diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbmwx" "b/data_all_eng_slimpj/shuffled/split2/finalzzbmwx" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbmwx" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nLeft Handed Materials (LHM) are a new kind of materials which were theoretically envisionned by Veselago \\cite{veselago} as early as in 1967. Such materials have simultaneously negative relative permittivity ($\\varepsilon_{\\mathrm{r}}$) and negative relative permeability ($\\mu_{\\mathrm{r}}$). This theoretical curiosity became a real field of research in 2000 after Pendry showed the potential of LHM to overcome the diffraction limit \\cite{pendry_prl00} and Smith \\textit{et al.} \\cite{smith_nr} proposed a first realization for such extra-ordinary materials based on periodic lattices combining Split Ring Resonators (SRR: Concentric annular rings with splits) and wires. The latter work can be considered as the experimental foundation of LHM (as the first experimental evidence of negative refraction) and it is based on a theoretical study by Pendry \\textit{et al.} which shows that negative permittivity could be obtained by a periodic arrangement of parallel wires and that a periodic lattice of SRR had a negative magnetic response around its resonance frequency \\cite{pendry_ieee99}.\n\n\\noindent In the recent years, there has been a keen interest in wave propagation in periodically structured media \\cite{wavesPC}. Investigation of photonic crystals has paved the way to the theoretical prediction and experimental realization of photonic band gaps \\cite{yablono87,john,krauss,zengerle87,gralak2000,notomi2002,luo2002} i.e. ranges of frequencies for which light, or a light polarization, is disallowed to propagate. Soon after, the focus has been extended to the study of acoustic waves in periodic media, and the existence of phononic band gaps has been verified both theoretically and experimentally \\cite{2-ross1,2-ross2,acoustic,zhang2004,hu2004,feng2006}. Recently, the interest was even extended to the study of different types of waves, e.g. liquid surface waves \\cite{chou97,chou98,mciver2000,torres2000,hu2003} or biharmonic waves \\cite{ross-platonic,sasha-prsa,farhat-apl,farhat-epl,farhat-prl2009} in perforated thin plates. It has been shown that complete bandgaps also exist for these waves when propagating through a periodic lattice of vertically standing rods or over a periodically perforated thin plate \\cite{torres2000,farhat-apl}. In addition, many interesting phenomena have been reported, including negative refraction \\cite{veselago,pendry2000,maystre2004,guenneau2005,smith2000,ramak2005}, the superlensing effect and cloaking \\cite{schurig,norris,farhatprl,farhatwm, farhatpre}. The essential condition for the AANR effect is that the constant frequency surfaces (EFS: equifrequency surfaces) should become convex everywhere about some point in the reciprocal space, and the size of these EFS should shrink with increasing frequency \\cite{zengerle87,gralak2000,notomi2002}.\\\\\n\n\\noindent In this paper, we focus on the application of split ring resonator (SRR) structures \\cite{pendry_ieee99,Seb_SRR,guenneau_physb06} to the domain of elastic waves. We first derive the homogenized governing equations of bending waves propagating within a thin-plate with a doubly periodic square array of freely vibrating holes shaped as SRR, from the generalized biharmonic equation, and an asymptotic analysis involving three scales (one for the thickness of the thin-cut of each SRR, one for the array-pitch, and one for the wave wavelength). We then present an analysis of dispersion curves. To do this, we set the spectral problem for the biharmonic operator within a doubly periodic square array of SRR, homogeneous stress-free boundary conditions are prescribed on the contour of each resonator and the standard Floquet-Bloch conditions are set on the boundary of an elementary cell of the periodic structure. Such a structure presents an elastic bandgap at low frequencies. It turns out that the asymptotic analysis of our structure allows us to get analytically the frequency of the first localized mode and then the frequency of the first band gap.\nThe aim of our work is actually to demonstrate the AANR effect at low frequencies for elastic thin perforated plates as well as their superlensing properties. Ultra-refraction is also considered and shows the versatility and power of using such structured media to realize new functionalities for surface elastic waves.\n\n\\section{Homogenization of a thin-plate with an array of stress-free SRR inclusions near resonance}\n\nThe equations for bending of plates can be found in many textbooks \\cite{timoshenko,graff}.\nThe wavelength $\\lambda$ is supposed to be large enough compared to the thickness\n of the plate $H$ and small compared to its in-plane dimension $L$, i.e. $H \\ll \\lambda \\ll L$.\nIn this case we can adopt the hypothesis of the theory of Von-Karman\n\\cite{timoshenko,graff}. In this way, the mathematical setup is essentially two-dimensional,\nthe thickness $H$ of the plate appearing simply as a parameter in the governing equation.\n\nWe would like to homogenize a periodically structured thin-plate involving resonant elements. The resonances are associated with fast-oscillating displacement fields in thin-bridges of perforations shaped as split ring resonators (SRR), and we filter these oscillations by introducing a third scale in the usual two-scale expansion. We start with the Kirchhoff-Love equation and we consider an open bounded region $\\Omega_f\\in\\mathbb{R}^2$. This region is e.g. a slab lens consisting of a square array of SRR shaped as the letter $C$.\n\n\\noindent When the bending wave penetrates the structured area $\\Omega_f$ of the plate\nwhose geometry is shown in Figs. \\ref{fig6}(c)-(d), it\nundergoes fast periodic oscillations. To filter these oscillations,\nwe consider an asymptotic expansion of the associated vertical\ndisplacement $U_\\eta$ solution of the biharmonic equation given in (\\ref{bihaeta}) in terms of a\nmacroscopic (or slow) variable ${\\bf x}=(x_1,x_2)$\nand a microscopic (or fast) variable ${\\bf\nx}_\\eta={\\bf x}\/\\eta$, where $\\eta$ is a small positive\nreal parameter.\n\n\\begin{figure}[]\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics[angle=0]{Fig1.eps}}\n\\caption{(a) Geometry of a split ring resonator C consisting of a disc $\\Sigma$ connected to a thin ligament $\\Pi_\\eta$ of length $l$ and thickness $\\eta h$ where $0<\\eta\\ll 1$ in a unit cell Y; (b) Helmholtz resonator consisting of a mass connected to a wall via a spring which models resonances of SRR in (a); (c) Doubly periodic square lattice of SRR with the first Brillouin zone $\\Gamma$XM\nin reciprocal space; (d) Geometry of the thin plate of thickness $H$, with a source on the left side of a platonic crystal (PC) slab occupying\nthe region $\\Omega_f$.}\n\\label{fig6}\n\\end{center}\n\\end{figure}\n\n\n\n\\noindent With all the above assumptions, the out-of-plane\ndisplacement ${\\bf u}_\\eta=(0,0,U_\\eta(x_1,x_2))$ in the\n$x_3$-direction (along the vertical axis) is solution of\n(assuming a time-harmonic dependence $\\exp(-i\\omega t)$\nwith $\\omega$ the angular wave frequency):\n\\begin{equation}\n\\begin{array}{ll}\n\\displaystyle{\\frac{\\partial^2}{\\partial x_1\\partial x_1}\\left({{D}_\\eta}\\left(\\frac{\\partial^2}{\\partial x_1\\partial x_1}+\\nu_\\eta \\frac{\\partial^2}{\\partial x_2\\partial x_2}\\right)\\right)U_\\eta} \\nonumber \\\\\n+\\displaystyle{\\frac{\\partial^2}{\\partial x_2\\partial x_2}\\left({{D}_\\eta}\\left(\\frac{\\partial^2}{\\partial x_2\\partial x_2}+\\nu_\\eta \\frac{\\partial^2}{\\partial x_1\\partial x_1}\\right)\\right)U_\\eta} \\nonumber \\\\\n+\\displaystyle{2\\frac{\\partial^2}{\\partial x_1\\partial x_2}\\left({D}_{\\eta}\\left(1-\\nu_\\eta\\right)\\frac{\\partial^2}{\\partial x_1\\partial x_2}\\right)U_\\eta\n-\\,\\beta_\\eta^4\\,U_\\eta}\n=0 \\; ,\n\\end{array}\n\\label{bihaeta}\n\\end{equation}\ninside the heterogeneous isotropic region $\\Omega_f$ (the platonic crystal, PC, in Fig. \\ref{fig6}(d)), where\n$$D_\\eta=D(\\frac{{\\bf x}}{\\eta}) \\; , \\; \\nu_\\eta=\\nu(\\frac{{\\bf x}}{\\eta}) \\; \\hbox{ and }\n\\; \\beta^4_\\eta=\\beta_0^4\\rho(\\frac{{\\bf x}}{\\eta}) \\; , $$\nare nondimensionalized spatially varying parameters related to the flexural rigidity of the plate, its Poisson ratio and the\nwave frequency, respectively. In most cases, $D$ and $\\nu$ take piecewise constant values, with\n$D>0$ and $-1\/2<\\nu<1\/2$. Note that $\\beta_0^2=\\omega\\sqrt{\\,\\rho_0 H\/D_0}$, where $D_0$ is the\nflexural rigidity of the plate outside the platonic crystal, $\\rho_0$ its density\nand H its thickness.\n\nRemark that (\\ref{bihaeta}) is written in weak form and we notably retrieve the classical boundary conditions for a\nhomogeneous plate with stress-free inclusions (vanishing of bending moments and shearing stress for vanishing\n$D_\\eta$ and $\\nu_\\eta$ in the soft phase) \\cite{graff}.\nSince there is only one phase in the problem which we consider (homogeneous medium outside freely-vibrating inclusions),\nit is also possible to recast (\\ref{bihaeta}) as\n\\begin{equation}\n(\\sqrt{D_\\eta}\\nabla^2 U_\\eta+\\beta_\\eta^2)(\\sqrt{D_\\eta}\\nabla^2 U_\\eta-\\beta_\\eta^2) U_\\eta=0 \\; , \\hbox{ in\n$\\Omega_f\\setminus\\overline{\\Theta_\\eta}$} \\; , \\;\n\\label{6-helmholtz}\n\\end{equation}\nsince $D_\\eta$ vanishes inside the inclusions $\\Theta_\\eta=\\bigcup_{i\\in\\mathbb{Z}^2}\\{ \\eta (i+C) \\}$ and it is a constant in the matrix.\nBear in mind that the number of SRR in $\\Omega_f$ is an integer which scales as $\\eta^{-2}$.\nNote also that the vanishing of bending moment and shearing stress deduced from (\\ref{bihaeta}) requires that\n\\begin{equation}\n\\begin{array}{ll}\n\\left((1+\\nu_\\eta)\\left(\\frac{\\partial^2}{\\partial x_1^2}+\\frac{\\partial^2}{\\partial x_2^2}\\right)\n+2(1-\\nu_\\eta)\\left(\\frac{\\partial^2}{\\partial x_1\\partial x_2}\\right)\\right)U_\\eta=0 \\; ,\n\\nonumber \\\\\n\\left((3-\\nu_\\eta)\\left(\\frac{\\partial^3}{\\partial x_1^3}+\\frac{\\partial^3}{\\partial x_2^3}\\right)\n+(1+\\nu_\\eta)\\left(\\frac{\\partial^3}{\\partial x_1^2\\partial x_2}\n+\\frac{\\partial^3}{\\partial x_1\\partial x_2^2}\\right)\\right)U_\\eta=0 \\; ,\n\\end{array}\n\\end{equation}\nAt the boundary $\\partial\\Theta_\\eta$ of $\\Theta_\\eta$, which is consistent with our former work on thin perforated plates \\cite{farhat-epl}.\n\n\\noindent In the present case, perforations are shaped as split ring resonators, and each SRR $C$ can be modeled as\n\\begin{equation}\nC=\\{a < \\sqrt{x_1^2+x_2^2} < b\\} \\setminus\\overline{\\Pi_\\eta} \\;\n\\end{equation}\nwhere $a$ and $b$ are functions of variables $x_1,x_2$,\nunless the ring is circular and\n\\begin{equation}\n\\Pi_\\eta = \\Bigl \\{ (x_1,x_2) \\, : \\, 0 0$, $a(x)$ is a $T$-periodic sign-changing (i.e.~indefinite) weight function and \n$g(u)$ is a nonlinear term satisfying $g(0) = 0$.\nAmong other results, we proved therein that a two-solution theorem holds for the $T$-periodic boundary value problem associated with equation \\eqref{eq-ode}: more precisely, for weight functions $a(x)$ satisfying the mean value condition $\\int_{0}^{T} a(x)\\,\\mathrm{d}x < 0$\nand for a large class of nonlinear terms $g(u)$ which are superlinear at zero (namely, $g(u)\/u \\to 0$ for $u \\to 0^{+}$), two positive $T$-periodic solutions of \\eqref{eq-ode} exist, whenever the parameter $\\lambda$ is large enough (see \\cite[Theorem~3.1]{BoFe-PP} for the precise statement of this result). We refer the reader to the introduction of \\cite{BoFe-PP} for several comments about this solvability pattern, arising as a result of a delicate interplay between the behaviors of the nonlinear differential operator driving equation \\eqref{eq-ode} and the nonlinear term $a(x)g(u)$ when $u \\to +\\infty$.\n\nThe technical tool used in \\cite{BoFe-PP} to prove the above mentioned result is topological degree theory in Banach spaces, along a line of research started in \\cite{FeZa-15jde} and later developed and applied in several different situations (cf.~\\cite{Fe-18}), always dealing with nonlinear BVPs for semilinear equations of the type\n\\begin{equation}\\label{eq-ode2}\nu'' + q(x)g(u) = 0,\n\\end{equation}\nwhere $q(x)$ is an indefinite weight function. As well known, the first step within this approach is the formulation of the differential equation as a nonlinear functional equation in a Banach space: this can be done in a standard way when considering equation \\eqref{eq-ode2}, since the differential operator $Lu = -u''$ is a (linear) Fredholm operator of index zero. Then, depending on the invertibility\/non-invertibility of $L$ (and, hence, on the boundary conditions), either classical Leray--Schauder degree theory or Mawhin's coincidence degree theory apply. \n\nAs far as the strongly nonlinear equation \\eqref{eq-ode} is concerned, different strategies can be followed to achieve this goal. In \\cite{BoFe-PP}, we chose to write \\eqref{eq-ode} as the equivalent planar system\n\\begin{equation*}\nu' = \\frac{v}{\\sqrt{1+v^2}}, \\qquad v' = -\\lambda a(x)g(u),\n\\end{equation*}\nin order to directly apply coincidence degree theory in the product space, as proposed in the recent paper \\cite{FeZa-17tmna}.\nThis approach, which looks very natural when dealing with the periodic problem, has the drawback of not being suited for other boundary conditions. In particular, in spite of the well-known strong analogies existing in this setting between the periodic and the Neumann boundary value problem (see, for instance, \\cite{BoZa-15}), the possibility of proving the Neumann counterpart of the result in \\cite{BoFe-PP} is not discussed therein.\n\nThe aim of this brief paper is to provide a positive answer to this question. \nMore generally, we deal with the Neumann boundary value problem for the PDE version of equation \\eqref{eq-ode}, namely\n\\begin{equation}\\label{eq-main-pde}\n\\begin{cases}\n\\, \\mathrm{div}\\,\\Biggl{(} \\dfrac{\\nabla u}{\\sqrt{1- | \\nabla u |^{2}}}\\Biggr{)} + \\lambda a(|x|) g(u) = 0, & \\text{in $B$,} \\\\\n\\, \\partial_{\\nu}u=0, & \\text{on $\\partial B$,}\n\\end{cases}\n\\end{equation}\nwhere $B$ is a ball of the $N$-dimensional Euclidean space and $a(\\vert x \\vert)$ is a (sign-changing) radial weight function. \n\nIn this framework, we prove the following two-solution theorem for positive radial solutions of \\eqref{eq-main-pde}.\n\n\\begin{theorem}\\label{th-main}\nLet $N \\geq 1$ be an integer and let $B \\subseteq \\mathbb{R}^N$ be an open ball of center the origin and radius $R>0$.\nLet $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ be an $L^{1}$-function such that\n\\begin{itemize}[leftmargin=30pt,labelsep=12pt,itemsep=5pt]\n\\item [$(a_{*})$]\nthere exist $m\\geq 1$ closed and pairwise disjoint intervals $I^{+}_{1},\\ldots,I^{+}_{m}$ in $\\mathopen{[}0,R\\mathclose{]}$ such that\n\\begin{align*}\n&\\qquad\\qquad a(r)\\geq0, \\; \\text{ for a.e.~$r\\in I^{+}_{i}$,} \\quad a\\not\\equiv0 \\; \\text{ on $I^{+}_{i}$,} \\quad \\text{for $i=1,\\ldots,m$,} \\\\\n&\\qquad\\qquad a(r)\\leq0, \\; \\text{ for a.e.~$r\\in \\mathopen{[}0,R\\mathclose{]}\\setminus\\bigcup_{i=1}^{m}I^{+}_{i}$;}\n\\end{align*}\n\\item [$(a_{\\#})$] $\\displaystyle \\int_{B} a(|x|) \\,\\mathrm{d}x < 0$.\n\\end{itemize}\nLet $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ be a continuous function satisfying \n\\begin{itemize}[leftmargin=30pt,labelsep=12pt,itemsep=5pt]\n\\item [$(g_{*})$] $g(0)=0$ and $g(u)>0$, for all $u > 0$;\n\\item [$(g_{0})$] $\\displaystyle \\lim_{u\\to 0^{+}} \\dfrac{g(u)}{u} = 0$ and $\\displaystyle \\lim_{\\substack{u\\to0^{+} \\\\ \\omega\\to1}}\\dfrac{g(\\omega u)}{g(u)}=1$;\n\\item [$(g_{\\infty})$] $\\displaystyle\\lim_{\\substack{u\\to+\\infty \\\\ \\omega\\to1}}\\dfrac{g(\\omega u)}{g(u)}=1$.\n\\end{itemize}\nThen, there exists $\\lambda^{*}>0$ such that for every $\\lambda>\\lambda^{*}$ there exist at least two positive radial solutions of problem \\eqref{eq-main-pde}.\n\\end{theorem}\n\nNotice that no Sobolev subcriticality assumptions are required for the nonlinear term: for instance, the function $g(u) = u^{p}$ enters the setting of Theorem~\\ref{th-main} for every $p > 1$. The only restrictions on the growth at infinity come from assumption\n$(g_{\\infty})$: as an example, a function behaving, for $u$ large, like $e^u$ cannot be treated. Observe that a dual condition is required also at $u = 0$; for some comments about these assumptions (of regular-oscillation type) we refer to \\cite[p.~452--453]{BoFeZa-16}.\n\n\\begin{figure}[!htb]\n\\begin{tikzpicture}[scale=1]\n\\begin{axis}[\n tick label style={font=\\scriptsize},\n axis y line=left, \n axis x line=middle,\n xtick={0.359781, 1.39176, 2.60244, 4.3119 ,5},\n ytick={-1,0,1},\n xticklabels={,,,,$5$},\n yticklabels={$-1$,$0$,$1$},\n xlabel={\\small $|x|$},\n ylabel={\\small $a(|x|)$},\nevery axis x label\/.style={\n at={(ticklabel* cs:1.0)},\n anchor=west,\n},\nevery axis y label\/.style={\n at={(ticklabel* cs:1.0)},\n anchor=south,\n},\n width=5.5cm,\n height=4.5cm,\n xmin=0,\n xmax=6,\n ymin=-1.5,\n ymax=1.5]\n\\addplot graphics[xmin=0,xmax=5,ymin=-1.1,ymax=1.1] {BoFe-fig01-a.pdf};\n\\end{axis}\n\\end{tikzpicture}\n\\qquad\n\\begin{tikzpicture}[scale=1]\n\\begin{axis}[\n tick label style={font=\\scriptsize,major tick length=3pt},\n scale only axis,\n enlargelimits=false,\n xtick={0, 0.359781, 1.39176, 2.60244, 4.3119 ,5},\n xticklabels={$0$,,,,,$5$},\n ytick={0,8},\n max space between ticks=50,\n minor y tick num=3, \n xlabel={\\small $|x|$},\n ylabel={\\small $u(|x|)$},\nevery axis x label\/.style={\nbelow,\nat={(1.9cm,0.1cm)},\n yshift=-8pt\n },\nevery axis y label\/.style={\nbelow,\nat={(0cm,1.4cm)},\n xshift=-3pt},\n y label style={rotate=90,anchor=south},\n width=3.8cm,\n height=2.8cm, \n xmin=0,\n xmax=5,\n ymin=0,\n ymax=8]\n\\addplot graphics[xmin=0,xmax=5,ymin=0,ymax=8] {BoFe-fig01-profile.pdf};\n\\end{axis}\n\\end{tikzpicture}\n\\vspace{10pt}\\\\\n\\includegraphics[width=0.46\\linewidth]{BoFe-fig01-graph1.pdf}\n\\includegraphics[width=0.46\\linewidth]{BoFe-fig01-graph2.pdf}\n\\caption{The picture shows the graph of the weight function $a(|x|)=(\\cos(||x|-5|^{3\/2}+1)$ in $\\mathopen{[}0,R\\mathclose{]}=\\mathopen{[}0,5\\mathclose{]}$ and the graphs of the two radial solutions of problem \\eqref{eq-main-pde} where $N=2$, $g(u)=u^{2}+u^{3}$ and $\\lambda=0.1$. We notice that the large solution appears quite ``sharp-cornered'', in agreement with the analysis in \\cite[Section~4]{BoFe-PP} and \\cite[Section~3.2]{BoGa-19ccm}.} \n\\label{fig-01}\n\\end{figure}\n\nThe proof of our main result is based again on topological degree theory, but, of course, with some differences with respect to the approach in \\cite{BoFe-PP}. In particular, after having converted the radial Neumann problem \\eqref{eq-main-pde} into the (singular) one-dimensional problem\n\\begin{equation}\\label{eq-main}\n\\begin{cases}\n\\, \\Biggl{(} r^{N-1}\\dfrac{u'}{\\sqrt{1-(u')^{2}}}\\Biggr{)}' + \\lambda r^{N-1} a(r) g(u) = 0,\\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation}\nwe write it as a fixed point equation in the Banach space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$, by exploiting a strategy comparable to the one used, for instance, in \\cite{BeJeMa-10,GaMaZa-93} (see also Remark~\\ref{rem-2.1}). At this point, Leray--Schauder degree theory can be applied: the computation of the degree on three different balls of the space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ leads to the result, similarly as in \\cite{BoFe-PP,BoFeZa-16}.\n\nThe paper is organized as follows. In Section~\\ref{section-2}, we describe the abstract setting and we present two lemmas for the computation of the degree. We give here all the details of the arguments, since it seems that no appropriate reference exists for the auxiliary results that we need. In Section~\\ref{section-3}, we give the proof of Theorem~\\ref{th-main}, by showing the applicability of the above mentioned lemmas, under the present assumptions. Compared to the case $N=1$, some technical difficulties arise when $N \\geq 2$. First, the convexity (respectively, concavity) of the solutions is not a priori prescribed by the negativity (respectively, positivity) of the weight function $a(r)$. Second, \nwhen the weight function is positive near the center of the ball, as expected, some further care is needed in the estimates in order to exclude the possible appearance of a ``sudden loss of energy'' for the solutions (cf.~\\cite{GaMaZa-97}). We stress that, due to the peculiar features of the Minkowski curvature operator, this can be successfully done even with no subcriticality assumptions on the nonlinear term.\n\n\n\\section{The abstract setting}\\label{section-2}\n\nIn this section we present an abstract formulation of the problem and we prove some preliminary results based on the theory of the topological degree.\n\n\\subsection{The fixed point reduction}\\label{section-2.1}\n\nThroughout this section, we deal with the boundary value problem \n\\begin{equation}\\label{eq-phi}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + r^{N-1} f(r,u) = 0, \\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation} \nwhere $N \\geq 1$ is an integer,\n\\begin{equation*}\n\\varphi(s) = \\frac{s}{\\sqrt{1-s^2}}, \\quad s \\in \\mathopen{]}-1,1\\mathclose{[},\n\\end{equation*}\nand $f \\colon \\mathopen{[}0,R\\mathclose{]} \\times \\mathbb{R} \\to \\mathbb{R}$ is an $L^{1}$-Carath\\'{e}odory function (i.e.~$f(\\cdot,u)$ is measurable for every $u \\in \\mathbb{R}$, $f(r,\\cdot)$ is continuous for a.e. $r \\in \\mathopen{[}0,R\\mathclose{]}$, and for every $K > 0$\nthere exists $\\eta_k \\in L^1(0,R)$ such that $|f(r,u)| \\leq \\eta_k(r)$ for a.e. $r \\in \\mathopen{[}0,R\\mathclose{]}$ and for every \n$| u| \\leq K$). Let us recall that a solution of \\eqref{eq-phi} is a continuously differentiable function $u\\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$, with $u'(0) = u'(R) = 0$ and $|u'(r)| < 1$ for every $r \\in \\mathopen{]}0,R\\mathclose{[}$, such that the map $r \\mapsto r^{N-1}\\varphi(u')$ is absolutely continuous on $\\mathopen{[}0,R\\mathclose{]}$ and the differential equation in \\eqref{eq-phi} is satisfied almost everywhere (see also Remark \\ref{regolarita}).\n\nIt is convenient for the sequel to embed problem \\eqref{eq-phi} into the two-parameter family of problems\n\\begin{equation}\\label{eq-phi-theta}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\vartheta r^{N-1} \\bigl{[} f(r,u) + \\alpha v(r) \\bigr{]} = 0, \\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation} \nwhere $\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$, $\\alpha\\geq0$ and $v(r)$ is a fixed Lebesgue integrable function.\nNotice that \\eqref{eq-phi} is exactly \\eqref{eq-phi-theta} with $\\vartheta=1$ and $\\alpha=0$.\n\nOur aim is to write \\eqref{eq-phi-theta} as a fixed point problem in the Banach space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ of continuous functions in $\\mathopen{[}0,R\\mathclose{]}$, endowed with the usual norm $\\|\\cdot\\|_{\\infty}$. Accordingly, for $\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$ and $\\alpha\\geq0$, we introduce the operator\n\\begin{equation*}\nT_{\\vartheta,\\alpha} \\colon \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}) \\to \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}),\n\\end{equation*}\ndefined as\n\\begin{equation}\\label{operator}\n\\begin{aligned}\n(T_{\\vartheta,\\alpha} u) (r) &= u(0) - \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} \\bigl{[} f(\\zeta,u(\\zeta)) + \\alpha v(\\zeta) \\bigr{]} \\,\\mathrm{d}\\zeta\n \\\\ &\\quad + \\int_{0}^{r} \\varphi^{-1} \\biggl{(} -\\dfrac{\\vartheta}{\\zeta^{N-1}} \\int_{0}^{\\zeta}\\xi^{N-1} \\bigl{[} f(\\xi,u(\\xi)) + \\alpha v(\\xi) \\bigr{]}\\,\\mathrm{d}\\xi \\biggr{)} \\,\\mathrm{d}\\zeta,\n\\end{aligned}\n\\end{equation}\nfor every $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$.\nNotice that the definition of $T_{\\vartheta,\\alpha}$ is well-posed, since the function\n\\begin{equation*}\n\\mathopen{]}0,R\\mathclose{]} \\ni \\zeta \\mapsto \\dfrac{1}{\\zeta^{N-1}} \\int_{0}^{\\zeta}\\xi^{N-1} \\bigl{[} f(\\xi,u(\\xi)) + \\alpha v(\\xi) \\bigr{]}\\,\\mathrm{d}\\xi\n\\end{equation*}\ncontinuously extends up to $\\zeta=0$. For future convenience, we also observe that $T_{\\vartheta,\\alpha} u\\in\\mathcal{C}^{1}(\\mathopen{[}0,R\\mathclose{]})$ with\n\\begin{equation}\\label{diff-operator}\n(T_{\\vartheta,\\alpha} u)' (r) = \n\\begin{cases}\n\\, \\displaystyle \\varphi^{-1} \\biggl{(} -\\dfrac{\\vartheta}{r^{N-1}} \\int_{0}^{r}\\xi^{N-1} \\bigl{[} f(\\xi,u(\\xi)) + \\alpha v(\\xi) \\bigr{]}\\,\\mathrm{d}\\xi \\biggr{)}, &\\text{if $r\\in\\mathopen{]}0,R\\mathclose{]}$,} \\\\\n\\, 0, &\\text{if $r=0$,}\n\\end{cases}\n\\end{equation}\nfor every $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$.\n\nThe following result holds true.\n\n\\begin{theorem}\\label{th-operator}\nLet $f \\colon \\mathopen{[}0,R\\mathclose{]} \\times \\mathbb{R} \\to \\mathbb{R}$ be an $L^{1}$-Carath\\'{e}odory function, $v \\in L^1(0,R)$, \n$\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$, and $\\alpha\\geq0$. Then, the operator $T_{\\vartheta,\\alpha}$ defined in \\eqref{operator} is completely continuous. Moreover, $u(r)$ is a solution of \\eqref{eq-phi-theta} if and only if $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ is a fixed point of $T_{\\vartheta,\\alpha}$.\n\\end{theorem}\n\n\\begin{proof}\nThe continuity of the operator $T_{\\vartheta,\\alpha}$ follows straightforwardly by observing that $T_{\\vartheta,\\alpha}$ is a composition of continuous maps. We claim that $T_{\\vartheta,\\alpha}$ sends bounded sets into relatively compact sets. As a well-known consequence of Ascoli--Arzel\\`{a} theorem, this is true if we prove that $\\{T_{\\vartheta,\\alpha}u_{n}\\}_{n}$ and $\\{(T_{\\vartheta,\\alpha}u_{n})'\\}_{n}$ are uniformly bounded for every bounded sequence $\\{u_{n}\\}_{n}$ in $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$. This easily follows from the regularity assumptions on $f(r,u)$ and $v(r)$.\n\nWe prove the equivalence between the Neumann boundary value problem \\eqref{eq-phi-theta} and the fixed point problem $u=T_{\\vartheta,\\alpha}u$.\nLet $u(r)$ be a solution of \\eqref{eq-phi-theta}, hence $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$. By integrating the equation in \\eqref{eq-phi-theta} in $\\mathopen{[}0,r\\mathclose{]}$, recalling that $u'(0)=0$, and dividing by $r^{N-1}$, we obtain\n\\begin{equation}\\label{eq-phi-th2.1}\n\\varphi(u'(r)) = -\\dfrac{\\vartheta}{r^{N-1}} \\int_{0}^{r} \\zeta^{N-1} \\bigl{[} f(\\zeta,u(\\zeta)) + \\alpha v(\\zeta) \\bigr{]} \\,\\mathrm{d}\\zeta, \\quad \\text{for all $r\\in\\mathopen{]}0,R\\mathclose{]}$.}\n\\end{equation}\nSince $u'(R)=0$ we deduce that\n\\begin{equation}\\label{eq-averR}\n\\dfrac{1}{R^{N-1}} \\int_{0}^{R}\\zeta^{N-1} f(\\zeta,u(\\zeta))\\,\\mathrm{d}\\zeta =0.\n\\end{equation}\nBy applying $\\varphi^{-1}$ to \\eqref{eq-phi-th2.1} and integrating in $\\mathopen{[}0,r\\mathclose{]}$, we thus deduce that $u=T_{\\vartheta,\\alpha}u$.\n\nOn the other hand, let $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ be such that $u=T_{\\vartheta,\\alpha}u$. Therefore, we have $u\\in\\mathcal{C}^{1}(\\mathopen{[}0,R\\mathclose{]})$ and $u'$ is given by the right-hand side of formula \\eqref{diff-operator}. In particular, $u'(0)=0$ and $|u'(r)|<1$ for every $r\\in\\mathopen{]}0,R\\mathclose{[}$. \nNext, by computing $u(0)=(Tu)(0)$, we obtain that \\eqref{eq-averR} holds and so $u'(R)=0$.\nBy computing $\\varphi(u')$, multiplying by $r^{N-1}$, and differentiating, we finally obtain that $u(r)$ solves \\eqref{eq-phi-theta}. The proof is thus completed.\n\\end{proof}\n\n\\begin{remark}\\label{regolarita}\nLet us notice that, for every solution $u(r)$ of \\eqref{eq-phi-theta}, from \\eqref{eq-phi-th2.1} it follows that $\\varphi(u')$ is absolutely continuous on $\\mathopen{[}0,R\\mathclose{]}$. Therefore, since $\\varphi^{-1}$ is smooth, $u'$ is absolutely continuous as well.\nIn particular, if both $f(r,u)$ and $v(r)$ are continuous functions, then $u \\in \\mathcal{C}^2(\\mathopen{[}0,R\\mathclose{]})$ (cf.~\\cite[Remark~3.3]{CoCoRi-14}).\n$\\hfill\\lhd$\n\\end{remark}\n\nAs a consequence of Theorem~\\ref{th-operator}, if $\\Omega\\subseteq \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ is an open and bounded set such that\n\\begin{equation*}\nu \\neq T_{\\vartheta,\\alpha} u, \\quad \\text{for all $u\\in\\partial\\Omega$,}\n\\end{equation*}\nthe Leray--Schauder degree $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{\\vartheta,\\alpha},\\Omega,0)$ is well-defined (we refer to \\cite{De-85} for a classical reference about topological degree theory).\n\n\\begin{remark}\\label{rem-2.1}\nDealing with the (one-dimensional) periodic boundary value problem\n\\begin{equation}\\label{eq-per}\n\\begin{cases}\n\\, \\bigl{(} \\varphi(u')\\bigr{)}' + f(r,u) = 0, \\\\\n\\, u(0) = u(R), \\; u'(0) = u'(R),\n\\end{cases}\n\\end{equation} \na similar strategy can be followed in order to obtain a functional analytic formulation. Precisely, it can be seen (cf.~\\cite{BeMa-07,MaMa-98}) that $u(r)$ is a solution of \\eqref{eq-per} if and only if \n$u \\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ is a fixed point of the operator $\\widetilde T \\colon \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}) \\to \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ defined as \n\\begin{align*}\n (\\widetilde T u) (r) &= u(0) -\\int_{0}^{R} f(\\zeta,u(\\zeta)) \\,\\mathrm{d}\\zeta\n \\\\ &\\quad + \\int_{0}^{r} \\varphi^{-1} \\biggl{(} - \\int_{0}^{\\zeta} f(\\xi,u(\\xi))\\,\\mathrm{d}\\xi + Q_{\\varphi}\\left( \\int_0^{\\odot} \nf(\\eta,u(\\eta))\\,\\mathrm{d}\\eta\\right) \\biggr{)} \\,\\mathrm{d}\\zeta,\n\\end{align*}\nwhere, for each $h \\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$, $Q_\\varphi(h) = Q_\\varphi(h(\\odot)) \\in \\mathbb{R}$ is defined as the unique solution of the equation\n\\begin{equation*}\n\\int_0^R \\varphi^{-1}\\left( -h(r) + Q_\\varphi(h)\\right)\\,\\mathrm{d}r = 0.\n\\end{equation*}\nIt is worth noticing that, compared with the Neumann case, this formulation is slightly less transparent: indeed, due to the non-locality of the\nperiodic boundary conditions, the additional term $Q_\\varphi$ appears. \nAn alternative fixed point formulation for \\eqref{eq-per}, relying on a direct use of coincidence degree theory for the equivalent planar system\n$u' = \\varphi^{-1}(v)$, $v' = - f(r,u)$, has been recently proposed in \\cite{FeZa-17tmna}.\n$\\hfill\\lhd$\n\\end{remark}\n\n\n\\subsection{Two degree lemmas}\\label{section-2.2}\n\nTaking advantage of the abstract setting just presented, we now prove two lemmas for the computation of the degree on open balls $B(0,d)$ (with center $0$ and radius $d>0$) of the Banach space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ in the framework of the Neumann problem\n\\begin{equation}\\label{eq-phi-ag}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\lambda r^{N-1} a(r)g(u) = 0, \\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation} \nwhere $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ is an $L^1$-function, $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ is a continuous function satisfying $g(0) = 0$, and $\\lambda>0$. \nNotice that, to enter the setting of the previous section, we need for the nonlinearity appearing in the equation to be defined for every $u \\in \\mathbb{R}$; accordingly, we set\n\\begin{equation*}\nf(r,u) = \\begin{cases}\n\\, \\lambda a(r) g(u), & \\text{if $u\\geq0$,} \\\\\n\\, -u,& \\text{if $u<0$.} \n\\end{cases}\n\\end{equation*}\nObserve that, due to the assumptions on $a(r)$ and $g(u)$, the function $f(r,u)$ is $L^{1}$-Carath\\'{e}odory.\n\nRecalling the definition \\eqref{operator} of $T_{\\vartheta,\\alpha}$, we thus compute the Leray--Schauder degree of the map $\\mathrm{Id}-T_{1,0}$. Observe that, by standard maximum principle arguments (based on the monotonicity of the map $r \\mapsto r^{N-1}\\varphi(u'(r))$ when $u(r)<0$), $u = T_{1,0} u$ implies that $u(r)$ is a \\emph{non-negative} solution of \\eqref{eq-phi} and thus solves \\eqref{eq-phi-ag}.\n\nThe first lemma gives conditions for zero degree.\n\n\\begin{lemma}\\label{lem-deg0}\nLet $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ be an $L^1$-function, let $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ be a continuous function satisfying $g(0) = 0$, and $\\lambda>0$. \nLet $d>0$ and assume that there exists a non-negative function $v \\in L^1(0,R)$, with $v\\not\\equiv0$, such that the following properties hold:\n\\begin{itemize}\n\\item[$(H_{1})$]\nIf $\\alpha \\geq 0$ and $u(r)$ is a non-negative solution of\n\\begin{equation}\\label{eq-lem-deg0}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\lambda r^{N-1} a(r) g(u) + \\alpha r^{N-1} v(r) = 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nthen $\\|u\\|_{\\infty}\\neq d$.\n\\item[$(H_{2})$]\nThere exists $\\alpha_{0} \\geq 0$ such that problem \\eqref{eq-lem-deg0}, with $\\alpha=\\alpha_{0}$, has no non-negative solutions $u(r)$ with $\\|u\\|_{\\infty}\\leq d$.\n\\end{itemize}\nThen, it holds that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0) = 0$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, recalling Theorem~\\ref{th-operator}, we have that $u(r)$ is a solution of \n\\begin{equation}\\label{eq-lem-deg0f}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + r^{N-1} \\bigl{[} f(r,u) + \\alpha v(r) \\bigr{]}= 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nif and only if $u = T_{1,\\alpha} u$.\nBy maximum principle arguments, since $v \\geq 0$, every solution of \\eqref{eq-lem-deg0f} is non-negative and thus solves \\eqref{eq-lem-deg0}.\nTherefore, for every $\\alpha \\geq 0$, condition $(H_{1})$ ensures that \n\\begin{equation*}\nu \\neq T_{1,\\alpha} u, \\quad \\text{for all $u\\in \\partial B(0,d)$.}\n\\end{equation*}\nWe deduce that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,\\alpha},B(0,d),0)$ is well-defined for every $\\alpha \\geq 0$. As a final step, we apply the homotopy invariance property of the degree to conclude that\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0)=\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,\\alpha_{0}},B(0,d),0)=0,\n\\end{equation*}\nwhere the last equality follows from hypothesis $(H_{2})$.\n\\end{proof}\n\nThe second lemma ensures non-zero degree. Notice that here we need to assume further conditions on $a(r)$ and $g(u)$.\n\n\\begin{lemma}\\label{lem-deg1}\nLet $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ be an $L^1$-function satisfying $(a_{\\#})$, let $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ be a continuous function satisfying $(g_{*})$, and $\\lambda>0$. Let $d>0$ and assume that the following property holds:\n\\begin{itemize}\n\\item[$(H_{3})$]\nIf $\\vartheta\\in \\mathopen{]}0,1\\mathclose{]}$ and $u(r)$ is a non-negative solution of\n\\begin{equation}\\label{eq-lem-deg1}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\vartheta \\lambda r^{N-1} a(r) g(u) = 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nthen $\\|u\\|_{\\infty} \\neq d$.\n\\end{itemize}\nThen, it holds that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0) = 1$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, recalling Theorem~\\ref{th-operator}, we have that $u(r)$ is a solution of\n\\begin{equation}\\label{eq-lem-deg11f}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\vartheta r^{N-1} f(r,u) = 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nif and only if $u = T_{\\vartheta,0} u$.\nBy maximum principle arguments, every solution of \\eqref{eq-lem-deg11f} is non-negative and thus solves \\eqref{eq-lem-deg1}.\n\nHypothesis $(H_{3})$ implies that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{\\vartheta,0},B(0,d),0)$ is well-defined for every $\\vartheta\\in\\mathopen{]}0,1\\mathclose{]}$.\nLet us consider the case $\\vartheta=0$. The fixed point problem $u = T_{0,0} u$ reduces to\n\\begin{equation*}\nu(r) = (T_{0,0} u)(r) = u(0) - \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} f(\\zeta,u(\\zeta)) \\,\\mathrm{d}\\zeta, \\quad u\\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}),\n\\end{equation*}\nwhose solutions are constant functions.\nObserving that\n\\begin{equation*}\n(\\mathrm{Id}-T_{0,0}) u \\equiv - \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} f(\\zeta,s) \\,\\mathrm{d}\\zeta, \\quad \\text{for all $u\\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ with $u\\equiv s\\in\\mathbb{R}$,}\n\\end{equation*}\nwe consider the function\n\\begin{equation*}\nf^{\\#}(s) = \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} f(\\zeta,s) \\,\\mathrm{d}\\zeta =\n\\begin{cases}\n\\, -\\dfrac{R}{N} s, & \\text{if $s \\leq 0$,} \\\\\n\\, \\dfrac{\\lambda}{R^{N-1}} \\biggl{(}\\displaystyle \\int_{0}^{R} \\zeta^{N-1}a(\\zeta) \\,\\mathrm{d}\\zeta \\biggr{)} g(s), & \\text{if $s \\geq 0$.}\n\\end{cases}\n\\end{equation*}\nBy hypothesis $(a_{\\#})$ we deduce that\n\\begin{equation*}\n\\int_{0}^{R} r^{N-1} a(r) \\,\\mathrm{d}r = \\dfrac{1}{\\omega_{N}} \\int_{B}a(|x|) \\,\\mathrm{d}x < 0\n\\end{equation*}\n(where $\\omega_{N}$ is the measure of the unit sphere in $\\mathbb{R}^{N}$) \nand, consequently, we obtain that $f^{\\#}(s)s < 0$ for $s \\neq 0$.\nAs a consequence of the excision property, we can reduce the study of the degree of $\\mathrm{Id}-T_{0,0}$ in the set $B(0,d)\\cap\\mathbb{R}=\\mathopen{]}-d,d\\mathclose{[}$.\nSince $f^{\\#}(s)$ has no zeros on $\\partial(B(0,d)\\cap\\mathbb{R})=\\{\\pm d\\}$ and, more precisely, $f^{\\#}(d)<00$ be such that\n\\begin{equation*}\n\\varepsilon<\\dfrac{ |I^{+}_{i} |}{4} \\quad \\text{ and } \\quad \\int_{\\sigma_{i}+2\\varepsilon}^{\\tau_{i}-2\\varepsilon} r^{N-1} a(r)\\,\\mathrm{d}r > 0, \\quad \\text{for every $i = 1,\\ldots,m$,}\n\\end{equation*}\nand let us define\n\\begin{equation*}\n\\delta^{*} = \\dfrac{2^{N-1}\\varepsilon^{N}}{R^{N-1}}\n\\end{equation*}\nand\n\\begin{equation*}\n\\delta_{*} = \\min_{i=1,\\ldots,m} \n\\dfrac{\\delta^{*}}{1+2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{i}| \\dfrac{\\tau_{i}^{2N-2}}{\\sigma_{i}^{N-1}(2\\varepsilon)^{N}}},\n\\end{equation*}\nif $\\sigma_{1}\\neq0$, or \n\\begin{equation*}\n\\delta_{*} = \\min \\left\\{ \n\\dfrac{\\delta^{*}-\\gamma}{1+2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{1}| \\dfrac{\\tau_{1}^{2N-2}}{\\gamma^{N-1}(2\\varepsilon)^{N}}},\n\\min_{i=2,\\ldots,m} \\dfrac{\\delta^{*}}{1+2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{i}| \\dfrac{\\tau_{i}^{2N-2}}{\\sigma_{i}^{N-1}(2\\varepsilon)^{N}}} \\right\\}.\n\\end{equation*}\nif $\\sigma_{1}=0$, where $\\gamma=\\min\\{\\delta^{*},\\tau_{1}\\}\/2$. Notice that, in both cases, $\\delta_{*}\\in\\mathopen{]}0,\\delta^{*}\\mathclose{[}$.\nMoreover, let\n\\begin{equation*}\n\\lambda^{*} = \\max_{i=1,\\ldots,m}\\dfrac{2 R^{N-1}\\varphi(1\/2)}{ \\min \\bigl{\\{} g(u) \\colon u \\in \\mathopen{[} \\delta_{*},\\delta^{*} \\mathclose{]} \\bigr{\\}} \\displaystyle{\\int_{\\sigma_{i}+2\\varepsilon}^{\\tau_{i}-2\\varepsilon} r^{N-1} a(r)\\,\\mathrm{d}r}}.\n\\end{equation*}\nFrom now on, let $\\lambda > \\lambda^{*}$ be fixed.\n\n\\subsubsection*{Step~2. Computation of the degree in $B(0,\\delta^{*})$.}\nWe are going to apply Lemma~\\ref{lem-deg0} to the open ball $B(0,\\delta^{*})$ taking as $v(r)$ the indicator function of the set $\\bigcup_{i} I^{+}_{i}$.\n\nFirst we verify condition $(H_{1})$. We suppose by contradiction that there exist $\\alpha \\geq 0$ and a non-negative solution $u(r)$ to \\eqref{eq-lem-deg0} such that $\\|u\\|_{\\infty} = \\delta^{*}$. \n\n\\smallskip\n\\noindent\n\\textit{Claim~1.} There exists $i=\\{1,\\ldots,m\\}$ such that\n\\begin{equation}\\label{eq-cl1}\n\\max_{r\\in I^{+}_{i}} u(r) = \\delta^{*}.\n\\end{equation}\nSince $v\\equiv0$ on $\\mathopen{[}0,R\\mathclose{]}\\setminus \\bigcup_{i} I^{+}_{i}$, by conditions $(a_{*})$ and $(g_{*})$ we deduce that the map $r\\to r^{N-1}\\varphi(u'(r))$ is non-increasing on each interval $I^{+}_{i}$ and non-decreasing on each interval $J \\subseteq \\mathopen{[}0,R\\mathclose{]}\\setminus \\bigcup_{i} I^{+}_{i}$. \nWe show that\n\\begin{equation}\\label{eq-convexity}\n\\max_{r\\in J}u(r) = \\max_{r\\in\\partial J} u(r),\n\\end{equation}\nfor every interval $J \\subseteq \\mathopen{[}0,R\\mathclose{]}\\setminus \\bigcup_{i} I^{+}_{i}$.\nIndeed, let $J=\\mathopen{[}\\tau,\\sigma\\mathclose{]}$ and $\\hat{r}\\in\\mathopen{]}\\tau,\\sigma\\mathclose{[}$. If $u'(\\hat{r}) \\geq 0$, then $u'(r) \\geq 0$ for all $r\\in\\mathopen{[}\\hat{r},\\sigma\\mathclose{]}$, and so $u(\\hat{r})\\leq u(\\sigma)$. Analogously, if $u'(\\hat{r}) \\leq 0$, then $u'(r) \\leq 0$ for all $r\\in\\mathopen{[}\\tau,\\hat{r}\\mathclose{]}$, and so $u(\\hat{r})\\leq u(\\tau)$. Therefore, \\eqref{eq-convexity} holds. \n\nWe further observe that if $\\tau=0$ then $u'(\\tau)=0$ and so $\\max_{r\\in J}u(r) = u(\\sigma)$; if $\\sigma=R$ then $u'(\\sigma)=0$ and so $\\max_{r\\in J}u(t) = u(\\tau)$. As a consequence of this and \\eqref{eq-convexity}, \\eqref{eq-cl1} follows.\n\n\nFrom now on we focus on the behavior of $u(r)$ on $I^{+}_{i}=\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$. \n\n\\smallskip\n\\noindent\n\\textit{Claim~2.} It holds that\n\\begin{equation}\\label{estim-u'}\n|u'(r)| \\leq \\dfrac{\\tau_{i}^{N-1}}{(2\\varepsilon)^{N}} \\, u(r), \\quad \\text{for all $r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$.}\n\\end{equation}\nIndeed, let us fix $r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$. If $u'(r)=0$ then the estimate is obvious. If $u'(r)>0$, by using the monotonicity of the map $r\\to r^{N-1}\\varphi(u'(r))$, we have that \n\\begin{equation*}\nu'(\\xi) \\geq \\varphi^{-1} \\biggl{(} \\biggl{(}\\dfrac{r}{\\xi}\\biggr{)}^{\\!N-1} \\varphi(u'(r)) \\biggr{)} \\geq \\varphi^{-1} \\bigl{(} \\varphi(u'(r)) \\bigr{)} = u'(r), \\quad \\text{for all $\\xi\\in\\mathopen{]}\\sigma_{i},r\\mathclose{]}$.}\n\\end{equation*}\nBy integrating the above inequality in $\\mathopen{[}\\sigma_{i},r\\mathclose{]}$ we obtain \n\\begin{equation*}\nu(r)\\geq u(r)-u(\\sigma_{i})= \\int_{\\sigma_{i}}^{r} u'(\\xi)\\,\\mathrm{d}\\xi\n\\geq (r-\\sigma_{i}) u'(r)\\geq 2\\varepsilon u'(r) \\geq \\dfrac{(2\\varepsilon)^{N}}{\\tau_{i}^{N-1}} u'(r),\n\\end{equation*}\nwhere the last inequality follows from $2\\varepsilon<\\tau_{i}$. This implies~\\eqref{estim-u'}. As last case, if $u'(r)<0$, by arguing as above, we have that \n\\begin{align*}\n-u'(\\xi) &\\geq \\varphi^{-1} \\biggl{(} \\biggl{(}\\dfrac{r}{\\xi}\\biggr{)}^{\\!N-1} \\varphi(-u'(r)) \\biggr{)} \n\\geq -\\biggl{(}\\dfrac{r}{\\xi}\\biggr{)}^{\\!N-1} u'(r)\n\\\\\n& \\geq -\\biggl{(}\\dfrac{r}{\\tau_{i}}\\biggr{)}^{\\!N-1} u'(r), \\quad \\text{for all $\\xi\\in\\mathopen{[}r,\\tau_{i}\\mathclose{]}$,}\n\\end{align*}\nwhere we have used the oddness of $\\varphi$ and $\\varphi^{-1}$, and the elementary inequality\n\\begin{equation*}\n\\varphi^{-1} \\bigl{(} \\vartheta \\varphi(s) \\bigr{)} \\geq \\vartheta s, \\quad \\text{for all $\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$ and $s\\in\\mathopen{[}0,1\\mathclose{[}$,}\n\\end{equation*}\ncoming from the convexity of $\\varphi$ in $\\mathopen{[}0,1\\mathclose{[}$.\nThen, by integrating in $\\mathopen{[}r,\\tau_{i}\\mathclose{]}$ we obtain \n\\begin{equation*}\nu(r)\\geq u(r)-u(\\tau_{i}) = - \\int_{r}^{\\tau_{i}} u'(\\xi)\\,\\mathrm{d}\\xi \n\\geq -(\\tau_{i}-r) \\biggl{(}\\dfrac{r}{\\tau_{i}}\\biggr{)}^{\\!N-1}u'(r) \n\\geq - \\dfrac{(2\\varepsilon)^{N}}{\\tau_{i}^{N-1}} u'(r),\n\\end{equation*}\nfinally implying \\eqref{estim-u'}.\n\nFor further convenience, we observe that from \\eqref{estim-u'} we have in particular that\n\\begin{equation}\\label{u'_1_2}\n|u'(r)| \\leq \\dfrac{1}{2}, \\quad \\text{for all $r \\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$.}\n\\end{equation}\n\n\\smallskip\n\\noindent\n\\textit{Claim~3.} It holds that\n\\begin{equation}\\label{eq-delta}\n\\min_{r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}} u(r)\\geq \\delta_{*}.\n\\end{equation}\nTo show this, let $r^{*}\\in I^{+}_{i}$ and $\\check{r}\\in\\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$ be such that\n\\begin{equation*}\nu(r^{*}) = \\max_{r\\in \\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}} u(r) = \\delta^{*}, \\quad u(\\check{r}) = \\min_{r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}} u(r).\n\\end{equation*}\nThe case $r^{*}=\\check{r}$ is trivial. So we consider the two cases: $r^{*}<\\check{r}$ and $r^{*}>\\check{r}$. \n\nIf $\\sigma_{i} \\leq r^{*} < \\check{r}\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$, then $u'(r^{*}) \\leq 0$. \nUsing the monotonicity of $r\\mapsto r^{N-1} \\varphi(u'(r))$ in $\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$, we deduce that\n\\begin{equation}\\label{eq-monot1}\nu'(\\xi) \\geq \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\zeta}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(u'(\\zeta)) \\biggr{)}, \\quad \\text{for all $\\xi,\\zeta\\in \\mathopen{]}\\sigma_{i},\\tau_{i}\\mathclose{]}$ with $\\xi\\leq\\zeta$,}\n\\end{equation}\nand so, from \\eqref{eq-monot1} with $\\xi=r^{*}$, $u'(\\zeta) \\leq 0$ for all $\\zeta\\in \\mathopen{[}r^{*},\\check{r}\\mathclose{]}$, in particular $u'(\\check{r}) \\leq 0$. \nAssume now $i\\geq 2$, or $i =1$ and $\\sigma_{1}\\neq 0$. \nAn integration of \\eqref{eq-monot1} with $\\zeta=\\check{r}$ on $\\mathopen{[}r^{*},\\check{r}\\mathclose{]}$ leads to\n\\begin{align*}\n\\delta^{*} - u(\\check{r}) &= - \\int_{r^{*}}^{\\check{r}} u'(\\xi) \\,\\mathrm{d}\\xi \n\\leq \\int_{r^{*}}^{\\check{r}} \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\check{r}}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r})) \\biggr{)} \\,\\mathrm{d}\\xi\n\\\\ &\\leq \\int_{r^{*}}^{\\check{r}} \\biggl{(} \\dfrac{\\check{r}}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r})) \\,\\mathrm{d}\\xi\n\\leq (\\check{r}-r^{*}) \\biggl{(} \\dfrac{\\check{r}}{r^{*}}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r}))\n\\\\ &\\leq |I^{+}_{i}| \\biggl{(} \\dfrac{\\tau_{i}}{\\sigma_{i}}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r})).\n\\end{align*}\nUsing \\eqref{u'_1_2} together with the fact that\n\\begin{equation*}\n\\varphi(s) \\leq 2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)}s, \\quad \\text{for all $s\\in\\biggl{[}0,\\dfrac{1}{2}\\biggr{]}$,}\n\\end{equation*}\nand estimate \\eqref{estim-u'}, we obtain\n\\begin{equation*}\n\\delta^{*} - u(\\check{r}) \\leq 2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{i}| \\dfrac{\\tau_{i}^{2N-2}}{\\sigma_{i}^{N-1}(2\\varepsilon)^{N}} \\, u(\\check{r}).\n\\end{equation*}\nThen, \\eqref{eq-delta} holds.\nConsider now the case $i=1$ and $\\sigma_{1}=0$. Then, recalling that $u'(\\xi)\\leq 0$ on $\\mathopen{[}0,\\tau_{1}\\mathclose{]}$, we have $r^{*}=0$ and $\\check{r}=\\tau_{1}-2 \\varepsilon$. Now, arguing as above and using the fact that $|u'(\\xi)|<1$ on $\\mathopen{[}0,\\tau_{1}\\mathclose{]}$, we deduce\n\\begin{align*}\n\\delta^{*} - u(\\tau_{1}-2\\varepsilon) &= - \\int_{0}^{\\tau_{1}-2 \\varepsilon} u'(\\xi) \\,\\mathrm{d}\\xi \n= - \\int_{0}^{\\gamma} u'(\\xi) \\,\\mathrm{d}\\xi - \\int_{\\gamma}^{\\tau_{1}-2\\varepsilon} u'(\\xi) \\,\\mathrm{d}\\xi\n\\\\&\\leq \\gamma + \\int_{\\gamma}^{\\tau_{1}-2\\varepsilon} \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\tau_{1}-2\\varepsilon}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\tau_{1}-2\\varepsilon)) \\biggr{)} \\,\\mathrm{d}\\xi\n\\\\ &\\leq \\gamma + 2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{1}| \\dfrac{\\tau_{1}^{2N-2}}{\\gamma^{N-1}(2\\varepsilon)^{N}} \\, u(\\tau_{1}-2 \\varepsilon).\n\\end{align*}\nThen, \\eqref{eq-delta} holds.\n\nOn the other hand, if $\\tau_{i}\\geq r^{*} > \\check{r}\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$, then $u'(r^{*}) \\geq 0$. Notice that this can happen only when $i\\geq 2$, or $i=1$ and $\\sigma_{1}\\neq 0$.\nUsing the monotonicity of $r\\mapsto r^{N-1} \\varphi(u'(r))$ in $\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$, we deduce that\n\\begin{equation}\\label{eq-monot2}\n\\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\xi}{\\zeta}\\biggr{)}^{\\! N-1} \\varphi(u'(\\xi)) \\biggr{)} \\geq u'(\\zeta), \\quad \\text{for all $\\xi,\\zeta\\in \\mathopen{]}\\sigma_{i},\\tau_{i}\\mathclose{]}$ with $\\xi\\leq\\zeta$,}\n\\end{equation}\nand so, from \\eqref{eq-monot2} with $\\zeta=r^{*}$, $u'(\\xi) \\geq 0$ for all $\\xi\\in \\mathopen{[}\\check{r},r^{*}\\mathclose{]}$, in particular $u'(\\check{r}) \\geq 0$. \nAn integration of \\eqref{eq-monot2} with $\\xi=\\check{r}$ on $\\mathopen{[}\\check{r},r^{*}\\mathclose{]}$ and an application of \\eqref{estim-u'} lead to\n\\begin{align*}\n\\delta^{*} - u(\\check{r}) &= \\int_{\\check{r}}^{r^{*}} u'(\\zeta) \\,\\mathrm{d}\\zeta\n\\leq \\int_{\\check{r}}^{r^{*}} \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\check{r}}{\\zeta}\\biggr{)}^{\\! N-1} \\varphi(u'(\\check{r})) \\biggr{)} \\,\\mathrm{d}\\zeta\n\\\\ &\\leq \\int_{\\check{r}}^{r^{*}}\\varphi^{-1} \\bigl{(} \\varphi(u'(\\check{r})) \\bigr{)} \\,\\mathrm{d}\\zeta\n\\leq |I^{+}_{i}| \\dfrac{\\tau_{i}^{N-1}}{(2\\varepsilon)^{N}} \\, u(\\check{r})\n\\leq |I^{+}_{i}| \\dfrac{\\tau_{i}^{N-1}}{(2\\varepsilon)^{N}} \\dfrac{\\tau_{i}^{N-1}}{\\sigma_{i}^{N-1}} \\, u(\\check{r}).\n\\end{align*}\nThen, we have \\eqref{eq-delta}.\n\n\\smallskip\n\nWe are now ready to verify condition $(H_{1})$ of Lemma~\\ref{lem-deg0}. We integrate the equation in \\eqref{eq-lem-deg0} on $\\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$ and, recalling \\eqref{u'_1_2}, \\eqref{eq-delta} and that $\\varphi$ is odd, we obtain\n\\begin{equation*}\n\\lambda \\min \\bigl{\\{} g(u) \\colon u \\in \\mathopen{[}\\delta_{*},\\delta^{*} \\mathclose{]}\\bigr{\\}} \\int_{\\sigma_{i}+2\\varepsilon}^{\\tau_{i}-2\\varepsilon} r^{N-1}a(r)\\,\\mathrm{d}r \\leq 2 R^{N-1}\\varphi \\biggl{(} \\dfrac{1}{2}\\biggr{)}, \n\\end{equation*}\na contradiction with respect to $\\lambda > \\lambda^{*}$.\n\nAs for assumption $(H_{2})$, we integrate the equation in \\eqref{eq-lem-deg0} in $\\mathopen{[}0,R\\mathclose{]}$ and, passing to the absolute value, we deduce\n\\begin{equation*}\n\\alpha \\| r^{N-1}v \\|_{L^{1}} \\leq \\lambda R^{N-1} \\|a\\|_{L^{1}} \\max_{u \\in \\mathopen{[}0,\\delta^{*}\\mathclose{]}} g(u).\n\\end{equation*}\nA contradiction follows for $\\alpha$ sufficiently large. From Lemma~\\ref{lem-deg0} we thus obtain\n\\begin{equation}\\label{deg-B0rho*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,\\delta^{*})) = 0.\n\\end{equation}\n\n\\subsubsection*{Step~3. Fixing the constant $d^{*}$ and computation of the degree in $B(0,d^{*})$.}\nFirst, we claim that there exists $d^{*}\\in\\mathopen{]}0,\\delta^{*}\\mathclose{[}$ such that, for every $\\vartheta\\in \\mathopen{]}0,1\\mathclose{]}$, every non-negative solution $u(r)$ of \\eqref{eq-lem-deg1} with $\\|u\\|_{\\infty} \\leq d^{*}$ is such that $u\\equiv0$.\n\nWe assume, by contradiction, that there exists a sequence $\\{u_{n}\\}_{n}$ of non-negative solutions of \\eqref{eq-lem-deg1} for $\\vartheta=\\vartheta_{n}$ satisfying $0<\\|u_{n}\\|_{\\infty}=d_{n}\\to0$.\nWe define\n\\begin{equation*}\nv_{n}(r) = \\dfrac{u_{n}(r)}{d_{n}}, \\quad r\\in\\mathopen{[}0,R\\mathclose{]},\n\\end{equation*}\nand observe that $v_{n}(r)$ is a non-negative solution of the Neumann problem associated with\n\\begin{equation}\\label{eq-v_n}\n\\Biggl{(} \\dfrac{r^{N-1} v_{n}'}{\\sqrt{1-(u_{n}')^{2}}}\\Biggr{)}' + \\vartheta_{n} \\lambda r^{N-1} a(r) q(u_{n}(r)) v_{n} = 0,\n\\end{equation}\nwhere we set $q(u) = g(u)\/u$ for $u > 0$ and $q(0) = 0$. \nIntegrating equation \\eqref{eq-v_n} between $0$ and $r$ and dividing by $r^{N-1}$ we obtain that\n\\begin{equation*}\n\\dfrac{v_{n}'(r)}{\\sqrt{1-u_{n}'(r)^{2}}} = -\\dfrac{1}{r^{N-1}} \\vartheta_{n} \\lambda \\int_{0}^{r} \\xi^{N-1}a(\\xi)q(u_{n}(\\xi))v_{n}(\\xi) \\,\\mathrm{d}\\xi, \\quad \\text{for all $r \\in \\mathopen{]}0,R\\mathclose{]}$.}\n\\end{equation*}\nPassing to the absolute value we have\n\\begin{equation*}\n|v_{n}'(r)| \\leq \\dfrac{|v_{n}'(r)|}{\\sqrt{1-u_{n}'(r)^{2}}}\n\\leq \\lambda \\int_{0}^{R} |a(\\xi)| |q(u_{n}(\\xi))||v_{n}(\\xi)| \\,\\mathrm{d}\\xi, \\quad \\text{for all $r \\in \\mathopen{]}0,R\\mathclose{]}$.}\n\\end{equation*}\nTherefore, using the first condition in $(g_{0})$ and the fact that $\\| v_{n} \\|_{\\infty} \\leq 1$, we obtain that $v_{n}' \\to 0$ uniformly.\nAs a consequence, $v_{n} \\to 1$ uniformly in $\\mathopen{[}0,R\\mathclose{]}$, since\n\\begin{equation*}\n|v_{n}(r) - 1 | = |v_{n}(r) - v_{n}(\\hat{\\eta}_{n}) | \\leq \\int_{0}^{R} | v_{n}'(\\xi) | \\,\\mathrm{d}\\xi, \\quad \\text{for all $r \\in \\mathopen{[}0,R\\mathclose{]}$,}\n\\end{equation*}\nwhere $\\hat{\\eta}_{n}\\in \\mathopen{[}0,R\\mathclose{]}$ is such that $u_{n}(\\hat{\\eta}_{n}) = \\|u_{n}\\|_{\\infty} = d_{n}$.\nAn integration of equation \\eqref{eq-v_n} in $\\mathopen{[}0,R\\mathclose{]}$ gives\n\\begin{equation*}\n\\int_{0}^{R} r^{N-1} a(r) g(u_{n}(r))\\,\\mathrm{d}r = 0\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\int_{0}^{R} r^{N-1} a(r) g(d_{n})\\,\\mathrm{d}r + \\int_{0}^{R} r^{N-1} a(r)\\bigl{[}g(d_{n} v_{n}(r)) - g(d_{n})\\bigr{]}\\,\\mathrm{d}r=0.\n\\end{equation*}\nDividing by $g(d_{n}) > 0$, we have\n\\begin{equation*}\n0 < - \\int_{0}^{R} r^{N-1} a(r)\\,\\mathrm{d}r \\leq R^{N-1}\\|a\\|_{L^{1}} \\sup_{r\\in \\mathopen{[}0,R\\mathclose{]}}\\biggl{|}\\dfrac{g(d_{n} v_{n}(r))}{g(d_{n})} - 1\\biggr{|}.\n\\end{equation*}\nUsing the second condition in $(g_{0})$ and recalling that $v_{n} \\to 1$ uniformly, we find a contradiction. The claim is thus proved and we can fix $d^{*}\\in\\mathopen{[}0,\\delta^{*}\\mathclose{[}$.\n\nFinally, condition $(H_{3})$ of Lemma~\\ref{lem-deg1} is trivially satisfied for $d=d^{*}$, and therefore\n\\begin{equation}\\label{deg-B0d*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d^{*})) = 1.\n\\end{equation}\n\n\\subsubsection*{Step~4. Fixing the constant $D^{*}$ and computation of the degree in $B(0,D^{*})$.}\nFirst, we claim that there exists $D^{*}>\\delta^{*}$ such that, for every $\\vartheta\\in \\mathopen{]}0,1\\mathclose{]}$, every non-negative solution $u(r)$ of \\eqref{eq-lem-deg1} satisfies $\\|u\\|_{\\infty} < D^{*}$.\n\nWe assume, by contradiction, that there exists a sequence $\\{u_{n}\\}_{n}$ of non-negative solutions of \\eqref{eq-lem-deg1} for $\\vartheta=\\vartheta_{n}$ and $\\lambda=\\lambda_{n}$ satisfying $\\|u_{n}\\|_{\\infty}=D_{n}\\to+\\infty$. We proceed similarly to the previous step. We define\n\\begin{equation*}\nv_{n}(r) = \\dfrac{u_{n}(r)}{D_{n}}, \\quad r\\in\\mathopen{[}0,R\\mathclose{]},\n\\end{equation*}\nwhich solves the Neumann problem associated with equation \\eqref{eq-v_n}. Since $\\|u_{n}'\\|_{\\infty} \\leq 1$, we easily find $\\|v_{n}'\\|_{\\infty} \\to 0$ and, consequently, $v_{n} \\to 1$ uniformly in $\\mathopen{[}0,R\\mathclose{]}$ (proceeding as shown in Step~3).\nIntegrating equation \\eqref{eq-v_n} and dividing by $g(D_{n}) > 0$, we thus obtain\n\\begin{equation*}\n0 < - \\int_{0}^{R} r^{N-1}a(r)\\,\\mathrm{d}r \\leq R^{N-1}\\|a\\|_{L^{1}} \\sup_{r\\in \\mathopen{[}0,R\\mathclose{]}}\\biggl{|}\\dfrac{g(D_{n} v_{n}(r))}{g(D_{n})} - 1\\biggr{|}.\n\\end{equation*}\nUsing $(g_{\\infty})$, a contradiction easily follows. The claim is thus proved and we can fix $D^{*}>\\delta^{*}$.\n\nFinally, condition $(H_{3})$ of Lemma~\\ref{lem-deg1} is trivially satisfied for $d=D^{*}$, and therefore\n\\begin{equation}\\label{deg-B0D*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,D^{*})) = 1.\n\\end{equation}\n\n\\subsubsection*{Step~5. Concluding the proof.}\nStarting from the formulas \\eqref{deg-B0rho*}, \\eqref{deg-B0d*}, \\eqref{deg-B0D*} proved in the last three steps, we apply the additivity property of the coincidence degree to obtain\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,\\delta^{*}) \\setminus \\overline{B(0,d^{*})}) = -1\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,D^{*}) \\setminus \\overline{B(0,\\delta^{*})}) = 1.\n\\end{equation*}\nAs a consequence of the existence property of the degree, there exist two solutions $u_{s}\\in B(0,\\delta^{*}) \\setminus \\overline{B(0,d^{*})}$ and $u_{\\ell}\\in B(0,D^{*}) \\setminus \\overline{B(0,\\delta^{*})}$ of problem \\eqref{eq-phi}. Then, $u_{s}(r)$ and $u_{\\ell}(r)$ satisfy \n\\begin{equation*}\nd^{*} < \\| u_{s} \\|_{\\infty} < \\delta^{*} < \\| u_{\\ell} \\|_{\\infty} < D^{*}.\n\\end{equation*}\nBy maximum principle arguments, both these solutions are non-negative and hence they solve \\eqref{eq-main}.\nIt thus remains to show that they are positive, that is, $u(r) > 0$ for every $r \\in [0,R]$, for both $u = u_s$ and $u = u_{\\ell}$.\n\nBy contradiction, assume that $u(r_0) = 0$ for some $r_0 \\in [0,R]$. Then $u'(r_0) = 0$, coming from the boundary conditions if $r_0 = 0$ and from the fact that $u(r)$ is non-negative if $r_0 \\in (0,R]$.\n\nIf $r_0 = 0$, integrating the equation we find\n\\begin{equation*}\nu(r) = - \\int_{0}^r \\varphi^{-1}\\biggl{(}\\dfrac{1}{\\zeta^{N-1}} \\int_{0}^{\\zeta} \\xi^{N-1}a(\\xi)g(u(\\xi))\\,\\mathrm{d}\\xi \\biggr{)} \\,\\mathrm{d}\\zeta, \\quad \\text{for all $r \\in \\mathopen{[}0,R\\mathclose{]}$,}\n\\end{equation*}\nimplying, since $|\\varphi^{-1}(s)|\\leq |s|$ for all $s \\in \\mathbb{R}$, that\n\\begin{equation*}\n|u(r)| \\leq \\int_{0}^r \\int_{0}^{\\zeta} |a(\\xi)| |g(u(\\xi))|\\,\\mathrm{d}\\xi \\,\\mathrm{d}\\zeta \n\\leq R \\int_{0}^r |a(\\xi)| |g(u(\\xi))|\\,\\mathrm{d}\\xi, \n\\end{equation*}\nfor all $r \\in \\mathopen{[}0,R\\mathclose{]}$.\nRecalling that $g(u)\/u \\to 0$ for $u \\to 0^{+}$, we finally obtain \n\\begin{equation*}\n|u(r)| \\leq M R \\int_{0}^r |a(\\xi)| |u(\\xi)|\\,\\mathrm{d}\\xi , \\quad \\text{for all $r \\in \\mathopen{[}0,R\\mathclose{]}$,}\n\\end{equation*}\nwhere $M > 0$ is a suitable constant. By Gronwall's lemma, $u(r) = 0$ for every $r \\in \\mathopen{[}0,R\\mathclose{]}$, a contradiction.\n\nIn the case $r_0 \\in \\mathopen{]}0,R\\mathclose{]}$, the contradiction is reached again using the assumption $g(u)\/u \\to 0$ for $u \\to 0^{+}$, which as well known implies\n (together with the smoothness of $\\varphi^{-1}$)\nthat the only solution of the planar system\n\\begin{equation*}\nu' = \\varphi^{-1}\\biggl{(} \\dfrac{v}{r^{N-1}} \\biggr{)}, \\qquad v' = -r^{N-1} a(r) g(u),\n\\end{equation*}\nsatisfying the initial condition $(u(r_0),v(r_0)) = (0,0)$ is the trivial one.\n\\qed\n\n\\bibliographystyle{elsart-num-sort}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nStable broadband sources are essential for a variety of applications, from spectroscopy \\cite{Liu2013} to imaging \\cite{Kano2005} and nonlinear optical parametric amplifiers \\cite{Reed1995}. Supercontinuum (also known as white light) generation offers a promising tool for these multidisciplinary uses. Sapphire is regarded as the crystal of choice for visible supercontinuum generation \\cite{Yakovlev1994} and is the focus of our particular study. Supercontinua are generated in sapphire through the process of single filamentation. This general process encompasses a wide variety of linear and nonlinear effects including self-steepening, self-phase modulation, dispersion, four-wave mixing, Raman excitation, second and third harmonic generation, and plasma generation, absorption, and refraction \\cite{Alfano2006}. Since in the most general case it is diffcult to separate the exact contribution of each of these, individual optimization of each effect is neither particularly feasible nor extremely desirable. However, each of these effects can be highly influenced by the spatial distribution of the beam. One of the most basic effects of spatial shaping is to focus the beam tighter, thereby generating more plasma and influencing the resultant spectrum. Although studies with Bessel \\cite{Majus2013} and Laguerre-Gauss beams \\cite{Sztul2006a} have failed to find significant spectral deviations or improvements in effciency from the Gaussian regime, systematic spatial optimization has not been performed. \n\nPrevious work that has shown promise in this direction includes a study performed with microlenses generated via spatial light modulators \\cite{Borrego-Varillas2014}. Moreover, there have been several successful studies of spectral pump-beam optimization in filamentation \\cite{Thompson2017, Ackermann2006} and other nonlinear effects, such as the control of second harmonic generation \\cite{Thompson2016} and the enhancement of spontaneous Raman signals through a turbid medium \\cite{Thompson2016a}. We extend these results and methods to the theoretically challenging regime of supercontinuum generation by using the wavefront shaping algorithm to influence the supercontinuum spectrum.\n\n\n\nThere are several aspects of interest here, including reshaping the physics of filamentation and drawing attention to the potentials of non-Gaussian beams. More commercial applications are also possible. We envision that this work may lead to universally stronger seeds for spectroscopic applications that depend on nonlinear effects for a large signal-to-noise. Our preliminary results are a proof-of-concept that the idea of spatial optimization has merit and may be further expanded with additional work.\n\n\n\n\\section{Experimental Setup}\n\nThe experimental setup is depicted in Fig. \\ref{fig:setup}. We used a Ti:Sapphire regenerative amplifier (Coherent, Legend) to produce infrared ($\\lambda = 802$ nm) $35$ fs pulses with a $1$ kHz repetition rate and $4$ W average power that we attenuated to produce supercontinuum. We then investigated two regimes of supercontinuum generation: chirped and unchirped. In the first case, we added positive chirp by changing the grating distance within the compressor unit of the amplifier. This produced pulses of 900 fs FWHM duration, measured using a commercial autocorrelator (Pulse Check; A.P.E.).\n\n\n\\begin{figure\n\t\\centering\n\t\n\n\n\t\\includegraphics[width=0.5\\textwidth]{setup}\n\t\\caption{Setup for generating supercontinua (SC) from shaped pulses; a photograph of the generated SC is shown in the inset. The angle of the spatial light modulator (SLM) is greatly exaggerated. OAP stands for off-axis parabola and was used to collimate the SC spectrum after the crystal.}\n\t\\label{fig:setup}\n\t\n\n\\end{figure} \n\nDifferent powers\/pulse were needed to produce supercontinua in each regime: we used 5-6 $\\mu$J in the chirped and 1 $\\mu$J in the unchirped cases. In both regimes, we used a spatial phase-only light modulator (Hamamatsu; x10468; abbreviated SLM) to shape the originally Gaussian beam. The beam has a 15 mm diameter prior to being shaped by the SLM; this is the greatest we could expand the beam without clipping on the SLM screen. We then split the beam and focused part of it with a 5 cm focal length lens to generate a supercontinuum in a 3 mm thick single crystal c-cut sapphire plate (Newlight Photonics; SAP0030-C; Toronto, ON). The resultant diverging supercontinuum was collimated by an off-axis paraboloid to a 1-inch diameter. The pump beam was then filtered out via a 750 nm short pass filter (Semrock; FF01-745\/SP-25; Rochester, NY). The filtered beam was subsequently refocused with a 20 cm focal length lens into a multi-mode 600 $\\mu$m core fiber. Since the supercontinuum light should roughly focus to 5 $\\mu$m, the fiber core is of sufficient size to collect all the light and not be affected by any spatial phase changes the SLM adds. We checked this assumption by translating the fiber in x-y dimensions in the focal plane; this operation revealed no unmeasured light. Hence, we are confident that the VCSA algorithm is not optimizing the light-collection system.\n\nThe other part of the beam was sent through a long (2 m focal length) lens to be very loosely focused onto a CCD array (Spiricon; SP620U) and recorded by the computer. These images did not take part in the spatial optimization at all -- they are there to help visualize the effect of different phase maps on the beam's spatial profile in the focus. \n\n\\section{Optimization Details}\n\nFor all optimization regimes, we used a variation of the continuous sequential algorithm (abbreviated VCSA) \\cite{Vellekoop2008, Thompson2015}. The VCSA groups pixels on the SLM together and cycles through $2\\pi$ phase values in 8 steps. The algorithm then compares spectrometer output in a particular spectral range before and after adding different phase values. If the average of the spectrometer reading in that spectral range improves, then the algorithm keeps the phase value. This cycle is repeated three times and results averaged to minimize influence from shot-to-shot fluctuations and other noise. The algorithm then moves on to another pixel group and repeats the process. Each iteration takes 12 seconds, with the spectrometer integration time forming the largest limit on speed. \n\nFor all results given in this paper, we employed the ``spiral out\" method of this algorithm, which starts with large pixel groups (of 264 $x$ 300 pixels) in the center and spirals out to the edges, as in Figure \\ref{fig:algorithm}. It then starts a new stage at the center with smaller pixel groups (of 132 $x$ 150 pixels) and spirals out until it is forced to repeat itself with even smaller groups of pixels (of 72 $x$ 60). The final run consists of groups of 24 $x$ 24 pixels. In total, we let the algorithm optimize for roughly half an hour for the results given in this paper. We do not consider time to be a major limit in our experiment, as there are no discernible differences between spectra taken at the beginning of the day and those taken at the end. Further, for spectroscopic applications, it will not be necessary to quickly reoptimize the masks so long as the user takes care to produce a bank of working masks that they may easily switch between.\n\n\\begin{figure\n\t\\centering\n\t\n\t\n\t\n\n\t\\includegraphics[width=0.5\\textwidth]{algorithm_3}\n\t\\caption{The smaller the block the longer the program takes to finish; the process can be stopped at any time if the user is satisfied with the results. Our SLM is comprised of 792 $x$ 600 pixels total.}\n\t\\label{fig:algorithm}\n\t\n\t\n\t\n\\end{figure} \n\n\\section{Preliminary Results}\n\nUsing these methods, we were able to obtain a general 10\\% broadening of the spectral width of the supercontinuum generation for highly positively chirped pulses (900 fs), as shown in Fig. \\ref{fig:chirpedresults}. However, this regime is tricky to work with as the damage threshold for these focusing conditions in sapphire is near the critical power of self-focusing. This makes further optimization difficult but not impossible, potentially under different focusing conditions.\n\nFor 35 fs unchirped pulses, we discovered that it is possible to shift the supercontinuum spectral cutoff peak between 450 and 650 nm, as our preliminary results indicate in Figure \\ref{fig:results}. The region from 450-500 nm is completely absent in the supercontinuum spectrum generated without any phase mask applied and so represents a significant broadening ($>\\approx 20\\%$). In this case, the effect of the added phase mask on the supercontinuum spectrum is easily noticeable by eye and hence can not be due to any limitations in our light-collection system.\n\n Further, the phase masks shown in Figure \\ref{fig:results} generate the same spectrum from day-to-day without any special additional environmental control, making our experiment repeatable in a variety of conditions. However, the spatial profile of the shaped beam is very sensitive to the alignment of the pump beam on the SLM screen. This is because any displacement in this region will result in different parts of the beam obtaining different phase values, and hence not reproducing the original phase-optimized beam. In this case, each phase mask will need to be re-optimized to obtain a tailored spectrum.\n\n\\begin{figure\n\t\\centering\n\t\n\t\n\t\n\n\t\\includegraphics[width=0.5\\textwidth]{chirped_results2}\n\t\\caption{SC spectrum before (blue line) and after (red line, dash-dot) spatially optimizing the pump pulse. The range of optimization was 500--550 nm.}\n\t\\label{fig:chirpedresults}\n\t\n\t\n\t\n\t\n\t\n\\end{figure} \n\n\n\n\\begin{figure\n\t\\centering\n\t\n\t\n\t\n\n\t\\includegraphics[width=1\\textwidth]{dotdash_results2}\n\t\\caption{(a) Measured supercontinuum spectrum for different optimization regimes -- the supercontinuum cutoff peak is spectrally shifted as the spatial shape changes. Each entry in the legend corresponds to the optimization range of that particular run of the algorithm (i.e. for the second entry, the algorithm attempts to optimize the average spectrometer-measured counts in the range of 450 -- 500 nm). All spectrums were taken with the phase masks and profiles in (b). (b) SLM phase masks (top), beam profiles in the focus magnified approximately 40 times and with the left three profiles integrated 5x longer than the right-most profile (middle), and true-color photographs of the resultant supercontinuum (bottom; taken with a Sony DSLR camera) for different optimization regimes.}\n\t\\label{fig:results}\n\t\n\t\n\t\n\\end{figure} \n\n\\section{Conclusions}\n\nOur preliminary results indicate that spatial beam shaping has a substantial untapped potential in optimizing supercontinuum generation by enhancing a particular spectral region. We envision that this technique can dramatically improve the ability to tailor the supercontinuum spectrum for any particular application. For example, we can provide significantly stronger seed pulses for optical parametric amplifiers and substantially enhance signals in broadband coherent anti-Stokes Raman spectroscopy\/microscopy. An SLM provides a much more flexible platform, as compared to a micro-structured fiber, to tailor the spectral properties of the supercontinuum\n\\cite{Zheltikov2006}. The user will simply load the SLM with the phase mask for the particular spectral range they desire. By pre-generating optimal phase masks, the frequency can be tuned at the 10 Hz refresh rate of the SLM.\n\nIn the future, we envision that this will lead to higher available powers for various nonlinear spectroscopy experiments and hence a greater signal-to-noise, paving the way for future precision measurements. Further work will include explorations of the theoretical foundations of spatial effects in high-order nonlinear optical interactions, which we initiated in \\cite{Thompson2016}. We also plan thorough investigations of algorithmic shaping in the IR.\n\nThis research was partially supported by the NSF (Grant \\# PHY-1307153, DBI-1532188, and ECCS-1509268), the US Department of Defense (award FA9550-15-1-0517), the Welch Foundation (Awards No. A-1547 and No. A-1261), the Cancer Prevention Research Institute of Texas (grant RP160834), and the Office of Navel Research (Award No. N00014-16-1-3054).\n\n\\bibliographystyle{jmo_test2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe $\\ast$-involution, sometimes referred to as Kashiwara's involution, is an involution on the crystal $B(\\infty)$ that is induced from a subtle involutive antiautomorphism of $U_q(\\mathfrak{g})$. \nThe importance of $\\ast$ in the theory of crystal bases and their applications cannot be understated. Here are just a few of its applications.\n\\begin{enumerate}\n\\item Saito~\\cite{Sai94} used the involution during the proof that Lusztig's PBW basis has a crystal structure isomorphic to $B(\\infty)$, provided that $\\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra.\n\\item Kamnitzer and Tingley~\\cite{KT09} generalize the definition of the crystal commutor of Henriques and Kamnitzer \\cite{HK06} in terms of the $\\ast$-involution. This leads to a proof by Savage~\\cite{Sav09} that the category of crystals forms a coboundary category over any symmetrizable Kac-Moody algebra.\n\\item In affine type $A$, Jacon and Lecouvey~\\cite{JL09} prove that $\\ast$ coincides with the Zelevinsky involution~\\cite{MW86,Zel80} on the set of simple modules for the affine Hecke algebra.\n\\end{enumerate}\nSeveral combinatorial realizations of the $\\ast$-involution are known in the literature. For example, Lusztig~\\cite{Lusztig93} gave a description of the behavior of $\\ast$ on Lusztig's PBW basis in the finite types, Kamnitzer~\\cite{Kamnitzer07} showed that $\\ast$ acts on an MV polytope by negation, Kashiwara and Saito \\cite{KS97} gave a description of $\\ast$ in terms of quiver varieties~\\cite{KS97}, and Jacon and Lecouvey~\\cite{JL09} give a description of the involution in terms of the multisegment model. Such model-specific calculations of the $*$-crystal operators are important as, {\\it a priori}, the algorithm for computing the action of these operators is not efficient \\cite[Thm. 2.2.1]{K93} (see also \\cite[Prop. 8.1]{K95}).\n\n\nIn this paper, the authors continue their development of the rigged configuration model of $B(\\infty)$~\\cite{SalS15,SalS15II}. Rigged configurations are sequences of partitions, one for every node of the underlying Dynkin diagram, where each part is paired with an integer, satisfying certain conditions. These objects arose as an important tool in mathematical physics from the studies of the Bethe Ansatz by Kerov, Kirillov, and Reshetikhin~\\cite{KKR86,KR86}, and they have been shown to correspond to the action and angle variables of box-ball systems~\\cite{KOSTY06}. Additionally, rigged configurations have been used extensively in the theory of Kirillov-Reshetikhin crystals \\cite{HKOTT02,HKOTY99,OSS13,OSS03,OSS03II,Sakamoto14,S05,S06,SchillingS15,SW10,Scrimshaw15}. During the course of this study, a crystal structure was given to rigged configurations~\\cite{S06, SchillingS15}.\n\nOur description of $\\ast$ is as nice as one could hope: in contrast to the definition of $e_a$ and $f_a$ on rigged configurations, one interchanges ``label'' and ``colabel'' to obtain a definition of $e_a^*$ and $f_a^*$ (see Definition \\ref{def:RC_star_crystal_ops}). In turn, applying $\\ast$ to a rigged configuration replaces all labels with its corresponding colabels and leaves the partitions fixed (see Corollary~\\ref{cor:RC_star_involution}).\n\nThe method of proof applied here is to use a classification theorem of $B(\\infty)$ asserted by Tingley and Webster~\\cite{TW} by translating the $\\ast$-involution directly into the classification theorem of Kashiwara and Saito~\\cite{KS97} without the use of Kashiwara's embedding. This classification theorem requires several assertions to be satisfied, and proving these assertions hold in $\\RC(\\infty)$ with our new $\\ast$-crystal operators consumes most of Section \\ref{sec:*crystal}.\n\n\nThe (conjectural) bijection $\\Phi$ between $U_q'(\\mathfrak{g})$-rigged configurations and tensor products of Kirillov-Reshetikhin crystals~\\cite{OSS13, OS12, OSS03, OSS03II, OSS03III, S05, SchillingS15, SS2006, Scrimshaw15} is given roughly as follows. It removes the largest row with a colabel of 0, which is the minimal colabel, for each $e_a$ from $b$ to the highest weight element in $B(\\Lambda_1)$, where $b$ is the leftmost factor in the tensor product. \nLet $\\theta$ be the involution on $U_q'(\\mathfrak{g})$-rigged configurations which interchanges labels with colabels on classically highest weight $U_q'(\\mathfrak{g})$-rigged configurations and let $\\widetilde{\\ast}^L$ denote the involution which is the composition of Lusztig's involution and the map sending the result to the classically highest weight element~\\cite{S05, SS2006} (where it is also denoted by $\\ast$). It is known that $\\Phi \\circ \\theta = \\widetilde{\\ast}^L \\circ \\Phi$ on classically highest weight elements. In particular, the latter map reverses the order of the tensor product. Thus, given the description of the crystal commutor, our work suggests there is a strong link between the $\\ast$-involution and the bijection $\\Phi$. We hope this could lead to a more direct description of the bijection $\\Phi$, its related properties, and a (combinatorial) proof of the $X = M$ conjecture of~\\cite{HKOTT02, HKOTY99}.\n\nAnother model for $B(\\infty)$ uses marginally large tableaux, as developed by Hong and Lee \\cite{HL08, HL12}. It is known that the bijection $\\Phi$ mentioned above can be extended to a $U_q(\\mathfrak{g})$-crystal isomorphism between rigged configurations and marginally large tableaux~\\cite{SalS15III} when $\\mathfrak{g}$ is of finite classical type or type $G_2$. An ambitious hope of this paper is that it may lead to a description of the $\\ast$-crystal structure on marginally large tableaux. (In finite type $A$, this result is in~\\cite{CT15}.) However, this appears to be a hard problem as the bijection $\\Phi$ is highly recursive and depends on conditions on colabels, many of which can change under applying the $\\ast$-crystal operators.\n\nThere is also a model for $B(\\infty)$ using Littelmann paths constructed by Li and Zhang~\\cite{LZ11}. From~\\cite{PS15}, natural virtualization maps arise to the embeddings on the underlying geometric information. The virtualization map on rigged configurations is also quite natural, giving evidence that rigged configurations encode more geometry than their combinatorial origins and description suggests. This is also evidence that there exists a straightforward and natural explicit combinatorial bijection between rigged configurations and the Littelmann path model. Thus this work could potentially lead to a description of the $\\ast$-crystal on the Littelmann path model.\n\nIn a similar vein, the virtualization map is known to act naturally on MV polytopes~\\cite{JS15,NS08}, also reflecting the geometric information of the root systems via the Weyl group. This is evidence that there should be a natural explicit combinatorial bijection between MV polytopes and rigged configurations (and the Littelmann path model). Moreover, considering the $\\ast$-involution, which acts by negation on MV polytopes~\\cite{Kamnitzer07, Kamnitzer10}, this work gives further evidence that such a bijection should exist. Furthermore, this bijection would suggest a natural generalization beyond finite type, which the authors expect to recover the KLR polytopes of~\\cite{TW}.\n\nThis paper is organized as follows.\nIn Section~\\ref{sec:background}, we give the necessary background on crystals and the $\\ast$-involution.\nIn Section~\\ref{sec:RC}, we give background information on the rigged configuration model for $B(\\infty)$.\nIn Section~\\ref{sec:*crystal}, we give the proof of our main theorem and some consequences.\nIn Section~\\ref{sec:hw_crystals}, we give a description of highest weight crystals using the $\\ast$-crystal structure and describe the natural projection from $B(\\infty)$ in terms of rigged configurations.\n\n\n\n\n\n\n\n\n\\section{Crystals and the $\\ast$-involution}\n\\label{sec:background}\n\nLet $\\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with quantized universal enveloping algebra $U_q(\\mathfrak{g})$ over $\\mathbf{Q}(q)$, index set $I$, generalized Cartan matrix $A = (A_{ij})_{i,j\\in I}$, weight lattice $P$, root lattice $Q$, fundamental weights $\\{\\Lambda_i \\mid i \\in I\\}$, simple roots $\\{\\alpha_i \\mid i\\in I\\}$, and simple coroots $\\{h_i \\mid i\\in I\\}$. There is a canonical pairing $\\langle\\ ,\\ \\rangle\\colon P^\\vee \\times P \\longrightarrow \\mathbf{Z}$ defined by $\\langle h_i, \\alpha_j \\rangle = A_{ij}$, where $P^{\\vee}$ is the dual weight lattice.\n\nAn \\defn{abstract $U_q(\\mathfrak{g})$-crystal} is a set $B$ together with maps\n\\[\n e_i, f_i \\colon B \\longrightarrow B\\sqcup\\{0\\},\\qquad\n\\varepsilon_i,\\varphi_i\\colon B \\longrightarrow \\mathbf{Z} \\sqcup \\{-\\infty\\},\\qquad\n\\mathrm{wt}\\colon B \\longrightarrow P\n\\]\nsatisfying certain conditions (see \\cite{HK02,K95}). Any $U_q(\\mathfrak{g})$-crystal basis, defined in the classical sense (see \\cite{K91}), is an abstract $U_q(\\mathfrak{g})$-crystal. In particular, the negative half $U_q^-(\\mathfrak{g})$ of the quantized universal enveloping algebra of $\\mathfrak{g}$ has a crystal basis which is an abstract $U_q(\\mathfrak{g})$-crystal. We denote this crystal by $B(\\infty)$ (rather than the using the entire tuple $(B(\\infty),e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt})$), and denote its highest weight element by $u_\\infty$. As a set, one has\n\\[\nB(\\infty) = \\{ f_{i_d} \\cdots f_{i_2} f_{i_1} u_\\infty : i_1,\\dots,i_d \\in I, \\ d \\ge 0 \\}.\n\\]\nThe remaining crystal structure on $B(\\infty)$ is\n\\begin{align*}\n\\mathrm{wt}( f_{i_d} \\cdots f_{i_2} f_{i_1} u_\\infty) &= -\\alpha_{i_1}-\\alpha_{i_2}-\\cdots-\\alpha_{i_d} ,\\\\\n\\varepsilon_i(b) &= \\max\\{ k \\in \\mathbf{Z} : e_ib \\neq 0 \\}, \\\\\n\\varphi_i(b) &= \\varepsilon_i(b) + \\langle h_i,\\mathrm{wt}(b) \\rangle. \n\\end{align*}\nWe say that $b\\in B(\\infty)$ has \\defn{depth} $d$ if $b = f_{i_d} \\cdots f_{i_2} f_{i_1} u_\\infty$ for some $i_1,\\dots,i_d \\in I$.\n\n\n\nThere is a $\\mathbf{Q}(q)$-antiautomorphism $*\\colon U_q(\\mathfrak{g}) \\longrightarrow U_q(\\mathfrak{g})$ defined by\n\\[\nE_i \\mapsto E_i, \\qquad\nF_i \\mapsto F_i, \\qquad\nq \\mapsto q, \\qquad\nq^h \\mapsto q^{-h}.\n\\]\nThis is an involution which leaves $U_q^-(\\mathfrak{g})$ stable. Thus, the map $\\ast$ induces a map on $B(\\infty)$, which we also denote by $*$, and is called the \\defn{$\\ast$-involution} or \\defn{star involution} (and is sometimes known as Kashiwara's involution). Denote by $B(\\infty)^*$ the image of $B(\\infty)$ under $*$. \n\n\\begin{thm}[\\cite{K93,Lusztig90}]\nWe have\n$\nB(\\infty)^* = B(\\infty).\n$\n\\end{thm}\n\nThis induces a new crystal structure on $B(\\infty)$ with Kashiwara operators\n\\[\n e_i^* = * \\circ e_i \\circ *,\\ \\ \\ \\ \\ \\ \n f_i^* = * \\circ f_i \\circ *,\n\\]\nand the remaining crystal structure is given by\n\\[\n\\varepsilon_i^* = \\varepsilon_i \\circ * , \\qquad \\qquad\n\\varphi_i^* = \\varphi_i \\circ *,\n\\]\nand weight function $\\mathrm{wt}$, the usual weight function on $B(\\infty)$.\nAdditionally, for $b\\in B(\\infty)$ and $i\\in I$, define\n\\begin{equation}\n\\label{eq:jump}\n\\kappa_i(b) := \\varepsilon_i(b) + \\varepsilon_i^*(b) + \\langle h_i, \\mathrm{wt}(b)\\rangle.\n\\end{equation}\nThis was called the {\\it $i$-jump} in \\cite{LV11}.\n\n\n\n\n\n\nWe will appeal to the following statement from \\cite{CT15}, which was proven in a dual form in \\cite{TW} based on Kashiwara and Saito's classification theorem for $B(\\infty)$ from \\cite{KS97}. First, a \\defn{bicrystal} is a set $B$ with two abstract $U_q(\\mathfrak{g})$-crystal structures $(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt})$ and $(B,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt})$ with the same weight function. In such a bicrystal $B$, we say $b\\in B$ is a \\defn{highest weight element} if $e_ib = e_i^\\star b = 0$ for all $i \\in I$.\n\n\\begin{prop}\n\\label{prop:star_properties}\nFix a bicrystal $B$ with highest weight $b_0$ such the crystal data is determined by setting $\\mathrm{wt}(b_0) = 0$. Assume further that, for all $i \\neq j$ in $I$ and all $b\\in B$, \n\\begin{enumerate}\n\\item\\label{item:star1} $f_ib$, $f_i^\\star b \\neq 0$;\n\\item\\label{item:star2} $f_i^\\star f_jb = f_jf_i^\\star b$;\n\\item\\label{item:star3} $\\kappa_i(b) \\ge 0$;\n\\item\\label{item:star4} $\\kappa_i(b) = 0$ implies $f_ib = f_i^\\star b$;\n\\item\\label{item:star5} $\\kappa_i(b) \\ge 1$ implies $\\varepsilon_i^\\star(f_ib) = \\varepsilon_i^\\star(b)$ and $\\varepsilon_i(f_i^\\star b) = \\varepsilon_i(b)$;\n\\item\\label{item:star6} $\\kappa_i(b) \\ge 2$ implies $f_if_i^\\star b = f_i^\\star f_ib$.\n\\end{enumerate}\nThen \n\\[\n(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt}) \\cong (B,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt}) \\cong B(\\infty),\\]\nwith $e_i^\\star = e_i^*$ and $f_i^\\star = f_i^*$.\n\\end{prop}\n\nHowever, we will need to slightly weaken the assumptions of Proposition~\\ref{prop:star_properties}.\n\n\\begin{prop}\n\\label{prop:weaker_conditions}\nLet $(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt})$ and $(B^\\star,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt})$ be highest weight abstract $U_q(\\mathfrak{g})$-crystals with the same highest weight vector $b_0 \\in B \\cap B^\\star$, where the remaining crystal data is determined by setting $\\mathrm{wt}(b_0) = 0$. Suppose also that (\\ref{item:star1})--(\\ref{item:star6}) are satisfied. Then\n\\[\n(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt}) \\cong (B^\\star,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt}) \\cong B(\\infty),\n\\]\nwith $e_i^\\star = e_i^*$ and $f_i^\\star = f_i^*$.\n\\end{prop}\n\n\\begin{proof}\nWe prove that $B \\cap B^\\star$ is closed under $f_i$ and $f_i^\\star$ by using induction on the depth and making repeated use of conditions (\\ref{item:star1})--(\\ref{item:star6}) above. The base case is depth $0$, where we just have $b_0$. Suppose all elements of depth at most $d$ in $B$ and $B^\\star$ are in $B \\cap B^\\star$. Next, fix some $b \\in B \\cap B^\\star$ at depth $d$. If $\\kappa_i(b) = 0$, then $f_i b = f_i^\\star b$ for all $i\\in I$. If $i \\neq j$, then $f_i^\\star f_j b' = f_j f_i^\\star b'$. Hence $f_i^* b \\in B \\cap B^\\star$, and $f_j b \\in B \\cap B^\\star$ by our induction assumption and that $B$ (resp., $B^\\star$) is closed under $f_j$ (resp., $f_i^\\star$). A similar argument shows that $f_i b \\in B \\cap B^\\star$ if $\\kappa_i(b) \\geq 1$ since $\\kappa_i(b') \\geq 2$. Therefore $B = B \\cap B^\\star = B^\\star$ since $B \\cap B^\\star$ is closed under $f_i$ and $f_j^\\star$ and generated by $b_0$ (along with $B$ and $B^\\star$). Thus the claim follows by Proposition~\\ref{prop:star_properties}.\n\\end{proof}\n\n\n\n\n\\section{Rigged configurations}\n\\label{sec:RC}\n\nLet $\\mathcal{H} = I \\times \\mathbf{Z}_{>0}$. A rigged configuration is a sequence of partitions $\\nu = (\\nu^{(a)} \\mid a \\in I)$ such that each row $\\nu_i^{(a)}$ has an integer called a \\defn{rigging}, and we let $J = \\bigl(J_i^{(a)} \\mid (a, i) \\in \\mathcal{H} \\bigr)$, where $J_i^{(a)}$ is the multiset of riggings of rows of length $i$ in $\\nu^{(a)}$. We consider there to be an infinite number of rows of length $0$ with rigging $0$; i.e., $J_0^{(a)} = \\{0, 0, \\dotsc\\}$ for all $a \\in I$. The term rigging will be interchanged freely with the term \\defn{label}. We identify two rigged configurations $(\\nu, J)$ and $(\\widetilde{\\nu}, \\widetilde{J})$ if \n\\[\nJ_i^{(a)} = \\widetilde{J}_i^{(a)}\n\\]\nfor any fixed $(a, i) \\in \\mathcal{H}$. Let $(\\nu, J)^{(a)}$ denote the rigged partition $(\\nu^{(a)}, J^{(a)})$.\n\nDefine the \\defn{vacancy numbers} of $\\nu$ to be \n\\begin{equation}\n\\label{eq:vacancy}\np_i^{(a)}(\\nu) = p_i^{(a)} = - \\sum_{(b,j) \\in \\mathcal{H}} A_{ab} \\min(i, j) m_j^{(b)},\n\\end{equation}\nwhere $m_i^{(a)}$ is the number of parts of length $i$ in $\\nu^{(a)}$. The \\defn{corigging}, or \\defn{colabel}, of a row in $(\\nu,J)^{(a)}$ with rigging $x$ is $p_i^{(a)} - x$. In addition, we can extend the vacancy numbers to\n\\[\np_{\\infty}^{(a)} = \\lim_{i\\to\\infty} p_i^{(a)} = - \\sum_{b \\in I} A_{ab} \\lvert \\nu^{(b)} \\rvert\n\\]\nsince $\\sum_{j=1}^{\\infty} \\min(i,j) m_j^{(b)} = \\lvert \\nu^{(b)} \\rvert$ for $i \\gg 1$. Note this is consistent with letting $i = \\infty$ in Equation~\\eqref{eq:vacancy}.\n\nLet $\\RC(\\infty)$ denote the set of rigged configurations generated by $(\\nu_{\\emptyset}, J_{\\emptyset})$, where $\\nu_{\\emptyset}^{(a)} = 0$ for all $a \\in I$, and closed under the crystal operators as follows.\n\n\\begin{dfn}\n\\label{def:RC_crystal_ops}\nFix some $a \\in I$, and let $x$ be the smallest rigging in $(\\nu,J)^{(a)}$.\n\\begin{itemize}\n\\item[\\defn{$e_a$}:] If $x =0$, then $e_a(\\nu, J) = 0$. Otherwise, let $r$ be a row in $(\\nu, J)^{(a)}$ of minimal length $\\ell$ with rigging $x$. Then $e_a(\\nu, J)$ is the rigged configuration which removes a box from row $r$, sets the new rigging of $r$ to be $x+1$, and changes all other riggings such that the coriggings remain fixed.\n\n\\item[\\defn{$f_a$}:] Let $r$ be a row in $(\\nu, J)^{(a)}$ of maximal length $\\ell$ with rigging $x$. Then $f_a(\\nu, J)$ is the rigged configuration which adds a box to row $r$, sets the new rigging of $r$ to be $x-1$, and changes all other riggings such that the coriggings remain fixed.\n\\end{itemize}\n\\end{dfn}\n\n\nWe define the remainder of the crystal structure on $\\RC(\\infty)$ by\n\\begin{gather*}\n\\varepsilon_a(\\nu, J) = \\max \\{ k \\in \\mathbf{Z} \\mid e_a^k(\\nu, J) \\neq 0 \\}, \\hspace{20pt} \\varphi_a(\\nu, J) = \\inner{h_a}{\\mathrm{wt}(\\nu,J)} + \\varepsilon_a(\\nu, J),\n\\\\ \\mathrm{wt}(\\nu, J) = -\\sum_{a \\in I} \\lvert \\nu^{(a)} \\rvert \\alpha_a.\n\\end{gather*}\nFrom this structure, we have $p_\\infty^{(a)} = \\inner{h_a}{\\mathrm{wt}(\\nu,J)}$ for all $a\\in I$.\n\n\\begin{thm}[{\\cite{SalS15, SalS15II}}]\n\\label{thm:binf_isomorphism}\nLet $\\mathfrak{g}$ be of symmetrizable type. Then $\\RC(\\infty) \\cong B(\\infty)$ as $U_q(\\mathfrak{g})$-crystals.\n\\end{thm}\n\n\\begin{prop}[{\\cite{SalS15, S06}}]\n\\label{prop:ep_phi}\nLet $(\\nu, J) \\in \\RC(\\infty)$ and fix some $a \\in I$. Let $x$ denote the smallest label in $(\\nu,J)^{(a)}$. Then we have\n\\[\n\\varepsilon_a(\\nu, J) = -\\min(0, x) \\hspace{40pt} \\varphi_a(\\nu, J) = p_{\\infty}^{(a)} - \\min(0, x).\n\\]\n\\end{prop}\n\nIt is a straightforward computation from the vacancy numbers to show that\n\\begin{equation}\n\\label{eq:convexity_exact}\n\\inner{h_a}{\\lambda} - \\sum_{b \\in I} A_{ab} m_i^{(b)} = -p_{i-1}^{(a)} + 2 p_i^{(a)} - p_{i+1}^{(a)}.\n\\end{equation}\nFrom this, we obtain the well-known convexity properties of the vacancy numbers.\n\n\\begin{lemma}[Convexity]\n\\label{lemma:convexity}\nIf $m_i^{(a)} = 0$, then we have\n\\[\n2 p_i^{(a)} \\geq p_{i-1}^{(a)} + p_{i+1}^{(a)}.\n\\]\nMoreover, $p_{i-1}^{(a)} \\geq p_i^{(a)} \\leq p_{i+1}^{(a)}$ if and only if $p_{i-1}^{(a)} = p_i^{(a)} = p_{i+1}^{(a)}$.\n\\end{lemma}\n\nIn the sequel, we will refer to this lemma simply as convexity as we will frequently use it.\n\n\n\n\n\n\n\\section{Star-crystal structure}\n\\label{sec:*crystal}\n\n\\begin{dfn}\n\\label{def:RC_star_crystal_ops}\nFix some $a \\in I$, and let $x$ be the smallest \\emph{co}rigging in $(\\nu,J)^{(a)}$.\n\\begin{itemize}\n\\item[\\defn{$e_a^*$}:] If $x =0$, then $e_a(\\nu, J) = 0$. Otherwise let $r$ be a row in $(\\nu, J)^{(a)}$ of minimal length $\\ell$ with corigging $x$. Then $e_a(\\nu, J)$ is the rigged configuration which removes a box from row $r$ and sets the new corigging of $r$ to be $x+1$.\n\n\\item[\\defn{$f_a^*$}:] Let $r$ be a row in $(\\nu, J)^{(a)}$ of maximal length $\\ell$ with corigging $x$. Then $f_a(\\nu, J)$ is the rigged configuration which adds a box to row $r$ and sets the new colabel of $r$ to be $x-1$.\n\\end{itemize}\n\\end{dfn}\n\nIf $e_a^*$ removes a box from a row of length $\\ell$ in $(\\nu, J)$, then the the vacancy numbers change by the formula\n\\begin{equation}\n\\label{eq:change_vac_e}\n\\widetilde{p}_i^{(b)} = \\begin{cases}\np_i^{(b)} & \\text{if } i \\leq \\ell, \\\\\np_i^{(b)} + A_{ab} & \\text{if } i > \\ell.\n\\end{cases}\n\\end{equation}\nOn the other hand, if $f_a^*$ adds a box to a row of length $\\ell$, then the vacancy numbers change by\n\\begin{equation}\n\\label{eq:change_vac_f}\n\\widetilde{p}_i^{(b)} = \\begin{cases}\np_i^{(b)} & \\text{if } i < \\ell, \\\\\np_i^{(b)} - A_{ab} & \\text{if } i \\geq \\ell.\n\\end{cases}\n\\end{equation}\nSimilar equations hold for $e_a$ and $f_a$ respectively. So the riggings of unchanged rows are changed according to Equation~\\eqref{eq:change_vac_e} and Equation~\\eqref{eq:change_vac_f} under $e_a$ and $f_a$, respectively. \n\n\\begin{remark}\nBy Equation~\\eqref{eq:change_vac_e} and Equation~\\eqref{eq:change_vac_f}, the crystal operators $e_a$ and $f_a$ preserve all colabels of $(\\nu, J)$ other than the row changed in $(\\nu, J)^{(a)}$.\n\\end{remark}\n\n\\begin{ex}\n\\label{ex:running}\nConsider type $D_4$ with Dynkin diagram\n\\[\n\\begin{tikzpicture}[xscale=2,yscale=.75]\n\\node[circle,fill,scale=.45,label={below:$1$}] (1) at (0,0) {};\n\\node[circle,fill,scale=.45,label={below:$2$}] (2) at (1,0) {};\n\\node[circle,fill,scale=.45,label={right:$3$}] (3) at (2,1) {};\n\\node[circle,fill,scale=.45,label={right:$4.$}] (4) at (2,-1) {};\n\\path[-]\n (1) edge (2)\n (2) edge (3)\n (2) edge (4);\n\\end{tikzpicture}\n\\]\nLet $(\\nu,J)$ be the rigged configuration\n\\begin{align*}\n(\\nu, J) \n&= f_2^*f_3^* f_1^* f_2^* f_2^* f_4^* f_3^* f_1^* f_2^* (\\nu_{\\emptyset}, J_{\\emptyset}) \\\\\n&=\n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{3,1}{-2,-1}{-3,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\end{align*}\nThen\n\\[\nf_2^*(\\nu, J) = \\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{4,1}{-3,-1}{-5,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\]\n\\end{ex}\n\nLet $\\RC(\\infty)^*$ denote the closure of $(\\nu_{\\emptyset}, J_{\\emptyset})$ under $f_a^*$ and $e_a^*$. We define the remaining crystal structure by\n\\begin{gather*}\n\\varepsilon_a^*(\\nu, J) = \\max \\{ k \\in \\mathbf{Z} \\mid (e_a^*)^k(\\nu, J) \\neq 0 \\}, \n\\hspace{20pt} \n\\varphi_a^*(\\nu, J) = \\inner{h_a}{\\mathrm{wt}(\\nu,J)} + \\varepsilon_a^*(\\nu, J),\n\\\\ \\mathrm{wt}(\\nu, J) = -\\sum_{a \\in I} \\lvert \\nu^{(a)} \\rvert \\alpha_a.\n\\end{gather*}\n\n\\begin{remark}\n\\label{remark:duality}\nWe will say an argument holds by duality when we can interchange:\n\\begin{itemize}\n\\item ``label'' and ``colabel'';\n\\item $e_a$ and $e_a^*$;\n\\item $f_a$ and $f_a^*$.\n\\end{itemize}\nFor an example, compare the proof of Proposition~\\ref{prop:ep_phi_star} with~\\cite[Thm.~3.8]{Sakamoto14}.\n\\end{remark}\n\n\\begin{lemma}\nThe tuple $(\\RC(\\infty)^*, e_a^*, f_a^*, \\varepsilon_a^*, \\varphi_a^*, \\mathrm{wt})$ is an abstract $U_q(\\mathfrak{g})$-crystal.\n\\end{lemma}\n\n\\begin{proof}\nThe proof that $(\\RC(\\infty)^*, e_a^*, f_a^*, \\varepsilon_a^*, \\varphi_a^*, \\mathrm{wt})$ is an abstract $U_q(\\mathfrak{g})$-crystal is dual to that $\\RC(\\infty)$ is an abstract $U_q(\\mathfrak{g})$-crystal under $e_a$ and $f_a$ in~\\cite[Lemma~3.3]{SalS15}.\n\\end{proof}\n\n\n\\begin{prop}\n\\label{prop:ep_phi_star}\nLet $(\\nu, J) \\in \\RC(\\infty)$ and fix some $a \\in I$. Let $x$ denote the smallest colabel in $(\\nu,J)^{(a)}$. Then we have\n\\[\n\\varepsilon_a^*(\\nu, J) = -\\min(0, x), \\hspace{40pt} \\varphi_a^*(\\nu, J) = p_{\\infty}^{(a)} - \\min(0, x).\n\\]\n\\end{prop}\n\n\\begin{proof}\nThe following argument for $\\varepsilon_a^*$ is essentially the dual to that given in~\\cite[Thm.~3.8]{Sakamoto14}. We include it here as an example of Remark~\\ref{remark:duality}.\n\nIt is sufficient to prove $\\varepsilon_a^*(\\nu, J) = -\\max(0, x)$ since $p_{\\infty}^{(a)} = \\inner{h_a}{\\mathrm{wt}(\\nu, J)}$. If $x \\geq 0 = \\varepsilon_a^*(\\nu, J)$, then $e_a^*(\\nu, J) = 0$ by definition. Thus we proceed by induction on $\\varepsilon_a^*(\\nu,J)$ and assume $x < 0$. Let $(\\nu',J') = e_a^*(\\nu, J)$ and $y'$ denote the resulting colabel from a colabel $y$. In particular, we have $x' = x + 1$ and all other colabels follow Equation~\\eqref{eq:change_vac_e}. Next, let $y$ denote the colabel of a row of length $j$. For $j < \\ell$, we have $y > x$ (equivalently $y \\geq x - 1$) because we chose $\\ell$ as large as possible. Thus $y' = y$, and hence $y' = y \\geq x + 1 = x'$. For $j \\geq \\ell$, we have $y \\geq x$ by the minimality of $x$ and $y' = y + 2$. Hence, $y' = y + 2 \\geq x + 1 = x'$, and so $\\varepsilon_a^*(\\nu', J') = \\varepsilon_a^*(\\nu, J) - 1$ as desired.\n\\end{proof}\n\n\nThe rest of this section will amount to showing that Conditions~(\\ref{item:star1})--(\\ref{item:star6}) of Proposition~\\ref{prop:weaker_conditions} hold.\nNote that using Proposition \\ref{prop:ep_phi_star} and \\cite[Prop. 4.2]{SalS15}, we can rewrite Equation \\eqref{eq:jump} as \n\\begin{equation}\n\\label{eq:RCjump}\n\\begin{aligned}\n\\kappa_a(\\nu,J) &= -\\min(0,x_\\ell) - \\min(0,x_c) + \\langle h_a , \\mathrm{wt}(\\nu,J) \\rangle, \\\\\n&= -\\min(0,x_\\ell) - \\min(0,x_c) + p_\\infty^{(a)},\n\\end{aligned}\n\\end{equation}\nwhere $x_\\ell$ and $x_c$ are the smallest label and colabel, respectively, in $(\\nu,J)^{(a)}$.\n\n\n\\begin{lemma}\n\\label{lemma:kappa0}\nFix $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$. Assume $\\kappa_a(\\nu, J) = 0$. Then $f_a(\\nu, J) = f_a^*(\\nu, J)$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $f_a$ adds a box to a row of length $i$ with rigging $x$. Recall that $x = -\\varepsilon_a(\\nu, J)$. Suppose the longest row of $\\nu^{(a)}$ has length $\\ell > i$ and let $x_\\ell$ denote any rigging of the longest row. Therefore, we have $x_\\ell > x$ by the definition of $f_a$, and we have $p_\\ell^{(a)} \\leq p_{\\infty}^{(a)}$ by convexity. Thus from the definition of $\\varepsilon_a^*$, we have\n\\begin{equation}\n\\label{eq:contradiction_kappa0}\n\\varepsilon_a^*(\\nu, J) \\geq x_\\ell - p_\\ell^{(a)} > x - p_{\\infty}^{(a)} = -\\varepsilon_a(\\nu, J) - p_{\\infty}^{(a)}.\n\\end{equation}\nThis implies $\\varepsilon_a^*(\\nu, J) + \\varepsilon_a(\\nu, J) + p_{\\infty}^{(a)} > 0$, which is a contradiction. Therefore $f_a$ must add a box to one of the longest rows of $\\nu^{(a)}$. Moreover, if $p_\\ell^{(a)} < p_{\\infty}^{(a)}$, then Equation~\\eqref{eq:contradiction_kappa0} would still hold and result in a contradiction. Similar statements holds for $f_a^*$ by duality.\n\nTherefore $f_a$ and $f_a^*$ act on the longest row of $\\nu^{(a)}$ and $p_i^{(a)} = p_{i+1}^{(a)} = p_{\\infty}^{(a)}$. Let $x$ and $x^*$ denote the label of the row on which $f_a$ and $f_a^*$ act, respectively. Both of these labels decrease by $1$ after applying $f_a$ and $f_a^*$, respectively, by Equation~\\eqref{eq:colabel_change} and Equation~\\eqref{eq:label_change}, respectively. So it is sufficient to show $x = x^*$. Note that $x \\leq x^*$ as the smallest colabel is the one with the largest rigging. Suppose $x < x^*$, then we have\n\\[\n\\varepsilon_a^*(\\nu, J) \\geq x^* - p_i^{(a)} > x - p_{\\infty}^{(a)} \\geq -\\varepsilon_a(\\nu, J) - p_{\\infty}^{(a)},\n\\]\nwhich is a contradiction. Therefore we have $f_a = f_a^*$.\n\\end{proof}\n\n\\begin{ex}\n\\label{ex:kappa0}\nLet $(\\nu,J)$ be the rigged configuration of type $D_4$ from Example \\ref{ex:running}. Then $\\kappa_2(\\nu,J) = 0$, and \nand \n\\[\nf_2(\\nu,J) = \n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{4,1}{-3,-1}{-5,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\]\nOne can check that this agrees with $f_2^*(\\nu,J)$ from Example \\ref{ex:running}.\n\\end{ex}\n\n\n\n\\begin{lemma}\n\\label{lemma:kappa1}\nFix $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$. Assume $\\kappa_a(\\nu, J) \\geq 1$. Then\n\\[\n\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J),\n\\hspace{40pt}\n\\varepsilon_a(f_a^*(\\nu,J)) = \\varepsilon_a(\\nu,J).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $x^c$ denote the smallest colabel of $(\\nu,J)^{(a)}$.\nLet $x$ and $i$ denote the rigging and length of the row on which $f_a$ acts. By the minimality of $x^c$, we have $x_c := p_i^{(a)} - x \\geq x^c$. Note that the colabel of the row after the application of $f_a$ becomes\n\\begin{equation}\n\\label{eq:new_colabel}\n\\widetilde{x}_c := p_{i+1}^{(a)} - 2 - (x - 1) = p_{i+1}^{(a)} - x - 1,\n\\end{equation}\nwhich implies that\n\\begin{equation}\n\\label{eq:colabel_change}\n\\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1.\n\\end{equation}\n\nThe remainder of the proof will be split into two cases: $x^c = p_i^{(a)} - x$ and $x^c < p_i^{(a)} - x$.\n\n\\case{$x^c = p_i^{(a)} - x$}\n\nFirst, consider the case $p_{i+1}^{(a)} \\leq p_i^{(a)}$. We also assume there exists a row of length $\\ell > i$ of $\\nu^{(a)}$, and let $x_\\ell$ denote the rigging of that row. Thus $x < x_\\ell$ and $x^c \\leq p_\\ell^{(a)} - x_\\ell$ by the definition of $f_a$ and the minimality of $x^c$. Hence \n\\[\np_i^{(a)} - x = x^c \\leq p_\\ell^{(a)} - x_\\ell < p_\\ell^{(a)} - x,\n\\]\nwhich is equivalent to $p_i^{(a)} < p_\\ell^{(a)}$. It must be the case, then, that $p_{i+1}^{(a)} > p_i^{(a)}$ by convexity, which is impossible. Thus the longest row of $\\nu^{(a)}$ must be of length $i$. Convexity implies $p_i^{(a)} = p_{i+1}^{(a)} = p_\\infty^{(a)}$, which results in\n\\begin{align*}\n\\kappa_a(\\nu, J) & = p_{\\infty}^{(a)} - \\min(x, 0) - \\min(x^c, 0)\n\\\\ & = p_i^{(a)} - \\min(x, 0) - \\min\\bigl(p_i^{(a)} - x, 0\\bigr).\n\\end{align*}\nSince $x$ was the rigging chosen by $f_a$, we must have $x \\leq 0$. Additionally, if $p_i^{(a)} - x = x^c \\leq 0$, we have\n\\[\n1 \\leq \\kappa_a(\\nu, J) = p_i^{(a)} - x - (p_i^{(a)} - x) = 0,\n\\]\nwhich is a contradiction. Thus $x^c \\geq 1$, which implies $\\varepsilon_a^*(\\nu, J) = 0 = \\varepsilon_a^*\\bigl(f_a(\\nu, J))$ since $\\widetilde{x}_c \\geq 0$.\n\nNext if $p_{i+1}^{(a)} = p_i^{(a)} + 1$, then $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$ since $\\widetilde{x}_c = x^c$, all other coriggings are fixed, and $\\widetilde{x}_c = x_c$ by Equation~\\eqref{eq:colabel_change} .\n\nSo we now assume $p_{i+1}^{(a)} \\geq p_i^{(a)} + 2$, which implies $\\widetilde{x}_c > x^c$. If there is another row with a corigging of $x^c$ or $x^c \\geq 0$, then $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$. So assume $f_a$ acts on the only row with a corigging of $x^c < 0$. Note that $p_i^{(a)} - x = x^c < 0$ and $x \\leq 0$ implies $p_i^{(a)} < 0$.\n\nWe have $m_i^{(a)} = 1$ as, otherwise, we would either have a second corigging of $x^c$ or a smaller corigging from the minimality of $x$. Thus, by Equation~\\eqref{eq:convexity_exact}, \n\\begin{align*}\n-2 - \\sum_{b \\neq a} A_{ab} m_i^{(b)} &= -p_{i-1}^{(a)} + 2p_i^{(a)} - p_{i+1}^{(a)} \\\\\n&\\leq -p_{i-1}^{(a)} + 2p_i^{(a)} - p_i^{(a)} - 2 \\\\\n&= -p_{i-1}^{(a)} + p_i^{(a)} - 2.\n\\end{align*}\nSince $m_i^{(b)} \\geq 0$ and $-A_{ab} \\geq 0$ for all $a \\neq b$, we then have\n\\[\n0 \\leq -p_{i-1}^{(a)} + p_i^{(a)},\n\\]\nor, equivalently, $p_{i-1}^{(a)} \\leq p_i^{(a)}$. If $i$ is the length of the smallest row, then $0 \\leq p_{i-1}^{(a)} \\leq p_i^{(a)} < 0$ by convexity, which is a contradiction. Thus let $x_\\ell$ denote rigging of the longest row in $\\nu^{(a)}$ such that $\\ell < i$, and by convexity, we have $p_\\ell^{(a)} \\leq p_i^{(a)}$. By the definition of $f_a$, we have $x_\\ell \\geq x$. Thus, we have $p_\\ell^{(a)} - x_\\ell \\leq p_i^{(a)} - x$. However, by the unique minimality of $x^c$, we have $p_\\ell^{(a)} - x_\\ell > x^c = p_i^{(a)} - x$. This is a contradiction. Therefore $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$.\n\n\\case{$x^c < p_i^{(a)} - x$}\n\nAssume $\\varepsilon_a^*\\bigl(f_a(\\nu,J)\\bigr) \\neq \\varepsilon_a^*(\\nu, J)$. Then\n\\begin{equation}\n\\label{eq:difference_epsilon_change}\np_{i+1}^{(a)} - p_i^{(a)} - 1 < x^c+ x - p_i^{(a)} < 0,\n\\end{equation}\nas, otherwise, the new corigging is not smaller than the minimal corigging (i.e., $\\widetilde{x}_c < x^c$), which occurs on a different row and does not change under $f_a$. We rewrite Equation~\\eqref{eq:difference_epsilon_change} as\n\\begin{equation}\n\\label{eq:rewrite_less_than}\np_{i+1}^{(a)} - 1 < x^c + x < p_i^{(a)}.\n\\end{equation}\nSuppose there exists a row of length $\\ell > i$ in $\\nu^{(a)}$. Then $x_\\ell > x$ and $x^c \\leq p_\\ell^{(a)} - x_\\ell$. Therefore,\n\\begin{equation}\n\\label{eq:double_inequality}\np_{i+1}^{(a)} - \\ell < p_\\ell^{(a)} - x_\\ell + x < p_\\ell^{(a)},\n\\end{equation}\nwhich implies $p_{i+1}^{(a)} \\leq p_j^{(a)}$ for all $\\ell \\geq j > i$ by convexity. Note that Equation~\\eqref{eq:double_inequality} implies that $\\ell > i+1$ since, otherwise, we would have $p_{i+1}^{(a)} < p_{i+1}^{(a)}$. We also have $p_{i+1}^{(a)} \\leq p_j^{(a)}$ for all $j > i$ if there does not exist a row of length $\\ell > i$. Since there does not exist a row of length $i+1$, we must have $p_i^{(a)} \\leq p_{i+1}^{(a)} \\leq p_{i+2}^{(a)}$ by convexity. Yet, Equation~\\eqref{eq:rewrite_less_than} implies\n\\[\np_{i+1}^{(a)} < p_i^{(a)} \\leq p_{i+1}^{(a)},\n\\]\nbut this is a contradiction. Therefore $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$.\n\\end{proof}\n\n\n\\begin{ex}\n\\label{ex:kappa1}\nAgain, let $(\\nu,J)$ be the rigged configuration of type $D_4$ from Example \\ref{ex:running}. Then $\\varepsilon_3(\\nu,J) = 0$, $\\varepsilon_3^*(\\nu,J) = 1$, and $\\kappa_3(\\nu,J) = 1$. We have\n\\begin{align*}\nf_3(\\nu,J) &=\n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{3,1}{-1,-1}{-1,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{3}{-1}{-2}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}\\\\\nf_3^*(\\nu,J) &= \n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{3,1}{-2,-1}{-2,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{3}{0}{-2}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\end{align*}\nThen $\\varepsilon_3^*\\bigl(f_3(\\nu,J)\\bigr) = 1$ and $\\varepsilon_3\\bigl(f_3^*(\\nu,J)\\bigr) = 0$.\n\\end{ex}\n\n\n\\begin{lemma}\n\\label{lemma:kappa2}\nFix $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$. Assume $\\kappa_a(\\nu, J) \\geq 2$. Then\n\\[\nf_af_a^*(\\nu, J) = f_a^* f_a(\\nu, J).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSuppose $f_a$ (resp., $f_a^*$) acts on row $r$ of length $i$ (resp., row $r^*$ of length $i^*$) with rigging $x$ (resp., $x^*$). Without loss of generality, let $r$ (resp., $r^*$) be the northernmost such row in the diagram of $\\nu^{(a)}$. Let $x_c^* = p_{i^*}^{(a)} - x^*$ and $x_c = p_i^{(a)} - x$. Note that $x \\leq x^*$ and $x_c^* \\leq x_c$. Applying $f_a^*$, the new rigging (and the only changed rigging) is\n\\begin{equation}\n\\label{eq:label_change}\n\\widetilde{x}^* = x^* + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1.\n\\end{equation}\nRecall that Equation~\\eqref{eq:colabel_change} gives the new corigging (and only changed corigging)\n\\[\n\\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1\n\\]\nafter applying $f_a$. We split the proof into three cases: the first two are cases in which $r \\neq r^*$ and the last is when $r = r^*$.\n\n\\case{$f_a$ acts on row $r \\neq r^*$ in $f_a^*(\\nu, J)$}\n\nSuppose $f_a f_a^*(\\nu, J) \\neq f_a^* f_a(\\nu, J)$. This is equivalent to $f_a^*$ acting on row $r' \\neq r^*$ in $f_a(\\nu, J)$. Note that we must have $r' = r$, since $f_a$ preserves all other colabels. From Equation~\\eqref{eq:colabel_change}, we must have $p_{i+1}^{(a)} < p_i^{(a)} + 2$, as otherwise $\\widetilde{x}_c > x_c \\geq x_c^*$, which would imply $r = r' = r^*$ and be a contradiction. Next, consider when $p_{i+1}^{(a)} = p_i^{(a)} + 1$. Thus we have $\\widetilde{x}_c = x_c$. Since $r' \\neq r^*$, we must have $i = i^*$ and $x_c^* = \\widetilde{x}_c = x_c$. However, this contradicts the assumption $r \\neq r^*$ as we have $x = x^*$.\n\nHence $p_{i+1}^{(a)} \\leq p_i^{(a)}$. Suppose $i$ is the length of the longest row of $\\nu^{(a)}$. Then $p_i^{(a)} = p_{i+1}^{(a)} = p_{\\infty}^{(a)}$ by convexity. Moreover, we have $\\widetilde{x}_c = x_c - 1 = x_c^*$ since $r, r' \\neq r^*$. Note that since $f_a$ (resp., $f_a^*$) acts on $r$ (resp., $r^*$), we must have $x \\leq 0$ (resp., $x_c^* \\leq 0$). Therefore, we have\n\\begin{align*}\n2 \\leq \\kappa_a(\\nu, J) & = p_i^{(a)} - \\min(x, 0) - \\min(x_c^*, 0)\n\\\\ & = p_i^{(a)} - x - x_c^*\n\\\\ & = p_i^{(a)} - x - (p_i^{(a)} - x - 1) = 1,\n\\end{align*}\nwhich is a contradiction.\n\nSuppose there exists a row $r_\\ell$ of length $\\ell > i$ in $\\nu^{(a)}$. Let $x_\\ell$ denote the rigging of $r_\\ell$, and note $x < x_\\ell$ by our assumption. Therefore, we have\n\\[\np_{i+1}^{(a)} - x - 1 = \\widetilde{x}_c \\leq x_c^* \\leq p_\\ell^{(a)} - x_l < p_\\ell^{(a)} - x,\n\\]\nwhich implies $p_{i+1}^{(a)} \\leq p_\\ell^{(a)}$. Assume there exists a row of length $i+1$ in $\\nu^{(a)}$ with rigging $x_{i+1}$. It follows that\n\\[\np_{i+1}^{(a)} - x > p_{i+1}^{(a)} - x_{i+1} \\geq x_c^* = p_{i^*}^{(a)} - x^*,\n\\]\nwhich is equivalent to\n\\[\np_{i+1}^{(a)} - p_{i^*}^{(a)} > x - x^*.\n\\]\nFurthermore,\n\\[\np_{i+1}^{(a)} - x - 1 = \\widetilde{x}_c \\leq x_c^* = p_{i^*}^{(a)} - x^*,\n\\]\nwhich results in\n\\begin{equation}\n\\label{eq:changing_f_star_inequality}\nx - x^* \\geq p_{i+1}^{(a)} - p_{i^*}^{(a)} - 1.\n\\end{equation}\nAdditionally, Equation~\\eqref{eq:changing_f_star_inequality} is necessarily a strict inequality if $i^* > i$ because it must be the case that $\\widetilde{x}_c < x_c^*$.\nHence\n\\[\np_{i+1}^{(a)} - p_{i^*}^{(a)} > x - x^* \\geq p_{i+1}^{(a)} - p_{i^*}^{(a)} - 1,\n\\]\nwhich is a contradiction for $i^* > i$ as the right inequality becomes a strict inequality.\n\nNext, note that $\\widetilde{x}^* = x^* + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 \\geq x$ since $f_a$ acts on $r$. Hence \n\\begin{equation}\n\\label{eq:unchanged_f_inequality}\np_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 \\geq x - x^*,\n\\end{equation}\nwhich is a strict inequality for $i \\leq i^*$. Thus \n\\[\np_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 \\geq x - x^* \\geq p_{i+1}^{(a)} - p_{i^*}^{(a)} - 1,\n\\]\nor, equivalently, $p_{i^*+1}^{(a)} \\geq p_{i+1}^{(a)}$. Since $i \\leq i^*$, we have $p_{i^*+1}^{(a)} > p_{i+1}^{(a)}$, and hence $i = i^*$ cannot occur\n\nNow suppose $i^* < i$. Therefore $x_c < p_{i+1}^{(a)} - x_{i+1}$, which implies\n\\[\np_{i+1}^{(a)} - x - 1 = \\widetilde{x}_c \\leq x_c^* < p_{i+1}^{(a)} - x_{i+1} < p_{i+1}^{(a)} - x.\n\\]\nSo $p_{i+1}^{(a)} < p_{i+1}^{(a)}$, which is a contradiction.\n\nFinally, if there does not exist a row of length $i+1$, then $p_i^{(a)} = p_{i+1}^{(a)} = p_\\ell^{(a)}$ by convexity, so the argument given above will still yield a contradiction. Hence, $f_a f_a^*(\\nu,J) = f_a^* f_a(\\nu, J)$.\n\n\\case{$f_a$ acts on row $r' \\neq r$ in $f_a^*(\\nu, J)$}\n\nNote that $r' = r^*$, where $r^* \\neq r$, as $f_a^*$ fixes all other riggings. So from Lemma~\\ref{lemma:kappa1}, we have\n\\[\n\\widetilde{x}^* = -\\varepsilon_a\\bigl(f_a^*(\\nu, J)\\bigr) = -\\varepsilon_a(\\nu, J) = x,\n\\]\nand hence $i \\leq i^*$. Therefore $x < x^*$ as $x = x^*$ implies $r = r^*$. Thus,\n\\[\nx = \\widetilde{x}^* = x^* + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 > x + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1,\n\\]\nand hence $p_{i^*+1}^{(a)} \\leq p_{i^*}^{(a)}$. Dually, $f_a^*$ acts on row $r$ in $f_a(\\nu, J)$ since this would contradict $f_a$ acting on $r^* \\neq r$ from the previous case. Similarly, \n\\[\n\\widetilde{x}_c = -\\varepsilon_a^*\\bigl(f_a^*(\\nu, J)\\bigr) = -\\varepsilon_a^*(\\nu, J) = x_c^*\n\\]\nby the dual version of Lemma~\\ref{lemma:kappa1}, implying $i^* \\leq i$. Hence, $i = i^*$ and\n\\[\np_i^{(a)} - x^* = x_c^* = \\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1 = - x + p_{i+1}^{(a)} - 1,\n\\]\nwhich yields $x - x^* = p_{i+1}^{(a)} - p_i^{(a)} - 1$. Thus\n\\begin{align*}\nx & = \\widetilde{x}^* = x^* + p_{i+1}^{(a)} - p_i^{(a)} - 1\n\\\\ p_i^{(a)} - x^* = x_c^* & = \\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1 = p_{i+1}^{(a)} - x - 1 \\leq p_i^{(a)} - x - 1,\n\\end{align*}\nwhich implies $x - x^* \\leq -1$. Hence,\n\\[\np_i^{(a)} - p_{i+1}^{(a)} = x^* - x - 1 \\leq -2,\n\\]\nand this contradicts $0 \\leq p_i^{(a)} - p_{i+1}^{(a)}$. \n\n\n\\case{$r = r^*$}\n\nFrom $x = x^*$ and $i = i^*$, we have \n\\begin{align*}\n\\widetilde{x}^* & = x^* + p_{i+1}^{(a)} - p_i^{(a)} - 1 = x + p_{i+1}^{(a)} - p_i^{(a)} - 1,\n\\\\ \\widetilde{x}_c & = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1 = p_i^{(a)} - x + p_{i+1}^{(a)} - p_i^{(a)} - 1 = p_{i+1}^{(a)} - x - 1,\n\\\\ x_c^* & = x_c = p_i^{(a)} - x.\n\\end{align*}\n\nIf $p_{i+1}^{(a)} \\leq p_i^{(a)} + 1$, then $\\widetilde{x}^* \\leq x$ and $\\widetilde{x}_c \\leq x_c^*$. Hence, $f_a$ and $f_a^*$ select row $r$ in $f_a^*(\\nu, J)$ and $f_a(\\nu, J)$, respectively, and so we have $f_a f_a^*(\\nu, J) = f_a^* f_a(\\nu, J)$. Next, consider the case when $p_{i+1}^{(a)} \\geq p_i^{(a)} + 2$. Then $\\widetilde{x}^* > x$ and $\\widetilde{x}_c > x_c^*$. If $m_i^{(a)} \\geq 2$, then there exists a row $r' \\neq r$ such that $x^* = x$ and $x_c^* = x_c$. So $f_a$ and $f_a^*$ select row $r'$ in $f_a^*(\\nu, J)$ and $f_a(\\nu, J)$, respectively, and thus we have $f_a f_a^*(\\nu, J) = f_a^* f_a(\\nu, J)$.\nIf $m_i^{(a)} = 1$, then, as in Lemma~\\ref{lemma:kappa1}, we have\n\\[\n-2 - \\sum_{b \\neq a} A_{ab} m_i^{(b)} = -p_{i-1}^{(a)} + 2p_i^{(a)} - p_{i+1}^{(a)} \\leq -p_{i-1}^{(a)} + p_i^{(a)} - 2.\n\\]\nfrom Equation~\\eqref{eq:convexity_exact}. This implies $p_{i-1}^{(a)} \\leq p_i^{(a)}$. We consider the case when $r$ is the smallest row of $\\nu^{(a)}$, which implies $r$ is the unique row with rigging $x$ and corigging $x_c^*$. Moreover, we have $0 \\leq p_{i-1}^{(a)} \\leq p_i^{(a)}$ by convexity. Because we are acting on $r$ by $f_a$ and $f_a^*$, we have $x \\leq 0$ and $x_c^* \\leq 0$. Hence,\n\\[\n0 \\leq p_i^{(a)} \\leq p_i^{(a)} - x = x_c^* \\leq 0 \\implies \n0 = p_i^{(a)} = x = x_c^*. \n\\]\nTherefore $f_a$ (resp., $f_a^*$) acts on a row of length $0$ in $f_a^*(\\nu, J)$ (resp., $f_a(\\nu, J)$) as all other riggings (resp., coriggings) are positive. Moreover, the resulting rigging is $-1$ in both cases, and so $f_a f_a^*(\\nu, J) = f_a^* f_a(\\nu, J)$.\n\nNow assume there exists a row $r_\\ell$ of length $\\ell < i$ in $\\nu^{(a)}$, and without loss of generality, suppose $\\ell$ is maximal. Let $x_\\ell$ denote the rigging of $r_\\ell$, and by the definition of $f_a$ and $f_a^*$, we have $x_\\ell \\geq x$ and $p_\\ell^{(a)} - x_\\ell \\geq x_c^* = p_i^{(a)} - x$. By convexity, we have $p_\\ell^{(a)} \\leq p_{i-1}^{(a)} \\leq p_i^{(a)}$. Therefore, we have\n\\[\np_i^{(a)} - x \\leq p_\\ell^{(a)} - x_\\ell \\leq p_i^{(a)} - x,\n\\]\nand so $p_i^{(a)} - x = p_\\ell^{(a)} - x_\\ell$. Moreover, if $p_\\ell^{(a)} < p_i^{(a)}$, then we have $x_\\ell < x$, which cannot occur, and hence we also have $x = x_\\ell$. Therefore $f_a$ and $f_a^*$ acts on $r_\\ell$ in $f_a^*(\\nu, J)$ and $f_a(\\nu, J)$, respectively. Thus we have $f_a f_a^*(\\nu, J) = f_a^*f_a(\\nu, J)$.\n\\end{proof}\n\n\n\\begin{ex}\n\\label{ex:kappa2}\nContinuing our running example, let $(\\nu,J)$ be the rigged configuration of type $D_4$ from Example \\ref{ex:running}. Then $\\kappa_4(\\nu,J) = 2$ and \n\\[\nf_4^*f_4(\\nu,J) = f_4f_4^*(\\nu,J) = \n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=5cm]\n \\rpp{3,1}{-1,-1}{-1,-1}\n \\end{scope}\n \\begin{scope}[xshift=11.5cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=16.5cm]\n \\rpp{3}{-1}{-2}\n \\end{scope}\n\\end{tikzpicture}.\n\\]\nWith $a=4$, we have the diagram\n\\[\n\\begin{tikzpicture}[xscale=2,yscale=1.5,font=\\normalsize]\n\\node (t) at (0,0) {$(\\nu,J)$};\n\\node (s) at (-1,-1) {$f_4^*(\\nu,J)$};\n\\node (a) at (1,-1) {$f_4(\\nu,J)$};\n\\node (ss) at (-2,-2) {$f_4^*f_4^*(\\nu,J)$};\n\\node (aa) at (2,-2) {$f_4f_4(\\nu,J)$};\n\\node (as) at (0,-2) {$f_4^*f_4(\\nu,J) = f_4f_4^*(\\nu,J)$};\n\\node (sss) at (-2,-3) {$\\substack{f_4^*f_4^*f_4^*(\\nu,J) \\\\ = f_4f_4^*f_4^*(\\nu,J)}$};\n\\node (asa) at (0,-3) {$\\substack{f_4^*f_4^*f_4(\\nu,J) \\\\ = f_4^*f_4f_4^*(\\nu,J) \\\\ = f_4f_4^*f_4(\\nu,J) \\\\ = f_4f_4f_4^*(\\nu,J)}$};\n\\node (aaa) at (2,-3) {$\\substack{f_4^*f_4f_4(\\nu,J) \\\\ = f_4f_4f_4(\\nu,J)}$};\n\\foreach \\x in {-2,0,2}\n {\\node at (\\x,-4) {$\\vdots$};}\n\\path[->]\n (t) edge (s)\n (t) edge (a)\n (s) edge (ss)\n (s) edge (as)\n (a) edge (aa)\n (a) edge (as)\n (ss) edge (sss)\n (as) edge (asa)\n (aa) edge (aaa)\n (sss) edge (-2,-3.85)\n (asa) edge (0,-3.85)\n (aaa) edge (2,-3.85);\n\\end{tikzpicture}\n\\]\nAs discussed in \\cite[Cor. 2.8]{CT15}, $\\kappa_4(\\nu,J)$ counts how many times one must apply either $f_4$ or $f_4^*$ to $(\\nu,J)$ to reach a point where $f_4$ and $f_4^*$ have the same affect.\n\\end{ex}\n\n\n\n\n\\begin{thm}\nLet $e_a$ and $f_a$ be the crystal operators given by Definition~\\ref{def:RC_crystal_ops}, and let $e_a^*$ and $f_a^*$ be given by Definition~\\ref{def:RC_star_crystal_ops}. Then we have\n\\[\ne_a^* = * \\circ e_a \\circ *, \\ \\ \\ \\ \\ \\\nf_a^* = * \\circ f_a \\circ *.\n\\]\n\\end{thm}\n\n\\begin{proof}\nWe show the conditions of Proposition~\\ref{prop:star_properties} hold for $\\RC(\\infty)$ with the given crystal operations. Fix some $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$.\n\nWe first note the fact that $f_a(\\nu, J)$, $f_a^*(\\nu, J) \\neq 0$ follows immediately from the definitions. So we have Condition~(\\ref{item:star1}). Now let $b \\in I$. As $f_b$ acts on labels and preserves colabels in $(\\nu,J)^{(k)}$, for $k \\neq b$ in $I$, and $f_a^*$ acts on colabels and preserves labels in $(\\nu,J)^{(k)}$, for $k \\neq a$ in $I$, it follows that $f_a^* f_b (\\nu, J) = f_b f_a^* (\\nu, J)$ for all $a \\neq b$. Hence Condition~(\\ref{item:star2}) is satisfied.\n\nLemma~\\ref{lemma:kappa0} implies Condition~(\\ref{item:star4}).\n\nLemma~\\ref{lemma:kappa1} implies Condition~(\\ref{item:star5}).\n\nLemma~\\ref{lemma:kappa2} implies Condition~(\\ref{item:star6}).\n\nThus it remains we prove Condition~(\\ref{item:star3}), that $\\kappa_a(\\nu,J) \\ge 0$. We prove this by induction on the depth of $(\\nu,J)$. Observe that $\\kappa_a(\\nu_\\emptyset,J_\\emptyset) = 0$, which is our base case. Now suppose $\\kappa_a(\\nu,J) \\ge 0$ for all $(\\nu,J) \\in \\RC(\\infty)$ at depth at most $d$. It suffices to show that $\\kappa_a\\bigl(f_a(\\nu,J)\\bigr) \\ge 0$ and $\\kappa_a\\bigl(f_a^*(\\nu,J)\\bigr) \\ge 0$.\n\nNote that all labels, except for the row of $\\nu^{(a)}$ at which the box was added, possibly change by adding $-A_{ab}$ under $f_a$ by Equation~\\eqref{eq:change_vac_f}. Additionally, $p_{\\infty}^{(b)}$ changes by $-A_{ab}$. Thus for $b \\neq a$ a label, and hence possibly $-\\varepsilon_b(\\nu, J)$, increases by $-A_{ab}$ and the colabels, and hence $-\\varepsilon_b^*(\\nu, J)$, stay fixed. Therefore, by the above and its dual, for $a \\neq b$, we have\n\\[\n\\kappa_b\\bigl(f_a(\\nu, J)\\bigr), \\kappa_b\\bigl(f_a^*(\\nu, J)\\bigr) \\geq \\kappa_b(\\nu, J) \\geq 0,\n\\]\nsince $-A_{ab} \\geq 0$. Now it is sufficient to show that $\\varepsilon_a^*(\\nu, J)$ increases by at least $1 - \\kappa_a(\\nu, J)$ because $\\varphi_a(\\nu, J) = p_{\\infty}^{(a)} + \\varepsilon_a(\\nu, J)$ decreases by $1$ after the application of $f_a$. If $\\kappa_a(\\nu, J) = 0$, then Lemma~\\ref{lemma:kappa0} gives $f_a(\\nu, J) = f_a^*(\\nu, J)$, and so $\\varepsilon_a^*(\\nu, J)$ is increased by $1$. By Lemma~\\ref{lemma:kappa1}, we have $\\varepsilon_a^*(\\nu, J) = \\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr)$ when $\\kappa_a(\\nu, J) \\geq 1$. Note that by our assumption, this is all possible values. Therefore, both $\\kappa_a\\bigl(f_a(\\nu, J)\\bigr), \\kappa_a\\bigl(f_a^*(\\nu, J)\\bigr) \\geq 0$, as required.\n\nThus Conditions~(\\ref{item:star1})--(\\ref{item:star6}) are satisfied. Moreover, $\\RC(\\infty)$ and $\\RC(\\infty)^*$ are both generated by the highest weight element $(\\nu_{\\emptyset}, J_{\\emptyset})$. Hence $\\RC(\\infty) = \\RC(\\infty)^*$ and the result follows from Proposition~\\ref{prop:weaker_conditions}.\n\\end{proof}\n\nNow from Definition~\\ref{def:RC_crystal_ops} and Definition~\\ref{def:RC_star_crystal_ops}, we have that the $*$-involution is given as follows.\n\n\\begin{cor}\n\\label{cor:RC_star_involution}\nThe $*$-involution on $\\RC(\\infty)$ is given by replacing every rigging $x$ of a row of length $i$ in $(\\nu, J)^{(a)}$ by the corresponding corigging $p_i^{(a)} - x$ for all $(a, i) \\in \\mathcal{H}$.\n\\end{cor}\n\nLet $\\mathfrak{g}$ and $\\widehat{\\mathfrak{g}}$ be symmetrizable Kac-Moody algebras such that there exists a folding of the Dynkin diagram of $\\mathfrak{g}$ to the Dynkin diagram of $\\widehat{\\mathfrak{g}}$ with corresponding index sets $I$ and $\\widehat{I}$, respectively. Consider the map $\\phi \\colon \\widehat{I} \\searrow I$ induced by such a Dynkin diagram folding and consider a sequence $(\\gamma_a \\in \\mathbf{Z}_{>0})_{a \\in I}$ such that the map $\\Psi \\colon P \\longrightarrow \\widehat{P}$ given by\n\\[\n\\Lambda_a \\mapsto \\gamma_a \\sum_{b \\in \\phi^{-1}(a)} \\Lambda_b\n\\]\nalso satisfies\n\\[\n\\alpha_a \\mapsto \\gamma_a \\sum_{b \\in \\phi^{-1}(a)} \\alpha_b.\n\\]\nThis induces a \\defn{virtualization map} $v$ of $B(\\infty)$ of type $\\mathfrak{g}$ to that of type $\\widehat{\\mathfrak{g}}$. In particular, on $\\RC(\\infty)$, the image $(\\widehat{\\nu}, \\widehat{J})$ of a rigging configuration $(\\nu, J)$ is given by\n\\[\n\\widehat{m}_{\\gamma_a i}^{(b)} = m_i^{(a)},\n\\hspace{30pt}\n\\widehat{J}_{\\gamma_a i}^{(b)} = \\gamma_a J_i^{(a)},\n\\]\nfor all $b \\in \\phi^{-1}(a)$. We refer the reader to~\\cite{OSS03III, SalS15III, SchillingS15} for more details.\n\n\\begin{cor}\nLet $v$ be a virtualization map on $B(\\infty)$ of type $\\mathfrak{g}$ to $\\widehat{\\mathfrak{g}}$. Then\n\\[\n\\ast \\circ v = v \\circ \\ast.\n\\]\n\\end{cor}\n\n\\begin{proof}\nThis follows from the fact\n$\n\\widehat{p}_{\\gamma_a i}^{(b)} = \\gamma_a p_i^{(a)}\n$\nfor all $b \\in \\phi^{-1}(a)$ and Corollary~\\ref{cor:RC_star_involution}.\n\\end{proof}\n\n\n\n\n\n\\section{Highest weight crystals}\n\\label{sec:hw_crystals}\n\nWe wish to classify the subcrystal of $\\RC(\\infty)$ which is isomorphic to $B(\\lambda)$ with respect to the $\\ast$-crystal structure. In particular, defining $B(\\lambda)$ requires the additional condition that $\\varphi_a^*(\\nu, J) = \\max\\{k \\in \\mathbf{Z} \\mid (f_a^*)^k(\\nu, J) \\neq 0\\}$. For example, the condition $\\varphi_a(\\nu,J) = \\max\\{ k \\in \\mathbf{Z} \\mid f_a^k(\\nu,J) \\neq 0 \\}$ means, for all riggings $x$ corresponding to a row of length $i$ in $\\nu^{(a)}$, we have $x \\leq p_i^{(a)}$. If we consider the natural dual to this, we have $p_i^{(a)} - x \\leq p_i^{(a)}$, or equivalently $x \\geq 0$. We show this is the correct condition by proving the dual version of~\\cite[Thm.~6.1]{SalS15}.\n\n\nFor any $\\lambda \\in P^+$, we define\n\\[\n\\RC(\\lambda) := \\{ (\\nu, J) \\in \\RC(\\infty) \\mid \\max J_i^{(a)} \\leq p_i^{(a)}(\\nu; \\lambda) \\text{ for all } (a, i) \\in \\mathcal{H} \\},\n\\]\nwhere\n\\begin{equation}\n\\label{eq:vacancy_numbers}\np_i^{(a)}(\\nu; \\lambda) := \\inner{h_a}{\\lambda} - \\sum_{b \\in I} A_{ab} \\sum_{j \\in \\mathbf{Z}_{>0}} \\min(i,j) m_j^{(b)}.\n\\end{equation}\nNote that Equation~\\eqref{eq:vacancy_numbers} differs from Equation~\\eqref{eq:vacancy} by\n\\[\np_i^{(a)}(\\nu) + \\inner{h_a}{\\lambda} = p_i^{(a)}(\\nu; \\lambda).\n\\]\nWhen there is no danger of confusion, we will simply write $p_i^{(a)} = p_i^{(a)}(\\nu; \\lambda)$.\n\nWe consider a crystal structure on $\\RC(\\lambda)$ as that inherited from $\\RC(\\infty)$ under the natural projection except with $\\mathrm{wt}(\\nu, J) = \\lambda - \\sum_{a \\in I} \\lvert \\nu^{(a)} \\rvert \\alpha_a$.\n\n\\begin{thm}[{\\cite{SalS15,SalS15II,S06}}]\nWe have\n$\n\\RC(\\lambda) \\cong B(\\lambda).\n$\n\\end{thm}\n\nUsing the $\\ast$-crystal structure, we easily obtain~\\cite[Prop.~8.2]{K95}. (We refer the reader to \\cite{K95} or \\cite{HK02} for an exposition on the tensor product of crystals. Note that we are using the opposite, anti-Kashiwara, convention. The precise definition in this setting may be found, for example, in \\cite{SalS15}.)\n\n\\begin{prop}\nLet $\\lambda \\in P^+$. Then we have\n\\[\n\\RC(\\lambda) \\cong \\{ t_\\lambda \\otimes (\\nu,J) \\in T_\\lambda \\otimes \\RC(\\infty) \\mid \\varepsilon_a^*(\\nu,J) \\le \\langle h_a , \\lambda \\rangle \\text{ for all } a \\in I\\}.\n\\]\n\\end{prop}\n\n\\begin{proof}\nFix some $(\\nu, J) \\in \\RC(\\infty)$. Let $x$ be a rigging of a row of length $i$. We have\n\\[\n\\inner{h_a}{\\lambda} \\geq \\varepsilon_a^*(\\nu, J) = -\\min(0, p_i^{(a)} - x)\n\\]\nif and only if\n\\[\np_i^{(a)} + \\inner{h_a}{\\lambda} \\geq x.\n\\]\nRecall that the left-hand side is the vacancy numbers in $\\RC(\\lambda)$ by Equation~\\eqref{eq:vacancy_numbers}, and so we have the defining relation for $\\RC(\\lambda)$.\n\\end{proof}\n\nBy letting $\\pi_{\\lambda} \\colon B(\\infty) \\longrightarrow B(\\lambda)$ be the natural projection, we can rephrase the last proposition as\n\\[\n\\tau(b^*) = \\varepsilon(b), \\qquad \\qquad \\varepsilon(b^*) = \\tau(b),\n\\]\nwhere $\\varepsilon(b) = \\sum_{a \\in I} \\varepsilon_a(b) \\Lambda_a$ and $\\tau(b) = \\min \\{ \\lambda \\mid \\pi_{\\lambda}(b) \\in B(\\lambda) \\}$. In~\\cite{SalS15II}, $\\tau$ was called the difference statistic and can be explicitly given on rigged configurations by\n\\[\n\\tau(\\nu, J) = \\sum_{a \\in I} \\min_{i \\in \\mathbf{Z}_{>0}} \\{p_i^{(a)} - \\max J_i^{(a)} \\} \\Lambda_a.\n\\]\n\n\n\n\n\nNow we formalize the dual version of $\\RC(\\lambda)$.\n\n\\begin{dfn}\nLet $\\RC(\\lambda)^*$ denote the closure of $(\\nu_{\\emptyset}, J_{\\emptyset})$ under $e_a^*$ and the following modified $f_a^*$, both using the vacancy numbers given by Equation~\\eqref{eq:vacancy_numbers} to determine the colabels. Consider $f_a^*$ as in Definition~\\ref{def:RC_star_crystal_ops} except define $f_a^*(\\nu, J) = 0$ if in the result, there exists a rigging $x < 0$.\n\\end{dfn}\n\nNote that the condition that $x \\geq 0$ is equivalent to $p_i^{(a)} - x \\leq p_i^{(a)}$. Hence, by duality, the proof of~\\cite[Lemma~3.6]{S06} holds, and we obtain the following.\n\n\\begin{lemma}\n\\label{lemma:lower_regular}\nLet $(\\nu, J) \\in \\RC(\\lambda)^*$. Then\n\\[\n\\varphi_a^*(\\nu, J) = \\max\\{ k \\in \\mathbf{Z} \\mid (f_a^*)^k (\\nu, J) \\neq 0 \\}\n\\]\nfor all $a \\in I$.\n\\end{lemma}\n\nLet $\\RC_{\\lambda}(\\infty)^* = T_{\\lambda} \\otimes \\RC(\\infty)$ with the $\\ast$-crystal structure. Let $C = \\{c\\}$ be the crystal given by\n\\[\n\\mathrm{wt}(c) = 0, \\qquad\n\\varphi_a^*(c) = \\varepsilon_a^*(c) = 0, \\qquad\nf_a^*c = e_a^*c = 0,\n\\]\nfor all $a \\in I$. Nakashima \\cite[Thm. 3.1]{N99} has shown that the connected component generated by $c \\otimes t_{\\lambda} \\otimes u_{\\infty}$ is isomorphic to $B(\\lambda)$.\n\nIn~\\cite{SalS15}, the map $\\psi_{\\lambda,\\mu} \\colon \\RC(\\lambda) \\longrightarrow \\RC(\\mu)$, for $\\lambda \\leq \\mu$ in $P^+ \\sqcup \\{\\infty\\}$, is the identity map on rigged configurations. This follows because $e_a$ and $f_a$ are determined by the riggings alone, not the vacancy numbers, and so preserving the labels is sufficient to show $\\psi_{\\lambda,\\mu}$ commutes with the crystal operators. However, for the $\\ast$-crystal structure, we need to preserve coriggings, and as such, we need to take into account the shift in vacancy numbers. Thus, define a map $\\psi_{\\lambda,\\mu}^* \\colon \\RC(\\lambda)^* \\longrightarrow \\RC(\\mu)^*$ as the identity on the partitions but with new riggings\n\\[\nx' = x + \\langle h_a , \\mu - \\lambda\\rangle,\n\\]\nwhere we make the convention that $\\inner{h_a}{\\infty} = 0$. Note that $\\psi_{\\lambda,\\mu}^*$ commutes with the crystal operators (however, it only becomes a crystal embedding after an appropriate tensor product is taken to shift weights).\n\nWith this modification, Proposition~\\ref{prop:ep_phi_star}, and Lemma~\\ref{lemma:lower_regular}, we have the dual argument of~\\cite[Thm.~6.1]{SalS15}. \n\n\\begin{thm}\nLet $C_{\\emptyset}^*$ denote the connected component of $C \\otimes \\RC_{\\lambda}(\\infty)^*$ generated by $c \\otimes (\\nu_{\\emptyset}, J_{\\emptyset})$. The map $\\Psi \\colon C_{\\emptyset}^* \\longrightarrow \\RC(\\lambda)^*$ given by \n\\[\nc \\otimes (\\nu_{\\lambda}, J_{\\lambda}) \\mapsto (\\psi_{\\lambda,\\infty}^*)^{-1}(\\nu_{\\lambda}, J_{\\lambda})\n\\]\nis a weight-preserving bijection which commutes with $e_a^*$ and $f_a^*$ for every $a\\in I$.\n\\end{thm}\n\n\\begin{cor}\nLet $\\mathfrak{g}$ be of symmetrizable type. Then $\\RC(\\lambda)^* \\cong B(\\lambda)$.\n\\end{cor}\n\nHence, we can now construct an explicit crystal isomorphism $\\RC(\\lambda)^* \\cong \\RC(\\lambda)$ by passing through $\\RC(\\infty)$.\n\n\\begin{cor}\nLet $\\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $\\lambda \\in P^+$. Define $\\Xi \\colon \\RC(\\lambda) \\longrightarrow \\RC(\\lambda)^*$ by $\\Xi(\\nu, J) = (\\nu, J')$, where the resulting riggings are\n\\[\nx' = x + \\inner{h_a}{\\lambda}.\n\\]\nThen $\\Xi$ is a crystal isomorphism.\n\\end{cor}\n\n\\begin{proof}\nWe have $\\Xi = (\\psi_{\\lambda,\\infty}^*)^{-1} \\circ \\psi_{\\lambda,\\infty}$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}