
\end{eqnarray}
where $g(s):=\partial_x^2\delta\phi(x)/(M_s^2v)$.
In terms of the dimensionless functions $f(s)$ and $g(s)$ the inequality~\eqref{ineq-2} becomes
\begin{align}
	\left|\frac{\partial_x^2}{M_s^2}\delta\phi\right| \ll |\delta\phi|
		\quad\Leftrightarrow\quad
	|g(s)| \ll |f(s)|\,.
\end{align}
Therefore, we have to analyze the function
\begin{align}
	h(s):=\frac{g(s)}{f(s)}
				=\frac{\lambda v^2}{2M_s^2}
	 					\frac{\dps\frac{{\rm d}^2}{{\rm d}s^2}\qty[\frac{1}{\cosh^2 s}\qty(\frac{3}{4}\log\frac{1+\tanh s}{1-\tanh s}-2\tanh s)]}
			 						{\dps\frac{1}{\cosh^2 s}\qty(\frac{3}{4}\log\frac{1+\tanh s}{1-\tanh s}-2\tanh s)}\,,
\end{align}
and check for which values of $s$ its modulus $|h(s)|$ is less than unity.
In the right panel of Fig.~\ref{fig:graph for verifivation of perturbative condition} we have shown the behavior of $h(s);$ we have only plotted the region $s\geq 0$ as $h(s)$ is an even function in $s$.
We can notice that $h(s)$ diverges at the point $s\sim 1.03402$ where $f(s)$ vanishes; therefore, in the proximity of this point the linearized solution $\delta\phi(x)$ might not be valid.
However, for $|s|\rightarrow \infty$ and $s\rightarrow 0$ the inequality~\eqref{ineq-2} can be satisfied:

\begin{figure}[t]
	\centering
	\includegraphics[scale=0.315]{validity-phi_L-delta-phi}\quad\includegraphics[scale=0.305]{validity-del-2-delta-phi}
	\caption{
			%
			(Left panel) behavior of $H(s)=f(s)/\tanh s$ as a function of $s=\sqrt{\lambda/2}vx.$ The modulus of the function is always less than unity, i.e. $|H(s)|< 1,$ supporting the validity of the linearized solution $\delta \phi (x) =v f(x)$ in Eq.~\eqref{eq:exact sol for delta phi}. $H(s)$ becomes smaller and smaller for larger values of $M_s.$
			(Right panel) behavior of the function $h(s)=g(s)/f(s).$
			As long as $|h(s)|\ll 1$ the linearized solution~\eqref{eq:exact sol for delta phi}  can be trusted as a good approximation of the true behavior of the nonlocal domain wall.
			We can notice that close to the asymptotics ($s\rightarrow \infty$) the function can be kept less than one, but there is a singularity at $s\sim 1.03402$ caused by the fact that $f(s)$ vanishes at this point.
            Moreover, the linearized approximation close to the origin and at infinity becomes better for larger values of $M_s.$
			In both panels we only showed the behavior for $s\geq 0$ because both functions $h(s)$ and $H(s)$ are even in $s.$
			We set $\lambda=2,$ $v=1$ and $M_s^2=14.3,$ which are consistent with the theoretical constraint $M_s^2\geq -\lambda v^2/W_0(-1/3e)$ in Eq.~\eqref{eq:neccessary cond for asymp sol}.
		}
	\label{fig:graph for verifivation of perturbative condition}
\end{figure}

\begin{align}
 	h(0)=\lim_{s\to0} h(s) = -7\frac{\lambda v^2}{M_s^2}\,,
 	\qquad
	h(\infty)=\lim_{s\to\infty} h(s) = 2\frac{\lambda v^2}{M_s^2}\,,\label{limits}
\end{align}
and by using the theoretical lower bound in Eq.~\eqref{eq:neccessary cond for asymp sol}, i.e. $\lambda v^2/M_s^2\leq-W_0^{-1}(-1/3e)\sim 0.14,$  it follows that both asymptotic limits are always less than unity, i.e. $|h(0)|< 0.98$ and $h(\infty)< 0.28,$ and the approximation becomes better for larger values of the scale of nonlocality $M_s$.

Let us now make two important remarks.

\paragraph{Remark 2.} In light of the remark at the end of Sec.~\ref{subsec-asym} we now understand that the linearized perturbative solution~\eqref{eq:exact sol for delta phi} would have not been valid if instead we had used $W_{-1}$ and the corresponding upper bound $M_s\leq -\lambda v^2/W_{-1}(-1/3e).$ In such a case a different domain wall solution with the same functional form of the asymptotic behavior is obviously guaranteed to exist, but we are not interested in it in the current work. Therefore, we emphasize again that we only work with a domain wall configuration consistent with the bound~\eqref{eq:neccessary cond for asymp sol}.

\paragraph{Remark 3.} We have noticed that the linearized approximation breaks down in the proximity of $s\sim 1.03402.$ It means that the boundary condition $\delta\phi(\pm\infty)=0$ imposed on the perturbation cannot be used to connect the behavior of the solution from $x=\pm\infty$ to $x=0$ because the boundary condition itself would break down. This might imply that apparently the linearized solution obtained in Eq.~\eqref{eq:exact sol for delta phi} can be trusted only close to the vacua, and that the above analysis is not enough to justify the behavior close to the origin.
However, by using a different and reliable perturbative expansion in the intermediate region around $s\sim 1.03402,$ and imposing junction conditions to glue different pieces of the solution defined in three different regions, we checked and confirmed that the behavior close to the origin found above is well justified. See App.~\ref{sec-corr} for more details.

\subsection{Estimation of width and energy} \label{subsec-order}

We now estimate the width and the energy per unit area of the nonlocal domain wall by performing an analogous analysis as the one made for the local two-derivative case in Sec.~\ref{sec-review}.

Let us approximate the field as $\phi\sim  v,$  and the gradient as $\del_x\sim R^{-1}$ where $R$ is the width of the wall.
From the expression of the energy-momentum  tensor~\eqref{final-stress-exp} (with $m^2=-\lambda v^2$), we can obtain the energy density of the wall:
\begin{align}
	\mathcal{E}(x)
		\equiv T_0^0(x)= -\frac{1}{2}\phi e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi
			+\frac{\lambda}{4}(\phi^4+v^4)\,.
	\label{eq:energy density for DW}
\end{align}
The next step would be to impose the balance between the kinetic and the potential energy in~\eqref{eq:energy density for DW} for the lowest-energy configuration, and solve the resulting equation for the width $R.$ The presence of the infinite-derivative operator makes the procedure less straightforward as compared to the local case because we should first understand how to estimate $e^{-\del_x^2/M_s^2},$ i.e. whether to replace the exponent with $-1/M_s^2R^2$ or $+1/M_s^2R^2.$

The strategy to follow in order to avoid any ambiguities is to Taylor expand, recast the infinite-derivative pieces in terms of an infinite number of squared quantities, replace $\del_x\sim 1/R,$ $\phi\sim v,$ and then re-sum the series.

By Taylor expanding in powers of $\partial_x^2/M_s^2$ the infinite-derivative terms in Eq.~\eqref{eq:energy density for DW}, and neglecting total derivatives, we can write
\begin{align}
\!\phi e^{-\del_x^2/M_s^2}\del_x^2\phi&= -\del_x\phi e^{-\del_x^2/M_s^2}\del_x\phi\nonumber\\[2mm]
&= -\left[(\del_x\phi)^2-\frac{1}{M_s^2}\del_x\phi\del_x^2\del_x\phi+\frac{1}{2!M_s^4}\del_x\phi\del_x^4\del_x\phi-\cdots+\frac{(-1)^n}{n!M_s^{2n}}\del_x\phi\del_x^{2n}\del_x\phi+\cdots\right]  \nonumber\\[2mm]
&=-\left[(\del_x\phi)^2+\frac{1}{M_s^2}(\del_x^2\phi)^2+\frac{1}{2!M_s^4}(\del_x^3\phi)^2+\cdots+\frac{1}{n!M_s^{2n}}(\del_x^{n+1}\phi)^2  +\cdots\right]\nonumber\\[2mm]
&= -\sum\limits_{n=0}^\infty \frac{1}{n!}\left(\frac{1}{M_s^2}\right)^n \left(\del_x^{n+1}\phi\right)^2\,,
\end{align}
and
\begin{equation}
\lambda v^2\phi e^{-\del_x^2/M_s^2}\phi=\lambda v^2\sum\limits_{n=0}^\infty \frac{1}{n!}\left(\frac{1}{M_s^2}\right)^n \left(\del_x^{n}\phi\right)^2\,.
\end{equation}
Then, by using $\phi\sim v$ and $\del_x\sim 1/R,$ we get
\begin{align}
\phi e^{-\del_x^2/M_s^2}\del_x^2\phi\sim -\frac{v^2}{R^2}\sum\limits_{n=0}^\infty\frac{1}{n!}\left(\frac{1}{M_s^2R^2}\right)^n = -\frac{v^2}{R^2}e^{1/(M_sR)^2}\,,
\end{align}
and
\begin{equation}
\lambda v^2\phi e^{-\del_x^2/M_s^2}\phi\sim\lambda v^4 e^{1/(M_sR)^2}\,.
\end{equation}
Thus, we have shown that the correct sign in the exponent when making the estimation is the positive one\footnote{To further remove any possible ambiguity and/or confusion, it is worth mentioning that the same result would have been obtained if we would have started with a positive definite expression for the kinetic energy, for instance with the expression $\del_x\phi e^{-\del_x^2/M_s^2}\del_x\phi=(e^{-\del_x^2/2M_s^2}\del_x\phi)^2\geq 0,$ where we integrated by parts and neglected total derivatives. Also in this case one can show (up to total derivatives) that $(e^{-\del_x^2/2M_s^2}\del_x\phi)^2=\sum_{k,l=0}^\infty 1/(k!\,l!) (1/2M_s^2)^{k+l}\left(\del_x^{(k+l+1)}\phi\right)^2\sim (v^2/R^2)\left[\sum_{k=0}^\infty 1/k!(1/2M_s^2R^2)^n\right]^2= (v^2/R^2)e^{1/(M_sR)^2}$.}.

To make more manifest the consistency with the low-energy limit $M_s \to \infty$, it is convenient to separate the kinetic and the potential contributions in~\eqref{eq:energy density for DW} as follows:
\begin{align}
\mathcal{E}(x)=&
    \qty[
        -\frac{1}{2} \phi e^{-\del_x^2/M_s^2}\partial_x^2 \phi
        - \frac{1}{2} \lambda v^2 \phi \qty(e^{-\del_x^2/M_s^2}-1) \phi
    ]
    + \qty[
        \frac{\lambda}{4} \qty(\phi^2-v^2)^2
    ]\,,
\end{align}
so that the balance equation between  kinetic and  potential energies reads
\begin{align}
    \frac{1}{2} \frac{v^2}{R^2} e^{1/(M_s R)^2} - \frac{1}{2} \lambda v^4 \qty(e^{1/(M_s R)^2}-1)
        \sim
    \frac{\lambda}{4} v^4\,.
    \label{eq:full-balance-eq}
\end{align}
We are mainly interested in the leading nonlocal correction, thus we expand for $M_sR\gg 1$ up to the first relevant nonlocal contribution:
\begin{eqnarray}
 \frac{1}{2}\frac{v^2}{R^2}\left(1+\frac{1}{M_s^2R^2}\right)-\frac{1}{2}\frac{\lambda v^4}{M_s^2R^2}\sim \frac{1}{4}\lambda v^4\nonumber
	\label{eq:full-balance-eq-leading}
\end{eqnarray}
The solution up to order $\mathcal{O}(1/M_s^2)$ is given by
\begin{eqnarray}
\frac{1}{R^2}\sim\frac{\lambda v^2}{2}\left(1+\frac{\lambda v^2}{2M_s^2} \right)\,,
\end{eqnarray}
from which we obtain
\begin{eqnarray}
	R\sim\sqrt{\frac{2}{\lambda }}\frac{1}{v}\left(1-\frac{\lambda v^2}{4M_s^2} \right)\,.
\end{eqnarray}
Therefore, the width of the nonlocal domain wall, $R$, turns out to be \textit{thinner} as compared to the local two-derivative case. This is consistent with both the asymptotic analysis in Sec.~\ref{subsec-asym} and with the linearized solution in Sec.~\ref{subsec-pert}. In fact, in the previous Subsections we found that the nonlocal configuration approaches the vacua $\pm v$ \textit{faster} as compared to the local case, i.e. the coefficient $B$ in Eq.~\eqref{eq:series expansion for coeff B-sqrt} is larger than the corresponding local one. Then, as shown in Eq.~\eqref{eq:tildeR}, $\widetilde{R}$ becomes smaller in the nonlocal case, which is consistent with the behaviour of $R$. That is, the coefficient $B$ and the width of the wall $R$ should be inversely proportional to each other; this means that if $B$ increases then $R$ must decrease, and indeed this is what we showed.

We can also estimate the energy per unit area
\begin{eqnarray}
	E&=& \int_\mathbb{R}{{\rm d}x}\qty[-\frac{1}{2}\phi e^{-\partial_x^2/M_s^2}(\partial_x^2+\lambda v^2)\phi
			+\frac{\lambda}{4}(\phi^4+v^4)] \nonumber\\[2mm]
	 &\sim& (\text{width of the wall}) \times (\text{energy density}) \nonumber\\[2mm]
		& \sim& R \times \lambda v^4 \nonumber\\[2mm]
		&\sim& \sqrt{\frac{\lambda}{2}}v^3\left(1-\frac{\lambda v^2}{4M_s^2}\right) \,,\label{energ-estim}
\end{eqnarray}
which is also \textit{decreased} as compared to the local case.

It is worth to emphasize that the expansion for small $\lambda v^2/M_s^2$ is well justified for the domain wall solution satisfying the bound in Eq.~\eqref{eq:neccessary cond for asymp sol} which was obtained by using the principal branch $W_0$ of the Lambert-W function.

\paragraph{Remark 4.} In Sec.~\ref{subsec-pert} we have estimated an additional scale $\ell$ in addition to the width ($\ell> R, \widetilde{R}$). In standard local theories all of the three scales are the same because there is only one physical scale, $\ell_{\rm L}=R_{\rm L}=\widetilde{R}_{\rm L} \sim (\sqrt{\lambda}v)^{-1}.$  In fact, in general $\ell$ and $R$ ($\widetilde{R}$) represent two different physical scales, and this becomes manifest in the nonlocal theory under investigation. The length scale $R$ ($\widetilde{R}$) is the one that contains the information about the size of the wall because it is proportional to $1/B,$ and it is related to how fast the field configuration approaches the vacuum. Whereas, the scale $\ell$ is related to how fast the field changes in the proximity of the origin, indeed it is inversely proportional to the gradient at $x=0$ (see Eqs.~\eqref{grad-zero} and~\eqref{ell-linearized}), and we have $\ell>\ell_{\rm L}.$ The difference between $\ell$ and $R$ ($\widetilde{R}$) is caused by the oscillatory behavior of the nonlocal solution around the local one.

\section{Comments on other topological defects} \label{sec-other}

So far we have only focused on the domain wall configuration. However, it would be very interesting if one could repeat the same analysis also in the case of other topological defects, like \textit{string} and \textit{monopole} which can appear in nonlocal models characterized by continuous-symmetry breaking. A full study of these topological defects in nonlocal field theories goes beyond the scope of this paper, however we can make some important comments.

First of all, the existence of such finite-energy configurations is always guaranteed by the non-trivial topological structure of the vacuum manifold.\footnote{Of course, they can exist only dynamically in the local case because of Derrick's theorem~\cite{Vilenkin:2000jqa}. But, in a nonlocal case, even Derrick's theorem might be circumvented. This issue will be left for a future work.}
Knowing that a solution must exist, then we could ask how some of their properties would be affected by nonlocality. Actually, a similar order-of-magnitude estimation as the one carried out in Sec.~\ref{subsec-order} can be applied to these other global topological defects. In particular, by imposing the balance between kinetic and potential energy one would obtain that the radius of both string and monopole are \textit{smaller} as compared to the corresponding ones in the local case.

We leave a more detailed investigation of higher dimensional topological defects,  including stabilizing gauge fields, for future tasks.

\section{Discussion \& conclusions}\label{sec-dis}

\paragraph{Summary.} In this paper, we studied for the first time topological defects in the context of the nonlocal field theories. In particular, we mainly focused on the domain wall configuration associated to the $\mathbb{Z}_2$-symmetry breaking in the simplest nonlocal scalar field theory with nonlocal differential operator $e^{-\Box/M_s^2}$. Despite the complexity of non-linear infinite-order differential equations, we managed to find an approximate analytic solution. Indeed, we were able to understand how nonlocality affects the behavior of the domain wall both asymptotically close to the vacua and around the origin.

Let us briefly highlight our main results:
\begin{itemize}

    \item We showed that the nonlocal domain wall approaches the asymptotic vacua $\pm v$ faster as compared to the local two-derivative case. We confirmed this feature in two ways: (i) studying the behavior of the solution towards infinity ($|x|\rightarrow \infty$); (ii) analyzing a linearized nonlocal solution found through perturbations around the local domain wall configuration.

    \item Such a faster asymptotic behavior also means that the width of the wall, $R$ ($\widetilde{R}$), is smaller than the corresponding local one. This physically means that the boundary separating two adjacent casually disconnected spatial regions with two different vacua (i.e. $+v$ and $-v$) becomes thinner as compared to the local case. We confirmed this property by making an order-of-magnitude estimation involving the balance equation between kinetic and potential energy. As a consequence, also the energy per unit area can be shown to be smaller.

    \item We noticed that the nonlocal domain wall has a very peculiar behavior around the origin, i.e. in the proximity of $x\sim 0$. We found that the linearized nonlocal solution, $\phi=\phi_{\rm L}+\delta \phi,$ oscillates around the local domain wall when going from $x=0$ to $|x|\rightarrow \infty.$ In other words, the perturbation $\delta\phi$ changes sign: when going from $x=0$ to $x=+\infty$ it is first negative and then positive, and vice-versa when going from $x=0$ to $x=-\infty$ . We confirmed the validity of the solution close to the origin in App.~\ref{sec-corr}.

    \item The specific nonlocal domain wall solution analyzed in this paper can exist only if the nonlocal scale $M_s$ satisfies the lower bound $M_s\gtrsim \sqrt{\lambda} v,$ namely if the energy scale of nonlocality is larger than the symmetry-breaking scale.

\end{itemize}

\paragraph{Discussion \& Outlook.}
Here we have only dealt with nonlocal field theories in flat spacetime without assuming any specific physical scenario. However, it might be interesting to understand how to embed our analysis in a cosmological context where we could expect also gravity to be nonlocal; see Refs.~\cite{Biswas:2005qr,Koshelev:2016xqb,Koshelev:2017tvv,Koshelev:2020foq} and references therein.

In particular, in Refs.~\cite{Koshelev:2017tvv,Koshelev:2020foq} inflationary cosmology in nonlocal (infinite-derivative) gravity was investigated, and the following experimental bound on the scale of nonlocality was obtained: $M_s\gtrsim H,$ where $H$ is the Hubble constant during inflation, i.e. $H\sim 10^{14}$GeV.
Our theoretical lower bound is consistent with the experimental constraint derived in~\cite{Koshelev:2017tvv,Koshelev:2020foq} for the gravity sector. Indeed, some symmetry breaking is expected to happen after inflation, i.e. at energies $v\lesssim H,$ which is consistent with the theoretical lower bound $M_s\gtrsim \sqrt{\lambda} v$ in Eq.~\eqref{eq:neccessary cond for asymp sol}.

Hence, in a cosmological context one would expect the following hierarchy of scales\footnote{We are implicitly assuming that there exists only one scale of nonlocality $M_s$ for both gravity and matter sectors.}:
\begin{equation}
M_s\gtrsim H\gtrsim v\,.\label{set-ineq}
\end{equation}
Very interestingly, this cosmological scenario can be used to rule out some topological-defect solutions in nonlocal field theory. For instance, in light of the discussions a the end of Sec.~\ref{subsec-asym} and Sec.~\ref{subsec-pert}, there must exist at least another domain wall configuration that is valid for $M_s^2\lesssim \lambda v^2.$ In such a case the set of inequalities~\eqref{set-ineq} would be replaced by $v\gtrsim M_s\gtrsim H,$ which implies that the symmetry breaking would happen before inflation. Thus, if we are interested in domain wall formation after inflation, then we can surely discard any configuration valid in the regime $M_s^2\lesssim \lambda v^2.$

It would be very interesting if one would consider other topological defects, like strings and monopoles, which can appear in nonlocal models characterized by continuous-symmetry breaking. In fact, global string might play important roles like axion emissions in an expanding universe. In the local case, the topological defects that are formed from global-symmetry breaking should be unstable because of Derrick's theorem~\cite{Vilenkin:2000jqa} which excludes the existence of stationary stable configurations in dimensions greater than one. Then, they can exist only dynamically e.g. in an expanding universe. However, Derrick's theorem might not apply to a nonlocal case thanks to the nonlocality. It would be also interesting to investigate whether such stationary stable configurations could exist in a nonlocal case or not.

As another potential future direction to follow we can consider another class of models characterized by gauge symmetries as well as global ones. In fact, among the possible physical applications that can be studied in relation to topological defects, we have gravitational waves, e.g. the ones emitted by cosmic strings. We would expect that the presence of nonlocality would change the dynamics in such a way to modify non-trivially the gravitational wave-form.
This type of investigations will provide powerful test-benches to test nonlocal field theories, and to further constrain the structure of the nonlocal differential operators in the Lagrangian and the value of the nonlocal scale. In this work we only focused on the simplest model with $F(-\Box)=e^{-\Box/M_s^2},$ but one could work with more generic operators. Actually, the class of viable differential operators is huge (e.g. see~\cite{Buoninfante:2020ctr}), and it would be interesting to reduce it by means of new phenomenological studies.

As yet another future work we also wish to consider other type of field theoretical objects. An important phenomenon is the \textit{false vacuum decay}~\cite{Coleman:1977py,Callan:1977pt,Coleman:1980aw} according to which the false vacuum -- which corresponds to a local minimum of the potential -- has a non-zero probability to decay through quantum tunneling into the true vacuum -- which corresponds to a global minimum. This tunneling process consists in interpolating between the false and the true vacuum through an instanton (a bounce solution).
It may be very interesting to generalize the standard analysis done for a local two-derivative theory to the context of nonlocal field theories, and to understand how nonlocality would affect this phenomenon, e.g. how the tunneling probability would change. Another interesting direction is to consider non-topological solitons like Q-balls and oscillons/I-balls. The existence of these objects is also related to the presence of a bounce solution. But, one should notice that, different from topological defects, the presence of these bounce solutions is not guaranteed in nonlocal theories. In fact, it is very difficult to guarantee the presence of such a bounce solution in a nonlocal theory, different from the local cases. Therefore, even if one would obtain (possible) approximate solutions somehow, one cannot make any argument based on such approximate solutions without the proof of the existence of exact bounce solutions. This is the reason why we dealt only with topological defects in this paper, and left a study of non-topological field theoretical objects for future work.

Finally, we should emphasize that non-linear and infinite-order differential equations are not only difficult to solve analytically but even numerically. Indeed, up to our knowledge no numerical technique to find domain wall solutions is currently known. Some techniques to solve nonlinear equations involving infinite-order derivatives have been developed in the last decades~\cite{Moeller:2002vx,Arefeva:2003mur,Volovich2003,Joukovskaya:2008cr,Calcagni:2008nm,Frasca:2020ojd}, but none of them seem to be useful for the type of field equations considered in this paper. Therefore, as a future task it will be extremely interesting to develop new numerical and analytic methods to find topological-defect solutions. This will also be important to investigate the stability of topological defects in nonlocal field theory, something that we have not done in this paper.

\subsection*{Acknowledgements}

The authors are grateful to Sravan Kumar for useful discussions.
Y.~M. acknowledges the financial support from Advanced Research Center for Quantum Physics and Nanoscience, Tokyo Institute of Technology.
M.~Y. acknowledges financial support from JSPS Grant-in-Aid
for Scientific Research No. JP18K18764, JP21H01080, JP21H00069.

redpajama_set_name: RedPajamaArXiv


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Range Rover Velar review: A handsome slab of British luxury
Undeniably beautiful, luxurious, and a true off-roader. But perhaps too much tech.
Sebastian Anthony - Sep 18, 2017 1:06 pm UTC
with 44 posters participating, including story author
The Range Rover Velar, with a special satin finish paint job. I think you'll agree the Velar looks the part of a £90,000 ($100,000+) car. All of the images in this story are super-high-res, so be sure to click to zoom in.
Pictured here is the First Edition model, with 22-inch alloys.
I'm not a fan of SUVs. I think they're big and cumbersome. They hog the road and barely squeeze down narrow roads. They consume too much fuel, they're not much fun to drive, and of course they cost much more than something sensible like a hatchback.
But then I climb up into the new Land Rover Range Rover Velar, sit down on my plush leather throne, push the start button, and all of those concerns just melt away. Maybe that's why SUVs are so popular right now; perhaps there are millions of people out there who hate SUVs until they actually sit in one.
Yes, the Velar still feels like you're driving a tank—but it's your tank, an exceptionally comfortable and powerful tank that will barrel down highways, bounce along B roads, and bound up a gravel escarpment with confident nonchalance.
Enlarge / If you want to drive a Velar in a disused quarry, you can. If you know someone with a disused quarry.
Yes, it's big and cumbersome—but the turning circle is decent, the bonnet is reasonably short, and there are various cameras, sensors, and driver assistance systems that all combine to make the Velar feel surprisingly sprightly.
Yes, the Velar consumes too much fuel—but Jaguar Land Rover has managed to produce a wonderful supercharged V6 power unit that provides an ample 380 horsepower, yet displaces just three litres and manages a decent real-world mileage of 24mpg (11.8 litres per 100km) through a mix of highway, village, and spirited country driving. The 510hp V8 on the Range Rover Sport, while undoubtedly more powerful and burbly, averages just 15mpg in real-world mixed driving.
Which leaves just the exorbitant price, and whether it's fun to drive. Oh, and the tech of course! Let's dive in.
Enlarge / Me, with the pastoral beauty of Norway sliding by in the background.
Sebastian Anthony
Driving the Range Rover Velar
You can tell that Jaguar Land Rover has tried to make the Velar an exciting and dynamic drive, with computer-controlled air suspension and locking differentials and in some cases a hugely powerful engine. And the Velar does feel good for an SUV. But no amount of techno-engineerological wizardry will ever make an SUV feel like a sporty sedan or coupé. That 380hp supercharged V6 engine roars as you floor the accelerator—it really does sound good!—but it never feels like you are actually moving fast, despite a 0-60mph time of almost five seconds.
Enlarge / Have a bit more Norway; it's really quite an attractive country.
Cornering is good, though. The Velar is light for an SUV (just 1900kg), and has a fairly low centre of gravity thanks to an aluminium space-frame and low roofline. Throw in the aforementioned fancy suspension, and no doubt some other computery bits behind the scenes, and the Velar really flies through corners. It doesn't feel top heavy, and there's hardly any roll.
The Velar's ride-quality is, as you'd expect, very smooth. You still feel the surface of the road—you don't just ride over the top of everything—but bumps and potholes and changes in camber are softened enough that every movement seems considered and controlled. The model I drove had giant arch-filling 22-inch alloys, which looked great but no doubt made the ride a little firmer than the 18- or 19-inch wheels that are also available.
The driving position is good but not great. As with most SUVs you feel high up and safe, but your forward vision is hampered by a large pillar, which is exacerbated by sitting quite far back from the windscreen. If you're at a junction, or turning a corner with a big kerb, there's a significant blindspot to be aware of. The view from the passenger seat is fantastic, but again there's a huge expanse of dashboard between you and the front of the car. The top-end leather interior of my Velar looks and feels luxurious.
You can off-road the Range Rover Velar—its all-terrain capabilities are in between the Evoque and the Sport—but very few owners will. The Velar has a max wading depth of 650mm and will easily take you up (and down) all but the steepest of hills. Four-wheel drive, torque vectoring, and an active rear locking differential keep the Velar moving impetuously forward.
Other things—the dual-zone climate control, heated massage seats, sunroof, adaptive cruise control, Wi-Fi Internet connectivity, and various other creature comforts—all work perfectly, as you'd expect from a luxury car that costs this much.
But the tech... ah. Alas, there we finally have a problem.
Sebastian Anthony Sebastian is the editor of Ars Technica UK. He usually writes about low-level hardware, software, and transport, but it is emerging science and the future of technology that really get him excited.
Email sebastian@arstechnica.co.uk // Twitter @mrseb

redpajama_set_name: RedPajamaCommonCrawl


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Kappsäcksproblemet är ett kombinatoriskt optimeringsproblem inom optimeringsläran. Namnet härstammar från den engelska beteckningen på problemet: knapsack problem.

Ursprungligen beskrevs ett problem där saker med olika volym ska packas ner i en kappsäck med begränsat utrymme så att samtliga saker inte kan packas ner. Sakerna anses också ha olika värden för personen som ska ta med sig kappsäcken. Frågan är vilka saker som ska packas ned i kappsäcken så att värdet av de nedpackade sakerna blir som störst.

Kappsäcksproblemet existerar inte bara som det klassiska problemet som beskrivits ovan, utan kan även hittas vid bland annat schemaläggning av flygplansrutter och produktionsplanering.

Exempel

Åsnedrivaren Åsneberg ska göra en resa mellan hamnstaden och en by i bergen. Det finns ett antal varor som de i byn betalar bra för. Byborna vill gärna ha guld, siden, salt och tvättsvamp. Åsneberg tjänar 500 kr/guldtacka, 200 kr/rulle siden, 150 kr/säcken salt och 100 kr/tvättsvamp. Åsneberg måste gå till en pålitlig guldsmed som inte förråder honom för rövarna som lurar längs vägen och smeden har bara två guldtackor. Åsneberg kan få köpa fyra rullar siden billigt från en sidenhandlare. Åsneberg kan köpa i princip hur mycket salt och tvättsvamp han vill. Hans åsna kan bära 80 kg packning, och packningens volym får inte överstiga 200 L.
Guldet väger 1 kg/tacka, siden 2 kg/rulle, saltet 25 kg/säck och tvättsvampen 0,5 kg/st.
Guldet tar 0,1 L/tacka, siden 5 L/rulle, saltet 20 L/säck och tvättsvampen 10 L/st.

Hjälp Åsneberg att tjäna så mycket som möjligt.

För att lösa problemet behöver vi en matematisk modell. Det första vi ska göra är att fundera på vad Åsneberg vill veta, det vill säga variabeldeklaration.
 antal guldtackor,  antal rullar siden,  antal säckar salt och  antal tvättsvampar.
Nästa steg är att bestämma vad vi ska optimera, d.v.s. målfunktion. I det här fallet är målfunktionen vad Åsneberg tjänar på de olika varorna. Tyvärr finns det också begränsningar som hindrar Åsneberg att tjäna hur mycket som helst på sin resa. Dessa begränsningar kallas bivillkor.

Målfunktionen ser ut som följande:

Det är brukligt att skala om målfunktioner för att underlätta räknandet. I det här fallet kan vi dividera med 10, värdet av optimum ändras (10 gånger) men det påverkar inte värdena på  … .

Vi har även bivillkor:

 (vad åsnan klarar av att bära i vikt)

 (vad åsnan klarar av att bära i volym)

 (antal tillgängliga guldtackor)

 (antal tillgängliga sidenrullar)

 (d.v.s. icke-negativa heltal)

Nu har vi optimeringsproblemet uttryckt i en matematisk modell och kan angripa det med lämplig metod, till exempel trädsökning.

En tillåten lösning på problemet är , ,  och . Denna lösning är bevisligen optimallösningen.

Källor
Kaj Holmberg, Kombinatorisk optimering med linjärprogrammering, 2006, Matematiska Institutionen Linköpings tekniska högskola.

Optimeringsproblem
NP-kompletta problem

redpajama_set_name: RedPajamaWikipedia


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text:
Hope Rescue at Gedrys Farm
May 28, 2013 By Angharad Leave a Comment
For a long time now I've wanted to get involved helping an animal charity, by offering kennel spaces during off peak times. I researched many charities, and although they all do a fantastic job, Hope Rescue really stood out for the amazing lengths they'll go to, to help stray dogs in the South Wales area. All the dogs we look after on behalf of Hope Rescue are fully vaccinated, and are all up to date with their worming and flea treatment.
About Hope Rescue
Hope Rescue is not a rescue in the conventional sense. Established in 2005, it is primarily a coordinating rescue and most of its work involves saving dogs from destruction or emergency situations, irrespective of age, breed or medical condition. Hope finds them spaces with reputable rescue organisations and then transports them to the rescues, often covering thousands of miles every month. Most of the dogs it helps are strays in Local Authority pounds which would otherwise be put to sleep once they have been held for the mandatory 7 days. In Wales, Hope commits to take all the stray dogs from 2 Local Authorities and also works with a number of others pounds, individual Dog Wardens and Rescues throughout the UK.
Hope uses boarding kennels in South Wales to hold dogs if there is a delay between their put to sleep date and a rescue space becoming available. Boarding fees are one of its highest expenses but emergency holding kennels are vital. Often there are not enough rescue spaces available, particularly for the bull breeds and other larger breeds, and so these kennels are literally life savers. In one month alone this year, the kennel fees cost £6,000 but without them, dogs would have been destroyed.
Whilst the main effort remains on the coordination of rescue places, Hope also takes in dogs for rehoming; these may include some of the larger breeds. It also takes in dogs from owners who find themselves in emergency situations and can no longer keep their dog.
Education and campaigning are also important aspects of Hope's work and includes promoting responsible dog ownership and awareness of welfare issues such as puppy farming, greyhound racing and breed specific legislation. It has an outreach programme that ranges from giving talks in schools and clubs, to helping dogs that live on the streets with their homeless owners.
Working with Hope Rescue
We house up to five dogs for Hope Rescue at any one time, whilst they are waiting to be fostered or taken on by a rescue organisation. We find this such an enjoyable experience here at Gedrys. Most of these dogs have not had the best start in life, and its wonderful taking the time to gain their trust and watch them flourish! Both I and my part-timer Marian are huge dog lovers, and this has lead to us becoming more involved with Hope Rescue than just giving them temporary shelter. I am currently fostering a Hope Rescue dog called Spy whilst he waits for his forever home, I am also taking part in the Caerphilly 10k run to help raise funds for Hope Rescue. A beautiful Doberman called Bella found herself in the care of Hope Rescue, they brought her to us, and she instantly won Marian's heart, and thus has found a lovely foster home with Marian and her family. Hope Rescue is a wonderful charity, and it's an absolute pleasure to be working with them. If you'd like to visit their website, to find out more and to see the dogs that are currently looking for a foster or forever homes, please click here.
Success stories – Hope Rescue dogs that have previously stayed with us
Hooch the American Bulldog was waiting patiently for over 6months in kennels, but has now moved to a foster home where he's being spoilt rotten whilst he waits for his forever home.
Dizzy the Rottweiler also had a long wait in kennels before being spotted by his new family. Dizzy didn't have the best start in life, but he's now been adopted into a lovely family where he lives on a farm with his two doggie sisters a rotti and mastiff, and Llamas too!
I am currently fostering Spy whilst he waits for his forever home, he too had a rough start and was skin and bone when he first came to our kennels. Despite everything Spy just loves human contact and snuggles, so much so we started spending our lunch hour together snuggled up in front of the gas fire. I was soon hooked and this is how I ended up fostering him.
Spy has now moved to his forever home where he's being spoilt rotten
Arnold, one of Hope's longest residents in emergency boarding kennels has finally moved to a foster home. He is a gorgeous, happy chappy, and is now getting to sample the the home comforts he's waited so patiently for.
We currently have 5 more Hope Rescue dogs staying with us waiting for a foster home. To see all the dogs currently in emergency boarding please click here.
Filed Under: News & offers


As much as I love wearing my off-the-shoulder and bell sleeved blouses, lets face it, most of my time is spent at home chasing around two toddlers, making those tops not the most practical options. I will still continue to wear them, because you know I'm obsessed with both trends, but on most days I'm going to reach for something that's comfortable, easy to move in, and that wont drape into my kid's food while I'm preparing them lunch. Let me introduce to you my new go-to top, the slouchy dolman tunic. I was recently contacted by the Mint Julep Boutique to try out their slouchy tunics, and ever since receiving them (I also got this pink color), I have not stopped wearing them! I love the oversized fit, allowing them to pair perfectly with both jeans and leggings, and the fact that they can be styled many different ways. I paired my taupe tunic with my faux leather pants and OTK boots for a slightly edgy and put together look, but have also been wearing them with skinny jeans and trainers. I'm also thinking that if I add a statement necklace, and switched out my boots for a pair of pumps, I'd have a cute (and comfy) dinner date look.
Not only do these slouchy tunics come in a variety of colors (new ones are added constantly), But I'm loving the price. They normally sell for $48, but Threads for Thomas readers can get them for $24! With the holidays fast approaching, I'm thinking these tunics will make a perfect gift!
How would you style this tunic?
Black suede over the knee boots and faux leather leggings: a perfect combo!
Perfect styled! Wouldn t do it any differently!
LOVE this slouchy tunic! I am with you about chasing around kids and dressing practical. You look casual and totally chic!
Such a great look! Slouchy tunics are one of my favorite things to wear too! I love that you made the look more dressy with the leggings and OTK boots! Also loving the Fall background in your pics too!
Ahhh I need this slouchy tunic - it looks amazing!
Rachael, this tunic is like a mom's dream! So cute and so easy and so comfy! And I live the blush combined with black. It keeps it looking super chic, too. Thanks for linking up with me On the Edge.
This is my favorite kind of weekend outfit!


text:
De Wilgen (Fries: De Wylgen) is een dorp in de gemeente Smallingerland, in de Nederlandse provincie Friesland.

Het ligt iets ten westen van Drachten en ten noorden van Boornbergum. Het dorp ligt aan de westzijde van de Drait (Nieuwe Drait/Oude Drait). In  telde het  inwoners. Samen met Smalle Ee en Buitenstverlaat heeft het sinds 1945 een dorpsbelangenvereniging.

Geschiedenis

Het dorp was lang een buurtschap van verspreide boerderijen. Later ontwikkelde zich een kleine kern. In 1580 werd de plaats al vermeld als De Wilgen. De plaatsnaam zou verwijzen naar het feit dat er veel wilgenbomen zouden hebben gestaan. De plaats heeft geen kerk gehad. Tot 1955 was het ook formeel een buurtschap van Boornbergum, waarna het de dorpsstatus heeft gekregen.

In de jaren 70 werd aan de Nieuwe Drait een kleine jachthaven aangelegd, ook aan de kant van Drachten kwam er een jachthaven. Op het eind van die eeuw werd een stuk van het dorpsgebied van Smalle Ee toegevoegd aan het dorpsgebied van De Wilgen. Op dat noordelijke stukje werd in het begin van de 21ste eeuw waterwijk De Zanding aangelegd. Er werd daarvoor veel grond weggraven. Het gaat om een uitbreiding van Drachten maar een grenscorrectie werd daarbij (nog) niet gedaan.

De buurtschap Buitenstverlaat aan de andere kant van de Drait wordt ook weleens als een buurtschap van De Wilgen gezien maar ligt en valt onder Drachten.

Sport
Sinds 1981 kent het dorp een eigen roeivereniging, De Dragt geheten.

Zie ook
 Lijst van gemeentelijke monumenten in De Wilgen

Plaats in Friesland
Geografie van Smallingerland


text:
import { TextAlignCenter24 } from "../../";

export = TextAlignCenter24;

redpajama_set_name: RedPajamaGithub


Worry no more people of Colchester, as our very own and favourite hygienist, Judith Griffiths talks you through the correct way to brush your teeth!
You may be asking yourself, "why is brushing your teeth correctly so important?" Well, the answer is simple! It ensures you maintain good oral hygiene, healthy gums and strong teeth and it's surprisingly easy when you know how.
Have a look at our specially designed video, it takes you through the necessary steps to brush your teeth properly in order to avoid such monstrous abominations as plaque, tooth decay and gum problems! So heed our word, watch the video and protect your pearly whites!


text:
Q: How to I pass Vector.<> from Flash with ExternalInterface I have a custom container (C#) for the Flash ActiveX control and am passing data back and forth.  Previously I would use ExternalInterface.call and pass an Array as a parameter. I would prefer to use the Vector class now that it is available, but it appears that when I do that the call is never made.
It is however made if it is embedded in IE. It appears that when in IE, Flash will send out JavaScript to execute rather than serializing to XML. My guess is that the Vector XML serialization isn't baked in, so Flash just ignores the call.
Anyone have any ideas? Other than just going back to using Array, I've already done that for now.

A: The docs note that:

Other built-in or custom classes -
  ActionScript encodes other objects as
  null or as an empty object. In either
  case any property values are lost.

It's not completely clear what that means, since custom classes are also Objects - I guess only vanilla objects count? But at any rate it looks like Vector falls into that "other built-in classes" category, so you'll need to either use Array, or re-cast to Array before you pass the data.

A: you can use arrays with [ArrayElementType("type")] instead. or write a serialization function for Vector

redpajama_set_name: RedPajamaStackExchange


text:
<?php

namespace Omnipay\SagePay\Message;

class ServerTokenRegistrationResponse extends Response
{
    public function isSuccessful()
    {
        return false;
    }

    public function isRedirect()
    {
        return isset($this->data['Status'])
            && in_array($this->data['Status'], array('OK', 'OK REPEATED'));
    }

    public function getRedirectUrl()
    {
        return isset($this->data['NextURL']) ? $this->data['NextURL'] : null;
    }

    public function getRedirectMethod()
    {
        return 'GET';
    }

    public function getRedirectData()
    {
        return [];
    }
}


Poppyfield Card Crafts : BLAST FROM THE PAST! - oops, I didn't set the time to publish this!
Lovely card a June. Sorry I cannot help with the sentiment.
Lovely card June! I loved the swirls, and still sneal them in now and again!!


In 1914, eccentric scientist Dr. Hugo Clock created a time machine with the plans of going back in time and preventing World War I from happening and preventing millions of needless deaths. However, his sneaky assistant, Dr. Klaus Scythe, hijacks his time machine with a plan to acquire an arsenal of history's most powerful weapons in order to rule the world. Now Dr. Clock must pursue Scythe through time and prevent him from unleashing the aerial war machines of the future on an unsuspecting era. This pulse-pounding 3D aerial action game plays across more than 19 levels and nine historical eras.


Ark survival Evolved Hack is considered by the different types of individuals all over the world. In the game, the players can enjoy action, survival and adventure content easily. With it, the players need be focused on several things. Some players are engaged in playing the game with careless nature. In this particular way, the players cannot make the things easier and play it in a better way. For the better results, the players are required to focus on some basic things. Now I'm going to mention these points.
First of all, the players need to be focused on the requirements. By it, the players are able to manage the activities of their character easily. Some players are not paying attention to these things until they do get warnings. You should not consider this particular way. It may become dangerous and the reason for unfavorable conditions at some time.
When it comes to collect meat for food requirements at that time target the fishes. The fishes are not able to fight back. As a result, the chances of getting damage by fish are equivalent to zero. Sometimes, the players can get fish meat from the ocean without putting efforts. It is available in the form of dead sharks or piranha.
The players should also put efforts for taming the dinosaurs properly. With it, they can achieve the in-game activities. With it, you are also able to use the dinosaurs as a tool and source of defense. Here question appears how to tam the dinosaurs. For such a task, the players have lots of options. They can tam the dinosaur by providing food. At once a dinosaur gets tamed after that you can ride on it or use it for hunting purposes.


text:
Q: Hadoop Streaming Job failed error in python From this guide, I have successfully run the sample exercise. But on running my mapreduce job, I am getting the following error
ERROR streaming.StreamJob: Job not Successful!
10/12/16 17:13:38 INFO streaming.StreamJob: killJob...
Streaming Job Failed!
Error from the log file
java.lang.RuntimeException: PipeMapRed.waitOutputThreads(): subprocess failed with code 2
at org.apache.hadoop.streaming.PipeMapRed.waitOutputThreads(PipeMapRed.java:311)
at org.apache.hadoop.streaming.PipeMapRed.mapRedFinished(PipeMapRed.java:545)
at org.apache.hadoop.streaming.PipeMapper.close(PipeMapper.java:132)
at org.apache.hadoop.mapred.MapRunner.run(MapRunner.java:57)
at org.apache.hadoop.streaming.PipeMapRunner.run(PipeMapRunner.java:36)
at org.apache.hadoop.mapred.MapTask.runOldMapper(MapTask.java:358)
at org.apache.hadoop.mapred.MapTask.run(MapTask.java:307)
at org.apache.hadoop.mapred.Child.main(Child.java:170)

Mapper.py
import sys

i=0

for line in sys.stdin:
    i+=1
    count={}
    for word in line.strip().split():
        count[word]=count.get(word,0)+1
    for word,weight in count.items():
        print '%s\t%s:%s' % (word,str(i),str(weight))

Reducer.py
import sys

keymap={}
o_tweet="2323"
id_list=[]
for line in sys.stdin:
    tweet,tw=line.strip().split()
    #print tweet,o_tweet,tweet_id,id_list
    tweet_id,w=tw.split(':')
    w=int(w)
    if tweet.__eq__(o_tweet):
        for i,wt in id_list:
            print '%s:%s\t%s' % (tweet_id,i,str(w+wt))
        id_list.append((tweet_id,w))
    else:
        id_list=[(tweet_id,w)]
        o_tweet=tweet

[edit] command to run the job:
hadoop@ubuntu:/usr/local/hadoop$ bin/hadoop jar contrib/streaming/hadoop-0.20.0-streaming.jar -file /home/hadoop/mapper.py -mapper /home/hadoop/mapper.py -file /home/hadoop/reducer.py -reducer /home/hadoop/reducer.py -input my-input/* -output my-output

Input is any random sequence of sentences.
Thanks,

A: You need to explicitly instruct that mapper and reducer are used as python script, as we have several options for streaming. You can use either single quotes or double quotes.
-mapper "python mapper.py" -reducer "python reducer.py"

or
-mapper 'python mapper.py' -reducer 'python reducer.py'

The full command goes like this:
hadoop jar /path/to/hadoop-mapreduce/hadoop-streaming.jar \
-input /path/to/input \
-output /path/to/output \
-mapper 'python mapper.py' \
-reducer 'python reducer.py' \
-file /path/to/mapper-script/mapper.py \
-file /path/to/reducer-script/reducer.py

A: Your -mapper and -reducer should just be the script name.
hadoop@ubuntu:/usr/local/hadoop$ bin/hadoop jar contrib/streaming/hadoop-0.20.0-streaming.jar -file /home/hadoop/mapper.py -mapper mapper.py -file /home/hadoop/reducer.py -reducer reducer.py -input my-input/* -output my-output

When your scripts are in the job that is in another folder within hdfs which is relative to the attempt task executing as "." (FYI if you ever want to ad another -file such as a look up table you can open it in Python as if it was in the same dir as your scripts while your script is in M/R job)
also make sure you have chmod a+x mapper.py and chmod a+x reducer.py

A: I ran into this error recently, and my problem turned out to be something as obvious (in hindsight) as these other solutions:
I simply had a bug in my Python code. (In my case, I was using Python v2.7 string formatting whereas the AWS EMR cluster I had was using Python v2.6).
To find the actual Python error, go to Job Tracker web UI (in the case of AWS EMR, port 9100 for AMI 2.x and port 9026 for AMI 3.x); find the failed mapper; open its logs; and read the stderr output.

A: Try to add
 #!/usr/bin/env python

top of your script.
Or,
-mapper 'python m.py' -reducer 'r.py'

A: make sure your input directory only contains the correct files

A: I too had the same problem
i tried solution of marvin W
and i also install spark , ensure that u have installed spark , not just pyspark(dependency) but also install the framework installtion tutorial
follow that tutorial

A: if you run this command in a hadoop cluster, make sure that python is installed in every NodeMnager instance.
#hadoop