diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlxin" "b/data_all_eng_slimpj/shuffled/split2/finalzzlxin" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlxin" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nA standard description of the World is usually presented in terms of the observable $3+1$ spatio-temporal dimensions. At the same time string theories have been developed, seeking to produce a consistent description of the Standard Model of physics including the phenomenon of gravity, which appear to be most consistent if large numbers of dimensions are postulated. A 26-dimensional space-time was deemed necessary for bosonic strings~\\cite{Lovelace1971} and a ten-dimensional one for type-II~\\cite{Gliozzi1976,Green1981} and heterotic strings~\\cite{Gross1985}. The latter theories are closely related to a mysterious theory called M-theory, which lives in 11 dimensions~\\cite{Duff1987}.\nIn contrast, classical physics requires 3 spatial dimensions, e.g. to accommodate Newton's inverse square law as argued already by Immanuel Kant. Ehrenfest has shown that atoms only exhibit stable orbits in a 3-dimensional space~\\cite{Ehrenfest1920}. These contradictions between requirements from classical and quantum physics for a 3-dimensional space and the possibility of a theory involving higher dimensions were already resolved in 1926 by Klein invoking the concept of compactification~\\cite{Klein1926}.\n\nIn the present study the accurate results from precision measurements on molecules are exploited to constrain existing theories on higher dimensions.\nFor molecular systems, state-of-the-art quantum level calculations of the molecular ions H$_2^+$, HD$^+$, and D$_2^+$, all fundamental three-particle Coulomb systems, have reached the precision that the uncertainty becomes limited by the precision at which values of the fundamental mass ratios $m_p\/m_e$ and $m_n\/m_p$ are known~\\cite{Korobov2009,Karr2011,Korobov2012}, although the recently improved determination of $m_p\/m_e$ \\cite{Sturm2014} demonstrates active progress on the experimental side.\nWhile experiments on the ro-vibrational spectrum of the H$_2^+$ isotopomer are still under way~\\cite{Karr2014} the small dipole moment of the HD$^+$ isotopomer has enabled the accurate study of electric-dipole-allowed transitions in various bands~\\cite{Koelemeij2007,Bressel2012,Koelemeij2012}.\n\nIn recent years great progress has also been made on the calculation of level energies in the neutral hydrogenic molecules. Accurate Born-Oppenheimer energies have been calculated for the electronic ground state of H$_2$, HD and D$_2$~\\cite{Pachucki2010}, as well as non-adiabatic interactions~\\cite{Pachucki2008,Pachucki2009} and relativistic and quantum electrodynamical (QED) corrections~\\cite{Pachucki2005,Pachucki2007}. Now a full set of ro-vibrational level energies of all quantum states up to the dissociation limit is available for all three isotopomers~\\cite{Piszcziatowski2009,Komasa2011}. These calculations on the ground electronic quantum levels were tested in experiments measuring the dissociation limits of H$_2$~\\cite{Liu2009}, D$_2$~\\cite{Liu2010}, and HD~\\cite{Sprecher2010}. Further they were compared to experimental values for the fundamental vibrational splitting in H$_2$ and the hydrogen isotopomers~\\cite{Dickenson2013,Niu2014}, to a measurement of the first overtone in H$_2$~\\cite{Campargue2012,Kassi2014} and D$_2$~\\cite{Kassi2012}, a measurement of the second overtone in H$_2$~\\cite{Hu2012,Tan2014}, and measurements of highly excited rotational levels in H$_2$~\\cite{Salumbides2011}. The results from the variety of experimental precision measurements on both the ionic and neutral hydrogen molecules are generally in excellent agreement with the QED-calculations, within combined uncertainty limits from theory and experiment.\n\nThe agreement between experiment and first-principles calculations on quantum level energies of molecules has inspired an interpretation of these data that goes beyond molecular physics. Since weak, strong and (Newtonian) gravitational forces have negligible contributions to their quantum level structure, electromagnetism is the sole force acting between the charged particles within light molecules, and QED is the fully-encompassing framework to perform calculations. This makes it possible to derive bounds on possible fifth forces between hadrons from molecular precision experiments compared with QED-calculations~\\cite{Salumbides2013,Salumbides2014}.\n\nTheories of higher dimensions were developed with the goal to resolve the hierarchy problem, i.e. the vast difference of scales between that of electro-weak unification (1 TeV) and that of the Planck scale (10$^{16}$ TeV), where gravity becomes strong. By permitting the leakage of gravity into higher dimensions while keeping the particles and the three forces of the Standard Model in 3+1 dimensions, and invoking a compactification range for the extra dimensions exceeding 3+1, two different testable theories were phrased by Arkhani-Hamed, Dimopoulos, and Dvali~\\cite{Hamed1998} and by Randall and Sundrum~\\cite{Randall1999,Randall1999b}. The mathematical formalisms of these theories can be applied to molecular physics test bodies, from which constraints on the compactification distances can be deduced for the former,\nwhile constraints on the brane separation or curvature can be derived for the latter. That is the subject of the present paper.\n\n\\section{The ADD-model}\n\\label{ADD-model}\nIt is the intention of the theory formulated by Arkani-Hamed, Dimopoulos, and Dvali~\\cite{Hamed1998}, referred to as ADD-theory, to establish an effective Planck scale to coincide with the electro-weak scale by allowing gravity to propagate in extra dimensions. The three forces of the Standard Model, tested at very short distances in particle and atomic physics experiments, are considered to act locally within a 3-brane (3 spatial dimensions and a time dimension) embedded in a higher dimensional bulk, where gravity may act allowing for gravitons to escape. By this means in ADD the hierarchy problem is nullified, and the so-called desert range between the electro-weak scale ($M_\\mathrm{EW}$) of 1 TeV and the Planck scale ($M_\\mathrm{Pl}$) of 10$^{16}$ TeV avoided. The extension of the extra dimensions is necessarily limited, in the case of flat metrics considered in ADD, since experiments of the Cavendish-type have proven that gravity obeys the Newtonian $1\/r$ potential beyond the range of 1 cm \\cite{Adelberger2003}. Hence, the extra dimensions are considered to be compactified within a range parameter $R_n$. While in principle the extra dimensions could exhibit differing range parameters, in the ADD-formalism and in the present analysis such difference is not made.\n\nThe Newtonian gravitational potential may be written as:\n\\begin{equation}\n V_\\mathrm{N}(r)= -G\\frac{m_1m_2}{r} = -\\frac{m_1m_2}{M_\\mathrm{Pl}^2} \\frac{1}{r} \\hbar c\n\\end{equation}\nwith the Planck mass defined as $M_{Pl}^2=\\hbar c\/G$ in SI units.\nIn the following discussions, we adopt the natural units $\\hbar=c=1$ and drop the $(\\hbar c)$-factor in the potentials.\nThe extra $n$ spatial dimensions proposed in the ADD theory result in a modification of Newtonian gravity, for distances shorter than the compactification length range, that is consistent with Gauss law:\n\\begin{equation}\n V_\\mathrm{ADD} = -\\frac{m_1m_2}{ M^{n+2}_{(4+n)}} \\frac{1}{r^{n+1}},\n\\label{ADD-short}\n\\end{equation}\nwhere the subscript 4 represents the known $(3+1)$ spacetime dimensions, and $M_{(4+n)}$ is the full higher-dimensional Planck mass.\nFor separations larger than the compactification length, $r > R_n$, the ADD potential should be in correspondence with the Newtonian $1\/r$-form\n\\begin{equation}\n V_\\mathrm{ADD} = -\\frac{ m_1 m_2 }{ M^{n+2}_{(4+n)}(R_n )^n } \\frac{1}{r}.\n\\label{ADD-long}\n\\end{equation}\nTo be more precise, $(R_n )^n$ should be the compactified volume of the extra dimensions $V_n$, thus a factor of order unity might be included for a specific compactification geometry.\n\nThe Planck mass $M_\\mathrm{Pl}$ is then related to the higher-dimensional mass $M_{(4+n)}$ via:\n\\begin{equation}\n M_\\mathrm{Pl}^2 = M^{n+2}_{(4+n)} (R_n)^n.\n\\label{Fund_M_Pl}\n\\end{equation}\nThus the fundamental mass $M_{(4+n)}$ may still be small and $M_\\mathrm{Pl}$ becomes large due to the compactified volume of extra dimensions.\nArkani-Hamed~\\emph{et al.}\\ have shown that if the fundamental mass is taken as $M_\\mathrm{EW}$ one extra dimension would have a range of order 10$^{10}$ km to account for the weakness of gravity. This is incompatible with experimental evidence. But for two extra dimensions $R_n$ would be of sub-millimeter size~\\cite{Hamed1998}, thus at a range where Newtonian gravity is not firmly tested.\nIn our present study we will not set a certain energy scale, and in particular we do not assume that $M_{4+n}\\sim M_\\mathrm{EW}$. Our goal is to constrain $R_n$ from molecular physics experiments without theoretical prejudice regarding the fundamental mass scale.\n\nWhile dealing with molecules the unit attraction of gravity can be chosen as that between two protons and a dimensionless gravitational coupling strength is defined as:\n\\begin{equation}\n \\alpha_G=Gm_p^2\/\\hbar c\n\\end{equation}\nNote that this particular choice of the gravitational coupling constant is equivalent to specifying $\\alpha_G = (m_p\/M_\\mathrm{Pl})^2 = 5.9\\times10^{-39}$.\nThen the Newtonian attraction between two particles consisting of $N_1$ and $N_2$ protons or neutrons ($m_n\\simeq m_p$ is adopted) can be written as:\n\\begin{equation}\n V_\\mathrm{N}(r)= -\\alpha_G N_1N_2 \\frac{1}{r}\n\\end{equation}\nFrom Eq.~(\\ref{Fund_M_Pl}), the ADD-potential of Eq.~(\\ref{ADD-short}) within the compactification radius $r < R_n$ may be rewritten as:\n\\begin{equation}\n V_\\mathrm{ADD}(r)= -\\alpha_G N_1N_2 R_n^n \\frac{1}{r^{n+1}},\n\\label{V_ADD}\n\\end{equation}\nwhile this potential reduces to normal Newtonian gravity $V_\\mathrm{N}$ for the range outside the compactification length range $r > R_n$.\n\nFor molecules this gravitational potential has an effect on the level energy of a molecular quantum state with wave function $\\Psi(r)$, to be written as an expectation value:\n\\begin{eqnarray}\n \\left &=& -\\alpha_G N_1N_2 \\left[ \\quad \\int_{R_n}^{\\infty} \\Psi^*(r) \\frac{1}{r} \\Psi(r) r^2 dr \\right. \\nonumber \\\\\n &&\\left. \\quad\\quad\\quad\\quad\\quad +R_n^n \\int_0^{R_n} \\Psi^*(r) \\frac{1}{r^{n+1}} \\Psi(r) r^2 dr \\quad \\right]\n\\label{Integrals}\n\\end{eqnarray}\nNote that the wave functions are given along a single coordinate $r$, i.e. the vibrational coordinate, that probes the gravitational forces between nucleons.\nHere the nuclear displacement is separated from electronic motion and the wave function $\\Psi(r)$ represents the probability that the nuclei in the molecule are at internuclear separation $r$.\nThe first integral term represents the ordinary gravitational attraction, which is for protons $8 \\times 10^{-37}$ times weaker than the electrostatic repulsion, and can therefore be neglected.\nThe second integral represents the effect of modified gravity and is evaluated using accurate wave functions for H$_2$.\nThe wave functions of the H$_2$ ground electronic state for the $v=0$ and 1 levels are shown in Fig.~\\ref{Wavefunctions}. In practice, the integration is performed up to $r=10$ \\AA\\ since the wave function amplitude is negligible beyond that. Also at shorter distances $r < 0.1$ \\AA\\ the wave function amplitude becomes negligible, for which reason the second integral in Eq.~(\\ref{Integrals}) converges without additional assumptions.\nThe HD$^+$ $v=0, J=2$ ground electronic state wave function is also displayed in Fig.~\\ref{Wavefunctions} showing the larger internuclear distance of the ion with respect to the neutral.\n\n\\begin{figure}\n\\begin{indented}\n\\item[]\\resizebox{0.8\\textwidth}{!}{\\includegraphics{fig1.eps}}\n\\caption{\\label{Wavefunctions} Wave functions for H$_2$ in the electronic ground state with $v=0, J=0$ and $v=1,J=0$, and for the HD$^+$ $v=0, J=2$ quantum state.}\n\\end{indented}\n\\end{figure}\n\nFor transitions between quantum states $\\Psi_1$ and $\\Psi_2$, as in spectroscopic transitions in molecules, a differential effect must be calculated:\n\\begin{equation}\n \\left<\\Delta V_\\mathrm{ADD}(n,R_n) \\right> = -\\alpha_G N_1N_2 R_n^n\n \\left[ \\left< {\\Psi_1} \\left| \\frac{1}{r^{n+1}} \\right|{\\Psi_1} \\right> - \\left< {\\Psi_2} \\left| \\frac{1}{r^{n+1}} \\right|{\\Psi_2} \\right> \\right]\n\\label{ADD-trans}\n\\end{equation}\nThis equation represents the expectation value for a high-dimensional gravity contribution to transitions in molecules. Here the ADD-expectation value is written explicitly as a function of the two relevant parameters:\nthe number $n$ of extra spatial dimensions and the compactification scale $R_n$.\nFrom Eq.~(\\ref{ADD-trans}), it is clear that a stronger effect can be expected if the difference in wave functions of the two states $\\Psi_1$ and $\\Psi_2$ is greater. For this reason, measurements on the dissociation limit in molecules, where $\\Psi_1$ is lowest energy bound state and $\\Psi_2$ is the non-interacting two-atom limit at $r=\\infty$, are the most sensitive probes.\n\n\\section{The Randall-Sundrum models}\n\nLet us now consider the Randall-Sundrum scenarios, RS-I and RS-II, to approach the physical description of extra dimensions in an alternative manner~\\cite{Randall1999,Randall1999b}. \nIn these scenarios, the particles and interactions of the Standard Model are confined in the SM-brane, separated by some distance $y_c$ from another (hidden) 3-brane along \\emph{one} extra dimension $y$. \nThe branes and the bulk are sources of gravity that were shown to produce an Anti-de-Sitter metric:\n\\begin{equation}\nds^2 = e^{-2 k |y|} \\eta_{\\mu\\nu}dx^\\mu dx^\\nu + dy^2,\n\\label{RS_metric}\n\\end{equation}\nwhere $\\exp(-k|y|)$ is a so-called warp factor and $k$ is the bulk curvature~\\cite{Randall1999}.\nThe warped metric differentiates the RS models from the ADD model with a flat metric where $k=0$.\nThus, the exponential warp factor in the RS scenarios solves the hierarchy problem alternatively, without requiring large extra dimensions as assumed in the ADD model.\n\nIn the RS scenarios the modified gravitational potential between two masses separated by a distance $r$ in the SM-brane can be expressed as:\n\\begin{equation}\n\tV_\\mathrm{RS}(r) = -G \\frac{m_1m_2}{r} \\left( 1 + \\Delta_\\mathrm{RS} \\right),\n\\end{equation}\nwhere $\\Delta_\\mathrm{RS}$ is the correction to the Newtonian potential.\nCallin~\\cite{Callin2004b} computed the potential in the framework of the RS-I scenario, obtaining for short distances:\n\\begin{equation}\n \\Delta_\\mathrm{RS-I}(r) \\simeq\n \\frac{4}{3\\pi kr}\\frac{1-e^{-2k y_c}}{1+\\frac{1}{3}e^{-2k y_c}}, \\quad kr \\ll 1 .\n\\label{RSI_potential_short}\n\\end{equation}\nHere one can distinguish two regimes, $ky_c \\ll 1$ and $ky_c \\gg 1$, with the result up to the leading order:\n\\begin{equation}\n\\renewcommand{\\arraystretch}{2}\n \\Delta_\\mathrm{RS-I}(r) \\simeq \\left\\{\n \\begin{array}{l l}\n \\frac{2y_c}{\\pi r} + ..., & \\quad ky_c \\ll 1,\\\\\n \\frac{4}{3 \\pi kr} + ..., & \\quad ky_c \\gg 1.\n \\end{array} \\right.\n\\renewcommand{\\arraystretch}{1}\n\\label{RSI_potential_ky}\n\\end{equation}\nIt turns out that the RS potential for long distances ($kr \\gg 1$) is not applicable to molecules and is not considered further.\n\nIn the RS-II scenario, the hidden 3-brane is chosen to be infinitely far ($y_c \\rightarrow \\infty$) from the SM-brane resulting in an effective model with a single 3-brane (SM-brane) in the bulk.\nThis solution thus offers the existence of extra dimensions that do not require compactification in contrast to the ADD model.\nFor short distances in the RS-II scenario, Callin and Ravndall \\cite{Callin2004a} obtained\n\\begin{equation}\n \\Delta_\\mathrm{RS-II}(r) = \\frac{4}{3\\pi kr} + ..., \\quad kr \\ll 1 ,\n \\label{RS_potential_leading_order}\n\\end{equation}\nfor the RS correction.\nNote the correspondence of Eq.~(\\ref{RS_potential_leading_order}) with that of Eq.~(\\ref{RSI_potential_ky}) for $ky_c \\gg 1$, which is expected since the latter RS-I condition implies the transition to RS-II at infinite brane separation.\n\nFrom these RS potential corrections, the expectation values of the leading-order shifts of transitions in molecules, in the short distance separation ($kr \\ll 1$) regime, are therefore:\n\\begin{equation}\n \\left<\\Delta V_\\mathrm{RS}(k) \\right> = \\alpha_G N_1N_2 \\mathcal{F} \\left(\\frac{4}{3\\pi k}\\right) \\left[ \\left< {\\Psi_1} \\left| \\frac{1}{r^2} \\right|{\\Psi_1} \\right> - \\left< {\\Psi_2} \\left| \\frac{1}{r^2} \\right|{\\Psi_2} \\right> \\right],\n\\label{RS-trans-short}\n\\end{equation}\nwhere $\\mathcal{F}=(1-e^{-2k y_c})\/(1+\\frac{1}{3}e^{-2k y_c})$ for RS-I and $\\mathcal{F}=1$ for RS-II.\nUsing these expressions, limits on the curvature $k$ or the brane separation $y_c$ based on molecular spectroscopy data can be derived.\n\n\\section{Constraints on higher dimensions from molecular data}\n\nIn the previous section, the expectation value for a higher-dimensional gravity contribution to a transition frequency in a molecule was presented for both ADD and RS approaches to higher dimensions.\nThis expectation value is interpreted as a contribution to the binding energy of molecules in certain quantum states.\nThis rationale will be used to derive constraints on characteristic parameters underlying the extra-dimensional theories, the compactifictaion range $R_n$ for the ADD scenario and the warp factor $k$ or brane separation $y_c$ for the RS scenario(s).\n\nIn Table~\\ref{data} a compilation is made of a comparison between theoretical and experimental values obtained in recent experiments for hydrogen neutral molecules and hydrogen molecular ions, and the stable isotopomers containing deuterons.\nRo-vibrational transitions in the ground electronic state are indicated by the change in vibrational quantum number $v$, while $D_0$ denotes the dissociation energy of the ground electronic state.\nIn the Table the agreement between theory and experiment is represented by the combined uncertainty $\\delta E$ with:\n\\begin{equation}\n \\delta E = \\sqrt{\\delta E_\\mathrm{exp}^2 + \\delta E_\\mathrm{theory}^2},\n\\label{Uncertainties}\n\\end{equation}\nwhere $\\delta E_\\mathrm{exp}$ and $\\delta E_\\mathrm{theory}$ signify uncertainties of theory and experiment.\nOn all but two cases the values for $\\delta E$ were found to be larger than the discrepancies between theory and experiment, denoted by $\\Delta E = E_\\mathrm{exp} - E_\\mathrm{theory}$, while the H$_{2}$\\ $v=0\\rightarrow1$ is within two standard deviations ($\\Delta E < 2\\,\\delta E$). From these results it is concluded that QED-theory for these molecular systems is in very good agreement with observations.\nRecent calculations by Korobov \\emph{et al.}~\\cite{Korobov2014} result in an increased discrepancy with the experimental results of Bressel \\emph{et al.}~\\cite{Bressel2012} at the level of 2.6 standard deviations, and we do not include the HD$^{+}$\\ $v=0\\rightarrow1$ values in the comparisons.\n\n\\begin{table}\n\\caption{\\label{data} Data from recent precision measurements of vibrational energy splittings as well as the dissociation energy $D_0$ in neutral and ionic molecular hydrogen and their isotopomers. Adapted from Ref.~\\cite{Salumbides2013} and updated with most recent data. $\\Delta E$ represents the deviation between theory and experiment, while $\\delta E$ represents the combined uncertainties, cf. Eq.~(\\ref{Uncertainties}).\n}\n\\begin{indented}\n\\lineup\n\\item[]\\begin{tabular}{@{}l@{\\hspace{10pt}}c@{\\hspace{15pt}}r@{.}l@{\\hspace{15pt}}r@{.}l@{\\hspace{15pt}}c}\n\\br\nspecies & transition & \\multicolumn{2}{l}{$\\Delta E$ (cm$^{-1}$)} & \\multicolumn{2}{l}{$\\delta E$ (cm$^{-1}$)} & Ref. \\\\\n\\mr\nH$_{2}$\t&$v=0\\rightarrow1$ \t&0&000\\,24\t&0&000\\,17\t&\\cite{Dickenson2013,Niu2014} \\\\\n\t&$v=0\\rightarrow2$\t&0&000\\,4 \t&0&002\\,0\t&\\cite{Campargue2012,Kassi2014} \\\\%Q(1)\n\t&$v=0\\rightarrow3$\t&-0&000\\,6\t&0&002\\,5\t&\\cite{Hu2012,Tan2014} \\\\%S(0)\n\t&$D_0$\t\t\t&0&000\\,0\t&0&001\\,2\t&\\cite{Liu2009} \\\\\n\\mr\nHD\t&$v=0\\rightarrow1$ \t&0&000\\,11\t&0&000\\,23\t&\\cite{Dickenson2013,Niu2014} \\\\\n\t&$D_0$\t\t\t&0&000\\,9\t&0&001\\,2\t&\\cite{Sprecher2010} \\\\\n\\mr\nD$_{2}$\t&$v=0\\rightarrow1$ \t&-0&000\\,02\t&0&000\\,17\t&\\cite{Dickenson2013,Niu2014} \\\\\n\t&$v=0\\rightarrow2$\t&-0&000\\,5\t&0&001\t\t&\\cite{Kassi2012} \\\\%S(0)\n\t&$D_0$\t\t\t&0&000\\,5\t&0&001\\,1\t&\\cite{Liu2010} \\\\\n\\mr\nHD$^{+}$\t&$v=0\\rightarrow1$ \t&-0&000\\,005\\,2\t&0&000\\,002\\,0 \t&\\cite{Bressel2012, Korobov2014} \\\\%(v=0,N=0)-(1,1)\n\t&$v=0\\rightarrow4$ \t& 0&000\\,009\t&0&000\\,017\t&\\cite{Koelemeij2007} \\\\%(v=0,N=2)-(4,3)\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\nThe agreement between theory and experiment for molecular systems is now translated into a constraining relation for higher dimensions in the ADD framework:\n\\begin{equation}\n \\left<\\Delta V_\\mathrm{ADD}(n,R_n)\\right> \\, < \\delta E .\n\\label{ADD-constrain-relation}\n\\end{equation}\nAs a first example we take the measurement on the fundamental vibration in the H$_2$ molecule. This is one of the most accurately measured numbers in neutral molecules, while also the QED-calculations for this fundamental rotationless transition are more accurate by an order of magnitude with respect to the absolute binding energies, because of cancellation of errors for non-rotating molecules~\\cite{Dickenson2013}.\nConstraints on $R_n$ can be derived via:\n\\begin{equation}\n (R_n)^n < \\frac{\\delta E}{\\alpha_G N_1N_2 \\Delta}\n\\label{Rn-constraint}\n\\end{equation}\nwith $\\Delta$ the difference in expectation values over the wave function densities between $v=0$ and $v=1$ vibrational states in the molecule:\n\\begin{equation}\n \\Delta = \\left[ \\left< \\frac{1}{r^{n+1}} \\right>_{\\Psi_1} - \\left< \\frac{1}{r^{n+1}} \\right>_{\\Psi_0} \\right]\n\\end{equation}\n\nThe wave functions for the lowest vibrational states, in the case of H$_2$ and for $J=0$, as obtained from \\emph{ab initio} calculations~\\cite{Piszcziatowski2009,Komasa2011} are plotted in Fig.~\\ref{Wavefunctions}. Since the wave functions are located in the same region of space the fundamental vibrational transition in the hydrogen molecule ($v=0 \\rightarrow v=1$) probes only a differential effect.\nThe resulting constraints on $R_n$ from the measurement of the fundamental vibration in the H$_2$ molecule\nfor the range of extra dimensions $n=1-8$ are presented in Fig.~\\ref{Limits-H2}.\nThe sloping lines in Fig.~\\ref{Limits-H2} represent calculated $V_\\mathrm{ADD}\/V_\\mathrm{N}$ for different $n$ and $R_n$ values.\nThe horizontal dashed line $\\delta E\/V_\\mathrm{N}$ indicates limits from molecular spectroscopy.\nHence, for certain numbers of extra dimensions $n$, $R_n$ is constrained to be less than the value where the $V_\\mathrm{ADD}\/V_\\mathrm{N}$ and $\\delta E\/V_\\mathrm{N}$ intersect in the graph.\nConstraints on $R_n$, obtained from a comparison with the fundamental vibrational transition of H$_2$ are presented in Table~\\ref{Constraints-R}.\n\n\\begin{figure}\n\\begin{indented}\n\\item[]\\resizebox{0.8\\textwidth}{!}{\\includegraphics{fig2.eps}}\n\\caption{(Color online) Limit on the compactification range $R_n$ as derived from the measurement of the fundamental vibration in the H$_2$ molecule~\\cite{Dickenson2013} in comparison with the ADD-formalism.}\n\\label{Limits-H2}\n\\end{indented}\n\\end{figure}\n\\begin{table}\n\\caption{\\label{Constraints-R}Constraints on the size $R_n$ of compactified dimensions (in units of m) as derived from a number of molecular features: (i) the fundamental ($0\\rightarrow1$) vibration in H$_2$, (ii) the dissociation limit $D_0$ of H$_2$, (iii) the dissociation limit of D$_2$, and (iv) the (4-0) R(2) ro-vibrational transition in HD$^+$. The constraints are derived within the ADD-framework assuming that $n$ extra dimensions are of equal size. The corresponding higher-dimensional Planck length $R_{\\mathrm{Pl},(4+n)}$ (in units of m) and Planck mass $M_{(4+n)}$ (in units of GeV) are also tabulated, where the smallest values for $R_{\\mathrm{Pl},(4+n)}$ and the highest value for $M_{(4+n)}$ is taken from the examples.}\n\\footnotesize\n\\begin{indented}\n\\lineup\n\\item[]\\begin{tabular}{@{}l@{\\hspace{17pt}}l@{\\hspace{12pt}}l@{\\hspace{12pt}}l@{\\hspace{12pt}}l@{\\hspace{20pt}}l@{\\hspace{12pt}}l}\n\\br\n $n$ & \\multicolumn{4}{c}{$R_n$} & $R_{\\mathrm{Pl},(4+n)}$ & $M_{(4+n)}$ \\\\\n\\mr\n & H$_2$ (1-0) & H$_2$ $D_0$ & D$_2$ $D_0$ & HD$^+$ (4-0) & (m) & (GeV) \\\\\n\\mr\n2 & $2.2\\times10^{ 4}$ & $1.0\\times10^{ 4}$ & $4.8\\times10^{ 3}$ & $2.8\\times10^{ 3}$ & $2.1\\times10^{-16}$ & $9.3\\times10^{-1}$\\\\\n3 & $7.7\\times10^{-1}$ & $1.9\\times10^{-1}$ & $1.2\\times10^{-1}$ & $1.0\\times10^{-1}$ & $3.0\\times10^{-15}$ & $6.5\\times10^{-2}$\\\\\n4 & $1.1\\times10^{-3}$ & $8.5\\times10^{-4}$ & $5.9\\times10^{-4}$ & $7.0\\times10^{-4}$ & $1.8\\times10^{-14}$ & $1.1\\times10^{-2}$\\\\\n5 & $3.3\\times10^{-5}$ & $3.2\\times10^{-5}$ & $2.4\\times10^{-5}$ & $3.1\\times10^{-5}$ & $5.8\\times10^{-14}$ & $3.4\\times10^{-3}$\\\\\n6 & $3.4\\times10^{-6}$ & $3.7\\times10^{-6}$ & $2.9\\times10^{-6}$ & $3.0\\times10^{-6}$ & $1.4\\times10^{-13}$ & $1.4\\times10^{-3}$\\\\\n7 & $6.9\\times10^{-7}$ & $7.8\\times10^{-7}$ & $6.4\\times10^{-7}$ & $6.3\\times10^{-7}$ & $2.8\\times10^{-13}$ & $7.1\\times10^{-4}$\\\\\n\\br\n\\end{tabular}\n\\end{indented}\n\\end{table}\n\nThe experimental as well as the theoretical results for the fundamental vibration in the hydrogen molecule are known to the 10$^{-4}$ cm$^{-1}$\\ level, an order of magnitude more accurate than the values for the binding energies~\\cite{Dickenson2013,Niu2014}. However, for a comparison of dissociation limits it is no longer a small difference along the internuclear coordinate axis that is probed, but the difference between the 1 \\AA\\ molecular scale and infinite atomic separation. The expectation value for the ADD-contribution to the binding energy of the lowest bound state in the H$_2$ molecule, or the $D_0$ binding energy, is:\n\\begin{equation}\n \\left<\\Delta V_\\mathrm{ADD}(n,R_n)\\right> = \\alpha_G N_1N_2 R_n^n \\left< \\frac{1}{r^{n+1}} \\right>_{\\Psi_0}\n\\end{equation}\nBy comparing to the experimental findings on $D_0$(H$_2$) \\cite{Liu2009} this leads to another set of constraints on $R_n$ for $n$ extra dimensions, which are also listed in Table~\\ref{Constraints-R}.\n\nThe method was further applied to the fundamental vibration of HD and D$_2$, where the experimental and theoretical uncertanties are similar to those in H$_2$. Although the heavier masses of the isotopomers improve the constraints obtained from H$_2$, as expected from Eq.~(\\ref{Rn-constraint}), the HD and D$_2$ fundamental vibration constraints are still less stringent compared to that from the H$_2$ dissociation limit. \nThe results obtained for D$_2$ dissociation energy~\\cite{Liu2010} lead to the tightest constraints on $R_n$ from the neutrals as listed in Table~\\ref{Constraints-R}, which scale by a factor $(\\sfrac{1}{4})^{1\/n}$ relative to H$_2$ due to the mass difference.\n\nThe experimental accuracy for the HD$^+$ molecular ion transitions is an order magnitude better than the corresponding neutral molecule system that stems mostly from the possibility of trapping the ionic species.\nThe theoretical calculation for the three-body HD$^+$ level energies is also more accurate than those of the neutral molecular hydrogen.\nHowever, the internuclear separation of HD$^+$ ($\\sim1.1$ \\AA) is greater than that of neutral hydrogen molecules ($\\sim0.76$ \\AA) as shown in Fig.~\\ref{Wavefunctions}.\nThus the neutrals are inherently more sensitive as the wave functions probe shorter internuclear distances compared to their ionic counterparts.\nThe constraints for $R_n$ derived from the HD$^+$ ($v=0,J=2\\rightarrow v=4,J=3$) ro-vibrational transition from Koelemeij et al.~\\cite{Koelemeij2007} are listed in Table~\\ref{Constraints-R}.\nIn the table, the $R_n$ constraints from D$_2$ $D_0$ are the most stringent for $n=4,5,6$ extra dimensions while the constraints from HD$^+$ are the most constraining for $n=2,3,7$. \nThe higher-dimensional Planck mass $M_{(4+n)}$ and corresponding Planck length $R_{\\mathrm{Pl},(4+n)}$ derived from the tightest $R_n$ constraints obtained in this study are also listed in Table~\\ref{Constraints-R}.\n\nSimilarly, we derive constraints pertaining to corrections in the RS scenario with one extra dimension, and the combined uncertainty $\\delta E$ for a specific molecular transition\n\\begin{equation}\n \\left<\\Delta V_\\mathrm{RS}(k) \\right> < \\delta E.\n\\end{equation}\nUsing the combined uncertainties for the $D_0$(D$_2$) study, we present constraints for the RS schemes.\nFor the RS-I scenario in the short distance ($kr \\ll 1$) regime, we obtain constraints for the brane separation of $y_c<1\\times 10^{18}$ m in the limit $ky_c\\ll 1$ using Eq.~(\\ref{RSI_potential_ky}). In the limit $ky_c\\gg 1$ in Eq.~(\\ref{RSI_potential_ky}), a constraint for the inverse of the curvature of $1\/k < 2\\times 10^{18}$ m is obtained.\nFor the RS-II model, we obtain constraints for the inverse of the curvature $1\/k < 2\\times 10^{18}$ m for $kr \\ll 1$ from Eq.~(\\ref{RS-trans-short}).\n\n\\section{Comparison with other constraints}\nThe constraints obtained from molecular systems probe length scales in the order of Angstroms.\nThis complements bounds probing subatomic to astronomical length scales obtained from other studies using distinct methodologies.\nLength scales of several hundred nanometers to microns are probed in Casimir-force studies using cantilevers~\\cite{Zuurbier2011} or atomic-force microscopy~\\cite{Banishev2013}.\nThe micrometer to millimeter range are probed in torsion-balance type experiments, with the tightest constraint obtained by Kapner \\emph{et al.}~\\cite{Kapner2007} for a single extra dimension of $R_1 < 4.4 \\times 10^{-5}$ m.\nThe centimeter to meter separations are accessed by Cavendish- or E\\\"{o}tv\\\"{o}s-type investigations in the laboratory, while astronomical scales can be probed in satellite or planetary orbits that also serve to constrain the universality of free fall and deviations from the gravitational inverse-square law~\\cite{Adelberger2003}.\nConstraints for the RS-theories are obtained by Iorio~\\cite{Iorio2012} using data from orbital motions of satellites or astronomical objects, with the tightest constraint for the inverse of the curvature of $1\/k < 5$ m obtained from the motion of the GRACE satellite.\nThe latter constraint is in the $kr \\gg 1$ regime of Eq.~(\\ref{RS_potential_leading_order}) and probes a different distance range to that of molecules ($kr \\ll 1$).\n\nPrecision spectroscopies of hydrogen~\\cite{Biraben2009,Parthey2010} and muonic atoms~\\cite{Pohl2010,Antognini2013} have been interpreted along the same lines in terms of the ADD-model~\\cite{Li2007} resulting in typical constraints of $R_3 < 10^{-5}$ m.\nThe interpretation is not straightforward because of the proton size puzzle~\\cite{Pohl2013}; in fact, the argument has been turned around, where the existence of extra dimensions are instead invoked as a possible solution to the puzzle~\\cite{Wang2013}. \nIn the treatment of atoms, some assumptions had to be made on the wave function density at $r=0$, typical for the $s$-states involved, causing problems in calculating the second integral of Eq.~(\\ref{Integrals}) over the \\emph{electronic} wave function that has a significant wave function amplitude at $r=0$ in atoms.\nNote that these difficulties are absent in molecules, as the molecular wave function probes the 0.1 - 5 Angstrom distance range.\n\nTo probe length scales in the subatomic range, one is ultimately limited by the increasing contributions from nuclear structure and the strong interaction, e.g. in neutron scattering studies~\\cite{Nesvizhevsky2008}.\nIn contrast to QED calculations, the most accurate lattice-QCD calculation of light hadron masses only achieves relative accuracies in the order of a few percent.\nNevertheless, the smaller nucleon size presents higher sensitivity to effects of ADD-type interactions, and constraints for the size of extra dimensions may be extracted.\nThe general method for molecules presented here may be applied to a comparison of \\emph{ab initio} lattice-QCD calculations with the measurements of light hadron masses.\nThe corresponding QCD test probes length scales of the size of a nucleon at $\\sim 10^{-15}$ m.\nThe \\emph{ab initio} calculations of D{\\\"u}rr \\emph{et al.}~\\cite{Durr2008} for the nucleon mass are estimated to be accurate to around 50 MeV\/$c^2$ while the experimental mass values are accurate to 20 eV\/$c^2$.\nThe calculated nucleon mass, with 3\\% relative accuracy, is the isospin average of proton $m_p$ and neutron $m_n$ masses, while $m_p$ is known experimentally to be $0.1\\%$ smaller than $m_n$.\nConstraints based on experimental nucleon masses and QCD theory have not been explored, but we produce here a rough first estimate by assuming that the three constituent quarks each have an effective mass that is $\\sfrac{1}{3}$ of the nucleon mass, and have separation distances $\\sim r_p$.\nAnalogous to Eq.~(\\ref{ADD-constrain-relation}) for QED interactions in molecules, the expectation value for an ADD contribution on the mass of the proton can be written as $V_\\mathrm{ADD}(p)\/c^2 < \\delta m_p$, yielding a bound for the case of $7$ extra dimensions of $R_7 < 2.4\\times10^{-10}$ m.\n\nIn high-energy particle collisions, higher-dimensional gravitons may be produced that could escape into the bulk, leading to events with missing energy in (3+1)-dimensional spacetime~\\cite{Hamed1998,Mirabelli1999,Giudice1999}.\nBased on this premise of an energy loss mechanism the phenomenology of the SN 1987A supernova was investigated, imposing limits on extra dimensions of $R_2 < 3\\times 10^{-6}$ m, $R_3 < 4\\times 10^{-7}$ m, and $R_4 < 2\\times 10^{-8}$ m~\\cite{Cullen1999}.\nSimilarly, from a missing energy analysis of proton colliding events at LHC, a constraint for $R_2<3.2\\times10^{-4}$ m can be extracted from the $M_{4+n} = 1.93$ TeV bound for $n=2$ given in Ref.~\\cite{Aad2013}.\nFor comparison, the Planck energy scale in $(4+n)$ dimensions in Table~\\ref{Constraints-R} turns out to be in the range between $1-1000$ MeV, but are derived from a completely independent methodology. \nAlso for $n>2$, the bounds derived from LHC are nominally more stringent than those from molecules.\nHowever, additional assumptions beyond the ADD potential in Eq.~(\\ref{ADD-short}), e.g. the fundamental quantization of gravity, the existence and propagation of gravitons in ($4+n$) dimensions and postulating the existence of massive new particles, are necessary for an effective theory~\\cite{Mirabelli1999,Giudice1999} to interpret the LHC missing energy signals. Such assumptions are not\nneeded for the molecular physics bounds, which are not sensitive to physics at very short distances.\n\n\\section{Conclusion and outlook}\n\nThe alternative approaches to constraining compactification radii for extra dimensions, partially surveyed here, are all complementary as they probe different length and energy scales.\nSome approaches serve to produce tighter limits, however, often at the expense of additional assumptions.\nIn the present study, a constraint is derived on compactification scales of extra dimensions from precision measurements on molecules, leading to straightforward interpretations.\nMolecules, in particular the lightest ones as neutral and ionic molecular hydrogen, exhibit wave functions representing the internuclear distances, with amplitudes confined to the range $0.1-5$ \\AA.\nCurrent state-of-the-art experiments on neutral molecular hydrogen determine vibrational splittings on the order of $10^{-4}$ cm$^{-1}$, or 3 MHz~\\cite{Dickenson2013}. Since the lifetimes of ro-vibrational quantum states in H$_2$ are of the order of $10^6$ s~\\cite{Black1976}, measurements of vibrational splittings on the order of $10^{14}$ Hz could in principle be possible at more than 20-digit precision, which leaves room for improvement ``at the bottom\" of over 10 orders of magnitude, if experimental techniques can be developed accordingly. Similar improvements in theory would make these molecular systems an ideal test ground for constraining or detecting higher dimensions, as well as fifth forces~\\cite{Salumbides2013}.\nAfter having performed a 15-digit accuracy calculation on Born-Oppenheimer energies~\\cite{Pachucki2010}, calculations of strongly improved accuracy have just been published~\\cite{Pachucki2014}, while improved calculations of non-adiabatic corrections are underway~\\cite{Pachucki2015}.\nImmediate improvements, based on existing technologies, on the experimental accuracies of the dissociation limits in the neutral hydrogen and its isotopomers~\\cite{Sprecher2011} and the spectroscopy of HD$^+$~\\cite{Karr2014,Tran2013,Schiller2014}, were discussed recently.\n\n\\ack\nThis work was supported by the Netherlands Foundation for Fundamental Research of Matter (FOM) through the program ``Broken Mirrors \\& Drifting Constants\".\nB.~Gato-Rivera and A.N.~Schellekens have been partially supported by funding from the Spanish Ministerio de Economia y Competitividad, Research Project FIS2012-38816, and by the Project CONSOLIDER-INGENIO 2010, Programme CPAN (CSD2007-00042).\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe H\\'enon problem, introduced in the 70's for the study of star clusters, see \\cite{H}, is\n\\begin{equation} \\label{H}\n\\left\\{\\begin{array}{ll}\n-\\Delta u = |x|^{\\alpha}|u|^{p-1} u \\qquad & \\text{ in } B, \\\\\nu= 0 & \\text{ on } \\partial B,\n\\end{array} \\right.\n\\end{equation} \nwhere $B$ stands for the unitary ball in ${\\mathbb R}^N$ with $N\\ge 2$ and the exponent ${\\alpha}$ is positive.\nHere we have written the power-type nonlinearity in its odd formulation since we are interested in both positive and sign-changing solutions.\nFor ${\\alpha}=0$ \\eqref{H} gives back the Lane-Emden problem \n\\begin{equation} \\label{LE}\n\\left\\{\\begin{array}{ll}\n-\\Delta u = |u|^{p-1} u \\qquad & \\text{ in } B, \\\\\nu= 0 & \\text{ on } \\partial B.\n\\end{array} \\right.\n\\end{equation} \nSome of the results we present here are new also for the latter, and since our techniques allow to deal with both problems simultaneously we shall include the case ${\\alpha}=0$ in the reasoning.\n\nIt is well known that, for ${\\alpha}>0$ fixed, the H\\'enon problem \\eqref{H} admits solutions, and in particular radial solutions, for every $p\\in(1,p_{{\\alpha}})$, being \n\\[ p_{{\\alpha}}=\\begin{cases} \\infty & \\text{ in dimension } N=2, \\\\ \\frac{N+2+2{\\alpha}}{N-2} & \\text{ in dimension } N\\ge 3 .\\end{cases} \\]\nThe same holds when ${\\alpha}=0$, i.e. for the Lane-Emden equation \\eqref{LE}, and in this case the threshold exponent for the existence of solutions coincides with the critical Sobolev exponent $p_0=\\frac{N+2}{N-2}$ in dimension $N\\ge 3$.\nIn that range of existence, for any given $m\\ge 1$ there is exactly one couple of radial solutions of \\eqref{H} which have exactly $m$ nodal zones, they are classical solutions and they are one the opposite of the other (see \\cite{Ni, BWi, NN}, or also \\cite{AG-sing-2}). \n\nSuch radial solutions are the only possible ones only in the framework of positive solutions and Lane-Emden equation, where the celebrated symmetry result by Gidas, Ni and Niremberg \\cite{GNN} holds. \nIt is well known that the H\\'enon problem in the ball has also nonradial positive solutions, and the literature on this subject is rich. First \\cite{SSW} showed that the minimal energy solution is nonradial when ${\\alpha}$ is large and $p$ is subcritical. After multi-peak solutions have been constructed by finite-dimensional reduction methods under various incidental assumptions, we mention \\cite{EPW, P, PS, HCZ} among others.\nNonradial solutions have also been produced by variational methods as in \\cite{S, BS, AG-N=2}, after imposing some constrains on the symmetries of the solutions, and by bifurcation methods in \\cite{AG14, FN17}.\n\n\nComing to nodal solutions, considerations based on the Morse index yield that the minimal energy solution is nonradial for every ${\\alpha}\\ge 0$. Indeed the minimal energy nodal solution has Morse index 2 by \\cite{BW}, while the Morse index of nodal radial solutions is greater, see \\cite{AP, AG-sing-2}. \nSign-changing multi-bubble solutions have been produced by finite-dimensional reduction methods, we can quote \\cite{BMP, EMP, BDP} for the Lane-Emden problem and \\cite{ZY} for the H\\'enon problem in the disk. They are very different from the radial ones since their nodal surfaces intersect the boundary of the ball.\nAnother interesting paper by Gladiali and Ianni \\cite{GI} showed the existence of solutions to the Lane-Emden equation which are nonradial but \\textquotedblleft quasi-radial\\textquotedblright, in the sense that their nodal lines are the boundary of nested domains contained in the disc. Some of these quasi-radial solutions are produced as least energy nodal solutions in symmetric spaces, some others by bifurcation w.r.t.~the parameter $p$.\nThe approach of least energy solutions in symmetric space has been extended also to the H\\'enon equation in \\cite{AG-N=2, Ama}, always in dimension $N=2$. \nConcerning the H\\'enon equation in dimension $N\\ge 3$, in the subcritical case a very recent paper by K\\\"ubler and Weth \\cite{KW} produced an infinite number of nonradial solutions by bifurcation w.r.t.~the parameter ${\\alpha}$, by a fine description of the profile of the radial solutions and of the distribution of their negative eigenvalues as ${\\alpha}\\to\\infty$. Such nonradial solutions are called by the authors \\textquotedblleft almost radial\\textquotedblright\\ because their nodal surfaces are homeomorphic to spheres. Of course, also the solutions produced by bifurcation arguments in \\cite{GI} are of the same kind.\n\n\nHere we aim to obtain nonradial bifurcation w.r.t. the parameter $p\\in (1,p_{{\\alpha}})$, for any given value of ${\\alpha}>0$ (and also ${\\alpha}=0$, as far as sign-changing solutions are concerned),\nso we must take into account also the supercritical case.\nThe Morse index of radial solutions when the parameter $p$ approaches the supremum of the existence range has been recently computed in four different papers (\\cite{DIP-N=2, DIP-N>3} concerning the Lane-Emden problem in dimension $N=2$ and $N\\ge 3$ respectively, and \\cite{AG-N=2, AG-N>3} for the H\\'enon problem), while when $p$ is close to $1$ it has been characterized in terms of the zeros of suitable Bessels function in \\cite{Ama}.\nStarting from these computations we see that for the positive solution to the H\\'enon equation the Morse index for $p$ close to $1$ is lower than at the supremum of the existence range, and the same holds for nodal solutions in dimension $N=2$, while in dimension $N\\ge 3$ the inequality is reversed. \nAlthough there are still nontrivial difficulties in deducing actual bifurcation: no variational structure\ncan be used to handle supercritical values of $p$ and only an odd change in the Morse index can produce a bifurcation\nresult. When dealing with the positive solutions, the first eigenvalue alone plays a role and this ensures that the kernel of the linearized operator contains exactly a one-dimensional subspace of the $O(N\\!-\\!1)$-invariant functions, and this observation was crucial in both \\cite{AG14} and \\cite{FN17}.\nFor nodal solutions, instead, the structure of the kernel is highly nontrivial. \nWe handle this situation by turning to the notion of degree and index of fixed points in cones introduced by Dancer in \\cite{D83}. This approach has already been applied to the Lane-Emden problem in an annulus, see \\cite{D92}, and then extended to higher dimension and to sign-changing solutions in \\cite{AG-bif}. \nIt can be applied also to the H\\'enon equation because the exact computations in \\cite{AG-N=2, AG-N>3, Ama} rely on a characterization of the Morse index in terms of a singular Sturm-Liouville problem from \\cite{AG-sing-1}, which allows to describe in full details the kernel of the linearized operator.\nFurthermore this tool provides a detailed bifurcation analysis also for positive solutions, and in the subcritical case, since it gives informations about the symmetries of the bifurcating solutions and the global properties of the branches.\n\n\\\n\nThis paper is organized as follows. In Section \\ref{sec:stat} we outline the positive cones that we will use and the main bifurcation results that we are going to prove. Section \\ref{sec:prel} deals with the Morse index: after recalling its characterization by means of the singular eigenvalues and the exact computations performed in the aforementioned papers, we check that the Morse changes across the range $p\\in (1,p_{{\\alpha}})$.\nNext in Section \\ref{sec:bif} the main results are proved, by taking advantage of the previous discussion on the Morse index and adapting that arguments to compute the index of fixed points in cones. \n\n\\section{Statement of the main results}\\label{sec:stat}\n\nWe adopt the \tspherical coordinates in ${\\mathbb R}^{N}$ given by $(r,\\theta,\\varphi)$ with $r=|x|\\in[0,+\\infty)$, $\\theta\\in[-\\pi,\\pi]$, $\\varphi =(\\varphi_1,\\dots\\varphi_{N\\!-\\!2})\\in (0,\\pi)^{N-2}$ \tso that\n\\[\\begin{array}{ll}\nx_1=r \\cos\\theta \\prod\\limits_{h=1}^{N\\!-\\!2}\\sin\\varphi_h , \\qquad &\nx_2= r \\sin\\theta \\prod\\limits_{h=1}^{N\\!-\\!2}\\sin\\varphi_h , \\\\\nx_{k}= r \\cos \\varphi_{k\\!-\\!2} \\prod\\limits_{h=k-1}^{N\\!-\\!2}\\sin\\varphi_h \\ \\mbox{ as } k=3,\\dots N-1 , \\quad &\nx_{N} = r \\cos \\varphi_{N\\!-\\!2}. \n\\end{array}\\]\nIn particular for any $x\\neq 0$, $(\\theta, \\varphi)$ are the coordinates of $x\/|x| \\in \\mathbb S_{N-1}$.\nNext for any natural number $n$ we introduce the spaces\n\\begin{align}\\label{H1n}\nH^1_{0,n} : = & \\big\\{u\\in H^1_0(B) \\, : \\, u(r,\\theta,\\varphi) \\hbox{ is even and } {2\\pi}\/n \\hbox{ periodic w.r.t. } \\theta , \\\\ \\nonumber \n& \\qquad \\qquad \\qquad \\hbox{ for every } r\\in (0,1) \\text{ and } \\varphi \\in (0,\\pi)^{N-2} \\big\\}, \\\\\n\\label{Xn}\nX_n : = & \tH^1_{0,n} \\cap C^{1,\\gamma}(B), \n\\intertext{and the positive cones already used in \\cite{AG-bif}, i.e.}\n\\label{Kn}\nK_n : = & \\big\\{u\\in X_n \\, : \\, \\hbox{is nonincreasing w.r.t.~ } \\theta\\in (0,\\pi\/n), \\\\ \\nonumber\n& \\qquad \\qquad \\hbox{ for every } r\\in (0,1) \\text{ and } \\varphi \\in (0,\\pi)^{N-2}\\big\\}.\n\\end{align}\nNotice that radial functions belong to $K_n$ for every $n$.\nOn the other side, only in dimension $N=2$ the intersection between two different cones reduces to the radial functions alone.\nInstead in dimension $N\\ge 3$ it contains also nonradial functions that do not depend on the angle $\\theta$.\n\nThroughout the paper we will take the exponent ${\\alpha}$ as fixed and write $\\mathcal{S}^m$ for the curve of radial solutions to \\eqref{H} with $m$ nodal zones, precisely\n\\begin{align}\\label{Sn}\n\\mathcal{S}^m = \\big\\{ (p, u_p) \\in (1, p_{{\\alpha}})\\times C^{1,\\gamma}(B) \\, : & \\ u_p \\text{ is the radial solution to \\eqref{H} } \\\\ \\nonumber \n& \\text{with $m$ nodal zones and } u_p(0)>0 \\big\\} . \\end{align}\nWe will show that a continuum of nonradial solutions in $K_n$ detaches from the curve $\\mathcal S^m$, for some integers $n$ depending on the exponent ${\\alpha}$ and the number of nodal zones $m$. To this aim we introduce the set\n \\begin{equation}\\label{Sigman}\n \\Sigma_n^m = {\\mathcal Cl} \\big\\{ (p, u) \\in (1,p_{{\\alpha}})\\times K_n\\setminus \\mathcal{S}^m \\, : \\, u \\mbox{ solves \\eqref{H}} \\big\\},\n\\end{equation}\nwhere the closure is meant according to the natural norm in $(1,p_{{\\alpha}})\\times C^{1,\\gamma}(B)$. Remark that the set $\\Sigma^m_n$ contains also the curves of radial functions $\\mathcal S^{m'}$ with $m'\\neq m$, but of course $\\mathcal S^m$ and $\\mathcal S^{m'}$ are separated. So we say that a couple $(p_n, u_{p_n}) \\in \\mathcal S^m \\cap \\Sigma_n^m $ is a nonradial bifurcation point, meaning that in every neighborhood of $(p_n,u_{p_n})$ in the product space $(1,p_{{\\alpha}})\\times C^{1,{\\gamma}}_0( B )$ there exists a couple $(q,v)$ such that $v$ is a nonradial solution of \\eqref{H} related to the exponent $q$. \nIn this case we set\n\\begin{equation}\\label{cn}\n\\mathcal C^m_n \\ \\mbox{ the closed connected component of $\\Sigma^m_n$ containing $(p_n,u_{p_n})$}\n\\end{equation}and we shall refer it as the \\textquotedblleft branch\\textquotedblright departing from $(p_n, u_{p_n}) $, with a little misuse of language.\nWe will also write $[t]$ and $\\lceil t\\rceil$, respectively, for the floor and the ceiling of a real number $t$, i.e.\n\\[ [t]=\\max \\left\\{ n\\in\\mathbb Z \\, : \\, n \\le t\\right\\} , \\quad \\lceil t\\rceil= \\min\\left\\{n\\in\\mathbb Z \\, : \\, n \\ge t \\right\\} .\\]\nEventually the same reasoning enables us to prove several bifurcation results. First we produce $\\lceil\\frac{{\\alpha}}{2}\\rceil$ global branches of positive nonradial solutions, precisely\n\n\\begin{theorem}[Bifurcation from positive solutions]\\label{teo:bif-H-1}\n\t\t\tIn any dimension $N\\ge 2$ and for every ${\\alpha}>0$, there are at least $\\lceil\\frac{{\\alpha}}{2}\\rceil$ different points along the curve $\\mathcal S^1$ where a nonradial bifurcation occurs.\n\t\t\tMore precisely for every $n=1, \\dots \\lceil\\frac{{\\alpha}}{2}\\rceil$ there exists a nonradial bifurcation point $(p_n,u_{p_n})\\in \\mathcal S^1 \\cap \\Sigma_n^1$ and the respective branch $\\mathcal C^1_n$ has the following global properties\n\t\t\t\\begin{enumerate}[i)]\n\t\t\t\t\\item $\\mathcal C^1_n$ is made up of positive solutions and unbounded, i.e. it contains a sequence $(p_k, u_k)$ with $\\|u_k\\|_{C^{1,\\gamma}}\\to \\infty$ or $p_k\\to p_{{\\alpha}}$.\n\t\t\t\t\\item In dimension $N=2$ the branches are separated, in the sense that their intersection contains at most isolated points along the curve of positive radial solutions $\\mathcal S^1$.\n\t\t\t\t\\item\tIn dimension $N\\ge 3$ two different branches can only have in common couples $(p, v)$, where $v$ are positive solutions to \\eqref{H} which\tdo not depend on the angle $\\theta$, and their overlapping can even make up a continuum. \t\n\t\t\t\\end{enumerate}\n\\end{theorem}\n\nIn the disc solutions enjoying the same symmetry properties\nhave been produced in \\cite{EPW}\nby the Lyapunov-Schmidt reduction method, and in \\cite{AG-N=2} by minimizing the energy associated to \\eqref{H} in the space $H^1_{0,n}$. In this last paper it has been proved that such \\textquotedblleft least energy $n$-invariant solutions\\textquotedblright are nonradial and different one from another at least for $p\\in (p_n,+\\infty)$, with $p_n$ the same exponent appearing here. On the other hand, they are certainly radial for $p$ close to one, thanks to the uniqueness result in \\cite{AG-bif}. It is therefore natural to think that the branches of bifurcating solutions shown by Theorem \\ref{teo:bif-H-1} are made up by these least energy $n$-invariant solutions, and so they do exist for every $p\\in (p_n, \\infty)$, and are separated. \n\\\\\nIn higher dimension Theorem \\ref{teo:bif-H-1} improves the bifurcation result obtained in \\cite{AG14}, which holds for ${\\alpha}\\in(0,1]$ and produces only one branch of nonradial solutions.\nNonradial solutions with similar symmetries have been produced by the finite-dimensional reduction method: in particular \\cite{PS} concerns the slightly subcritical case and exhibits solutions which blow up when $p$ approaches the critical Sobolev exponent, while \\cite{HCZ} proves the existence also in the critical case. \nBesides nonradial solutions do exist also for $p$ close to $p_{{\\alpha}}$, as showed in \\cite{FN17}. It is very likely that some of the nonradial solutions found in Theorem \\ref{teo:bif-H-1} coincide with the ones in \\cite{FN17}, where the specular viewpoint (bifurcation w.r.t. ${\\alpha}$) is adopted.\n\n\\\n\nComing to nodal solutions, the asymptotic Morse index and consequently the number of nonradial branches depend on the dimension. We therefore state the bifurcation results separately.\n\\\\\nIn the plane the set $\\Sigma^2_n$ is nonempty at least for $n=\\left[\\frac{2+{\\alpha}}{2}\\beta+1\\right], \\dots \\left\\lceil\\frac{2+{\\alpha}}{2}\\kappa-1\\right\\rceil$, where $\\beta\\approx 2,\\!305$ and $\\kappa \\approx 5,\\!1869 $ are fixed numbers related to the computation of the Morse index at $p$ next to 1 and at infinity, respectively, whose characterization is recalled in Section \\ref{sec:prel}. Precisely we have\n\n\\begin{theorem}[Bifurcation from nodal solutions in dimension $N=2$]\\label{teo:bif-H-m-N=2}\n\t\tConsider problem \\eqref{H} in dimension $N=2$. \n\t\t For every ${\\alpha}\\ge 0$ there are at least $\\left\\lceil\\frac{2+{\\alpha}}{2}\\kappa-1\\right\\rceil -\\left[\\frac{2+{\\alpha}}{2}\\beta\\right] $ different points along the curve $\\mathcal S^2$ where nonradial bifurcation occurs.\n\t\t More precisely for every $n=\\left[\\frac{2+{\\alpha}}{2}\\beta+1\\right], \\dots \\left\\lceil\\frac{2+{\\alpha}}{2}\\kappa-1\\right\\rceil$ there exists a nonradial bifurcation point $(p_n,u_{p_n})\\in \\mathcal S^2 \\cap \\Sigma_n^2$ and the respective branches $\\mathcal C^2_n$ have the following properties\n\t\\begin{enumerate}[i)]\t\n\t\t\\item There is a ball $\\mathcal B$ in $(1,\\infty)\\times C^{1,\\gamma}(B)$ centered at $(p_n,u_{p_n})$ such that $\\mathcal C^2_n\\cap \\mathcal B\\setminus\\{(p_n, u_{p_n})\\}$ is made up of nonradial solutions with $2$ nodal zones, one of which contains $x=0$ and is homeomorphic to a disc.\n\t\t\\item Every branch\tcontains a sequence $(p_k, u_k)$ with either $\\|u_k\\|_{C^{1,\\gamma}}\\to \\infty$, or $p_k\\to \\infty$, or possibly $p_k\\to 1$ and $u_k$ converges to an eigenfunction of \n\\begin{equation}\\label{prima-autof-weight} \n\\left\\{\n\\begin{array}{ll}\n-\\Delta \\omega= \\mu |x|^{{\\alpha}} \\omega & \\text{ in } B, \\\\\n\\omega = 0 & \\text{ on } \\partial B , \n\\end{array}\\right.\n\\end{equation} \twhich belongs to $K_n$.\t\n\\item Two different branches can only have radial solutions in common. \nPrecisely $\\mathcal C^2_n \\cap \\mathcal C^2_{n'} \\cap \\mathcal S^2$ contains at most isolated points, and if there is some $m\\ge 3$ such that $\\mathcal C^2_n \\cap \\mathcal C^2_{n'} \\cap \\mathcal S^m$ is nonempty, then $\\mathcal S^m \\subset \\mathcal C^2_n \\cap \\mathcal C^2_{n'} $.\n\\end{enumerate}\t\n\\end{theorem}\n\t\nThe possibility that $p_k\\to 1$ but $u_k$ stays bounded remains open because the uniqueness of nodal solutions does not hold either in a neighborhood of $p=1$, see \\cite[Theorem 1.3]{Ama}. \nConcerning property {\\it iii)}, i.e. the possible overlapping of two different branches, we are not aware of any technique which enables to capture the formation of further nodal zones and\/or a secondary bifurcation. Consequently a nonradial branch could, in principle, touch another radial curve $\\mathcal S^m$ with $m\\ge 3$, and then incorporate it because of the way in which $\\Sigma^2_n$ and $\\mathcal C^2_n$ have been defined.\n\nTheorem \\ref{teo:bif-H-m-N=2} applies also to ${\\alpha}=0$, i.e.~to the Lane-Emden equation, giving back \\cite[Theorem 1.2]{GI} since in this particular case $\\left[\\frac{2+{\\alpha}}{2}\\beta+1\\right]= 3$ and $\\left\\lceil\\frac{2+{\\alpha}}{2}\\kappa-1\\right\\rceil=5$.\n\\\\\nFor ${\\alpha}>0$ it is worth comparing this existence result with the ones in \\cite{Ama} and in \\cite{AG-N=2}, both concerning the least energy $n$-invariant nodal solutions, that we denote hereafter by $U_{p,n}$.\nFor $n=1,\\dots \\left\\lceil\\frac{2+{\\alpha}}{2}\\beta-1\\right\\rceil$, $U_{p,n}$ is nonradial for both $p$ close to 1 and large.\nIt seems that in this case $U_{p,n}$ is nonradial for every $p>1$ and the curve $p\\mapsto U_{p,n}$ does not intersect the curve of radial solutions. This is certainly true for $n=1$, i.e. the least energy nodal solution.\nConversely for $n=\\left[\\frac{2+{\\alpha}}{2}\\beta+1\\right], \\dots \\left\\lceil \\frac{2+{\\alpha}}{2}\\kappa-1\\right\\rceil$, \\cite[Proposition 4.10]{Ama} and \\cite[Theorem 1.6]{AG-N=2} yield that $U_{p,n}$ are radial for $p$ close to 1, and then nonradial (and different one from another) when $p$ is large. Therefore\nthe curves $p\\mapsto U_{p,n}$ coincide with the one of radial solutions for $p\\in (1,p_n)$, and then they give rise to the nonradial bifurcation stated by Theorem \\ref{teo:bif-H-m-N=2}. \n\nOnly bifurcation from the curve $\\mathcal S^2$ is taken into account, since the behaviour of nodal solutions as $p\\to \\infty$ is known only in the case of two nodal zones. When this paper was already finished we came to know that a very recent preprint by Ianni and Saldana \\cite{IS} describes the asymptotic profile of every radial solutions. Starting from this it is possible, in principle, to compute exactly their Morse index and then the same arguments used here produce bifurcation also in the general case.\n\n\\\n\nIn dimension $N\\ge 3$ the set $\\Sigma^m_n$ is nonempty at least for $n=2+ \\big[\\frac{{\\alpha}}{2}\\big]$, $\\dots$ $n_{{\\alpha}}^m$,\nwhere the number $n_{{\\alpha}}^m\\ge 2(m-1)+ [{\\alpha}(m-1)]$ is characterized later on in Remark \\ref{n-def} and can be numerically computed.\n\n\\begin{theorem}[Bifurcation from nodal solutions in dimension $N\\ge 3$]\\label{teo:bif-H-m}\n\tConsider problem \\eqref{H} in dimension $N\\ge3$. For every ${\\alpha}\\ge 0$ and $m\\ge 2$, at least $2m-3+ [{\\alpha}(m-1)]- [{\\alpha}\/2]$ different nonradial bifurcations take place along the curve $\\mathcal S^m$.\n\tMore precisely for every $n=2 + \\big[\\frac{{\\alpha}}{2}\\big] , \\dots n_{{\\alpha}}^m $ there exists a nonradial bifurcation point $(p_n,u_{p_n})\\in \\mathcal S^m \\cap \\Sigma_n^m$ and the respective branches $\\mathcal C^m_n$ have the following properties\n\\begin{enumerate}[i)]\t\n\t\\item There is a ball $\\mathcal B$ in $(1,p_{{\\alpha}})\\times C^{1,\\gamma}(B)$ centered at $(p_n,u_{p_n})$ such that $\\mathcal C^m_n\\cap \\mathcal B\\setminus\\{(p_n, u_{p_n})\\}$ is made up of nonradial solutions with $m$ nodal zones, one of which contains $x=0$ and is homeomorphic to a ball, while the other ones are homeomorphic to spherical shells.\n\t\\item Every branch contains a sequence $(p_k, u_k)$ with either $\\|u_k\\|_{C^{1,\\gamma}}\\to \\infty$, or $p_k\\to p_{{\\alpha}}$, or possibly $p_k\\to 1$ and $u_k$ converges to an eigenfunction of \\eqref{prima-autof-weight} \twhich belongs to $K_n$.\n\t\\item The intersection between two different branches, if non-empty, is made up of nodal solutions which do not depend by the angle $\\theta$.\n\\end{enumerate}\t\n\\end{theorem}\n\n\t\nThe branches of nodal bifurcating solution in dimension $N\\ge 3$ can overlap along radial solutions with a different number of nodal zones, but also along nonradial solutions that do not depend by the angle $\\theta$.\n\\\\\nThe statement of Theorem \\ref{teo:bif-H-m} is new also in the simpler case ${\\alpha}=0$, to the author's knowledge.\nFor the reader's convenience, we state separately the bifurcation result concerning the Lane-Emden equation. \n\n \t\\begin{theorem}[Bifurcation for the Lane Emden equation in dimension $N\\ge 3$]\\label{teo:bif-LE}\n\t\tConsider problem \\eqref{LE} in dimension $N\\ge3$. For every $m\\ge 2$ the curve $\\mathcal S^m$ bifurcates at $2m-3$ points, at least.\n\t\tMore precisely for every $n=2 , \\dots n^m_0 $ there exists a nonradial bifurcation point $(p_n,u_{p_n})\\in \\mathcal S^m \\cap \\Sigma_n^m$ \n\t\tand the continuum detaching at $(p_n, u_{p_n}) $, i.e. $\\mathcal C^m_n$ has the following\n\t\\begin{itemize}\t\n\t\t\\item Local property: there is a ball $\\mathcal B$ in $(1,p_{0})\\times C^{1,\\gamma}(B)$ centered at $(p_n,u_{p_n})$ such that $\\mathcal C^m_n\\cap \\mathcal B\\setminus\\{(p_n, u_{p_n})\\}$ is made up of nonradial solutions with $m$ nodal zones, one of which contains $x=0$ and is homeomorphic to a ball, while the other ones are homeomorphic to spherical shells,\n\t\\item Global property: every branch contains a sequence $(p_k, u_k)$ with either $\\|u_k\\|_{C^{1,\\gamma}}\\to \\infty$, or $p_k\\to p_{0}$, or possibly $p_k\\to 1$ and $u_k$ converges to an eigenfunction of \n\t\t\\begin{equation}\\label{prima-autof} \n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t-\\Delta \\omega= \\mu \\, \\omega & \\text{ in } B, \\\\\n\t\t\\omega = 0 & \\text{ on } \\partial B ,\n\t\t\\end{array}\\right.\n\t\t\\end{equation}\t\n\t\twhich belongs to $K_n$.\n\t\\item Separation property: the intersection between two different branches, if non-empty, is made up of nodal solutions which do not depend by the angle $\\theta$.\n\t\\end{itemize}\t\n\t\\end{theorem}\nThere is numerical evidence that $ n^m_0= 2(m-1)$ in any dimension $N\\ge 3$, so that Theorem \\ref{teo:bif-LE} provides exactly $2m-3$ branches of nonradial solutions. In particular, in the case of $2$ nodal zones, there should be only one branch in dimension $N\\ge 3$, while 3 different branches have been produced in dimension $N=2$. The planar case indeed differs from the other ones, as already observed in several occasions.\n\n\\\n\nLet us mention in passing that the number of nonradial branches produced in Theorems \\ref{teo:bif-H-1}, \\ref{teo:bif-H-m-N=2} and \\ref{teo:bif-H-m} goes to infinity when $\\alpha\\to \\infty$, which is consistent with the specular study (bifurcation w.r.t. ${\\alpha}$) performed in \\cite{KW}.\n\n\n\\section{Preliminaries on the computation of the Morse index}\\label{sec:prel}\n\nTo emphasize the dependence on the exponent $p\\in (1, p_{{\\alpha}})$, we take the exponent ${\\alpha}\\ge 0$ and the number of nodal zones $m$ as fixed and denote by $u_p$ the unique radial solution to \\eqref{H} with $m$ nodal zones which is positive at the origin.\nWe also write \n\\begin{align}\n\\label{linearized}\nL_{p} \\psi &=-\\Delta \\psi-p |x|^{\\alpha} |u_p|^{p-1}\\psi, \\\\\n\\label{forma-quadratica}\n{\\mathcal Q}_p(\\psi)& =\\int_\\Omega \\left(|\\nabla \\psi|^2 -p|x|^{\\alpha} |u_p|^{p-1}\\psi^2\\right) dx\n\\end{align}\nfor the linearized operator at $u_p$ and the related quadratic form, respectively. They will be considered on the space $H^1_0(B)$, or in one of its subspaces specified case-by-case.\n\\\\\nThe Morse index, that we denote hereafter by $m(u_p)$, is the maximal dimension of a subspace of $H^1_0(B)$ in which the quadratic form ${\\mathcal Q}_p$ \n\tis negative defined, or equivalently the number of the negative eigenvalues of \n\t\\begin{equation}\\label{standard-eig-prob}\n\tL_p \\psi = \\Lambda \\psi , \\qquad \\psi \\in H^1_0(B).\n\t\\end{equation}\nFor radial solutions one can also look at the radial Morse index, denoted by $m_{\\text{\\upshape rad}}(u_p)$, i.e. the number of the negative eigenvalues of for \\eqref{standard-eig-prob} whose relative eigenfunction is $H^1_{0,{\\text{\\upshape rad}}}(B)$, the subspace of $H^1_0(B)$ given by radial functions. \n\\\\\nAs explained in full details in \\cite{AG-sing-1}, this matter can be regarded through a {\\it singular} eigenvalue problem associated to the linearized operator $L_p$, which has to be handled in weighted Lebesgue and Sobolev spaces\n\\begin{align*}\n{\\mathcal L} & = \\{\\omega: B\\to {\\mathbb R}\\, : \\, \\omega\/|x| \\in L^2(B)\\}, \\qquad \\mathcal{H}_0 =H_0^1(B)\\cap \\mathcal L .\n\\end{align*}\nThe Morse index (on $H^1_0(B)$ as well as on some of its subspaces) turns out to be equal to the number of the negative eigenvalues of \n\\begin{equation}\\label{singular-eig-prob}\nL_p \\psi = \\widehat\\Lambda \\psi \/ {|x|^2} , \\qquad \\psi \\in {\\mathcal H}_0(B).\n\\end{equation}\nConcerning radial solutions to the H\\'enon problem, it turns helpful the transformation \n\t\\begin{equation}\\label{transformation-henon}\nt=r^{\\frac{2+{\\alpha}}{2}} , \\qquad w(t)=u(r) ,\n\\end{equation} \nintroduced in \\cite{GGN}, or a slight variation of it\n\t\\begin{equation}\\label{transformation-henon-no-c}\nt=r^{\\frac{2+{\\alpha}}{2}} , \\qquad v(t)= \\left(\\frac{2}{2+{\\alpha}}\\right)^{\\frac{2}{p-1}} u(r),\n\\end{equation} \n which map radial solutions to \\eqref{H} into solutions of one-dimensional problems\n\\begin{equation}\\label{LE-radial}\n\\begin{cases}\n- \\left(t^{M-1} w^{\\prime}\\right)^{\\prime}= \\left(\\frac{2}{2+{\\alpha}}\\right)^2 t^{M-1} |w|^{p-1} w , \\qquad & 0k} & -\\frac{2N-2+{\\alpha}}{2+\\alpha} <{\\nu}_m(p) <0 . &\n\\end{align}\nMoreover the Morse index of $u_p$ is given by\n\t\\begin{align}\\label{tag-2-H}\n\tm(u_p) & =\\sum\\limits_{i=1}^{m} \\sum\\limits_{j=0}^{\\lceil J_i -1\\rceil} N_j ,\n\t\\end{align}\n\t\\begin{tabular}{ll} \n\twhere & $\\lceil s \\rceil = \\{\\min n\\in \\mathbb Z \\, : \\, n\\ge s\\}$ denotes the ceiling function and \\\\\n& $J_i(p)=\\frac{2+{\\alpha}}{2} \\left(\\sqrt{\\left(\\frac{N-2}{2+{\\alpha}}\\right)^2- \\nu_i(p)}-\\frac{N-2}{2+{\\alpha}}\\right)$, \n\t\\end{tabular}\n\n\\noindent being \t$\tN_j=\\frac{(N+2j-2)(N+j-3)!}{(N-2)!j!}$ the multiplicity of the eigenvalue \n\t${\\lambda}_j=j(N+j-2)$ for the Laplace-Beltrami operator in the sphere ${\\mathbb S}_{N\\!-\\!1}$.\n\\end{proposition}\n\nAfterward, the Morse index has been exactly computed at the ends of the existence range by computing the limits of the eigenvalues $\\nu_i(p)$.\nThe paper \\cite{Ama} dealt with $p$ close to 1, and we need to introduce some more notation to recall the obtained result. For every $\\beta\\ge 0$ we write $\\mathcal J_{\\beta}$ for the Bessel function of first kind \n\\[{\\mathcal J}_{\\beta}(r) = r^{\\beta}\\sum\\limits_{k=0}^{+\\infty} \\dfrac{(-1)^k}{k!\\Gamma(k+1+\\beta)} \\left(\\frac{r}{2}\\right)^{2k}, \\quad r\\ge 0 , \\]\nand $z_i(\\beta)$ for the sequence of its positive zeros.\nSince the map $\\beta\\mapsto z_i(\\beta)$ is continuous and increasing, for every fixed integer $m$ there exist $\\beta_i=\\beta_i(\\alpha, N,m)$ such that \n\\begin{equation}\\label{betai-def} \\begin{split}\n z_i(\\beta_i) \\text{ (the $i^{th}$ zero of the Bessel function ${\\mathcal J}_{\\beta_i}$)} \\\\\n\\text{ coincides with $z_m\\Big(\\frac{N-2}{2+\\alpha}\\Big)$ (the $m^{th}$ zero of ${\\mathcal J}_{\\frac{N-2}{2+\\alpha}}$)}.\n \\end{split}\\end{equation}\nIt is clear that \n\\[ \\beta_1>\\beta_2 \\dots >\\beta_m=\\frac{N-2}{2+\\alpha} .\\]\n\nIn \\cite[Proposition 3.3 and Theorem 1.2]{Ama} it is proved that \n\\begin{theorem}\\label{teo:mi-p=1} \t\n\t Let ${\\alpha}\\geq 0$ and $u_p$ be a radial solution to \\eqref{H} with $m$ nodal zones.\n\t Then \n\t \\begin{align}\\label{nup=1}\n\t\\lim\\limits_{p\\to 1} \\nu_i(p) = \\left(\\frac{N-2}{2+\\alpha}\\right)^2 -\\beta_i^2 \\quad \\text{ as } i=1,\\dots m.\n\t \\end{align}\n\tAfter there exists $\\bar p =\\bar p({\\alpha})>1$ such that for $p\\in(1,\\bar p)$ the Morse index of $u_p$ is given by\n\t\\begin{equation}\\label{mi-p=1}\n\tm(u_p)= 1+\\sum_{i=1}^{m-1}\\sum _{j=0}^{\\left\\lceil \\frac{(2+{\\alpha})\\beta_i -N}{2} \\right\\rceil} N_j \n\t\\end{equation}\n\tif $\\alpha \\neq {\\alpha}_{\\ell,n} = (2n+N)\/\\beta_{\\ell} -2$, and it is estimated by \n\t\\begin{equation}\\label{mi-p=1brutta} \\begin{split}\t1+\\sum_{i=1}^{m-1}\\sum _{j=0}^{\\left\\lceil \\frac{(2+{\\alpha})\\beta_i -N}{2} \\right\\rceil} N_j \\le m(u_p)\t\\le 1 +\\sum_{i=1}^{m-1}\\sum _{j=0}^{\\left\\lceil \\frac{(2+{\\alpha})\\beta_i -N}{2} \\right\\rceil} N_j + \\sum\\limits_{\\ell} N_{1+\\frac{(2+{\\alpha})\\beta_{\\ell} -N}{2}} .\t\\end{split}\t\\end{equation}\n\tif $\\alpha = {\\alpha}_{\\ell,n}$ for some $\\ell$ and $n$.\n\\end{theorem}\n\n\nThe situation at the supremum of the existence range changes drastically depending if the dimension is $N=2$ or greater.\nThe Morse index in dimension $N\\ge 3$ is computed in \\cite{AG-N>3} extending some previous results on the Lane-Emden problem in \\cite{DIP-N>3}; precisely \\cite[Propositions 3.3, 3.10 and Theorem 1]{AG-N>3} state that\n\\begin{theorem}\\label{teo:mi-p=palpha} \t\n\tLet ${\\alpha}\\geq 0$ and $u_p$ be a radial solution to \\eqref{H} with $m$ nodal zones in dimension $N\\ge 3$.\n\tThen \n\t\\begin{align}\\label{nu-p=palpha}\n\t\\lim\\limits_{p\\to p_{{\\alpha}}} \\nu_i(p) = - \\frac{2N-2+{\\alpha}}{2+\\alpha} \\quad \\text{ as } i=1,\\dots m.\n\t\\end{align}\n\tAfter there exists $p^{\\star}=p^{\\star}({\\alpha})\\in (1,p_{{\\alpha}})$ such that the Morse index of $u_p$ is given by\n\t\\begin{equation}\\label{mi-p=palpha}\n\tm(u_p)= \\sum\\limits_{j=1}^{\\left\\lceil \\frac{{\\alpha}}{2}\\right\\rceil} N_j+ (m-1)\\sum _{j=0}^{\\left[\\frac{2+{\\alpha}}{2}\\right]} N_j \n\t\\end{equation}\nfor $p\\in(p^{\\star}, p_{{\\alpha}})$.\n\\end{theorem}\n\nIn dimension $N=2$ only the Morse index of the least energy radial solution (i.e. the positive one) and of the least energy nodal radial solution (i.e. the one with two nodal zones) are known. They have both been computed in the paper \\cite{AG-N=2}, where it is shown that\n \n \\begin{theorem}\\label{teo:mi-p=infty-1} \t\n\tLet ${\\alpha}\\geq 0$ and $u_p$ be a positive radial solution to \\eqref{H} in dimension $N=2$.\n\tThen \n \t\\begin{align}\\label{nu-p=infty-1}\n \t\\lim\\limits_{p\\to \\infty} \\nu_1(p) = - 1 ,\n \t\\end{align}\n \tand there exists $p^{\\star} > 1$ such that for $p > p^{\\star}$ the Morse index of $u_p$ is given by\n \t\\begin{equation}\\label{mi-p=infty-1}\n \tm(u_p)= 1 + 2 \\left\\lceil \\frac{{\\alpha}}{2}\\right\\rceil .\n \t\\end{equation}\n \\end{theorem}\n\n \\begin{theorem}\\label{teo:mi-p=infty-2} \t\n\tLet ${\\alpha}\\geq 0$ and $u_p$ be a radial solution to \\eqref{H} with 2 nodal zones in dimension $N=2$, then \n \t\\begin{align}\\label{nu-p=infty-2}\n \t\t\\lim\\limits_{p\\to \\infty} \\nu_1(p) = - \\kappa^2 \\ \\text{ with } \\kappa\\approx 5,\\!1869 \\qquad \n \t\t\\lim\\limits_{p\\to \\infty} \\nu_2(p) = - 1 .\n \t\\end{align}\n \tMoreover there exists $p^{\\star} > 1$ such that for $p > p^{\\star}$ the Morse index of $u_p$ is given by\n \t\\begin{align}\\label{mi-p=infty-2}\n \t& m(u_p) =2\\left\\lceil\\frac {2+{\\alpha}}2\\kappa\\right\\rceil + 2\\left\\lceil\\frac{\\alpha}{2}\\right\\rceil \n \t\\intertext{when ${\\alpha}\\neq {\\alpha}'_n= 2(n\/{\\kappa}-1)$, while when ${\\alpha}={\\alpha}'_n$ it holds} \\label{mi-p=infty-2-brutta}\n \t(2+{\\alpha}) \\kappa + 2\\left\\lceil\\frac{\\alpha}{2}\\right\\rceil & \\le m(u_p)\\le (2+{\\alpha}) \\kappa+ 2\\left\\lceil\\frac{\\alpha}{2}\\right\\rceil+ 2 .\n \t\\end{align}\n \\end{theorem}\n \nAn analogous result for the radial solution to the Lane-Emden problem with two nodal zones has been obtained in \\cite{DIP-N=2}.\n\n\\\n\nComparing \\eqref{mi-p=1} with \\eqref{mi-p=palpha} or \\eqref{mi-p=infty-1} one sees that the positive radial solution has Morse index $1$ when $p$ is close to $1$, and greater than $1$ when $p$ is at the opposite end of the existence range (for ${\\alpha}>0$).\n\\\\\nIt is not hard to see that in dimension $N=2$ the solution with 2 nodal zones shares the same behaviour, for every ${\\alpha}\\ge 0$.\nIndeed in this case formulas \\eqref{mi-p=1} and \\eqref{mi-p=1brutta}, respectively, simplify into\n\\\nm(u_p)= 2 \\left\\lceil \\frac{2+{\\alpha}}{2}\\beta \\right\\rceil \n\\\nif $\\alpha \\neq {\\alpha}_{n} = 2(n+1)\/\\beta -2$, or\n\\\n(2+{\\alpha})\\beta \\le m(u_p)\t\\le (2+{\\alpha})\\beta + 2\n\\\nif $\\alpha = {\\alpha}_{n}$ for some for some integer $n$.\nSo remembering that $\\lceil t \\rceil < t+1$ we have\n\\[ m(u_p)\t\\le (2+{\\alpha})\\beta + 2 \\quad \\text{ for } p\\in (1, \\bar p) \\]\nfor every ${\\alpha}\\ge 0$. Furthermore the parameter $\\beta$ turns out to be \n\\begin{equation}\\label{beta} \n \\beta\\approx 2,\\!305\n \\end{equation}\nas noticed in \\cite{Ama}.\nTherefore taking $q>p^{\\star}$ we deduce from \\eqref{mi-p=infty-2}, \\eqref{mi-p=infty-2-brutta} that \n\\begin{align*} \nm(u_q) -m(u_p) & \\ge 2 \\left\\lceil\\frac{2+{\\alpha}}{2}\\kappa\\right\\rceil + 2\\left\\lceil\\frac{{\\alpha}}{2} \\right\\rceil - (2+{\\alpha})\\beta-2\n\\intertext{and since clearly $ \\lceil t \\rceil \\ge t $ we have}\n& \\ge (2+{\\alpha})(\\kappa -\\beta) +{\\alpha} -2 \\ge 2(\\kappa-\\beta -1) > 2.\n\\end{align*}\n\n\nIn higher dimensions, the approximation of the parameters $\\beta_i$ appearing in \\eqref{mi-p=1} can be numerically performed after having chosen a specific value for $\\alpha$, which fixes the baseline Bessel function ${\\mathcal J}_{\\frac{N-2}{2+{\\alpha}}}$. To have an overall picture it can be useful to establish some estimate.\nWe report here the elementary proof of an estimate of the Bessel zeros that contributes to this aim. \n\n\\begin{lemma}\\label{betap=1-}\n\tFor all $\\beta>0$ and $i, m$ integers with $i< m$ we have \n\t\\begin{align}\\label{est-bessel} z_{i}(\\beta+2(m-i)) \\frac{N-2}{2+{\\alpha}} + 2 (m-i) \\ \\text{ as } i=1,\\dots m-1 \n\\end{equation}\nand there is $\\bar p =\\bar p({\\alpha})>1$ such that the Morse index of $u_p$ is estimated from below by\n\\begin{align} \\label{morsep=1-}\nm(u_p)& \\ge 1 + \\sum\\limits_{i=1}^{m-1}\\sum\\limits_{j=0}^{[(2+{\\alpha})(m-i) ]} N_j \n\\\\ \\label{morsep=1-bis}\n& = m+ \\sum\\limits_{k=1}^{m-1}(m-k) \\sum\\limits_{j=1+[(2+{\\alpha}) (k-1)]}^{[(2+{\\alpha}) k]} N_{j} \n\\\\ \\label{morsep=1-ter}\n& \\ge m+ \\sum\\limits_{k=1}^{m-1}(m-k) \\sum\\limits_{j=1+(2+[{\\alpha}])(k-1)}^{(2+[{\\alpha}])k} N_{j} \n\\end{align}\nfor $p\\in(1, \\bar p)$.\n\\end{proposition}\n\\begin{proof}\n \\eqref{unico-controllo} is an immediate consequence of Lemma \\ref{betap=1-} since $\\beta_m= \\frac{N-2}{2+{\\alpha}}$ and the map $\\beta \\mapsto z_i(\\beta)$ is increasing.\nIn particular the index $J_i(p)$ appearing in \\eqref{tag-2-H} satisfy $J_i(p) > (2+{\\alpha})(m-i)$ in a right neighborhood of $p=1$, and \n\t \\eqref{morsep=1-} follows.\n\t \\\\\n\t\tNext, \n\t\t\t\\[ \\begin{split}\n\t\t1 + \\sum\\limits_{i=1}^{m-1}\\sum\\limits_{j=0}^{[(2+{\\alpha})(m-i) ]} N_j = m + \\sum\\limits_{i=1}^{m-1}\\sum\\limits_{j=1}^{[(2+{\\alpha})(m-i) ]} N_j\n\t\t= m + \\sum\\limits_{i=1}^{m-1}\\sum\\limits_{k=1}^{m-i}\\sum\\limits_{j=1+[(2+{\\alpha})(k-1)]}^{[(2+{\\alpha})k ]} N_j\\\\\n\t\t= m + \\sum\\limits_{k=1}^{m-1}\\sum\\limits_{i=1}^{m-k}\\sum\\limits_{j=1+[(2+{\\alpha})(k-1)]}^{[(2+{\\alpha})k ]} N_j\n\t\t= m + \\sum\\limits_{k=1}^{m-1}(m-k)\\sum\\limits_{j=1+[(2+{\\alpha})(k-1)]}^{[(2+{\\alpha})k ]} N_j ,\n\t\t\\end{split}\\]\n\t\t\t which is \\eqref{morsep=1-bis}.\n\t\t\t \n\t\t\\\\\n\t\tMoreover, as clearly $[(2+{\\alpha}) k]\\ge \\left(2+[{\\alpha}]\\right) k$, we have\n\t\t\\begin{align*}\n\t\t(m-k) \\sum\\limits_{j=1+[(2+{\\alpha})(k\\!-\\!1)]}^{[(2+{\\alpha}) k]} N_j = (m-k)\\sum\\limits_{j=1+[(2+{\\alpha})(k\\!-\\!1)]}^{(2+[{\\alpha}])k} N_j + (m-k) \\sum\\limits_{j=1+(2+[{\\alpha}])k}^{[(2+{\\alpha})k] } N_j \\\\\n\t\t\\underset{k'=k+1}{=} (m-k)\\sum\\limits_{j=1+[(2+{\\alpha})(k\\!-\\!1)]}^{(2+[{\\alpha}])k} N_j + (m-k'+1) \\sum\\limits_{j=1+(2+[{\\alpha}])(k'-1)}^{[(2+{\\alpha})(k'-1)] } N_j \\\\\n\t\t\\ge (m-k)\\sum\\limits_{j=1+[(2+{\\alpha})(k\\!-\\!1)]}^{(2+[{\\alpha}])k} N_j + (m-k') \\sum\\limits_{j=1+(2+[{\\alpha}])(k'-1)}^{[(2+{\\alpha})(k'-1)] } N_j \n\t\t\\end{align*}\n\t\tHence\n\t\t\t\\[ \\begin{split} m(u_p)\\ge m + \\sum\\limits_{k=1}^{m-1}(m-k)\\sum\\limits_{j=1+[(2+{\\alpha})(k-1)]}^{[(2+{\\alpha})k ]} N_j \n\t\t\\\\\t\\ge m + \\sum\\limits_{k=1}^{m-1}(m-k)\\sum\\limits_{j=1+[(2+{\\alpha})(k-1)]}^{(2+[{\\alpha}])k } N_j \n\t\t\t+ \\sum\\limits_{k=2}^{m-1}(m-k)\\sum\\limits_{j=1+(2+[{\\alpha}])(k-1)}^{[(2+{\\alpha})(k-1)] } N_j \\\\\n\t\t\t= m + (m-1) \\sum\\limits_{j=1}^{2+[{\\alpha}] } N_j + \\sum\\limits_{k=2}^{m-2}(m-k)\\sum\\limits_{j=1+(2+[{\\alpha}])(k-1)}^{(2+[{\\alpha}])k } N_j ,\n\t\t\\end{split}\\]\n\t\twhich is \\eqref{morsep=1-ter}.\n\n\t\\end{proof}\n\n\\remove{\\begin{remark}[Lane-Emden equation]\\label{morseLEp=1}\nThere is numerical evidence that, beneath \\eqref{est-bessel}, it also holds\n\\begin{align}\\label{est-bessel+} \nz_{i}(\\beta+2(m-i)+1)>z_{m}(\\beta) .\n\\end{align}\nIn that case \\eqref{unico-controllo} is improved to\n\\begin{equation}\\label{unico-controllo+}\n2 (m-i)+ 1 >\\beta^m_i - \\frac{N-2}{2+{\\alpha}} > 2 (m-i) \\ \\text{ as } i=1,\\dots m-1 .\n\\end{equation}\nConsequently \\eqref{Jp=1} gives\n\\begin{equation}\\label{unico-controllo+J}\n(2+{\\alpha}) (m-i)+ \\frac{2+{\\alpha}}{2} > \\lim\\limits_{p\\to 1}J^m_i(p) > (2+{\\alpha}) (m-i) \\ \\text{ as } i=1,\\dots m-1 .\n\\end{equation}\nIf ${\\alpha}=0$ it follows that $\\lim\\limits_{p\\to 1} \\lceil J^m_i(p) -1\\rceil = 2(m-i)$,\nso that \\eqref{morsep=1} simplifies into\n\\begin{align} \\label{morsep=1le}\nm(u^m_p) = m+\\sum\\limits_{i=1}^{m-1}(m-i)(N_{2i-1}+N_{2i})\n\\end{align}\nand the estimate in \\eqref{morsep=1-} is attained.\nMoreover in dimension $2$ \\eqref{morsep=1le} generalizes the computation in \\cite{GI} (concerning $m=2$) to\n\\[ \tm(u^m_p) = m(2m-1) .\t\\]\n\\end{remark}\n\n\tIn the general case $\\alpha>0$ the situation is more tangled and an explicit computation of the Morse index can not be deduced in this way, because the estimate \\eqref{unico-controllo+J} does not single $\\lim\\limits_{p\\to 1}\\lceil J^m_i(p) -1\\rceil$ out.\nSeemingly the inequality in \\eqref{morsep=1-} is strict and the presence of the nonautonomous coefficient $|x|^{\\alpha}$ causes a more relevant increasing of the Morse index. This is confirmed by the planar case, where we have noticed that $m(u^2_p) = 8$ near at $p=1$ for ${\\alpha}$ in a left neighborhood of $1$, while estimate \\eqref{morsep=1-} only gives $m(u^2_p) \\ge 6$.\n\nAlthough not optimal, the estimate \\eqref{morsep=1-} is sufficient to show that, for every fixed ${\\alpha} \\ge 0$ and $m\\ge 2$, the Morse index changes in the range of existence of solutions, and this will be crucial in Section \\ref{sec:bif} to prove bifurcation.}\n\nWe therefore see that, in dimension $N\\ge 3$, the Morse index of nodal radial solutions for $p$ close to $p_{{\\alpha}}$ is smaller than the one for $p$ is close to $1$.\n\n\t\\begin{corollary}\\label{cambio-morse-N>3}\n\tIn dimension $N\\ge 3$, for every value of ${\\alpha}\\ge 0$ and $m\\ge 2$ there exist $1<\\bar p< \\bar q < p_{{\\alpha}}$ such that\n\twe have\n\t\\[ m(u_p) > m(u_q) \\qquad \\text{ as } 10 .\n\t\\end{split}\\end{equation}\n\t\\eqref{suff} can be proved by induction on the number of nodal zones $m\\ge 2$, taking advantage from the fact that in dimension $N\\ge 3$ the multeplicity $N_j$ increases with $j$, i.e.\n\t\\begin{equation}\\label{mult-incr}\n\tN_{j+1} > N_j \\quad \\text{ as \\ } \\ j\\ge 1.\n\t\\end{equation}\n\t\\\\\n\tWe first check \n\t\\[ h(2)=\\sum\\limits_{j=1}^{2+[{\\alpha}]} N_{j} - 2 \\sum\\limits_{j=1}^{1+\\left[{\\alpha}\/{2}\\right]} N_j = \\sum\\limits_{j=2+[{\\alpha}\/2]}^{2+[{\\alpha}]} N_{j} - \\sum\\limits_{j=1}^{1+\\left[{\\alpha}\/{2}\\right]} N_j \n\t>0 .\\]\n\tWhen ${\\alpha}\\in[0,2)$, $[{\\alpha}]\\ge [{\\alpha}\/2]=0$ and we have $h(2)\\ge N_2- N_1 >0$.\n\t\\\\\n\tOtherwise if $\\alpha\\ge 2$ then $[{\\alpha}\/2]\\ge 1$ and \\eqref{mult-incr} yields\n\t\\begin{align*} h(2) & >\\left([{\\alpha}]-[{\\alpha}\/2]\\right) N_{2+[{\\alpha}\/2]} - [{\\alpha}\/2] N_{1+[{\\alpha}\/2]} \n\t\\intertext{and since $[{\\alpha}]\\ge 2 [{\\alpha}\/2]$ we have}\n\t& \\ge [{\\alpha}\/2] \\left(N_{2+[{\\alpha}\/2]} - N_{1+[{\\alpha}\/2]} \\right)> 0\n\t\\end{align*}\n\tby using \\eqref{mult-incr} once more.\n\t\\\\\n\tAfter we take that $h(m)>0$ for some $m\\ge 2$ and deduce that also $h(m+1)>0$.\n\tLet us compute\n\t\\begin{align*}\n\t\\sum\\limits_{i=1}^{m}(m+1-i) \\sum\\limits_{j=1+(i-1)(2+[{\\alpha}])}^{i(2+[{\\alpha}])} N_{j} = \\sum\\limits_{i=1}^{m-1}(m-i) \\sum\\limits_{j=1+(i-1)(2+[{\\alpha}])}^{i(2+[{\\alpha}])} N_{j} + \\sum\\limits_{j=1}^{(2+[{\\alpha}])m} N_{j} ,\n\t\\end{align*}\n\thence\n\t\\[ h(m+1) = h(m) + \\sum\\limits_{j=1}^{(2+[{\\alpha}])m} N_{j} - \\sum\\limits_{j=1}^{1+[{\\alpha}\/2]} N_{j} > \\sum\\limits_{j=2+[{\\alpha}\/2]}^{(2+[{\\alpha}])m} N_{j} >0 ,\\]\n\tand this concludes the proof.\n\\end{proof}\n\n\n\tIn the next section we will see that the changes in the Morse index caused by the first singular eigenvalue $\\nu_1(p)$ play a crucial role in establishing bifurcation results. Therefore the parameter $\\beta_1$ deserves a special attention, in particular the integer number \n\t\\begin{equation}\\label{n-def}\n\tn_{{\\alpha}}^m : = \\left\\lceil \\frac{(2+{\\alpha})\\beta_1 -N}{2}\\right\\rceil ,\n\t\\end{equation}\nwhich is characterized by the double inequality\n\\begin{equation}\\label{n-def-char}\nz_1\\left(\\frac{ 2 n_{{\\alpha}}^m+N-2}{2+{\\alpha}} \\right) < z_m\\left(\\frac{N-2}{2+{\\alpha}} \\right) \\le z_1\\left(\\frac{2n_{{\\alpha}}^m+N}{2+{\\alpha}} \\right) .\n\\end{equation}\nOnce that the dimension $N$, the exponent ${\\alpha}$ and the number of nodal zones $m$ have been fixed, the number $n^m_{{\\alpha}}$ can be easily computed by using iteratively the function \\texttt{Besselzero} in MathLab, for instance. Besides it is already known by \\eqref{unico-controllo} that \n\t\\begin{equation}\\label{n-unico-controllo} n_{{\\alpha}}^m \\ge 2(m-1) + [ {\\alpha}(m-1)] .\n\t\\end{equation} \nFor ${\\alpha}=0$ (Lane-Emden equation) there is numerical evidence that for every $N$\n\\[ z_1\\left(2(m-1) + \\frac{N-2}2 \\right) < z_m\\left(\\frac{N-2}2 \\right) \\le z_1\\left(2(m -1) +\\frac{N}2 \\right) , \\]\nso that $n_{0}^m = 2(m-1)$ indeed.\n\n\\section{Global bifurcation}\\label{sec:bif}\n\n\nHere we prove the bifurcation results stated in Section \\ref{sec:stat}. \nIt is well known that if $(p, u_p)$ is a bifurcation point in the curve $\\mathcal S^m$, then the solution $u_{p}$ has to be degenerate, which means that the linearized operator\n$L_{p}$ defined in \\eqref{linearized} has nontrivial kernel in $H^1_0(B)$, or equivalently ${\\Lambda}=0$ is an eigenvalue for \\eqref{standard-eig-prob}. \nIn Section \\ref{sec:prel} we have noticed that the Morse index changes within the interval $(1, p_{{\\alpha}})$, so that degeneracy values do exist.\nBesides we can not rely on any variational structure, since we aim to include also supercritical values of $p$, and bifurcation can be obtained only through an odd change of the Morse index. Hence a better knowledge of the kernel of $L_p$ is needed.\nBy \\cite[Theorem 1.3]{AG-sing-2} the radial solutions are radially nondegenerate, i.e.~the kernel of $L_{p}$ does not contain radial functions. Moreover, the degeneracy has been characterized in \\cite[Proposition 1.5]{AG-sing-1} in terms of the eigenvalues $\\nu_i(p)$ showing that\n\\begin{proposition}\\label{prop:degeneracy}\nLet \t$u_p$ be a radial solution to \\eqref{H} with $m$ nodal zones. It is degenerate if and only if\n\t\\begin{equation}\\label{non-radial-degeneracy-H}\n\t{\\nu}_i(p) = - \\left(\\frac{2}{2+{\\alpha}}\\right)^2 j (N-2+j) \\qquad \\mbox{ for some $i=1,\\dots m$ and $j\\ge 1$.}\n\t\\end{equation}\n\tBesides any function in the kernel of $L_{p}$ can be written according to the decomposition formula \n\t\\begin{equation}\\label{decomposition}\n \\phi(x)= \\psi_{i,p}(|x|^{\\frac{2+{\\alpha}}2})Y_j(x\/|x|) ,\\end{equation}\t\n where $\t\\psi_{i,p}$ is an eigenfunction for \\eqref{eigenvalue-problem} related to an eigenvalue $\\nu_i(p)$ satisfying \\eqref{non-radial-degeneracy-H}, and $Y_j$ stands for an eigenfunction of the eigenvalue $j (N-2+j)$ of the Laplace-Beltrami operator.\n\\end{proposition}\nFor a positive solution only the first eigenvalue $\\nu_1(p)$ plays a role and one can manage to obtain an odd change in the Morse index by restricting the attention to the subspace of $O(N-1)$-invariant functions, as in \\cite{AG14, FN17}.\nFor a nodal solution, instead, the equality \\eqref{non-radial-degeneracy-H} can hold for different values of $i$ and $j$ and \\eqref{decomposition} brings out that the kernel of $L_p$ has a complex structure.\nThis difficulty can be dealt with by turning to the notion of degree and index of fixed points in the positive cones introduced in Section \\ref{sec:stat}.\n\\remove{{\\color{cyan} introduced by Dancer in \\cite{D83}, which in addition gives some insights on the symmetries of the bifurcating solutions and the global properties of the branches, providing a more detailed bifurcation analysis also for positive solutions, and in the subcritical case. }\n\t\\\\\nTo go on some notation is needed.\nWe adopt the \tspherical coordinates in ${\\mathbb R}^{N}$ given by $(r,\\theta,\\varphi)$ with $r=|x|\\in[0,+\\infty)$, $\\theta\\in[-\\pi,\\pi]$, $\\varphi =(\\varphi_1,\\dots\\varphi_{N\\!-\\!2})\\in (0,\\pi)^{N-2}$ \tso that\n\t\\[\\begin{array}{ll}\n\tx_1=r \\cos\\theta \\prod\\limits_{h=1}^{N\\!-\\!2}\\sin\\varphi_h , \\qquad &\n\tx_2= r \\sin\\theta \\prod\\limits_{h=1}^{N\\!-\\!2}\\sin\\varphi_h , \\\\\n\tx_{k}= r \\cos \\varphi_{k\\!-\\!2} \\prod\\limits_{h=k-1}^{N\\!-\\!2}\\sin\\varphi_h \\ \\mbox{ as } k=3,\\dots N-1 , \\quad &\n\tx_{N} = r \\cos \\varphi_{N\\!-\\!2}. \n\t\\end{array}\\]\n\tIn particular for any $x\\neq 0$, $(\\theta, \\varphi)$ are the coordinates of $x\/|x| \\in \\mathbb S_{N-1}$.\n\tNext for any natural number $n$ we introduce the spaces\n\t\\begin{align*}\n\tH^1_{0,n} : = & \\big\\{u\\in H^1_0 \\, : \\, u(r,\\theta,\\varphi) \\hbox{ is even and } {2\\pi}\/n \\hbox{ periodic w.r.t. } \\theta , \\\\\n\t& \\qquad \\qquad \\hbox{ for every } r\\in (0,1) \\text{ and } \\varphi \\in (0,\\pi)^{N-2} \\big\\}, \\\\\n\tX_n : = & \tH^1_{0,n} \\cap C^{1,\\gamma}(B), \n\t\\intertext{and the positive cones already used in \\cite{AG-bif}, i.e.}\n\tK_n : = & \\big\\{u\\in X_n \\, : \\, \\hbox{ is nonincreasing w.r.t.~ } \\theta\\in (0,\\pi\/n), \\\\\n\t& \\qquad \\qquad \\hbox{ for every } r\\in (0,1) \\text{ and } \\varphi \\in (0,\\pi)^{N-2}\\big\\}.\n\t\\end{align*}\n$\\Sigma_n^m$ will stand for the closure of the set made up of nonradial solutions in the cone $K_n$, or rather of\n\t\\[ \\{ (p, u) \\in (1,p_{{\\alpha}})\\times K_n\\setminus \\mathcal{S}^m \\, : \\, u \\mbox{ solves \\eqref{H}} \\}. \n\t\\]\nWe will show that an unbounded continuum of nodal nonradial solutions in $\\Sigma^m_n$ detaches from the curve $\\mathcal S^m$, for some integers $n$ depending on the exponent ${\\alpha}$ and the number of nodal zones $m$. In that case, letting $(p_n, u_{p_n})$ be the bifurcation point, we denote by $\\mathcal C_n$ the closed connected component of $\\Sigma^m_n$ containing $(p_n,u_{p_n})$ and we shall refer it as a \\textquotedblleft branch\\textquotedblright departing from $(p_n, u_{p_n}) $, with a little abuse of language.\n\t\\\\}\n\n\t\tLetting $T$ be the operator\n\\[ T(p,v) : (1,p_{{\\alpha}})\\times C^{1,\\gamma}_0(B)\\longrightarrow C^{1,\\gamma}_0(B) , \\quad T(p,v)=(-\\Delta)^{-1}\\left(|v|^{p-1}v\\right), \\]\nit is clear that $\\mathcal S^m$ are curves of fixed points for $T$ and more generally $u$ solves \\eqref{H} when $u=T(p,u)$.\nMoreover\nminor variations on \\cite[Lemmas 2.2, 3.1]{AG-bif} allow seeing that \n\n\\begin{lemma} \n\tThe operator $T(p,\\cdot)$ maps both $X_n$ and $K_n$ into themselves.\n\t\\end{lemma}\n\nDenoting by $T'_{u}(p,\\cdot)$ the Fr\\'echet derivative of $T(p,\\cdot)$ computed at $u$, we say that $u$ is an isolated fixed point for $T(p,\\cdot )$ w.r.t. $X_n$ \nwhen $I-T'_{u}(p,\\cdot)$ is invertible in $X_n$, which is assured by the nondegeneracy of $u$. Starting from the characterization of degeneracy in Proposition \\ref{prop:degeneracy} one can see that radial solutions $u_p$ are isolated fixed points, except at most a discrete set of $p$. \nIt follows from a general regularity result.\n\n\\begin{lemma}\\label{lem:analiticita}\n\tThe maps $p\\mapsto \\nu_i(p)$ are analytic in $p$. \n\\end{lemma} \nWe do not report the details of the proof. For positive solutions to the H\\'enon equation it has been proved in \\cite[Proposition 4.1]{AG14}.\nFor sign changing solutions to the Lane-Emden equation in dimension $N\\ge 3$ the proof is contained in \\cite[Lemma 3.2]{DW}, and it has been adapted to the case $N=2$ in \\cite[Lemma 7.1]{GI}.\n\n\\\n\nOne can now compute the index of $u_p$ relative to the cone $K_n$, see \\cite{D83}, which will be denote by $\\mathrm{index}_{K_n} (p, u_p)$.\nIt is important to note that, also in the case of nodal solutions, the first singular eigenvalue determines by itself such index.\n\n\t\\begin{lemma}\\label{soloi=1}\n\t\tLet $p$ be such that $u_p$ is nondegenerate. Then \n\t\t\t\\\n\t\t\\mathrm{index}_{K_n}(p,u_p)=\\left\\{\\begin{array}{ll}\n\t\t0 & \\text{ if }\\nu_1(p)< -\\big(\\frac{2}{2+{\\alpha}}\\big)^2 n (N-2+n), \\\\[.2cm]\n\t\n\t\t{\\mathrm{deg}}_{X_n}(I-T(p,\\,))=\\pm 1\n\t\t& \\text{ if }\\nu_1(p)> -\\big(\\frac{2}{2+{\\alpha}}\\big)^2 n (N-2+n) . \\end{array}\\right.\n\t\t\\\n\t\\end{lemma}\nHere the symbol ${\\mathrm{deg}}_{X_n}(I-T(p,\\, ))$ stands for the Leray-Shauder degree of the operator $I-T(p,\\, )$ restricted at $X_n$, computed in a neighborhood of $(p, u_p)$ which does not contain nonradial solutions (this choice is possible since $u_p$ is nondegenerate by assumption).\n\t\\begin{proof}\n\t\tTheorem 1 in \\cite{D83} states that for isolated fixed points\n\t\t\\[\n\t\t\\mathrm{index}_{K_n}(p,u)= \\begin{cases}\n\t\t0 & \\hbox{ if } T'(p,u) \\mbox{ has the property ${\\alpha}$},\n\t\t\\\\[.2cm]\t{\\mathrm{deg}}_{X_n}(I-T(p,\\, ))=\\pm 1 & \\mbox{ otherwise.}\n\t\t\\end{cases}\\]\n\t\n\tIn this way, the prove reduces to show that the so-called {\\em property $\\alpha$} holds if and only $\\nu_1(p)+\\big(\\frac{2}{2+{\\alpha}}\\big)^2 n (N-2+n)<0$.\n\t\tSeveral characterizations of the {\\em property $\\alpha$} are provided in Lemma 3 and the following Remark in \\cite{D83}.\n\t\tTo state the one which will be used here we need the sets\n\t\t\\[\\begin{array}{ll}\n\t\tW^+ := &\\!\\! \\!\\! \\{v\\in X_n \\ u_p+\\gamma v\\in K_n\\ \\text{ for some }\\gamma>0\\}, \\\\\n\t\tW^0 := &\\!\\! \\!\\! \\{v\\in W^+_{u_p}\\ : -v\\in W^+_{u_p}\\}, \\\\\n\t\tV &\\!\\! \\!\\! \\mbox{ the orthogonal (in the $H^1_0$ sense) complement to $W^0$ in $X_n$.}\n\t\t\\end{array}\\]\n\t\tNotice that the functions in $W^0$ do not depend by the angle $\\theta$.\n\t\tNext, $T'$ has the {\\em property ${\\alpha}$} if \n\t\n\t\tthere exists $t\\in (0,1)$ such that the problem \n\t\t\\begin{equation}\\label{property-alpha}\n\t\t\\begin{cases} \n\t\t-\\Delta v = tp |x|^{{\\alpha}}|u_p|^{p-1}v & \\text{ in } B , \\\\\n\t\tv\\in \\overline{W^+}\\setminus W^0 & \n\t\t\\end{cases}\\end{equation}\n\t\thas a solution.\n\t\tWe follow the proof of \\cite[Theorem 1]{D92} and look at the family of eigenvalue problems\n\t\t\\begin{equation}\\label{lambdat}\n\t\t\\begin{cases} -\\Delta v -t p |x|^{{\\alpha}} |u_p|^{p-1}v = \\Lambda v & \\text{ in } B , \\\\\n\t\tv\\in V & \\end{cases}\n\t\t\\end{equation}\n\t\tand let $\\Lambda_t$ be its first eigenvalue. \n\t\tWhen $t=0$ \\eqref{lambdat} reduces to an eigenvalue problem for the Laplacian and certainly ${\\Lambda}_0>0$.\n\t\tWhen $t=1$, instead, \\eqref{lambdat} gives back the eigenvalue problem \\eqref{standard-eig-prob}, but only eigenfunctions in $V$ matter.\n\t\tFurthermore the variational characterization yields that the first eigenvalue $\\Lambda_t$ is strictly decreasing w.r.t.~$t$.\n\t\t\\\\\n\t\tIf $T'$ has the property $\\alpha$, then $\\Lambda_t\\le 0$ for some $t<1$ and therefore $\\Lambda_1<0$. This in turn means that the eigenvalue problem \\eqref{standard-eig-prob} has a negative eigenvalue with related eigenfunction in $V$ and then $\\nu_i(p)+ \\big(\\frac{2}{2+{\\alpha}}\\big)^2 j (N-2+j)<0$ for some $i=1,\\dots m$ and $j$ such that the related spherical harmonic belongs to $V$, by the characterization in \\cite[Theorem 1.4]{AG-sing-1}.\nTaking advantage from the description of the spherical harmonics given in the proof of Theorem 1.1 in \\cite{AG-bif}, one sees that $j$ must be a multiple of $n$ and so, in particular, $\\nu_1(p)+ \\big(\\frac{2}{2+{\\alpha}}\\big)^2 n (N-2+n)<0$.\n\t\t\\\\\n\t\tOn the other hand if $\\nu_1(p)+\\big(\\frac{2}{2+{\\alpha}}\\big)^2 n (N-2+n)<0$ we let $\\psi$ be the first radial eigenfunction for \\eqref{eigenvalue-problem} and $Y_n$ the spherical harmonic related to $n (N-2+n)$ belonging $V$ (which does exist for what we have said before). Now $v (r,\\theta,\\phi)= \\psi(r^{\\frac{2+{\\alpha}}{2}}) \\, Y_n(\\theta, \\phi)$ is in $V$ and an easy computation shows that \n\t\t\\begin{align*}\n\t\t\\int_B\\left(|\\nabla v|^2 - p|x|^{{\\alpha}}|u_p|^{p-1} v^2\\right) dx = \n\t\n\n\t\t\\\\\t\t=\t\\int_0^1 r^{N-1} \\left[ \\left(\\frac{d}{dr}\\psi\\big(r^{\\frac{2+{\\alpha}}{2}}\\big)\\right)^2 - p r^{{\\alpha}} |u_p|^{p-1} \\left(\\psi(r^{\\frac{2+{\\alpha}}{2}}) \\right)^2 \\right] dr \n\t\t\\int_{\\mathbb S_{N\\!-\\!1}} Y_n^2 d\\sigma(\\theta,\\varphi) \\\\\n\t+ \t\\int_0^1 r^{N-3} \\left(\\psi\\big(r^{\\frac{2+{\\alpha}}{2}}\\big) \\right)^2 dr \n\t\\int_{\\mathbb S_{N\\!-\\!1}} | \\nabla Y_n|^2 d\\sigma(\\theta,\\varphi) \n\t\t\\\\\n\t\t\t= \\int_0^1 r^{N-1+{\\alpha}} \\left[ \\left(\\frac{2+{\\alpha}}{2}\\right)^2 \\left(\\psi'(r^{\\frac{2+{\\alpha}}{2}}) \\right)^2 - p |u_p|^{p-1} \\left(\\psi(r^{\\frac{2+{\\alpha}}{2}}) \\right)^2 \\right] dr\n\t\t\\int_{\\mathbb S_{N\\!-\\!1}} Y_n^2 d\\sigma(\\theta,\\varphi) \\\\\n\t\t+ \t\\int_0^1 r^{N-3} \\left(\\psi\\big(r^{\\frac{2+{\\alpha}}{2}}\\big) \\right)^2 dr \n\t\\int_{\\mathbb S_{N\\!-\\!1}} | \\nabla Y_n|^2 d\\sigma(\\theta,\\varphi) \n\t\t\t\\intertext{using the change of variable $t=r{\\frac{2+{\\alpha}}{2}}$ and the notation in \\eqref{ap}}\n\t\t\t =\t\\frac{2+{\\alpha}}{2} \\int_0^1 t^{M-1} \\left[ \\left(\\psi'(t) \\right)^2 - a_p(t) \\left(\\psi(t) \\right)^2 \\right] dt\n\t\t\\int_{\\mathbb S_{N\\!-\\!1}}Y_n^2 d\\sigma(\\theta,\\varphi) \\\\\n\t\t\t\t+ \t\\frac2{2+{\\alpha}}\t\\int_0^1 t^{M-3} \\psi^2(t) \\, dt \n\t\t\t\\int_{\\mathbb S_{N\\!-\\!1}} | \\nabla Y_n|^2 d\\sigma(\\theta,\\varphi) \n\t\t\\intertext{and as $\\psi$ solves \\eqref{eigenvalue-problem} and $Y_n$ is a spherical harmonics we get}\n\t\t = \\frac{2+{\\alpha}}{2} \\nu_1(p)\\int_0^1 t^{M-3} \\psi^2(t) \\, dt\n\t\t \\int_{\\mathbb S_{N\\!-\\!1}} Y_n^2 d\\sigma(\\theta,\\varphi) \\\\\n\t\t + \\frac{2}{2+{\\alpha}}\t n (N-2+n)\t\t\\int_0^1 t^{M-3} \\psi^2(t) \\, dt \n\t\t \\int_{\\mathbb S_{N\\!-\\!1}} Y_n^2 d\\sigma(\\theta,\\varphi) \n\t\t\t\\\\\t= \\frac{2+{\\alpha}}{2} \\left( \\nu_1(p)+\\Big(\\frac{2}{2+{\\alpha}}\\Big)^2 n (N-2+n)\\right)\\int_0^1 t^{M-3} \\psi^2(t) \\, dt\n\t\t\\int_{\\mathbb S_{N\\!-\\!1}} Y_n^2 d\\sigma(\\theta,\\varphi) \\\\\n\t\t= \\left( \\nu_1(p)+\\Big(\\frac{2}{2+{\\alpha}}\\Big)^2 n (N-2+n)\\right) \\int_B \\frac{v^2}{|x|^2} dx <0 .\n\t\t\t \t\\end{align*}\n\t\tHence the first eigenvalue ${\\Lambda}_1$ is negative, and since ${\\Lambda}_0>0$ there exists $t\\in(0,1)$ such that $\\Lambda_t=0$, which means that $T'$ has the property $\\alpha$.\n\t\\end{proof}\n\nRelying on Lemma \\ref{soloi=1} one can see that a sufficient condition for bifurcation is \n\t\t\\begin{equation}\\label{lem:cedelbuono}\t\\left( \\lim\\limits_{p\\to 1}\\nu_1(p)+\\Big(\\frac{2}{2\\!+\\!{\\alpha}}\\Big)^2 n(N\\!-\\!2\\!+\\!n)\\right) \\left( \\lim\\limits_{p\\to p_{{\\alpha}}}\\nu_1(p)+\\Big(\\frac{2}{2\\!+\\!{\\alpha}}\\Big)^2 n(N\\!-\\!2\\!+\\!n)\\right)< 0,\t\n\t\t\\end{equation}\n\tfor some integer $n$.\n\t\n\t\n\t\n\n\\begin{proposition}\\label{prop:bif} \n\t If $n$ is an integer which fulfills \\eqref{lem:cedelbuono}, then there exists at least one $p_n \\in (1, p_{{\\alpha}})$ such that $(p_n, u_{p_n})$ is a nonradial bifurcation point and \n the branch $\\mathcal C_n$ defined according to \\eqref{cn} is global, in the sense that it contains a sequence $(p_k,u_k)$ with\n\\begin{enumerate}[i)]\n\t\\item either $\\|u_k\\|_{C^{1,\\gamma}(B)}\\to +\\infty$,\n\t\\item or $p_k\\to p_{{\\alpha}}$, \n\t\\item or $p_k\\to 1$.\n\t\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nUnder assumption \\eqref{lem:cedelbuono}, and thanks to Lemma \\ref{lem:analiticita}, there exists at least one (and an odd number of) $\\bar p\\in (1, p_{{\\alpha}})$ and $\\delta>0$ such that \n\\[\\begin{array}{c}\n{\\nu}_1(\\bar p) = - \\left(\\frac{2}{2+{\\alpha}}\\right)^2 n (N-2+n) , \\\\\n\\left({\\nu}_1(\\bar p\\!-\\!\\delta) + \\big(\\frac{2}{2+{\\alpha}}\\big)^2 n (N\\!-\\!2\\!+\\!n) \\right) \\left({\\nu}_1(\\bar p\\!+\\!\\delta) + \\big(\\frac{2}{2+{\\alpha}}\\big)^2 n(N\\!-\\!2\\!+\\!n) \\right) <0 , \n\\\\\n\\nu_{i}(p)\\neq - \\left(\\frac{2}{2+{\\alpha}}\\right)^2 j (N-2+j) \n\\end{array} \t\\]\nfor every $i=1,\\dots m$, $j\\ge 0$, and $p\\in(\\bar p-\\delta,\\bar p+\\delta)$, $p\\neq \\bar p$.\n\\\\\nLemma \\ref{soloi=1} then implies that the Leray Schauder degree in the cone $K_n$ changes and the remaining of the proof follows as in \\cite[Theorem 1.2]{AG-bif}.\nSee also \\cite{Gla-glob}, where a more detailed proof is given in the case of positive solutions.\n\\end{proof}\n\nWe are now ready to prove the bifurcation results stated in Section \\ref{sec:stat}.\nConcerning positive solutions, we have already pointed out that the Morse index near at $p_{{\\alpha}}$ is strictly greater than the one near at $1$, for every ${\\alpha}>0$.\nSince only the first eigenvalue $\\nu_1(p)$ is negative and gives a contribution to the Morse index, it is clear that there exists at least one value of $n$ such that \\eqref{lem:cedelbuono} holds. Let us complete the proof of Theorem \\ref{teo:bif-H-1}.\n\n\\begin{proof}[Proof of Theorem \\ref{teo:bif-H-1}]\nFirst, we check that \\eqref{lem:cedelbuono} is fulfilled for every integer $n=1,\\dots \\lceil{{\\alpha}}\/{2}\\rceil$.\nRecalling that \t$ \\lim\\limits_{p\\to 1}\\nu_1(p) =0$ by \\eqref{nup=1} (since for positive solutions $\\beta_1=\\frac{N-2}{2+{\\alpha}}$), it is equivalent to see that \n\\[\\lim\\limits_{p\\to p_{{\\alpha}}}\\nu_1(p) < - \\Big(\\frac{2}{2+{\\alpha}}\\Big)^2 n(N-2+n)\\] \nfor $1\\le n<\\frac{2+{\\alpha}}{2}$, i.e.\n\\[ \\lim\\limits_{p\\to p_{{\\alpha}}}\\nu_1(p) \\ge - \\frac{2N-2+{\\alpha}}{2+{\\alpha}} .\\]\nBut \\eqref{nu-p=palpha} and \\eqref{nu-p=infty-1} state that equality holds in any dimension $N\\ge 2$.\n\nTherefore Proposition \\ref{prop:bif} gives the first part of the claim. \nAs for property {\\it i)}, every branch $\\mathcal C_n$ must be composed of nonnegative solutions by continuity, so that maximum principle ensures that or they are positive, or they are identically zero. But this last occurrence is not allowed since the trivial solution is isolated.\nBesides the same Proposition \\ref{prop:bif} states that $\\mathcal C_n$ contains a sequence $(p_k, u_{k})$ such that either $\\|u_k\\|\\to \\infty$, or $p_k\\to p_{{\\alpha}}$, or $p_k\\to 1$.\nMoreover the occurrence $p_k\\to 1$ is forbidden by the uniqueness of positive solutions for $p$ close to $1$ in \\cite[Theorem 3.1]{AG-bif} (which can be easily extended also to dimension $N=2$).\n\\\\\nAs for the possible intersection between two branches $\\mathcal C_n$ and $\\mathcal C_{n'}$, it has to be composed by $(p, v)$ such that $v\\in K_n\\cap K_{n' }$ is a positive solution to \\eqref{H}.\nIn dimension $N=2$ $K_n\\cap K_{n' }$ reduces to radial function, and therefore $v=u_p$ is a radial positive solution to \\eqref{H}, which has to be degenerate and therefore isolated.\nIn dimension $N\\ge 3$, instead, $K_n\\cap K_{n' }$ contains also functions which are nonradial, but do not depend by the angle $\\theta$.\n\t\\end{proof}\n\n\n\\\n\nAfter we deal with bifurcation from nodal solutions, and we begin by examining the planar case.\n\n\\begin{proof}[Proof of Theorem \\ref{teo:bif-H-m-N=2}]\nFirst, we compute the values of the integer $n$ for which \\eqref{lem:cedelbuono} holds. \nThanks to \\eqref{nup=1} and \\eqref{nu-p=infty-2}, it means that\n\\[ -\\beta^2= \\lim\\limits_{p\\to 1} \\nu_1(p) > -\\left(\\frac{2n}{2+{\\alpha}}\\right)^2> \\lim\\limits_{p\\to \\infty} \\nu_1(p)=-\\kappa^2,\\] \nwhich is clearly equivalent to $\\frac{2+{\\alpha}}{2}\\beta -\\left(\\frac{2}{2+{\\alpha}}\\right)^2n(N-2+n) > \\lim\\limits_{p\\to 1} \\nu_1(p) .\\]\n\tBy \\eqref{nup=1} and \\eqref{nu-p=palpha} it means that \n\t\\[ \\frac{2N-2+{\\alpha}}{2+{\\alpha}} < \\left(\\frac{2}{2+{\\alpha}}\\right)^2 n(N-2+n) < {\\beta}_1^2 - \\left(\\frac{N-2}{2+{\\alpha}}\\right)^2 ,\\]\nwhich can be rearranged into\n\t\t\\[ 1 + 2\\frac{N-2}{2+{\\alpha}}< \\left(\\frac{2n}{2+{\\alpha}}\\right)^2 + 2 \\frac{2n}{2+{\\alpha}}\\frac{N-2}{2+{\\alpha}} < {\\beta}_1^2 - \\left(\\frac{N-2}{2+{\\alpha}}\\right)^2 ,\\]\n\t\twhich in turn, after adding the term $\\Big(\\frac{N-2}{2+{\\alpha}}\\Big)^2$ to every member and extracting square roots, becomes \n\t\t\t\\[ \\frac{N+{\\alpha}}{2+{\\alpha}} < \\frac{N-2+2n}{2+{\\alpha}} < {\\beta}_1 ,\\]\n\t\ti.e.\n\t\t\t\\[ \\frac{2+{\\alpha}}{2}< n< \\frac{2+{\\alpha}}{2} \\beta_1 - \\frac{N-2}{2} \\]\n\t\tas claimed.\tIt means that \\eqref{lem:cedelbuono} is fulfilled by $n= 2 + \\left[\\frac{{\\alpha}}{2}\\right], \\dots n_{\\alpha}^m$, where $n_{\\alpha}^m$ is defined in \\eqref{n-def}.\n Remembering that $n_{\\alpha}^m \\ge 2(m-1)+ [ {\\alpha} (m-1)$, see \\eqref{n-unico-controllo}, one can see that \n \\eqref{lem:cedelbuono} holds for at least $2m-3 + [{\\alpha}(m-1)] - \\left[\\frac{{\\alpha}}{2}\\right] $ different values of $n$.\n\n\t\n\tEventually the conclusion follows by Proposition \\ref{prop:bif}\t, arguing as in the proof of Theorem \\ref{teo:bif-H-m-N=2}. The only difference stands in the possible overlapping between branches, i.e. property {\\it iii)}, which in higher dimension can also contain nonradial solutions which do not depend by the angle $\\theta$. \n\\end{proof}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe problem of how a large animal develops from a single cell has been\na central problem in biology. A somewhat simpler, but nontrivial,\nproblem is how a baby animal grows into an adult, increasing the\ntotal body mass by two or three orders of magnitude, while different\nparts of the body keep roughly same proportions during growth. We\nwill refer to this property as \\textit{proportionate growth}. Our work\nhas been motivated by trying to construct minimal physical models with\nthis property.\n\nA simple example of proportionate growth in a non-biological context\nis a dew drop on a windowpane. Its shape may be approximately\ndescribed as a spherical frustum, where the contact angle with the\nglass surface is determined by the surface tension. As it takes water\nfrom the super-saturated air in the surrounding, it grows in size, and\nshows nearly proportionate growth. However, it is not easy to\nconstruct models showing proportionate growth in patterns with\nsubstructures. In fact, no other model of growth studied in\nphysics literature so far, shows\nthis property. In the well studied Eden model \\cite{eden},\ndiffusion-limited aggregation \\cite{dla}, invasion percolation\n\\cite{invasion}, or the Kardar-Parisi-Zhang type models \\cite{kpz},\ngrowth occurs in the outer ``active regions'', whereas the inner core once\nformed, remains essentially frozen afterwards.\n\nIt seems reasonable that a proportionate growth would require some\ncentral regulation or a long-distance communication and coordination\nbetween different parts of the structure. For an animal growth this is\ncertainly true. The growth is orchestrated by the turning on and off\nof different regulatory enzymes and chemicals, ultimately determined\nby the genetic program encoded in the cell's DNA. It is interesting\nthat, such growths can be achieved in a model system with components of\nmuch lower complexity, \\textit{i.e.}, a cellular automaton model with only a few\nstates per site. In an earlier work \\cite{epl}, we have studied the Abelian\nSandpile Model (ASM) as\nthe prototypical model of proportionate growth. The\npatterns are formed by adding large number of sand grains at a single\nsite on a periodic initial configuration (also referred to as\n``background''), and letting the system relax to a stable configuration. We were able to characterize \nthe full pattern analytically in one simple case. The effect of adding absorbing\nsites, or multiple centers of growth was studied in \\cite{jsp}.\n\nReal biological growth occurs in a fairly noisy environment. While\ndeterministic cellular automaton models with simple toppling rules\ncannot be considered realistic biologically, it is still interesting\nto ask whether the patterns produced by growing sandpiles are robust\nagainst introduction of a small amount of noise. In the presence of\nnoise, an analytical study of this problem is quite difficult. The\ntechniques used in \\cite{epl} to characterize the pattern exactly, no\nlonger work, as they depend on the potential function in each patch\nbeing a quadratic function of the coordinates (see \\sref{sec:prel}).\nThe work reported here is exploratory, and mainly numerical. However, the\nfact that the patterns show some degree of robustness, even in the\npresence of noise, strongly suggests that this is not a special\nproperty related to the exact solubility of the ASM, and a more general\nmacroscopic description of pattern formation and pattern selection in\nthis problem, not requiring an exact solution, should be\npossible. We discuss how the ``least action principle'' for\nASMs could provide a possible framework for understanding pattern\nformation in our problem, as the variational formulation provides\na quantitative criterion for pattern selection.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=6.0cm]{fig1}\n\\caption{The F-lattice: A square lattice with directed edges whose\ndirections are assigned periodically as shown.}\n\\label{fig:flattice}\n\\end{center}\n\\end{figure}\n\nWe have studied the robustness of these patterns against different\ntypes of noises, namely, random fluctuations in the position of the point of\naddition of grains, disorder in the periodic background\nconfiguration of heights, and disorder in the connectivity of the\nunderlying background lattice. We find that the patterns show a varying\ndegree of robustness to addition of a small amount of noise in each\ncase. However, introducing stochasticity in the toppling rules seems\nto destroy the asymptotic pattern completely, even for a weak noise.\n\nThe spatial patterns formed in sandpile models were first studied by\nLiu \\textit{et.al.} \\cite{liu}. The asymptotic shapes of the boundaries of\nsandpile patterns produced by adding grains at single site on\ndifferent periodic backgrounds was discussed in a later work by Dhar \\cite{dhar99}. Borgne\n\\textit{et.al.} \\cite{borgne} obtained some bounds on the\nrate of growth of these boundaries, and later these bounds were\nimproved by Fey \\textit{et.al.} \\cite{redig} and Levine\n\\textit{et.al.} \\cite{lionel}. The first detailed analysis of\ndifferent periodic structures found in the sandpile patterns was discussed by\nOstojic \\cite{ostojic}. An extensive collection of centrally\nseeded sandpile patterns on different lattices, with high resolution\nimages, can be seen in\n\\cite{wilson}. Other special configurations in the ASMs, like the identity \\cite{borgne,identity,caracciolo} or the\nstable state produced from special unstable states \\cite{liu}, also show complex\nstructures , which share common features with the single\nsource patterns studied here.\n\nThis paper is organized as follows. In \\sref{sec:prel}, we define the\nmodels precisely, and introduce the scaled excess\ndensity function and the scaled toppling function. These functions give a quantitative\ncharacterization of the patterns. In \\sref{sec:addition}, we discuss\nthe robustness of the patterns against small fluctuations in the\nposition of the point of addition. In \\sref{sec:bkg}, we discuss the\neffect of noise in the background configuration. In\n\\sref{sec:quenched}, we discuss the effect of disorder in the underlying lattice on\nwhich the growth occurs. This is modeled by a quenched disorder in\nthe toppling rules. We find that, the patterns are quite sensitive to\nchanges in the toppling matrix. In \\sref{sec:manna}, we discuss the\neffect of noise due to fluctuations in the toppling process,\n\\textit{i.e.}, at each\ntoppling, there is a small probability that the grain transfer occurs\nin a direction not given by the toppling rule. We find that, even a\nvery small amount of noise in the toppling rules completely wipes out\nthe asymptotic pattern. Finally, in the concluding \\sref{sec:discussion}, we \ndiscuss the `least action principle' for the ASM, and suggest that\nit could provide a basic framework for understanding pattern formation in these systems.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.5cm]{fig2a}\n\\includegraphics[width=7.5cm]{fig2b}\n\\caption{The patterns formed in the ASM defined on the F-lattice with\ncheckerboard background of heights $0$ and $1$. These two patterns\ncorrespond to $N=80,000$ and $N=320,000$ grains, respectively. Color code:\nRed$=0$, White$=1$. For comparison, the size of the first pattern has been enlarged by a factor\ntwo.}\n\\label{fig:prop}\n\\end{center}\n\\end{figure}\n\n\\section{Preliminaries and definitions}\\label{sec:prel}\nFor our numerical studies, we have used two model systems (Here the\nterm ``model system'' is used as in biology literature: the fruit fly\nis a model animal, and biological functions in other animals are\nqualitatively similar).\n\nThe first model is defined on an infinite square lattice with directed\nedges, such that at each site there are two edges coming in, and two\ngoing out (\\fref{fig:flattice}). This directed square lattice is\ncalled the F-lattice. At each site $\\mathbf{x}$, there is a non-negative integer\nvariable $z\\left(\\mathbf{x}\\right)$, called the number of sand grains \nat $\\mathbf{x}$, also called the height of the sandpile\nat that site. Any site with height\ngreater than $1$ is said to be unstable, and it topples by\ntransferring exactly two grains in the direction of outward arrows\nfrom that site. We start with an initial configuration in which the\nheights $0$ and $1$ form a checkerboard pattern. At each time step, we\nadd a grain at the origin, and let the resulting configuration relax\nuntil all sites are stable. After $N$ grains have been added, with $N$\nlarge, we find that the heights form an intricate and beautiful pattern,\nwhose size grows as $N$ increases. The resulting patterns for $N=\n80,000$ and $320,000$ are shown in \\fref{fig:prop}. Note that, what\nappears to be solid red region in the figure due to low resolution, is\nactually a checkerboard pattern of alternate red and white squares.\nDetails may be seen by zooming in. We see that the two scaled patterns\nare the same, except that the smaller patches close to the center of the pattern are\nresolved better in the second.\n\nThe second model system is the ASM on an undirected infinite square\nlattice. We define the ASM on this lattice as follows: in a stable\nconfiguration, all sites have heights less than $4$. Any site where the height is\ngreater than $3$ is said to be unstable, and it relaxes by transferring $4$ sand\ngrains from that site, one to each of the four nearest neighbors. We\nstart with an initial configuration where the height at each site is\n$2$. The stable configuration after adding $N= 250,000$ grains at the origin is\nshown in \\fref{fig:btw}.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.0cm]{fig3}\n\\caption{A pattern produced on a square lattice with a background of\nheight $2$ at every site.}\n\\label{fig:btw}\n\\end{center}\n\\end{figure}\n\nOne can consider patterns obtained when the initial configuration has a\ndifferent periodic structure. For the undirected square lattice, when\nthe initial configuration is a periodic arrangement of heights, in which\neach site has height less than or equal to $2$, one finds a pattern in\nwhich the diameter of the pattern grows as $\\sqrt{N}$ \\cite{denboer}.\nFor the F-lattice, there are some backgrounds on which the growth\nof the pattern is faster than $\\sqrt{N}$ \\cite{triangular}. In all the cases studied\nso far, if there are no infinite avalanches, the patterns show\nproportionate growth. Although we do not have a rigorous proof of this\nimportant property, there is good numerical evidence, and we shall assume it in the following.\n\nA key observation is that, for large $N$, the patterns in\n\\fref{fig:prop} or \\fref{fig:btw} may be seen as a union of disjoint\npatches, each of which occupies a non-zero fraction of the area of the full\npattern, and the arrangement of heights inside a single patch is exactly\nperiodic. We denote the diameter of the pattern by $\\Lambda$, which\nmay be defined as the width of the smallest square enclosing the\npattern.\nWe define reduced coordinates $\\xi =x\/\\Lambda$ and $\\eta = y\/\\Lambda$. The\nlocal excess density of grains $\\Delta \\rho( \\xi,\\eta)$ is defined\nas the difference in the density of grains in the final and initial\npatterns, in a small neighborhood of the point $(\\xi,\\eta)$ in the reduced\ncoordinates. We specify the asymptotic pattern in the\nlimit of large $N$, by specifying the function $\\Delta \\rho(\n\\xi,\\eta)$. From the fact that inside each patch, there is a periodic\npattern of integer heights, it follows that the excess density $\\Delta\n\\rho(\\xi,\\eta)$ is a rational constant for each patch.\n\nIt is useful to define a function $\\Phi(\\xi,\\eta)$ as the\nscaled number of topplings at the site with reduced coordinates $(\\xi,\n\\eta)$. Let $T_N(x,y)$ be the number of topplings at site $(x,y)$, after\nadding $N$ grains and relaxing the system completely. We define the function\n\\begin{equation}\n\\Phi(\\xi,\\eta) = \\lim_{N \\rightarrow \\infty} \\frac{1}{\\Lambda^2}T_N([\\xi \\Lambda],[y \\Lambda]),\n\\end{equation}\nwhere $[x]$ denotes the integer nearest to $x$. The conservation of\nnumber of grains implies that the potential function satisfies the Laplace's\nequation \\cite{epl}\n\\begin{equation}\n\\nabla^2 \\Phi(\\xi,\\eta) = -\\delta(\\xi,\\eta) + \\Delta \\rho(\\xi,\\eta).\n\\end{equation}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.5cm]{fig4a}\n\\includegraphics[width=7.5cm]{fig4b}\n\\caption{The patterns produced on the F-lattice by adding sand grains at\nsites randomly chosen from a square region of area equal to $9\\%$ and\n$1\\%$, respectively, of that of the corresponding final patterns. The\ninitial configuration is with checkerboard distribution of\nheights $0$, and $1$. Color code: Red=0, white=1.}\n\\label{fig:30perc}\n\\end{center}\n\\end{figure}\nIn an electrostatic analogy, we can think of $\\Phi(\\xi,\\eta)$ as\nthe potential produced by a unit positive point charge at the origin,\nand an areal charge density $-\\Delta \\rho(\\xi,\\eta)$. We shall refer to\n$\\Phi(\\xi,\\eta)$ as the potential function hereafter. In each periodic patch,\nthe potential $\\Phi(\\xi,\\eta)$ is a quadratic function of the\ncoordinates $\\xi$ and $\\eta$ \\cite{ostojic}. For the pattern on the F-lattice, it was\nshown that the coefficients of the quadratic terms are simple\nrational numbers, while the linear terms can be determined by the\ncondition that the potential and its derivative are continuous\nfunctions at the boundaries where two patches meet. This allowed us to\ncharacterize the asymptotic pattern completely \\cite{epl}.\n\n\\section{Effect of fluctuations in the site of addition}\\label{sec:addition}\nThe patterns studied so far are produced by adding one grain at each time step, \nat a fixed site (the origin). Now, we\nconsider how these change when the site of addition fluctuates in time at\nrandom. To be more specific, we add $N$ grains by randomly\ndistributing them among sites within a small square centered at the\norigin. The size of the square is chosen to be of length $\\epsilon\n\\Lambda$, with $\\epsilon<1$. The background is a checkerboard\ndistribution of heights $1$ and $0$. \n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.0cm]{fig5}\n\\caption{A pattern similar to the one in \\fref{fig:30perc}, but this\ntime the grains are added uniformly, four at every site inside the\nsquare of width $160$ lattice units. The diameter of the full pattern\nis $592$. Color code: Red=0, White=1.}\n\\label{fig:uh}\n\\end{center}\n\\end{figure}\n\nWe have shown the pattern for $\\epsilon=0.3$ and $N=120,000$ in\nfigure \\ref{fig:30perc}a, and\nthe pattern corresponding to $\\epsilon=0.1$ and $N=75,000$ in figure\n\\ref{fig:30perc}b. We see that the patches away from the boundary of\nthe square region of addition, are identical to that of the\nsingle source pattern (compare with \\fref{fig:prop}). That is, the patches at\nthe outer layers in the pattern, are not much affected by this change. Only near the\ncenter, within a distance of order $\\epsilon$, we see a change.\nNear the center, the dense set of patches of decreasing size\nis smeared out, and there are no periodic quadratic patches left.\nHowever, there are new accumulation points of patches, which develop\nat the four corners of the square region of addition.\nAs more grains are added, finer patches appear in a way that their number\nwould become infinite in the asymptotic limit of large $N$. For small\n$\\epsilon$, the shape of the outer parts of the pattern shows only a\nweak dependence on its value.\n\nThe pattern changes in an interesting way, when the addition of\ngrains are not random. The pattern produced by uniformly adding\n$m=4$ grains at every site inside a square of size $2l\\times 2l$ is\nshown in figure \\ref{fig:uh}. The boundary of the square is indicated\nby a solid blue line. The pattern outside and away from the square, looks very\nsimilar to the pattern corresponding to random addition. The most significant change is\ninside the square, where there are well defined periodic patches.\nAgain, there is no accumulation point of tinier and\ntinier patches at the origin. As we increase $l$, keeping $m$ fixed, more and more patches appear near the corner of\nthe square. It seems that their number would tend to infinity for\nlarge $N$.\n\\section{Randomness in the initial configuration}\\label{sec:bkg}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.5cm]{fig6a}\n\\includegraphics[width=7.5cm]{fig6b}\n\\caption{The patterns produced on the F-lattice by adding $N=228,000$ and\n$N=896,000$ grains at a single site on a background of mostly checkerboard\ndistribution, except height $1$ at $1\\%$ and $10\\%$ of the sites,\nrespectively, are replaced by height zero. Color code: Red=0, White=1.}\n\\label{fig:idtype1}\n\\end{center}\n\\end{figure}\nThe patterns show a significant amount of robustness to the noise in\nthe background. The least effect of change in the background on\nF-lattice occurs if some heights $1$ are replaced by heights zero.\n\nThis is easy to see using the abelian property of the\nASM. Let $C$ be the initial height configuration and $D$ be the final configuration produced\nby adding $N$ grains at the origin. Consider a particular site $i$, which has\nheight $1$ in both $C$ and $D$. Let the configurations obtained from\n$C$ and $D$ by changing the height at this site from $1$ to $0$ be\ncalled $C'$ and $D'$, respectively. Then, if $C'$ and $D'$ contain no forbidden sub-configurations\n\\cite{dharphysica}, using the abelian property one can show that addition of $N$ grains in $C'$ would give relaxed configuration $D'$. Also the toppling function is same in both the\ncases. \n\nThus we expect that a very small concentration of $1$'s replaced by\n$0$'s will have only a small effect. \nThis expectation is verified numerically. The\npatterns on the F-lattice corresponding to backgrounds with $1\\%$ and $10\\%$ noise in\nthe background are shown in \\fref{fig:idtype1}a and\n\\fref{fig:idtype1}b, respectively. We see that the qualitative\nstructure and placement of different patches is unaffected in the\nlow noise case. In particular, there are only two types of patches, and the relative positions and\nsizes of the larger patches are not changed much. The excess density is uniform within\neach patch, and jumps sharply across clearly defined patch\nboundaries. The outer boundary of the pattern, separating the\nred region outside and the eight largest white patches, seems to remain a nearly\nperfect octagon, but with slightly rounded off corners. However, other\npatch boundaries are no longer straight lines in the presence of noise, and\na significant curvature in the patch boundaries is clearly seen in\npatterns for larger noise.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.0cm]{fig7a}\n\\includegraphics[width=7.0cm]{fig7b}\n\\caption{The patterns produced on a mostly checkerboard background,\nexcept heights at $1\\%$ and $10\\%$ sites, respectively, are flipped.\nColor code: Red=0, White=1.}\n\\label{fig:ID1}\n\\end{center}\n\\end{figure}\n\nEven for the relatively large noise value (\\Fref{fig:idtype1}b), the basic \neight petal structure of the pattern without noise is clearly visible. \nHowever with increase of defect density, the relative area of the\ndense patches (white color) decreases. Also the corners of the outer boundary of the\npattern becomes smoother, and for defect density close to $50\\%$, the\npattern becomes a circle with a single aperiodic patch inside.\n\nThe patterns are more sensitive to the changes in heights $0$'s to $1$'s. In\nfigures \\ref{fig:ID1}a and \\ref{fig:ID1}b, we have shown the resulting\npatterns when in the initial background pattern, the heights at a fraction $\\epsilon$ of randomly\nchosen sites are flipped from $1$ to $0$, and vice versa. The mean density of the background remains\n$1\/2$. The patterns correspond to $\\epsilon = 0.01$ and $0.1$,\nrespectively. In this case, the most noticeable qualitative\nchanges seems to be the fact that boundaries between patches are no\nlonger sharp, which makes even a precise definition of a patch\ndifficult.\n\nComparatively, the patterns in an ASM on a undirected square lattice are\nmore robust against addition of small amount of randomness in background. We introduce randomness\nin the uniform background of height $2$ by assigning each site\nheights $0,1$ or $2$ with probabilities $p\/2,p\/2$ and $1-p$, respectively,\nindependent of other sites. \n\nThe patterns corresponding to $p=0.01$ and $p=0.1$ are shown in figure\n\\ref{fig:btwID}a and \\ref{fig:btwID}b, respectively. These should be compared with the pattern produced\nby adding $N=250,000$ grains on a background of height $2$ at all\nsites, shown in figure \\ref{fig:btw}.\n\nAt $p=0.01$, the patches with height predominantly\n$3$, do not change much, except the presence of reduced heights at\nthe defect sites. This is easy to understand using an argument similar to the one\ngiven for the F-lattice pattern. The presence of defect sites inside\nthe rest of the regions, generates line-discontinuities in the heights, which washes out\nthe smaller features of the pattern. As the noise is increased, the number\nof defect lines increases, and the finer features are\nlost. Also the corners of the outer boundary become round. At\nnoise strength $p\\simeq 0.5$, the pattern becomes a circle with\nrandom distribution of heights of uniform density, inside.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.0cm]{fig8a}\n\\includegraphics[width=7.0cm]{fig8b}\n\\caption{The patterns produced on a square lattice with a background in\nwhich heights $z=2$ at most of the sites except, $1\\%$ and $10\\%$ of the sites,\nrespectively, are with random assignment of heights $0$ or $1$. Color\ncode: Red=0, White=1.}\n\\label{fig:btwID}\n\\end{center}\n\\end{figure}\n\nWe note that, other types of the randomness in the initial conditions can have different \nbehavior. For example, den Boer \\textit{et.al.} \\cite{denboer} have\nshown that if one adds an arbitrary small density\nof sites with height $3$, while all the other sites have height $2$ on the undirected square lattice,\none gets infinite avalanches for \\textit{finite} $N$, with probability $1$. \n\n\\section{Randomness in the Toppling matrix}\\label{sec:quenched}\n\nWe now consider the effect of disorder in the underlying lattice on which the growth occurs.\nWe have considered two types of disorders. \nThe first one is a bond disorder, where a randomly chosen fraction $\\epsilon$ \nof the bonds are removed. Here, no grain transfer can occur along these bonds. For the undirected square lattice case, to keep the conservation law of sand in the model,\nwe change the critical height at the end points of each such broken bond,\nso that a site becomes unstable when its height equals\nor exceeds its coordination number. The toppling rules are\ndeterministic, and the number of sand grains are conserved in a\ntoppling. However, the toppling matrix is no\nlonger translationally invariant.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.5cm]{fig9a}\n\\includegraphics[width=7.5cm]{fig9b}\n\\caption{The patterns produced on a square lattice with $1\\%$ and\n$10\\%$ of the edges, respectively, are broken. Initial configuration with all heights $2$.}\n\\label{fig:btwBB}\n\\end{center}\n\\end{figure}\n\nThe patterns corresponding to the undirected ASM on the square lattice with $1\\%$ and\n$10\\%$ broken bonds are shown in figure \\ref{fig:btwBB}. We see that\neven for $\\epsilon = 0.01$, the pattern has changed substantially. \nThe outermost patches, which, in the absence of noise, had all sites with height $3$, now show a large\nnumber of criss-crossing defect lines. Further, counting inwards from\noutside, one can clearly see at least\nthree or four more rings of patches. Fewer features are clearly\ndistinguishable, for larger noise.\nHowever, some remnant of the\ncharacteristic four-petal pattern of the noise-free case, can be\nclearly seen even for $\\epsilon=0.10$. \n\nThe second type of disorder that we considered, is a type of site-disorder. We consider the F-lattice, where \na small fraction $\\epsilon$ of the sites are chosen at random, and we change the direction of bonds \ngoing out (from up-down to left-right, and vice-versa). The critical\nheight remains unchanged, and is the same for all sites. However now,\nat each site the number of in-arrows is not necessarily equal to the\nnumber of out-arrows. As discussed by Karmakar \\textit{et.al.}\n\\cite{karmakar}, this is a relevant perturbation,\nand an arbitrarily small $\\epsilon$ changes the critical exponents of\nthe\navalanches. We find that the patterns in growing sandpiles are also unstable\nto even a little amount of this kind of randomness. The pattern corresponding\nto $p=0.01$ and $N =57,000$ is shown in figure \\ref{fig:rd}. This pattern is a circle\nwith no distinguishable structures inside.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.0cm]{fig10}\n\\caption{A pattern produced on a F-lattice in which $1\\%$ of the sites\nhas incoming and outgoing arrows switched. Color code: Red=0, White=1.}\n\\label{fig:rd}\n\\end{center}\n\\end{figure}\n\\section{Effect of randomness in the toppling}\\label{sec:manna}\nWe have also studied the effect of noise in the toppling rules. We\nhave considered the F-lattice.\nFor each toppling at a site, the direction of the outgoing grains\ndiffers from the direction of the outgoing arrows, with a probability\n$\\epsilon$.\nThe two grains go in the\ndirection of outgoing arrows with probability $1 -\\epsilon$, while they go\nin the direction of incoming arrows with probability $\\epsilon$. \nThe stochastic toppling rules take this modification of the ASM to the \nManna universality class, which is different from that of the\ndeterministic ASM with fixed toppling rules. In this case, we expect\nthat the pattern would be unstable against such perturbations. This\nis verified by our simulations. We simulated the pattern\nobtained by adding $57,000$ grains on the F-lattice with checkerboard\nbackground and $\\epsilon = 0.01$. The resulting pattern\nis a simple, nearly circular blob, with no other discernible features.\nIt is visually indistinguishable from the pattern in \\fref{fig:rd}.\n\n\\section{Discussion}\\label{sec:discussion}\nThe complicated and beautiful patterns produced in the growing\nsandpiles are the result of an interplay between macroscopic conservation laws\n(encoded in the Poisson's equation satisfied by the potential\nfunction) and the integer nature of the microscopic variables. This\nis not yet well understood. In fact, starting from the ASM rules, as yet we\ncan not prove even the existence of proportionate growth\nin the growing patterns.\n\nIn the presence of noise, an analytical study of this problem is even\nmore difficult. Clearly, the potential function is no longer\npiece-wise quadratic, and the analytical techniques used in\n\\cite{epl},\nto characterize the pattern exactly, no longer work. In fact, in\nfigures \\ref{fig:ID1} and \\ref{fig:btwBB}, there are no sharp patch\nboundaries, and perhaps one can not even give a clear definition of\n`patches', at all. The patterns are characterized by the nontrivial\nspatial dependence of the density function $\\Delta \\rho(\\xi,\\eta)$.\nThe pictures are reminiscent of Rayleigh-B\\'{e}nard convection patterns\n\\cite{rayleigh}, where a linear analysis about the uniform steady state\nshows that, in some regime of parameters, they\nbecome unstable to a class of space-dependent perturbations. In\nour numerical studies, we see that the featureless\ncircular-blob-pattern of growth at high noise levels, is unstable with\nrespect to\nsome characteristic low-wavelength density instabilities for low-noise\ndeterministic models. However, there is an important difference\nbetween these two cases. In the convection problem, the amplitude of\nthe perturbations grows in time exponentially till it reaches a\nsaturation value, determined by the nonlinear terms, whereas in the sandpile\nproblem, the notion of ``growth of amplitude in time'' is not well-defined.\n\nNevertheless, for the sandpile patterns there is the ``least action\nprinciple'', which is a variational principle that allows us to compare\ndifferent trial patterns, and select the pattern corresponding to the\nminimum action. Here `action' is measured in terms of the total number of \ntoppling events. The\nprinciple, in the ASM context, is informally stated as the lazy\nman's maxim: ``Don't do anything, unless you have to''. If we think of\ntopplings as dissipative events, it is similar in spirit to the\nprinciple of minimal heat production in resistor networks, or the\nminimum entropy production principle, often discussed in\nnon-equilibrium statistical physics. While the extent of validity of\nthe latter, in general, is not clearly established (see, for example,\nthe discussion in \\cite{jaynes}), for this special case of ASM's\nwith a threshold rule for topplings, given that the order of\ntopplings does not matter, the principle is easily proved, and is more or\nless built into the rules of evolution \\cite{denboer}.\n\nMore precisely, if one considers a starting configuration $C$ of an\nASM, with one or more unstable sites, then the toppling rules of the\nASM determine the stable final configuration $F$, uniquely. Suppose we\nmodify the toppling rules of the ASM by dropping the condition that a\ntoppling occurs only at sites where the height exceeds the threshold\nvalue, and allowing topplings at any site. For example, starting with a\nconfiguration of all sites with height zero on the undirected square\nlattice, a toppling at the origin would make the height there $-4$,\nand height at each of the neighbors, $+1$. Then, starting from $C$,\nthere is a large number of stable configurations reachable by\ntopplings. The `least action principle' for the ASM states\nthat, if we can reach a \\textit{stable} configuration $F'$ from $C$\nunder this less restrictive dynamics, the number of topplings\nrequired to reach $F'$ is greater than that required to reach $F$, for\nall $F' \\neq F$. \n\nThe variational principle allows us to compare different guesses for\nthe final configuration, and tells us which one is closer to the actual\npattern. The main difficulty in applying this principle, in practice, is\nthat the set of configurations, over which the extremization has to be\ndone, is all possible configurations {\\it reachable from the initial\nunstable pattern by topplings}. Characterizing this set is rather\ndifficult. However, one can restate this principle in terms of\nnon-negative integer toppling functions, $T_N(x,y)$. For any choice of\n$T_{N}\\left( x,y \\right)$, there is a well-defined, easily computed, final height\nconfiguration. Also, one can systematically improve on an initial\ntrial function, by performing more topplings at sites which are unstable in the final\nconfiguration, or by untoppling at sites where the heights are too low.\nThis has been shown to yield a very efficient algorithm for\ndetermining the final configuration for the related rotor-router\nmodel \\cite{tobias}.\n\nThere are other questions that have been addressed only partially in\nthis paper. Certainly, it would be useful to have a more quantitative\nstudy of the patterns using the toppling function. Numerical studies\nwith significantly larger $N$, can clarify whether or not the density\nfunction shows spatial discontinuities in the presence of low noise of the type shown in figures \\ref{fig:idtype1} and\n\\ref{fig:btwID}, and whether the excess density is exactly a constant\nwithin a patch. It is hoped that further work will clarify these\nissues.\n\nWe thank A. Libchaber for emphasizing the importance of introducing noise in our models, and Satya N. Majumdar for his\nconstructive comments on an earlier draft of this paper.\n\\section*{References}\n\\bibliographystyle{unsrt} \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Plain Language Summary}\n A large amount of images are retrieved by a specialized instrument designed\n to make observations in the ionosphere.\n These images are contaminated by instrumental noises.\n In order to recover useful signals from these noises, we train a deep\n learning model.\n A dataset containing the labeled signals are used for both training and \n validating the model performance.\n The desired signals are manually labeled using a labelling software.\n By comparing the model predictions with the labels, the\n results show that the model can well-identify the elongated, overlapping, \n or compact signals.\n The model is also capable of correcting some missing and incorrect labels.\n The performance of the model is sensitive to the data number of \n the corresponding labels fed to the model during training.\n The recovered useful signals are then used to estimate physical quantities\n which are important for the study of ionospheric physics.\n\n \\section{Introduction}\n The ionosphere is a region of ionized gases, plasmas,\n populating the upper atmosphere and thermosphere \\cite{intro}.\n The ionosphere consists of layers concentrated at specific heights.\n Radio waves propagate through the ionospheric layers at different group \n velocities and, hence, split into different wave modes\n according to the electron density, the magnetic field, etc.\n An experimental ground-based technique of ionosondes \n has been used for a long time to investigate the vertical profile of the \n ionospheric ionization represented by the density of free electrons, \n so-called electron content (EC).\n\n The data product of ionosonde measurements are ionograms,\n which exhibit signals deflected by the ionosphere at various\n virtual heights as a function of the sounding frequency.\n The virtual height of the deflection is obtained by assuming that the wave \n beams are propagating at the speed of light.\n The sounding frequency at which the virtual height rapidly increases is\n called the critical frequency, which also corresponds to the local\n maximum of the EC.\n In addition, the splitting in the sounding frequency between the signals\n of different wave modes is related to the local magnetic field.\n These ionogram parameters can be used for\n the true height analysis \\cite{true_height_1,true_height_2},\n estimating the magnetic field strength \\cite{handbook}, and modelling\n the electron density higher than the deflection height by the Chapman\n function \\cite{topside}.\n Furthermore, the stability of some ionogram interpretation\n algorithms \\cite{oxpoint_a,oxpoint_b} rely on the intersection point of\n the ordinary mode and the extraordinary mode signals.\n\n The ionograms from Hualien's\n Vertical Incidence Pulsed Ionospheric Radar (VIPIR) are\n featured by small and compact signals or thin and elongated signals.\n These ionograms are contaminated by stripe noises\n appearing in many frequency bands.\n \\citeA{snr} showed that the Hualien dataset has stronger\n noise signals compared with the Jicarmaca dataset \\cite{aeperu}.\n\n Thousands of measurements are produced by VIPIR per day.\n It is hard work and time-consuming for skilled researchers\n to recover ionospheric signals from such immense dataset.\n Therefore an automated method based on fuzzy logic \\cite{fuzzy}\n has been applied.\n In recent years, there are also deep learning techniques\n applied to the ionogram recovery \\cite{unetseg,dias,aeperu}.\n\n In this research, we implement a deep learning model to the\n Hualien VIPIR ionograms, and recover different ionogram signals.\n The data and the preprocessing of the dataset are presented in\n Section~\\ref{sec:data}.\n The deep learning model and the validation of the model are\n described in Section~\\ref{sec:model}.\n In Section~\\ref{sec:result}, \n we evaluate the performance of signal recovery (in \\ref{sec:recover}),\n and derive the ionogram parameters from the\n recovered ionograms (in \\ref{sec:param}).\n The results are discussed in Section~\\ref{sec:discuss}\n and summarized in Section~\\ref{sec:summary}.\n\n \\section{Data}\n \\label{sec:data}\n We use the ionograms acquired from the Hualien VIPIR digisonde\n operated at Hualien, Taiwan ($23.8973^\\circ$N, $121.5503^\\circ$E).\n The dataset contains $6131$ ionograms spanning from\n 2013\/11\/08 to 2014\/06\/29.\n Each ionogram covers a virtual height range up to 800km,\n and sounding frequency range from 1MHz to 22MHz,\n and the signal amplitude range up to 100 decibels (dB).\n\n To reduce the data size and remove the calibration signals,\n we reduce the size of ionograms to virtual height range from 66 to 600km,\n and sounding frequency from 1.58 to 20.25MHz.\n Different useful signals in each ionogram are manually identified and\n labeled into polygons using \\texttt{labelme} \\cite{labelme}.\n The polygons are rasterized into a binary array of dimension 800x1600x11,\n which correspond, respectively, to frequency, height and the label.\n The eleven labels of useful signals are defined as follows\n (see Figure~\\ref{fig:labeling}):\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=.8\\textwidth,clip,trim=0 0 0 0]{ionosphere.png}\n \\caption{A schematic diagram of ionospheric signals. Different signals are indicated by arrows with annotations (see the text for details).}\n \\label{fig:labeling}\n \\end{figure}\n\n \\begin{enumerate}\n \\item Eo: Ordinary signal of the E-layer, with the virtual height increasing as the sounding frequency increased.\n \\item Ex: Extra-ordinary signal of the E-layer, with the virtual height increasing as the sounding frequency increased.\n \\item Eso: Ordinary signal of the sporadic E-layer, with the virtual height decreasing as the sounding frequency increased.\n \\item Esx: Extra-ordinary signal of the sporadic E-layer, with the virtual height decreasing as the sounding frequency increased.\n \\item Es: Signal of the sporadic E-layer, with the virtual height constant as the sounding frequency increased.\n \\item F1o: Ordinary signal of the F1-layer, with the virtual height increasing as the sounding frequency increased.\n \\item F1x: Extra-ordinary signal of the F1-layer, with the virtual height increasing as the sounding frequency increased.\n \\item F2o: Ordinary signal of the F2-layer.\n \\item F2x: Extra-ordinary signal of the F2-layer.\n \\item F3o: Ordinary signal of the F3-layer, with the virtual height increasing as the sounding frequency increased.\n \\item F3x: Extra-ordinary signal of the F3-layer, with the virtual height increasing as the sounding frequency increased.\n \\end{enumerate}\n The $11$ signals shown in Figure~\\ref{fig:labeling} usually do not appear\n simultaneously in all ionograms, \n and some signals can be too faint to be labeled.\n The percentages of different labeled signals in our data set are\n 7.89\\% for Eo, 0.99\\% for Ex,\n 39.81\\% for Es, 13.34\\% for Eso, 10.59\\% for Esx,\n 39.19\\% for F1o, 26.06\\% for F1x,\n 94.65\\% for F2o, 88.90\\% for F2x,\n 0.13\\% for F3o and 0.16\\% for F3x.\n Eso, Esx, F3o, and F3x have very poor statistics.\n Therefore, we discarded F3o and F3x to increase the statistics of E layer,\n and combine Eso, Esx, and Es labels into the Esa label.\n As a result, we reduce $11$ labels into $7$ labels.\n The panels in Figure~\\ref{fig:lt_mm_7lab}a show the local time \n distribution of the occurrence of the $7$ labels in the bulge of the \n equatorial ionoization anomaly at Taiwan.\n During the studied time in Taiwan, Esa, F2o, and F2x labels have the \n highest occurrence rate, and Eo and Ex the lowest.\n Sporadic Esa and F2 (F2o and F2x) layers occur at all dayside\n local time hours.\n F1 layer (F1o and F1x) does not occur in the evening.\n The E layer (Eo and Ex) occurs mainly in the morning.\n The panels in Figure~\\ref{fig:lt_mm_7lab}b show the seasonal distribution \n of the occurrence of the layers.\n F2o and F2x as well as F1o and F1x appear in all seasons and have similar \n distributions.\n The sporadic layer Esa has the highest occurrence rate during\n the summer \\cite{e_summer}.\n Our focus is to recover the Esa, F2o, and F2x labels since the radio wave \n propagation in the Taiwan region is mostly affected by these layers,\n due to their high occurrence rate.\n\n \\begin{figure}\n \\includegraphics[width=\\textwidth,clip,trim=0 2cm 0 2cm]{dist_lt_mm.pdf}\n \\caption{The panels in (a) and in (b) are the local time distributions \n and seasonal distributions, respectively, of the 7 signal labels, as\n indicated above the corresponding panels. All panels follow the scale \n of the left most y-axis.}\n \\label{fig:lt_mm_7lab}\n \\end{figure}\n\n Finally, the dataset was split into ratios of 64\\%, 16\\% and 20\\%,\n respectively, for the training set, the validation set and the test set,\n resulting in 3925, 981, 1226 ionograms in each set.\n The coverage ratios of the seven labels in each set are\n shown in Table~\\ref{tab:datacov}.\n\n \\begin{table}\n \\centering\n \\begin{tabular}{*{8}{c}}\n \\hline\n & Eo & Ex & Esa & F1o & F1x & F2o & F2x \\\\\n \\hline\n Train (\\%) & 8.13 & 0.97 & 52.60 & 39.22 & 26.45 & 94.37 & 88.10 \\\\\n Validation (\\%) & 8.36 & 1.33 & 53.31 & 38.12 & 22.53 & 95.82 & 87.77 \\\\\n Test (\\%) & 6.77 & 0.82 & 52.04 & 39.97 & 27.65 & 94.62 & 88.34 \\\\\n \\hline\n \\end{tabular}\n \\caption{Coverage ratio of each signal label in percentage for \n training, validation and testing set.}\n \\label{tab:datacov}\n \\end{table}\n\n \\section{Methodology}\n \\label{sec:model}\n \\subsection{Deep Learning Model}\n The model employed for this study is Spatial-Attention U-Net (SA-UNet),\n developed by \\citeA{SDUNet,SAUNet}.\n The SA-UNet is featured by a U-Net \\cite{unet} architecture with a spatial\n attention module at the bottle-neck of the model structure.\n U-Net has been successful in classifying an image into different labels.\n With modifications to the U-Net, SA-UNet has been shown that it is capable \n of identifying vessels from the eyeball images \\cite{SAUNet}.\n The model implementation is also available on\n Github (\\url{https:\/\/github.com\/clguo\/SA-UNet}).\n Since vessels and ionogram traces are both tiny or elongated features,\n we consider that SA-UNet is suitable for the ionogram recovery.\n\n The architecture of the SA-UNet is shown in Figure~\\ref{fig:sa_unet}.\n When an ionogram is sent through the model,\n the convolution block extracts the features,\n and the pooling layer reduce the image size.\n The DropBlock \\cite{DropBlock1} randomly drops the image pixels to\n virtually increase the sample size, which can potentially reduce the\n overfitting.\n The spatial attention module rescales the features,\n so that the model can put more emphasis on important features.\n The features extracted are assembled and localized in the\n transpose convolution blocks.\n The skip-connections return the image size back to the original size.\n The last convolution layer outputs the probability of the\n seven signal labels at each pixel.\n The probability of each label is rounded to a binary value.\n Since the activation in the last layer is a sigmoid function,\n which does not normalize the output probability,\n the model retains the capability of predicting multiple labels\n for a same pixel.\n The drop rate of the DropBlock in the original SA-UNet is calculated\n as follows:\n \\begin{linenomath*}\n \\begin{eqnarray}\n \\gamma = \\frac{1 - P}{K^2}\\frac{H}{H - K + 1}\\frac{W}{W - K + 1},\n \\end{eqnarray}\n \\end{linenomath*}\n where $\\gamma$ is the drop rate,\n $K$ is the kernel size, $H$ is the height of the image,\n $W$ is the width of the image, and $P$ is the probability to keep the \n kernel.\n To reduce the computational complexity, we slightly modified the \n above formula of drop rate to the following:\n \\begin{linenomath*}\n \\begin{eqnarray}\n \\gamma = \\frac{1 - P}{K^2}.\n \\end{eqnarray}\n \\end{linenomath*}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth,clip,trim=0 10cm 0 0]{sa-unet.pdf}\n \\caption{The upper panel shows the architecture of the SA-UNet.\n The number on top of each rectangle (layer) indicates the \n dimensionality along the feature space of the input or the output.\n The lower panel shows the structure of the Spatial Attention module,\n which is the red arrow in the upper panel.\n \\texttt{N}, \\texttt{H}, \\texttt{W}, \\texttt{C} denote the \n dimensionality along the sample, height, width, and feature space, \n respectively.\n The asterisk and cross signs represent the convolution and the\n product of array elements, respectively.}\n \\label{fig:sa_unet}\n \\end{figure}\n\n \\subsection{Model Training, Evaluation and Optimization}\n In each training epoch, a mini-batch is fed into the model,\n and the loss is obtained by calculating the binary crossentropy between the\n ground truth and the prediction.\n The model weights are then updated by backpropagating the gradients \n obtained by the AMSGrad optimizer \\cite{amsgrad}.\n An epoch is completed after all mini-batches are used up.\n The learning rate is set to $0.001$ initially, and is halved until it \n reaches a minimum rate of $10^{-6}$ if no improvement of the loss of the \n validation data is found from the previous $5$ epochs.\n A final model is selected from all the epochs by manually checking the\n performance of the validation set.\n We found no significant improvement after $60$ epochs.\n\n We measure the model performance by the intersection over union (IoU).\n We modify the definition of IoU in order to apply this to $7$ labels.\n The IoU of each label $k$ in an individual ionogram $n$,\n $\\mathrm{IoU}_{n,k}$ is computed as the ratio of the total number of\n intersecting pixels of the ground truth and the prediction\n ($\\mathrm{I}_{n,k}\\equiv\\mathrm{truth}\\cap\\mathrm{prediction}$)\n over the total number of union pixels of ground truth and prediction\n ($\\mathrm{U}_{n,k}\\equiv\\mathrm{truth}\\cup\\mathrm{prediction}$).\n The mean IoU of each label $k$, $\\mathrm{IoU}_{k}$, is computed as the\n sum of the intersecting pixels over the sum of the union pixels of all\n ionograms.\n The equations for IoU$_{n,k}$ and mean IoU$_{k}$ are as follows:\n \\begin{linenomath*}\n \\begin{eqnarray}\n \\mathrm{IoU}_{n, k} &=& \\frac{\\mathrm{I}_{n,k}}{\\mathrm{U}_{n,k}},\\\\\n \\textrm{mean IoU}_k &=& \\frac{\\sum_{n=1}^{N}\\mathrm{U}_{n,k}\\cdot\n \\mathrm{IoU}_{n,k}}{\\sum_{n=1}^{N}\\mathrm{U}_{n,k}}, \\nonumber \\\\\n &=& \\frac{\\sum_{n=1}^{N}\\mathrm{I}_{n,k}}{\\sum_{n=1}^{N}\\mathrm{U}_{n,k}}.\n \\end{eqnarray}\n \\end{linenomath*}\n The mean IoU is calculated in such a way that we do not\n encounter zero-division for labels with both $I_{n,k}=0$ and $U_{n,k}=0$.\n An example of application of this technique is shown in\n Figure~\\ref{fig:echo_ffp_fp} (see Section~\\ref{sec:recover}).\n\n The model is optimized by tuning the hyperparameters.\n In this study, the hyperparameter includes the size of the convolution \n kernel, the keep probability of DropBlock, the size of the mini-batch.\n The default hyperparameters are kernel size of $3\\times3$,\n keep probability of $0.85$, and batch-size of $4$.\n To prevent a large searching grid, the hyperparameter is tuned individually\n while the others are fixed to their default values.\n The size of the convolution kernel varies between $3\\times3$, $5\\times5$,\n and $7\\times7$;\n the keep probability varies between $0.5$, $0.7$, and $0.85$;\n the size of the mini-batch varies between $1$, $2$, $4$, and $8$.\n This results in $10$ different combinations of hyperparameters.\n By manually comparing the IoUs of the validation data from the $10$ \n combinations of hyperparameters,\n we determine the optimal hyperparameters as kernel sizes of\n $7\\times7$, and the batch size of $4$.\n We found the optimal performance occurs at the $44$th epoch.\n\n \\section{Result}\n \\label{sec:result}\n \\subsection{Recovery of Ionogram Signals}\n \\label{sec:recover}\n\n The distribution of IoUs of each label in the test set is plotted\n in Figure~\\ref{fig:iou_dist} with the mean IoU printed in the corresponding\n panels.\n The figure shows that the most accurately recovered signals by\n SA-UNet are Esa, F2o, and F2x, with mean IoUs reaching approximately $0.7$.\n F1o and F1x are less well recovered, with mean IoU equal to\n $0.56$ and $0.39$, respectively.\n The near-zero mean IoU of Eo and Ex\n indicates that the model cannot identify these two signals.\n Comparing the mean IoUs of different labels and their coverage ratio in\n the training set (Table~\\ref{tab:datacov}) indicates that\n the two are highly related:\n The labels with coverage ratio over $50\\%$ are the best recovered\n labels, and the two labels, Eo and Ex, with the lowest coverage ratios\n are the worst recovered.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=.9\\textwidth]{nzIoUs.pdf}\n \\caption{Distribution of non-zero IoU of each label, and the \n corresponding mean IoU and median IoU.\n The vertical dashed lines in each panel mark the locations of\n the mean IoU and median IoU.\n }\n \\label{fig:iou_dist}\n \\end{figure}\n \n A notable amount of zero IoUs are observed.\n We divide these zero IoUs into three different types\n (see Table~\\ref{tab:zero_ious}).\n\n \\begin{table}\n \\centering\n \\begin{tabular}{*{8}{c}}\n \\hline\n & Eo & Ex & Esa & F1o & F1x & F2o & F2x\\\\\n \\hline\n Test set & 83 & 10 & 638 & 490 & 339 & 1160 & 1083\\\\\n Zero IoUs & 52 & 10 & 36 & 114 & 209 & 22 & 88\\\\\n FN & 28 & 10 & 14 & 3 & 11 & 3 & 1\\\\\n FFP & 17 & 0 & 9 & 16 & 95 & 9 & 52\\\\\n FP & 7 & 0 & 13 & 95 & 103 & 10 & 35\\\\\n \\hline\n \\end{tabular}\n \\caption{Rows from top to bottom are the number of instances in \n the test set, the number of instances with zero IoUs, \n the number of correctly labeled signals not identified by the model\n (FN); unlabled signals correctly identified by the model (FFP);\n and incorrectly predicted signals (FP).}\n \\label{tab:zero_ious}\n \\end{table}\n\n \\begin{enumerate}\n \\item\n Model fails to identify correctly labeled signals (false negative, FN):\n This happens when the model either fails to identify the signal or incorrectly identities it as other labels.\n Such situation commonly occurs for the signals with low coverage ratio,\n which can result in model being undertrained.\n In fact, all Ex instances in $10$ ionograms are identified as Eo label.\n\n \\item\n Model correctly identifies the signals that are not labeled\n (false false-positive, FFP):\n This happens when the model correctly identifies the signals that \n were not labeled or incorrectly labeled.\n The signals that are overlapping with other signals\n or are contaminated by strong noise\n are difficult for human to accurately label them.\n The correct identification of such signals by our model indicates its\n superior capability over human eye.\n One such example can be seen in Figure~\\ref{fig:echo_ffp_fp}a:\n there is a tiny Eo signal in the ionogram\n that was incorrectly labeled as Esa due to strong noise.\n The model correctly separates the signal into increasing (Eo) and \n the decreasing (Esa) part, \n resulting in zero IoU for Eo and lower IoU for Esa.\n \\item\n Model incorrectly identifies the signals that do not exist\n (false positive; FP):\n Such situation occurs when the model is overtrained to a specific label,\n thus trying to identify too many unrelated pixels as the label.\n For example, in panel (b) of Figure~\\ref{fig:echo_ffp_fp}, \n there are only F2o and F2x signals labeled in the ground truth\n data, but the model divides the beginning parts of F2o and F2x as\n F1o and F1x signals.\n \\end{enumerate}\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth,clip,trim=0 6.5cm 0 4.5cm]{out_example.pdf}\n \\caption{\n The panels from left to right are\n the ionograms, their corresponding ground truths, and \n model predictions.\n The IoU of each label is printed next to the panels of model prediction.\n The measurement time of each ionogram: \n (a) 2014\/06\/14 23:37:03, (b) 2013\/12\/01 00:55:03, \n (c) 2013\/12\/05 02:27:03, and (d) 2013\/12\/31 01:57:04.\n }\n \\label{fig:echo_ffp_fp}\n \\end{figure}\n\n In addition to different types of zero IoUs, mean IoU can also be\n decreased by echo signals being identified as the primary ones.\n Echo signals are the signals bouncing between the ground and the\n ionosphere.\n In the ionograms, they have similar shapes as the main signals,\n but appear at higher altitudes with weaker amplitudes.\n As shown in panel (c) of Figure~\\ref{fig:echo_ffp_fp}, the model correctly \n identifies the signals from the ionosonde measurement.\n However, in panel (d), the echo signals at higher altitude are also\n identified by the model as F2o and F2x labels,\n lowering their IoUs as a result.\n\n To investigate the local time and the seasonal dependency of the model\n performance,\n we consider the median value of IoU, and use the first and the third\n quartiles as the error bar.\n The panels in Figure~\\ref{fig:iou_lt}a show the local time variation \n of IoUs of Eo, Esa, F1o, F1x, F2o, and F2x of the test.\n Apart from 23:00, the median value of IoU of\n Esa, F2o, and F2x labels in general are independent of the local time.\n The panels in Figure~\\ref{fig:iou_lt}b show the seasonal variation of the \n IoUs.\n The IoUs of F1 (F2o) and F2 (F2o and F2x) layers have a tendency\n to decrease in the summer, while the IoU of Esa label increases.\n This could be caused by non-uniform statistics of the layers.\n Namely, sporadic Es has higher occurrence and intensity during\n summer months.\n Strong Es layer very often hide the F2 layer such that the statistics of \n those labels decrease in summer \\cite{snr}.\n\n \\begin{figure}\n \\includegraphics[width=.9\\textwidth,clip,trim=0 1cm 3.5cm 1.5cm]\n {iou_lt_mm.pdf}\n \\caption{The panels in (a) and in (b) are the local time and seasonal\n variations of IoUs, respectively, of the 7 signal labels, as\n indicated above the corresponding panels. All panels follow the scale\n of the left most y-axis.}\n \\label{fig:iou_lt}\n \\end{figure}\n\n\n \\subsection{Examination of Ionogram Parameters}\n \\label{sec:param}\n From the ionograms recovered by the deep learning model,\n we extract the virtual heights, the critical frequencies and\n the intersection frequencies by an automated procedure.\n In this study, we only present the parameters for F2 signals,\n because our model performs the best on F2 signals.\n The same analysis can also be applied to other signals.\n For a signal X $\\in\\left\\{\\mathrm{F2o, F2x}\\right\\}$ in the\n $n$-th recovered ionogram, $X_n(f_i, h'_j)$ at frequency $f_i$ and\n virtual height $h'_j$ is defined as:\n \\begin{linenomath*}\n \\begin{equation}\n X_{n}(f_i, h'_j) = \\left\\{\n \\begin{array}{l}\n 1,\\;\\mathrm{signal\\ of\\ X}; \\\\\n 0,\\;\\mathrm{otherwise.}\n \\end{array}\n \\right.\n \\end{equation}\n \\end{linenomath*}\n\n The virtual height of the F2 signal is defined as the minimum of $h'_j$\n where F2o$(f_i, h'_j)$ is non-zero:\n \\begin{linenomath*}\n \\begin{eqnarray}\n \\mathrm{h'F2}_n &=&\n \\min h'_j\\;\\mathrm{of}\\;\\mathrm{F2o}_n(f_i, h'_j), \\\\\n \\end{eqnarray}\n \\end{linenomath*}\n and the critical frequencies\n foF2 and fxF2 are defined as the maximum of $f_i$ \n at which F2o$(f_i, h'_j)$ and F2x$(f_i, h'_j)$ are non-zero:\n \\begin{linenomath*}\n \\begin{eqnarray}\n \\mathrm{foF2}_n &=& \\max f_i\\;\\mathrm{of}\\;\\mathrm{F2o}_n(f_i, h'_j), \\\\\n \\mathrm{fxF2}_n &=& \\max f_i\\;\\mathrm{of}\\;\\mathrm{F2x}_n(f_i, h'_j),\n \\end{eqnarray}\n \\end{linenomath*}\n and the intersection frequency between the\n F2o and the F2x signals, is defined as the first intersection point\n from the high frequency.\n \\begin{linenomath*}\n \\begin{equation}\n \\mathrm{Intersection\\ frequency(F2o, F2x)} = \n \\max{f_i \\; \\mathrm{ of } \\; (\\mathrm{F2o}_n \\cap \\mathrm{F2x}_n)}\n \\end{equation}\n \\end{linenomath*}\n \n \n \n \n \n \n For F2o and F2x signals with large overlapping area,\n the extracted intersection frequency is not reliable and cannot be used.\n We consider the overlapping area greater than $0.3$ of their combined area\n as highly overlapped.\n Out of $1226$ ionograms recovered from the test set,\n $702$ are below the threshold.\n One example is shown in Figure~\\ref{fig:intersect}a.\n The corresponding original ionogram is measured at 2013\/12\/05 02:27:03,\n the same timestamp as the ionogram in Figure~\\ref{fig:echo_ffp_fp}c.\n For this timestamp, the virtual height of F2 is $250.46$ km,\n the critical frequencies foF2 and fxF2 are $10.42$ MHz,\n and $11.14$ MHz, respectively, and the intersection frequency between\n the F2o signal and the F2x signal is $9.25$ MHz.\n\n The distributions of foF2 and fxF2 are plotted in\n Figure~\\ref{fig:intersect}b.\n It shows that the extracted foF2 varies from $4.07$ MHz to $19.36$ MHz.\n The distribution of the difference between fxF2 and foF2 (fxF2$-$foF2)\n is shown in Figure~\\ref{fig:intersect}c.\n It shows a distribution close to a Gaussian profile, with a tail on the\n left-hand side.\n The peak of the distribution is $0.63$ MHz, and the full-width half-maximum\n (FWHM) is $0.36$ MHz, making the uncertainty of the difference (FWHM$\/2$)\n to be $0.18$ MHz.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth,clip,trim=0 2.5cm 0 2.5cm]\n {intersect.pdf}\n \\caption{(a) The recovered ionogram at 2013\/12\/05 02:27:03 showing\n only F2o and F2x signals.\n The corresponding critical frequencies of F2o and F2x, and the\n intersection frequency between the two signals are $10.42$ MHz,\n $11.14$ MHz, and $9.25$ MHz, respectively, marked by the red, blue,\n and the black dashed line.\n (b) The distribution of foF2 (blue bars) and fxF2 (orange bars).\n (c) The distribution of fxF2 $-$ foF2. Red dashed line indicates\n the location of the maximum, and black dashed lines indicate the\n location of the half-maximum.}\n \\label{fig:intersect}\n \\end{figure}\n\n \\section{Discussion}\n \\label{sec:discuss}\n Studies have shown that the deep learning models are able to scale the\n ionograms automatically.\n The reported performance of the deep learning models are summarized\n and compared with our result in Table~\\ref{tab:ious_model}.\n \\citeA{aeperu} applied the Autoencoder model (denoted by the superscript a)\n to extract F region signals\n from Peru's ionograms, and obtained IoU$\\approx 0.6$ for the combined\n F layer signals (F1 $+$ F2), after fine-tuning the model parameters.\n The Fully Convolutional DenseNet model (denoted by the superscript b)\n used in \\citeA{snr} to recover\n the ionospheric signals obtained an IoU of nearly $0.6$ for Esa and\n F2o layers.\n \\citeA{unetseg} used multiple U-Net models (denoted by the superscript c)\n to scale E, F1, F2 layers,\n and obtained the dice-coefficient loss (DCL) of $0.16$, $0.17$, and $0.11$\n for E, F1 and F2 layers, respectively, in the ionograms of 64x48 pixels,\n and $0.22$, $0.23$, and $0.19$ in the ionograms of 192x144 pixels.\n Note that a smaller DCL indicates a better performance.\n Our SA-UNet model, applied to highly noisy ionograms of 800x1600 pixels,\n achieves an IoU $\\ge 0.7$ and DCL score $\\le 0.18$ for Esa and F2 layers.\n To provide additional information for interested readers, \n we also list the recall rates and precision rates achieved \n by our model, which are calculated\n by counting the pixel number of true positives, false positives, and\n false negatives.\n It should be noted that the ionograms \n used by \\citeA{aeperu} and \\citeA{unetseg}\n have been obtained either at middle latitudes or\/and in the remote regions\n with low man-made signals.\n In short, our study shows that the SA-UNet can improve the performance to\n IoU $\\approx 0.7$ and reduce DCL to below 0.18.\n\n \\begin{table}\n \\centering\n \\begin{tabular}{*{10}{c}}\n \\hline\n & Size (HxW) & Metric & Eo & Ex & Esa & F1o & F1x & F2o & F2x\\\\\n \\hline\n Autoencoder$^a$ & 256x208 & IoU & N\/A & N\/A & N\/A\n & \\multicolumn{4}{c}{0.60} \\\\\n \\hline\n FC-DenseNet$^b$ & 800x1600 & IoU & 0.00 & 0.00 & 0.56 & 0.47 & 0.33 & 0.59 & 0.48\\\\\n \\hline\n U-Net$^c$ & 64x48 & DCL & \\multicolumn{3}{c}{0.16}\n & \\multicolumn{2}{c}{0.17} & \\multicolumn{2}{c}{0.11}\\\\\n & 192x144 & DCL & \\multicolumn{3}{c}{0.22}\n & \\multicolumn{2}{c}{0.23} & \\multicolumn{2}{c}{0.19}\\\\\n \\hline\n SA-UNet & 800x1600 & IoU & 0.25 & 0.00 & 0.74 & 0.56 & 0.39\n & 0.73 & 0.70\\\\\n & & Recall & 0.28 & 0.00 & 0.85 & 0.73 & 0.51\n & 0.83 & 0.83 \\\\\n & & Precision & 0.68 & N\/A & 0.85 & 0.71 & 0.61 \n & 0.86 & 0.82 \\\\\n & & DCL & 0.61 & 1.00 & 0.15 & 0.28 & 0.44 \n & 0.16 & 0.18 \\\\\n \\hline\n \\end{tabular}\n \\caption{The reported performance of different models, and the \n corresponding size of the input ionograms in units of pixels.\n DCL denotes the dice-coefficient loss.\n All metrics are calculated pixel-wise.}\n \\label{tab:ious_model}\n \\end{table}\n\n The critical frequency foF2 is directly related to the maximum number\n density $N$ of electrons in F2 layer, and the difference between\n fxF2 and foF2 can provide an estimation of the local geomagnetic field\n $B$ \\cite{handbook}:\n \\begin{linenomath*}\n \\begin{eqnarray}\n N &=& 1.24\\times10^{10}\\times(\\mathrm{foF2} \/ \\mathrm{MHz})^2\n \\;[\\mathrm{m}^{-3}] \\\\\n B &\\approx& 0.71\\times(\\mathrm{fxF2} - \\mathrm{foF2})\\;[\\mathrm{G}].\n \\end{eqnarray}\n \\end{linenomath*}\n\n Our results show that the derived electron number density ranges from\n $1.66\\times10^{11}$ m$^{-3}$ to $3.75\\times10^{12}$ m$^{-3}$,\n and the magnetic field is $B \\approx 0.45\\pm0.13$ G.\n Since the local variation of the geomagnetic field is usually small,\n the large uncertainty of the magnetic field is likely caused by strong\n noises in the ionograms.\n We also find 76 recovered ionograms with foF2 $>$ fxF2.\n They are caused by both false positive and false negative predictions.\n\n The existence of echoes does not affect the extraction of\n critical frequencies.\n However, cares should be taken for extracting the critical\n virtual heights.\n In addition to the local geomagnetic field,\n the difference between foF2 and fxF2 can also be used to estimate\n foF2 (i.e. $\\mathrm{foF2} \\approx \\mathrm{fxF2} - 0.63\\,\\mathrm{MHz}$)\n in the case when fxF2 is obtained but F2o cannot be accurately recovered.\n\n The intersection frequency can be used for the verification of the ordinary\n and extra-ordinary signals from the F2 layer.\n Namely, the ascending branch of the ordinary signal should be situated\n at lower frequencies than the extra-ordinary one.\n This technique can provide a robust correction of inaccurate model\n predictions of the signals from the F2 layer and, thus,\n more accurate determination of the critical frequencies foF2 and fxF2.\n\n The low percentages of \n Eo and Ex labels in the training, validation and the test set \n could bias the model performance.\n In order to generalize the data representation,\n an k-fold or a leave-one-out cross validation may be applied \\cite{cv1,cv2}.\n Moreover, there are a few techniques which may reduce the \n class imbalance problem, such as undersampling\/oversampling to the \n training set \\cite{ousampling},\n loss weighted according to the amount of the labels \\cite{wtloss1,wtloss2},\n transfer learning from a larger dataset, or ensemble learning of multiple \n models, to name a few.\n\n It is important to note that the present dataset spans less than\n one year, and includes only ionospheric statistics at low latitudes\n under very dynamic region of the bulge of equatorial ionization anomaly.\n The ionosphere dynamics depends on latitude and varies with solar and \n geomagnetic activity.\n Hence, to enable the application of our model to other datasets,\n the model should be trained on the combination of different data set.\n Alternatively,\n the transfer learning technique can be applied to reduce\n the training time and improve the model performance on a different data set.\n\n \\section{Conclusions}\n \\label{sec:summary}\n In this study, we show that SA-UNet is capable of recovering\n F2o and F2x signals, and the Esa label (which is a combination of\n Eso, Esx, and Es signals) from the highly contaminated Hualien VIPIR\n measurements.\n While the performance of identifying Eo and Ex \n is poor due to the class imbalance, their low percentage \n in the statistics suggests that they have little effect \n on the ionosphere in the Taiwan region.\n By comparing the input ionogram, ground truth labeling,\n and the model prediction, our results show that the model is capable of \n recovering signals that are not labeled, due to contamination by strong \n noise or overlapping with another signal.\n The recovered ionograms can be further used in extracting the\n virtual height, the critical frequency and the intersection frequency of \n different labels, which are important in the determination of the electron \n density profile and the magnetic field of the ionosphere overhead.\n\n \\section{Open Research}\n The ionogram data and the model used in this study can be openly accessed on\n Kaggle.\n The link for the data is provided as follows:\n \\url{https:\/\/www.kaggle.com\/changyuchi\/ncu-ai-group-data-set-fcdensenet24}.\n The link for the model is provided as follows:\n \\url{https:\/\/www.kaggle.com\/guanhanhuang\/ncu-ai} \\cite{ncuai}.\n The model is constructed using Tensorflow 2.3 \\cite{tensorflow} and\n its Keras packages,\n and is trained on Kaggle under the GPU computation environment.\n\n \\acknowledgments\n This work is funded by the Ministry of Science and Technology of Taiwan\n under the grant number 109-2923-M-008-001-MY2 and 109-2111-M-008-002.\n The figures are made by using the Matplotlib package \\cite{matplotlib}.\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nConsider two monic, degree $n\\geq 1$ complex polynomials\n\\[ \\phi(z)=z^n+a_{n-1}z^{n-1}+\\cdots +a_1z+a_0\\]\nand \\[\\psi(z)=z^n+b_{n-1}z^{n-1}+\\cdots +b_1z+b_0.\\]\n\nA beautiful classical fact is that the condition for $\\phi$ and $\\psi$ to have a common root is polynomial in the coefficients $a_i$ and $b_j$. More precisely, $\\phi$ and $\\psi$ have a common\n root if and only if\n \\begin{equation}\n \\label{eq:res=0}\n\\Res(\\phi,\\psi):=\\Res(a_0,\\ldots a_{n-1},b_0,\\ldots ,b_{n-1})= 0\n\\end{equation}\nwhere $\\Res$ is the {\\em resultant}, given by\n\\[\n\\Res(\\phi,\\psi)=\\det\n\\left[\n\\begin{array}{cccccccc}\na_0&a_1& \\cdots &a_{n-1} & 1&0 & \\cdots &0 \\\\\n0&a_0&\\cdots&\\cdots&a_{n-1}& 1& \\cdots & 0\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots &\\vdots &\\vdots\\\\\nb_0&b_1&\\cdots&b_{n-1}&1&0&\\cdots &0\\\\\n0&b_0&\\cdots&\\cdots&b_{n-1}&1&\\cdots &0\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots& \\vdots\\\\\n0&0&\\cdots& b_0&\\cdots&\\cdots&b_{n-1}&1\n\\end{array}\n\\right]\n\\]\n\nThis is a homogeneous polynomial of degree $n$ in the $a_i$ and similarly in the\n$b_i$. It has integer coefficients. Fix a field $k$, and denote by $\\mathbb{A}^n$ the affine space over\n$k$. The resultant can be thought of as a map\n\\[\\Res:\\mathbb{A}^{2n}\\to\\mathbb{A}^1\\]\nfrom the space $\\mathbb{A}^{2n}$ of pairs of monic, degree $n$ polynomials to $k$. The {\\em resultant locus} $\\mathcal{M}_n:=\\mathbb{A}^{2n}\\setminus \\Res^{-1}(0)$ is a classically studied object. It is isomorphic to the moduli space of degree $n$ rational maps $\\mathbb{P}^1\\to\\mathbb{P}^1$ taking $\\infty$ to 1. Harder to understand is the {\\em ``resultant $=1$'' hypersurface} $\\Res_n:=\\Res^{-1}(1)$ in $\\mathbb{A}^{2n}$.\n\nSince the polynomial $\\Res$ has integer coefficients, we can extend scalars to $\\mathbb{C}$ and consider the complex points $\\Res_n(\\mathbb{C})$, and we can also reduce modulo $p$ for any prime $p$. This gives a variety defined over $\\mathbb{F}_p$, and\nfor any positive power $q=p^d$ we can consider both the $\\mathbb{F}_q$-points as well as the $\\overline{\\Fb}_q$-points of $\\Res_n$, where $\\overline{\\Fb}_q$ is the algebraic closure of $\\mathbb{F}_q$. Three of the most fundamental arithmetic invariants attached to a such a variety $\\Res_n$ are:\n\n\\begin{enumerate}\n\\item The cardinality $|\\Res_n(\\mathbb{F}_q)|$.\n\n\\item The \\'{e}tale cohomology $H_{{et}}^*(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$, where $\\ell$ is a prime not dividing $q$.\n\n\\item The eigenvalues of the (geometric) Frobenius morphism \\[\\Frob_q: H_{{et}}^*(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)\\to H_{{et}}^*(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell).\\]\n\\end{enumerate}\n\nOur main theorems compute the \\'{e}tale cohomology of $\\Res_n$ as well as the associated eigenvalues of Frobenius, building on the topological work of Segal and Selby \\cite{SS}. We then apply this to compute the cardinality of finite field versions of these moduli spaces; that is, of $\\Res_n(\\mathbb{F}_q)$ and $X_n(\\mathbb{F}_q)$, where $\\mathbb{F}_q$ is a finite field.\n\nThere is a canonical $\\mu_n$-action on $\\Res_n$; see Section \\ref{sec:action}. This induces a $\\mu_n$-action on $H_{{et}}^*(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$. Since $\\mathbb{Q}_\\ell$ has characteristic $0$, it follows that $H_{{et}}^*(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)\\otimes_{\\mathbb{Q}_\\ell}\\mathbb{C}$ decomposes into a direct sum of irreducible representations of $\\mu_n$. The irreducible representations of $\\mu_n$ are parametrized by integers $m$ with $0\\leq m0$, $H^{2i}_{{et}}(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$ is nonzero if and only if $i1$. Our analysis proceeds in a series of steps.\n\n\\paragraph{Step 1 (The $\\mu_n$-isotopic decomposition of $H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$): }\n\nFor each $i\\geq 0$ there is a decomposition of $H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$ into $\\mu_n$-isotypic components :\n\\begin{equation}\n\\label{eq:isotyp4}\nH_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)\\cong H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)^{\\mu_n}\\bigoplus H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q})^{\\mu_n^\\perp}\n\\end{equation}\nwhere \\[H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)^{\\mu_n^\\perp}\\otimes\\mathbb{C}:=\\bigoplus_{m=1}^{n-1} H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q})_m.\\]\nThe decomposition \\eqref{eq:isotyp4} is invariant under the action of $\\Frob_q$. However,\nfor $m>0$ the subspace $H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q})_m$ is in general {\\em not} $\\Frob_q$-invariant. In fact, this failure of invariance is at the crux of the proof of the theorem.\nWe begin by finding the $\\Frob_q$-invariant subspaces.\n\nTo this end, for any factor $a$ of $n$, define\n\\begin{equation*}\n \\mathcal{O}_{a}:=\\{ m ~|~ 1\\le m\\le n-1,~(m,n)=a\\}\n\\end{equation*}\nand define\n\\begin{equation*}\n H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q})_a:=\\bigoplus_{m\\in\\mathcal{O}_a} H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q})_m.\n\\end{equation*}\nFix $n\\geq 1$ and assume that $(q,n)=1$. We claim that for each $i\\geq 0$ the splitting\n\\begin{equation}\\label{eq:newdecomp}\n H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)^{\\mu_n^\\perp}\\otimes\\mathbb{C}\\cong\\bigoplus_{a\\mid n} H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q})_a\n\\end{equation}\nis $\\Frob_q$-equivariant. To see this, note that since $(q,n)=1$, multiplication of $q$ acts by an automorphism of $\\mathbb{Z}\/n\\mathbb{Z}$, and so preserves the order of elements in $\\mathbb{Z}\/n\\mathbb{Z}$. For $m\\in \\mathcal{O}_a$, the order of $m$ in $\\mathbb{Z}\/n\\mathbb{Z}$ equals $n\/a$. Thus\nthe order of $qm$ in $\\mathbb{Z}\/n\\mathbb{Z}$ is also $n\/a$, and thus the greatest common divisor of $qm \\ {\\rm mod}\\ n$ and $n$ equals $a$, proving the claim.\n\n\\paragraph{Step 2 (The $\\mu_n$-invariant part of \\boldmath$H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$): }\n\nFirst note that $\\Res-1$ is an irreducible polynomial. Indeed, $\\Res$ is an irreducible polynomial \\cite[Section 77]{Va} and\n\\begin{lemma}\n Let $\\Phi(x_1,\\ldots,x_n,y_1,\\ldots,y_m)$ be an irreducible, bi-homogeneous polynomial of bi-degree $(p,q)$. Then $\\Phi-1$ is irreducible.\n\\end{lemma}\n\\begin{proof}\n Suppose $\\Phi-1=PQ$. Without loss of generality, we can assume that $P$ and $Q$ are of total degrees $c$ and $d$. Write $P=P_0+P_1$ where $P_0$ is homogenous of total degree $c$ and $\\deg(P_1)1$. Over $\\mathbb{C}$, $\\pi_1(\\mathcal{M}_n\/\\mathbb{G}_m)\\cong\\pi_1(\\Res_n\/\\mu_n)\\cong\\mu_n$ since\n$\\pi_1(\\Res_n)=0$. Now use the fact that the $\\mu_n$ action on $H_{{et}}^j(\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$ is the restriction of the action of $\\mathbb{G}_m$ induced by left multiplication. Over $\\mathbb{C}$, since $\\mathbb{C}^*$ is connected, this action is trivial. Thus, after perhaps throwing away finitely many primes, naturality of base change (Theorem~\\ref{theorem:monopole:main}) implies that the $\\mu_n$ action on $H_{{et}}^j(\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)$ is trivial. We have thus shown:\n\\[\nE_2^{i,j}=\n\\left\\{\n\\begin{array}{ll}\nH_{{et}}^i({\\mathcal{M}}_n\/\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell(-j))\n& \\text{if $j=0,1$}\\\\\n0&\\text{else}\n\\end{array}\n\\right.\n\\]\nThe differential $d_2^{i,j}: E_2^{i,j}\\to E_2^{i+2,j-1}$ thus gives, for each $i\\geq 0$, a homomorphism\n\\[\n d_2^{i,1}:H_{{et}}^i({\\mathcal{M}}_n\/\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell(-j)))\\to H_{{et}}^{i+2}({\\mathcal{M}}_n\/\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell(1-j)).\n\\]\nSince $E_2^{i,j}=0$ for $i>1$ and $j<0$, the only nontrivial differentials occur on the $E_2$ page, and, for each $i>0$:\n\\begin{equation}\n \\label{eq:MnSS1}\n H_{{et}}^i({\\mathcal{M}}_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_{\\ell})\\cong \\ker(d_2^{i-1,1})\\oplus H_{{et}}^i({\\mathcal{M}}_n\/\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell(0))\/\\image(d_2^{i-1,1})\n\\end{equation}\nwhile $H_{{et}}^0({\\mathcal{M}}_n\/\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell(0))\\cong H_{{et}}^0({\\mathcal{M}}_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)\\cong \\mathbb{Q}_\\ell(0)$.\n\nEquation \\eqref{theorem:Mn:etalcoho} now gives that\n$H_{{et}}^i({\\mathcal{M}}_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_{\\ell}) \\cong \\mathbb{Q}_{\\ell}(-i)$ for $i=0,1$ and equals $0$ for $i>1$. Now, the target of $d$ on $E_2^{i,j}$ is $0$ for $i=4n,4n-1$, so these entries vanish. Working backwards, starting at $i=4n$ and working down to $i=1$, we can apply apply Equation \\eqref{eq:MnSS1} using that the left-hand side equals $0$, to conclude that $H_{{et}}^i({\\mathcal{M}}_n\/\\mathbb{G}_{m\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)=0$ for $i\\geq 1$. This concludes the computation of\n$H_{{et}}^i(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell)^{\\mu_n}$.\n\n\\paragraph{Step 3 (The $H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q})_a$): }In this step we analyze the individual summands $H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q})_a$ of the decomposition in Equation \\eqref{eq:newdecomp}. We will prove:\n\n\\[H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q})_a\n\\cong \\left\\{\n\\begin{array}{ll}\n\\left(\\bigoplus_{m\\in\\mathcal{O}_a} \\mathbb{Q}_\\ell(0)\\right)\\otimes\\mathbb{Q}_\\ell(a-n)\\otimes\\mathbb{C} &j-2(n-a)=0\\\\\n0&j\\neq 0\n\\end{array}\\right.\n\\]\n\nGiven this claim, the bijection\n\\begin{align*}\n \\mathcal{O}_a&\\to^\\cong\\{m'\\le \\frac{n}{a}~|~\\gcd(m',\\frac{n}{a})=1\\}\\\\\n m&\\mapsto\\frac{m}{a}\n\\end{align*}\nimplies that $H^i_{{et}}(\\Res_{n\/\\overline{\\Fb}_q};\\mathbb{Q}_\\ell))$ has rank $\\phi(\\frac{n}{n-\\frac{i}{2}})$. To prove the claim, recall that for $a|n$ we defined\n\\begin{equation*}\n \\mathcal{O}_{a}:=\\{ m ~|~ 1\\le m\\le n-1,~(m,n)=a\\}.\n\\end{equation*}\nFor any $m\\in \\mathcal{O}_a$ note that the order of $e^{2\\pi i m\/n}$ is $n\/a$. For each $a | n$, define\n\\[\n Y_{n,a}:=\\{\\frac{\\phi}{\\psi}\\in \\Res_n~:~\\psi(z)=\\chi(z)^{n\/a} \\ \\ \\text{for some $\\chi(z)\\in k[z],~\\deg(\\chi)=a$}\\}.\n\\]\nOver any field $K$ containing a \\emph{primitive} $n^{th}$ root of unity, Segal and Selby \\cite[Proposition 2.1]{SS} construct an isomorphism\\footnote{While Proposition 2.1 of \\cite{SS} is stated only over the field $\\mathbb{C}$, the proof works {\\it verbatim} over any field $K$ containing a primitive $n^{th}$ root of unity.}\n\\begin{equation}\\label{eq:SSiso}\n Y_{n,a\/K}\\cong \\mu_n\\times_{\\mu_a} (\\Res_{a\/K}\\times \\mathbb{A}_K^{n-a}).\n\\end{equation}\n\nIn fact, as we now show, these varieties are isomorphic over $K=\\mathbb{F}_q$ for any $q$.\n\n\\begin{proposition}\\label{prop:SS}\n The isomorphism \\eqref{eq:SSiso} is defined over $\\mathbb{F}_q$, i.e.\n \\begin{equation*}\n Y_{n,a}\\cong \\mu_n\\times_{\\mu_a} (\\Res_a\\times \\mathbb{A}^{n-a})\n \\end{equation*}\n as $\\mathbb{F}_q$-varieties.\n\\end{proposition}\n\\begin{proof}\n The homogeneity of the resultant implies that $\\Res(\\phi,\\chi^{n\/a})=\\Res(\\phi,\\chi)^{n\/a}$. Thus, for any $\\frac{\\phi}{\\psi}\\in Y_{n,a}$, $\\Res(\\phi,\\psi)$ is an $n\/a^{th}$ root of unity. Over $\\overline{\\Fb}_q$, this gives a decomposition\n \\begin{equation*}\n Y_{n,a}\\cong\\coprod_{\\lambda\\in\\mu_{n\/a}} Y_{n,a,\\lambda}\n \\end{equation*}\n where $Y_{n,a,\\lambda}=\\Res^{-1}(\\lambda)\\cap Y_{n,a}$. Following Segal and Selby, given $\\frac{\\phi}{\\chi^{n\/a}}\\in Y_{n,a}$ we can write\n \\begin{equation*}\n \\phi=\\phi_0\\chi+\\phi_1\n \\end{equation*}\n where $\\deg(\\phi_0)