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+{"text":"\\section{Introduction}\nRecently discovered topologically nontrivial phases have attracted many researchers and offered a new direction to modern physics\\cite{Hasan2010,Qi2011}.\nTopologically nontrivial phase and trivial phase, in the presence of time-reversal symmetry, are distinguished by the $Z_2$ invariant\\cite{Fu2007,Fu2007a}.\nStrong spin-orbit coupling is known to be essential to realize topological phases, since topological phases originate in the parity change in the lowest unoccupied band from even to odd induced by spin-orbit coupling.\nTopological phases are characterized by the gapless edge (surface) states which are protected by time-reveral symmetry.\nIn 3D topological insulators, the surface states are described by the two-component massless Dirac fermions.\nThe bulk states in such as Bi$_2$Se$_3$ are described by the four-component anisotropic massive Dirac fermions\\cite{Zhang2009}.\nIt is known that the surface states are robust against perturbation and disorder\\cite{Bardarson2007,Nomura2007}.\nWhat about against electron correlation,  i.e. Coulomb interaction?\nThis is a natural question, because it has been revealed that strong electron correlation is important in many systems and may induce novel phenomena.\n\nA novel Mott-insulating phase was found recently in an iridate\\cite{Kim2008}, a $5d$-electron system, and has gathered much attention.\nRemarkably, the phase is induced by the cooperation of strong spin-orbit coupling and strong electron correlation.\nEvolved by this discovery, many studies have been done intensively in systems where both spin-orbit coupling and electron correlation exist, for the search for novel phases induced by them.\nEspecially, it is of interest that topological phases such as the quantum spin Hall insulator\\cite{Shitade2009} and the Weyl semimetal\\cite{Wan2011} are predicted in iridates.\nThese results suggest that topological phases may emerge in strongly correlated $d$-electron systems.\nPreceding studies mainly focus on the competition between the spin or charge ordered phase and the topological phase in Hubbard-like models on honeycomb lattices\\cite{Raghu2008,Meng2010,Rachel2010,Varney2010,Hohenadler2011,Yamaji2011,Ruegg2011,Zheng2011,Yu2011}, other 2D lattices\\cite{Sun2009,Wen2010,Yoshida2012,Yoshida2012a,Hohenadler2012} and 3D lattices\\cite{Zhang2009a,Pesin2010,Mong2010,Kurita2011}.\nAnother study on the surface Dirac fermions shows that the Dirac fermions become massive with finite correlation strength due to the spotaneous magnetization\\cite{Baum2012}.\n\nOn the other hand, the electron correlation effect in graphene, a two-dimensional Dirac fermion system, has been studied widely.\nIn graphene in vacuum, the coupling constant becomes effectively large due to the small Fermi velocity. It has been predicted that a finite band gap is induced in charge neutral graphene in vacuum.\nIn such a case, the strong coupling lattice gauge theory is applied\\cite{Hands2008,Drut2009,Drut2009a,Armour2010,Drut2010,Araki2010,Araki2011,Araki2012,Buividovich2012}.\nThe chiral condensate is the order parameter for the insulator-semimetal transition in the lattice gauge theory.\nIt is noteworthy that lattice Monte Carlo studies show quantitatively correct critical value of the coupling strength below which the system becomes gapless\\cite{Drut2009,Drut2009a,Buividovich2012} (graphene on a SiO$_2$ substrate is conducting).\nThese results motivated us to do this study.\n\nIn this paper, we focus on the strong electron correlation effect in a 3D Dirac fermion system on a lattice which is a simple model describing a topologically nontrivial state.\nWe adopt $1\/r$ long-range Coulomb interaction as an interaction between the bulk electrons, because the screening effect in Dirac fermion systems is considered to be weak due to the vanishing of the density of states.\nThis situation is nothing but what is described by the U(1) lattice gauge theory.\nTherefore, we can perform the strong coupling expansion of the lattice gauge theory by assuming that the effective coupling constant is large.\nThe procedure is as follows.\nFirst we derive the effective action by the strong couling expansion.\nNext we calculate the effective potential (the free energy per unit volume at zero temperature) with the use of the Hubbard-Stratonovich transformation and the mean-field approximation.\nFinally we obtain the value of the chiral condensate as the stationary point of the effective potential.\nOur model, the Wilson fermions, breaks chiral symmetry by itself, and thus we cannot use the chiral condensate as the order parameter for the the insulator-semimetal transition.\nWe regard the chiral condensate as a correction to the bare mass.\n\nThe main purpose of this study is devided into two parts:\n(I) answer the question that whether the topological insulator phase survives at the limit of infinitely strong Coulomb interaction between the bulk electrons, or not. \nTo do this, we have to obtain the value of the chiral condensate, which corresponds to a correction to the bare mass, in the strong coupling limit.\n(II) search for the phase in which time-reversal and inversion symmetries are spontaneously broken due to electron correlation.\nSuch a phase, \"Aoki phase\" has been confirmed in the lattice quantum chromodynamics (QCD) with Wilson fermions\\cite{Aoki1984,Aoki1986,Sharpe1998} and was suggested recently in a mean-field study of Wilson fermions with the short-range interaction\\cite{Sekine2012}.\n\n\\section{Model}\nIt is known that the effective Hamiltonian of 3D topological insulators such as Bi$_2$Se$_3$ is described by the Wilson fermion\\cite{Zhang2009}:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{H}_{0}(\\bm{k})={\\sum}_j\\sin k_j\\cdot\\alpha_j+m(\\bm{k})\\beta,\\label{H_0}\n\\end{aligned}\n\\end{equation}\nwhere $m(\\bm{k})=m_0+r\\sum_j\\left(1-\\cos k_j\\right)$, $r>0$, $j\\ (=1,2,3)$ denotes spacial axis, and $\\alpha_j$, $\\beta$ are the Dirac gamma matrices given by\n\\begin{equation}\n\\begin{aligned}\n\\alpha_j=\n\\begin{bmatrix}\n0 & \\sigma_j\\\\\n\\sigma_j & 0\n\\end{bmatrix},\\ \\ \\ \\ \\\n\\beta=\n\\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix}.\n\\end{aligned}\n\\end{equation}\nThe energy of this system is measured in units of $v_{\\mathrm{F}}\/a$ with $v_{\\mathrm{F}}$ and $a$ being the Fermi velocity and the lattice constant, respectively.\nThe Hamiltonian (\\ref{H_0}) has time-reversal ($\\mathcal{T}$) symmetry and inversion ($\\mathcal{I}$) symmetry, i.e., $\\mathcal{T}\\mathcal{H}_{0}(\\bm{k})\\mathcal{T}^{-1}=\\mathcal{H}_{0}(-\\bm{k})$ and $\\mathcal{I}\\mathcal{H}_{0}(\\bm{k})\\mathcal{I}^{-1}=\\mathcal{H}_{0}(-\\bm{k})$ are satisfied, where $\\mathcal{T}=\\bm{1}\\otimes(-i\\sigma_2)\\mathcal{K}$ ($\\mathcal{K}$ is the complex conjugation operator) and $\\mathcal{I}=\\sigma_3\\otimes\\bm{1}$.\nIn the Hamiltonian (\\ref{H_0}), the spinor is written in the basis of $\\left[c^\\dag_{\\bm{k}A\\uparrow},c^\\dag_{\\bm{k}A\\downarrow},c^\\dag_{\\bm{k}B\\uparrow},c^\\dag_{\\bm{k}B\\downarrow}\\right]$, where $c^\\dag$ is the creation operator of an electron, $A$, $B$ denote two orbitals, and $\\uparrow$ ($\\downarrow$) denotes up- (down-) spin\\cite{Zhang2009}.\n\nIn the presence of time-reversal symmetry and inversion symmetry, the $Z_2$ invariant of the system is given by\\cite{Fu2007,Fu2007a}\n\\begin{equation}\n\\begin{aligned}\n(-1)^\\nu=\\prod_{i=1}^8\\left\\{-\\mathrm{sgn}\\left[m\\left(\\bm{\\Lambda}_i\\right)\\right]\\right\\}, \\label{Z2invariant}\n\\end{aligned}\n\\end{equation}\nwhere $\\bm{\\Lambda}_i$ are the eight time-reversal invariant momenta.\nIt is easily shown that if $0>m_0>-2r$ or $-4r>m_0>-6r$ ($m_0>0$, $-2r>m_0>-4r$, or $-6r>m_0$), the system is topologically nontrivial (trivial).\n\nLet us consider a strongly correlated topological insulator in the Euclidean spacetime, which is described by the Wilson fermions with $1\/r$ Coulomb interaction between the bulk electrons.\nWe start from the Euclidean action of (3+1)D Wilson fermion interacting with electromagnetic field on a lattice, which is given by\n\\begin{equation}\n\\begin{aligned}\nS_{F}=&-\\sum_{n,\\mu}\\left[\\bar{\\psi}_nP^-_\\mu U_{n,\\mu}\\psi_{n+\\hat{\\mu}} + \\bar{\\psi}_{n+\\hat{\\mu}}P^+_\\mu U^\\dag_{n,\\mu}\\psi_n\\right]\\\\\n&+(m_0+4r)\\sum_{n}\\bar{\\psi}_n \\psi_n,\\label{ActionQED}\n\\end{aligned}\n\\end{equation}\nwhere $P^\\pm_\\mu=(r\\pm \\gamma_\\mu)\/2$.\nHere $n=(n_0,n_1,n_2,n_3)$ denotes a site on a spacetime lattice and $\\hat{\\mu}$ ($\\mu=0,1,2,3$) denotes the unit vector along $\\mu$-direction.\n$U_{n,\\mu}$ is the link variable, which is defined by $U_{n,\\mu}=e^{iagA_{\\mu}(n+\\hat{\\mu}\/2)}$, where $A_\\mu=(A_0,\\bm{A})$ is the four-vector potential, $a$ is the lattice constant, and $g^2=e^2\/\\epsilon$ with $e$ and $\\epsilon$ being electric charge and the permittivity of the system, respectively.\nAlthough the timelike Wilson term (the term proportional to $r$) is introduced artificially to eliminate fermion doublers, the spatial Wilson terms have a physical meaning (arise due to strong spin-orbit coupling).\nIn this paper, according to the Hamiltonian (\\ref{H_0}), we adopt the Dirac representation in the Euclidean spacetime ($\\{\\gamma_\\mu,\\gamma_\\nu\\}=2\\delta_{\\mu\\nu}$):\n\\begin{equation}\n\\begin{aligned}\n\\gamma_0=\n\\begin{bmatrix}\n1 & 0\\\\\n0 & -1\n\\end{bmatrix},\\ \\ \\ \n\\gamma_j=\n\\begin{bmatrix}\n0 & -i\\sigma_j\\\\\ni\\sigma_j & 0\n\\end{bmatrix},\\ \\ \\ \n\\gamma_5=\n\\begin{bmatrix}\n0 & 1\\\\\n1 & 0\n\\end{bmatrix},\n\\end{aligned}\\label{gamma-matrices}\n\\end{equation}\nwhere $j=1,2,3$ and $\\sigma_j$ are the Pauli matrices.\n\nIn the case of 3D topological insulators, the Fermi velocity $v_{\\rm F}$ is about $3\\times 10^{-3}c$ where $c$ is the speed of light in vacuum.\nThen the interactions between the bulk electrons can be regarded as only the instantaneous Coulomb interaction ($A_j=0$) like in the case of graphene\\cite{Hands2008,Drut2009,Drut2009a,Armour2010,Drut2010,Araki2010,Araki2011,Araki2012,Buividovich2012}, so the action (\\ref{ActionQED}) is rewritten as\n\\begin{equation}\n\\begin{aligned}\nS_{F}=S_F^{(\\tau)}+S_F^{(s)}+(m_0+4r)\\sum_{n}\\bar{\\psi}_n \\psi_n,\\label{ActionTI}\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nS_F^{(\\tau)}&=-\\sum_{n}\\left[\\bar{\\psi}_nP^-_0 U_{n,0}\\psi_{n+\\hat{0}} + \\bar{\\psi}_{n+\\hat{0}}P^+_0 U^\\dag_{n,0}\\psi_n\\right]\\\\\nS_F^{(s)}&=-\\sum_{n,j}\\left[\\bar{\\psi}_nP^-_j\\psi_{n+\\hat{j}} + \\bar{\\psi}_{n+\\hat{j}}P^+_j\\psi_n\\right],\n\\end{aligned}\n\\right.\n\\end{equation}\nand $U_{n,0}=e^{i\\theta_n}\\ (-\\pi\\leq\\theta_n\\leq\\pi)$.\nThe Wilson fermions breaks chiral symmetry by itself (the terms proportional to $r$ and $m_0$), i.e., the action (\\ref{ActionTI}) is not invariant under the chiral transformation $\\psi\\rightarrow e^{i\\theta\\gamma_5}\\psi$.\nIn our model, chiral symmetry is equivalent to the symmetry of the pseudospin for two $p$-orbitals $A$ and $B$.\nThe pure U(1) gauge action on a lattice is given by\n\\begin{equation}\n\\begin{aligned}\nS_G=\\beta\\sum_n\\sum_{\\mu>\\nu}\\left[1-\\frac{1}{2}\\left(U_{n,\\mu\\nu}+U^\\dag_{n,\\mu\\nu}\\right)\\right],\n\\end{aligned}\n\\end{equation}\nwhere $\\beta=v_{\\rm F}\/g^2$. The plaquette contribution $U_{n,\\mu\\nu}$ is defined by\n\\begin{equation}\n\\begin{aligned}\nU_{n,\\mu\\nu}=U_{n,\\mu}U_{n+\\hat{\\mu},\\nu}U^\\dag_{n+\\hat{\\nu},\\mu}U^\\dag_{n,\\nu},\n\\end{aligned}\n\\end{equation}\nwhere $U_{n,j}=1$ in our case.\nThe total action on a lattice is written as\n\\begin{equation}\n\\begin{aligned}\nS=S_F+S_G.\n\\end{aligned}\n\\end{equation}\nThe dielectric constant $\\epsilon_r$ of Bi$_2$Se$_3$ is rather large\\cite{dielectricconstant-Bi2Se3} ($\\epsilon_r=\\epsilon\/\\epsilon_0\\approx 100$).\nThis means that the Coulomb interaction between the bulk electrons in Bi$_2$Se$_3$ is considered to be weak.\nIn fact, the value of $\\beta$ is approximated as\n\\begin{equation}\n\\begin{aligned}\n\\beta=\\frac{v_{\\rm F}\\epsilon_r}{4\\pi c}\\cdot\\frac{4\\pi\\epsilon_0\\hbar c}{e^2}\\approx 3,\n\\end{aligned}\n\\end{equation}\nand we cannot perform the strong coupling expansion in Bi$_2$Se$_3$.\nHowever, we think it would be important from a theorerical viewpoint to examine the strong electron correlation effect in Dirac fermion systems which describe topologically nontrivial states.\n\n\n\\section{Effective Action}\nLet us perform the strong coupling expansion.\nThe strong coupling expansion has  been often used in QCD\\cite{Kawamoto1981,Hoek1982,Drouffe1983,Nishida2004,Miura2009} where the coupling between fermions (quarks) and gauge fields (gluons) are strong.\nWe can carry out the $U_0$ integral by using the SU($N_c$) group integral formulae:\n\\begin{equation}\n\\begin{aligned}\n\\int dU1=1,\\ \\ \\int dUU_{ab}=0,\\ \\ \\int dUU_{ab}U_{cd}^\\dag=\\frac{1}{N_c}\\delta_{ad}\\delta_{bc}.\n\\end{aligned}\n\\end{equation}\nOur case corresponds to the case of $N_c=1$.\nIn the following, we derive the effective action $S_{\\rm eff}[\\psi,\\bar{\\psi}]$ by carrying out  the $U_0$ integral:\n\\begin{equation}\n\\begin{aligned}\nZ=\\int \\mathcal{D}[\\psi,\\bar{\\psi},U_0]e^{-S_F-S_G}=\\int \\mathcal{D}[\\psi,\\bar{\\psi}]e^{-S_{\\rm eff}}.\n\\end{aligned}\n\\end{equation}\n\nFirst we consider the strong coupling limit ($\\beta=0$). In this case, the timelike partition function is given by\n\\begin{equation}\n\\begin{aligned}\nZ^{(\\tau)}_{\\mathrm{SCL}}[\\psi,\\bar{\\psi}]=\\int \\mathcal{D}U_0e^{-S^{(\\tau)}_F}.\n\\end{aligned}\n\\end{equation}\nIntegration with respect to $U_0$ is carried out to be\n\\begin{equation}\n\\begin{aligned}\n&Z^{(\\tau)}_{\\mathrm{SCL}}=\\exp\\left[\\sum_n\\bar{\\psi}_nP^-_0 \\psi_{n+\\hat{0}}\\bar{\\psi}_{n+\\hat{0}}P^+_0 \\psi_n\\right].\\label{Z_SCL}\n\\end{aligned}\n\\end{equation}\nHere we have used the fact that the grassmann variables $\\psi$'s and $\\bar{\\psi}$'s satisfy $\\psi^2=\\bar{\\psi}^2=0$. We can rewrite this term as\n\\begin{equation}\n\\begin{aligned}\n\\bar{\\psi}_nP^-_0 \\psi_{n+\\hat{0}}\\bar{\\psi}_{n+\\hat{0}}P^+_0 \\psi_n= -\\mathrm{tr}\\left[M_nP^+_0M_{n+\\hat{0}}P^-_0\\right],\n\\end{aligned}\n\\end{equation}\nwhere we have defined $(M_n)_{\\alpha\\beta}=\\bar{\\psi}_{n,\\alpha}\\psi_{n,\\beta}$ and used $(P^\\pm_0)_{\\alpha\\beta}=(P^\\pm_0)_{\\beta\\alpha}$. The subscripts $\\alpha$ and $\\beta$ denote the component of spinors.\n\nNext we evaluate the term of the order of $\\beta$. \nIn order to evaluate the plaquette contributions from $S_G$, we use the cumulant expansion\\cite{Miura2009,Kubo1962}.\nLet us define an expectation value:\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle A\\right\\rangle&\\equiv \\frac{1}{Z^{(\\tau)}_{\\mathrm{SCL}}}\\int \\mathcal{D}U_0A[U_0]e^{-S^{(\\tau)}_F}.\n\\end{aligned}\n\\end{equation}\nThen using this definition, the full timelike partition function can be expressed as \n\\begin{equation}\n\\begin{aligned}\nZ^{(\\tau)}=\\int \\mathcal{D}U_0e^{-S^{(\\tau)}_F-S_G}=Z^{(\\tau)}_{\\mathrm{SCL}}\\left\\langle e^{-S_G}\\right\\rangle.\\label{Z-tau-full}\n\\end{aligned}\n\\end{equation}\nThe contribution from $S_G$ is given by\n\\begin{equation}\n\\begin{aligned}\n\\Delta S\\equiv -\\log\\left\\langle e^{-S_G}\\right\\rangle=-\\sum_{n=1}^{\\infty}\\frac{(-1)^n}{n!}\\left\\langle S_G^n\\right\\rangle_c,\n\\end{aligned}\n\\end{equation}\nwhere $\\left\\langle \\cdots\\right\\rangle_c$ is a cumulant.\nThe correction to the action up to $\\mathcal{O}(\\beta)$ is given by\n\\begin{equation}\n\\begin{aligned}\n\\Delta S&=\\left\\langle S_G\\right\\rangle_c=\\left\\langle S_G\\right\\rangle\\\\\n&=-\\frac{\\beta}{2}\\sum_n\\sum_{\\mu>\\nu}\\left\\langle U_{n,\\mu\\nu}+U^\\dag_{n,\\mu\\nu}\\right\\rangle.\\label{S-NLO}\n\\end{aligned}\n\\end{equation}\nThe expectation value of $U_{n,\\mu\\nu}$ is evaluated as follows\\cite{Miura2009}:\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle U_{n,\\mu\\nu}\\right\\rangle\\simeq \\int dU_{n,0} U_{n,\\mu\\nu}e^{-s^{(\\tau)}_P},\n\\end{aligned}\n\\end{equation}\nwhere $s^{(\\tau)}_P$ is the plaquette-related part of $S^{(\\tau)}_F$.\nWe see that the terms with $(\\mu,\\nu)=(i,j)$ become constant and find only $(\\mu,\\nu)=(j,0)$ terms to survive:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\left\\langle U_{n,j0}\\right\\rangle&=-\\mathrm{tr}\\left[V^+_{n,j}P^+_0V^-_{n+\\hat{0},j}P^-_0\\right],\\\\\n\\left\\langle U^\\dag_{n,j0}\\right\\rangle&=-\\mathrm{tr}\\left[V^-_{n,j}P^+_0V^+_{n+\\hat{0},j}P^-_0\\right],\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere we have defined $\\left(V^+_{n,j}\\right)_{\\alpha\\beta}=\\bar{\\psi}_{n,\\alpha}\\psi_{n+\\hat{j},\\beta}$ and $\\left(V^-_{n,j}\\right)_{\\alpha\\beta}=\\bar{\\psi}_{n+\\hat{j},\\alpha}\\psi_{n,\\beta}$.\n\nFinally, substituting Eqs. (\\ref{Z_SCL}) and (\\ref{S-NLO}) to Eq. (\\ref{Z-tau-full}), we obtain the effective action up to $\\mathcal{O}(\\beta)$:\n\\begin{equation}\n\\begin{aligned}\nS_{\\mathrm{eff}}=&(m_0+4r)\\sum_{n}\\bar{\\psi}_n\\psi_n-\\sum_{n,j}\\left[\\bar{\\psi}_nP^-_j\\psi_{n+\\hat{j}} + \\bar{\\psi}_{n+\\hat{j}}P^+_j\\psi_n\\right]\\\\\n&+\\sum_n\\mathrm{tr}\\left[M_nP^+_0M_{n+\\hat{0}}P^-_0\\right]\\\\\n&+\\frac{\\beta}{2}\\sum_{n,j}\\left\\{\\mathrm{tr}\\left[V^+_{n,j}P^+_0V^-_{n+\\hat{0},j}P^-_0\\right]+(V^+\\longleftrightarrow V^-)\\right\\}.\\label{Eff-Act}\n\\end{aligned}\n\\end{equation}\n\n\\section{Effective Potential and Chiral Condensate}\nIn this section, we derive the effective potential with the use of the extended Hubbard-Stratonovich transformation (EHS)\\cite{Miura2009,Araki2010,Araki2011,Araki2012}, and then we obtain the value of the chiral condensate as the stationary point of the effective potential.\nWe apply the EHS to the trace of arbitrary two matrices. Introducing two auxiliary fields $R$ and $R'$, we obtain\n\\begin{equation}\n\\begin{aligned}\n&e^{\\kappa\\mathrm{tr}AB}\\propto\\\\\n&\\hspace{0.3cm}\\int \\mathcal{D}[R,R'] \\exp\\left\\{-\\kappa\\sum_{\\alpha\\beta}\\left[(R_{\\alpha\\beta})^2+(R'_{\\alpha\\beta})^2\\rule{0pt}{3ex}\\right.\\right.\\\\\n&\\hspace{0.5cm}\\left.\\left.-(A_{\\alpha\\beta}+B^T_{\\alpha\\beta})R_{\\beta\\alpha}-i(A_{\\alpha\\beta}-B^T_{\\alpha\\beta})R'_{\\beta\\alpha}\\rule{0pt}{3ex}\\right]\\rule{0pt}{5ex}\\right\\},\\label{EHS}\n\\end{aligned}\n\\end{equation}\nwhere $\\kappa$ is a positive constant and the superscript $T$ denotes the transpose of a matrix.\nTwo auxiliary fields take the saddle point values $R_{\\alpha\\beta}=\\left\\langle A+B^T\\right\\rangle_{\\beta\\alpha}$\/2 and $R'_{\\alpha\\beta}=i\\left\\langle A-B^T\\right\\rangle_{\\beta\\alpha}$\/2, respectively. \nDefining $Q=R+iR'$ and $Q'=R-iR'$, Eq. (\\ref{EHS}) is rewritten as\n\\begin{equation}\n\\begin{aligned}\n&e^{\\kappa\\mathrm{tr}AB}\\propto\\\\\n&\\int \\mathcal{D}[Q,Q'] \\exp\\left\\{-\\kappa\\left[Q_{\\alpha\\beta}Q'_{\\alpha\\beta}-A_{\\alpha\\beta}Q_{\\beta\\alpha}-B^T_{\\alpha\\beta}Q'_{\\beta\\alpha}\\right]\\right\\},\\label{EHS-Q}\n\\end{aligned}\n\\end{equation}\nwith the saddle point values $Q_{\\alpha\\beta}=\\left\\langle B^T\\right\\rangle_{\\beta\\alpha}$ and $Q'_{\\alpha\\beta}=\\left\\langle A\\right\\rangle_{\\beta\\alpha}$.\n\n\\subsection{Effective Potential in the Strong Coupling Limit}\nWe consider to decouple the third term in the effective action (\\ref{Eff-Act}) to fermion bilinear form.\nTo do this, we set $(\\kappa, A, B)=(1, M_nP^+_0, -M_{n+\\hat{0}}P^-_0)$ in Eq. (\\ref{EHS-Q}). \nIn this case, the saddle point values are given by $Q_{\\alpha\\beta}=-\\left\\langle M_{n+\\hat{0}}P^-_0\\right\\rangle_{\\alpha\\beta}$ and $Q'_{\\alpha\\beta}=\\left\\langle M_nP^+_0\\right\\rangle_{\\beta\\alpha}$.\nHere let us assume that\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle M_n\\right\\rangle&=\\sigma e^{i\\theta\\gamma_5}=\\sigma (\\cos\\theta I+i\\sin\\theta\\gamma_5)\\\\\n&=\\sigma\n\\begin{bmatrix}\n\\cos\\theta & i\\sin\\theta\\\\\ni\\sin\\theta & \\cos\\theta\n\\end{bmatrix},\\label{M_n}\n\\end{aligned}\n\\end{equation}\nbecause we are now interested in the phase structure of the Wilson fermions interacting via the long-range Coulomb interaction in the strong coupling limit, i.e., the mass term (the terms proportional to the identity matrix) is important when determining the phase is whether  topologically trivial or nontrivial (see Eq. (\\ref{Z2invariant})).\nWe are also interested in the possibility of the existence of the symmetry broken phase (\"Aoki phase\") in this model.\nThus the pseudoscalar modes $i\\gamma_5$ should be taken into account.\nThen it follows that\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\left\\langle \\bar{\\psi}\\psi\\right\\rangle&=\\sigma\\cos\\theta\\equiv \\phi_\\sigma\\\\\n\\left\\langle \\bar{\\psi}i\\gamma_5\\psi\\right\\rangle&=\\sigma\\sin\\theta\\equiv \\phi_\\pi.\n\\end{aligned}\n\\right.\n\\end{equation}\nThe terms $\\left\\langle \\bar{\\psi}\\psi\\right\\rangle$ and $\\left\\langle \\bar{\\psi}i\\gamma_5\\psi\\right\\rangle$ decribe the chiral condensate and the condensate of pseudoscalar mode, respectively.\n\nThe Wilson fermions breaks chiral symmetry by itself (the terms proportional to $r$ and $m_0$).\nHence we cannot use the value of $\\left\\langle \\bar{\\psi}\\psi\\right\\rangle$ to determine the system is whether insulating or semimetallic, unlike in the case of graphene where chiral symmetry is not broken in the noninteracting limit. \nWe regard the value of the chiral condensate $\\left\\langle \\bar{\\psi}\\psi\\right\\rangle$ as a correction to the bare mass.\n\nSubstituting $(\\kappa, A, B)=(1, M_nP^+_0, -M_{n+\\hat{0}}P^-_0)$ to Eq. (\\ref{EHS-Q}), we obtain\n\\begin{equation}\n\\begin{aligned}\n&\\exp\\left\\{-\\sum_n\\mathrm{tr}\\left[M_nP^+_0M_{n+\\hat{0}}P^-_0\\right]\\right\\}\\\\\n&\\sim\\exp\\left\\{-\\sum_{n}\\left[(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2\\rule{0pt}{3ex}\\right.\\right.\\\\\n&\\left.\\left.\\hspace{0.35cm}+\\frac{1}{2}\\bar{\\psi}_n\\left[-(1-r^2)\\phi_\\sigma+i\\gamma_5^T(1+r^2)\\phi_\\pi\\right]\\psi_n\\rule{0pt}{3ex}\\right]\\rule{0pt}{4ex}\\right\\},\n\\end{aligned}\n\\end{equation}\nwhere we have applied the mean-field approximation for the chiral condensate and the condensate of pseudoscalar mode.\nThus the effective action in the strong coupling limit expressed by the two auxiliary fields $\\phi_\\sigma$ and $\\phi_\\pi$ is given by\n\\begin{equation}\n\\begin{aligned}\nS_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=&N_sN_\\tau\\left[(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2\\right]\\\\\n&+\\sum_k \\bar{\\psi}_k\\mathcal{M}(\\bm{k};\\phi_\\sigma,\\phi_\\pi)\\psi_k, \\label{S_eff_SCL}\n\\end{aligned}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{M}=&{\\sum}_j i\\gamma_j\\sin k_j+m_0+r\\left(4-{\\sum}_j\\cos k_j\\right)\\\\\n&-\\frac{1}{2}(1-r^2)\\phi_\\sigma+i\\gamma_5^T\\frac{1}{2}(1+r^2)\\phi_\\pi.\\label{M-SCL}\n\\end{aligned}\n\\end{equation}\nHere $N_s=V$ and $N_\\tau=1\/T$ with $V$ and $T$ being the volume and the temperature of the system, respectively and we have done the Fourier transform from $n=(n_0,\\bm{n})$ to $k=(k_0,\\bm{k})$.\n\nThe effective potential at zero temperature per unit spacetime volume is given by\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=-\\frac{1}{N_sN_\\tau}\\log Z(\\phi_\\sigma,\\phi_\\pi).\n\\end{aligned}\n\\end{equation}\nIntegration with respect to $\\psi$ and $\\bar{\\psi}$ is carried out by the formula $\\int D[\\psi,\\bar{\\psi}]e^{-\\bar{\\psi}\\mathcal{M}\\psi}=\\mathrm{det}\\mathcal{M}$. \nTherefore we need to calculate the determinant of $\\mathcal{M}$.\nFrom Eq. (\\ref{M-SCL}), the matrix $\\mathcal{M}$ is written explicitly as\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{M}&=\n\\begin{bmatrix}\n\\tilde{m}(\\bm{k})+r & \\sigma_j\\sin k_j+i\\frac{1+r^2}{2}\\phi_\\pi\\\\\n-\\sigma_j\\sin k_j+i\\frac{1+r^2}{2}\\phi_\\pi & \\tilde{m}(\\bm{k})+r\n\\end{bmatrix}\\\\\n&\\equiv\n\\begin{bmatrix}\nA & B\\\\\nC & D\n\\end{bmatrix},\n\\end{aligned}\\label{M}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\tilde{m}(\\bm{k})=m_0-\\frac{1-r^2}{2}\\phi_\\sigma+r{\\sum}_j\\left(1-\\cos k_j\\right). \\label{m_eff}\n\\end{aligned}\n\\end{equation}\nAs we see from Eq. (\\ref{m_eff}), the chiral condensate $\\phi_\\sigma$ corresponds to a correction to the bare mass $m_0$ in the original Hamiltonian (\\ref{H_0}).\nThat is, $m(\\bm{k})$ in the noninteracting Hamiltonian (\\ref{H_0}) changes to $\\tilde{m}(\\bm{k})$ in the strong coupling limit.\nThe term \"$r$\" of $\\tilde{m}(\\bm{k})+r$ in Eq. (\\ref{M}) originates in the timelike components of the action.\nAfter a straightforward calculation, we have\n\\begin{equation}\n\\begin{aligned}\n&\\mathrm{det}\\mathcal{M}=\\mathrm{det}A\\cdot\\mathrm{det}\\left(D-CA^{-1}B\\right)\\\\\n&=\\left[{\\sum}_j\\sin^2k_j+\\left[\\tilde{m}(\\bm{k})+r\\right]^2+\\frac{(1+r^2)^2}{4}\\phi_\\pi^2\\right]^2.\\label{detM}\n\\end{aligned}\n\\end{equation}\nThe same result can be derived by the formula $\\mathrm{det}\\mathcal{M}=\\sqrt{\\mathrm{det}(\\mathcal{M}\\mathcal{M}^\\dag)}$.\nFinally we arrive at the effective potential in the strong coupling limit:\n\\begin{equation}\n\\begin{aligned}\n&\\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2-2\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\\\\n&\\times\\log\\left[{\\sum}_j\\sin^2k_j+\\left[\\tilde{m}(\\bm{k})+r\\right]^2+\\frac{(1+r^2)^2}{4}\\phi_\\pi^2\\right].\\label{Feff-SCL}\n\\end{aligned}\n\\end{equation}\n\nThe values of $\\phi_\\sigma$ and $\\phi_\\pi$ are obtained by the stationary conditions $\\partial \\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)\/\\partial \\phi_\\sigma=\\partial \\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)\/\\partial \\phi_\\pi=0$.\nWhen $r=1$, Eq. (\\ref{Feff-SCL}) does not depend on $\\phi_\\sigma$.\nIn this case,  the stationary point is obtained by the following equation:\n\\begin{equation}\n\\begin{aligned}\n\\phi_\\sigma&=\\frac{1}{4N_sN_\\tau}\\frac{\\int \\mathcal{D}[\\psi,\\bar{\\psi},U_0]\\sum_n\\bar{\\psi}_n\\psi_n e^{-S}}{\\int \\mathcal{D}[\\psi,\\bar{\\psi},U_0]e^{-S}}\\\\\n&=-\\frac{1}{4N_sN_\\tau}\\frac{1}{Z}\\frac{d Z}{d m_0}\\\\\n&=\\frac{1}{4}\\frac{d\\mathcal{F}_{\\mathrm{eff}}}{d m_0}.\n\\end{aligned}\n\\end{equation}\nWhen $r>1$, the coefficient of the first term in Eq. (\\ref{Feff-SCL}), $1-r^2$, becomes negative and thus Eq. (\\ref{Feff-SCL}) does not have the stationary point.\nThis is because the logarithmic term is doninant when $\\phi_\\sigma$ is small and then $\\phi_\\sigma^2$ term becomes dominant as $\\phi_\\sigma$ gets larger.\nTherefore the condition that the coefficient of $\\phi_\\sigma^2$ must be positive is needed for Eq. (\\ref{Feff-SCL}) to have the stationary point.\nThis fact is consistent with the requirement of the reflection positivity of lattice gauge theories with Wilson fermions\\cite{Menotti1987}.\n\nIn the chiral limit ($r=m_0=0$), the effective potential is a function of only $\\sigma$, reflecting the chiral symmetry of the action.\nThis is understood as follows: in the chiral limit, the action is invariant under the chiral transformation $\\psi\\rightarrow e^{i\\theta\\gamma_5}\\psi$.\nThis transformation doesn't depend on the value of $\\theta$, and thus the effective potential also doesn't depend on it.\nNote that this effective potential corresponds to that of the staggered fermion (SF) model for graphene\\cite{Araki2010,Araki2011} except for an additional factor 4 by setting $r=0$ and changing from (3+1)D to (2+1)D:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi)=4\\mathcal{F}^{\\mathrm{SF}}_{\\mathrm{eff}}(\\phi_\\sigma,\\phi_\\pi).\n\\end{aligned}\n\\end{equation}\nThis result is reasonable, because the two cases describes the same system where the $2^3=8$ fermion doublers appear.\n\n\\subsection{Effective Potential Up to $\\bm{\\mathcal{O}(\\beta)}$}\nLet us evaluate the $\\mathcal{O}(\\beta)$ contribution to the effective potential.\nWe write the fourth term in the effective action (\\ref{Eff-Act}) as $\\Delta S_1+\\Delta S_2(\\equiv \\Delta S)$.\nThen we should choose such that $(\\kappa, A, B)=(\\beta\/2, V^+_{n,j}P^+_0, -V^-_{n+0,j}P^-_0)$ in Eq. (\\ref{EHS-Q}) for $\\Delta S_1$:\n\\begin{equation}\n\\begin{aligned}\ne^{-\\Delta S_1}\n\\propto&\\exp\\left\\{-\\frac{\\beta}{2}\\sum_{n,j}\\left[S_{\\alpha\\beta}S'_{\\alpha\\beta}-A_{\\alpha\\beta}S_{\\beta\\alpha}-B^T_{\\alpha\\beta}S'_{\\beta\\alpha}\\right]\\right\\}, \\label{Delta S_1}\n\\end{aligned}\n\\end{equation}\nwith the saddle point values $S_{\\alpha\\beta}=\\left\\langle B^T\\right\\rangle_{\\beta\\alpha}=-\\left\\langle V^-_{n+0,j}P^-_0\\right\\rangle_{\\alpha\\beta}$ and $S'_{\\alpha\\beta}=\\left\\langle A\\right\\rangle_{\\beta\\alpha}=\\left\\langle V^+_{n,j}P^+_0\\right\\rangle_{\\beta\\alpha}$.\nSimilarly, setting $(\\kappa, A, B)=(\\beta\/2, V^-_{n,j}P^+_0, -V^+_{n+0,j}P^-_0)$ in Eq. (\\ref{EHS-Q}) for $\\Delta S_2$, we obtain\n\\begin{equation}\n\\begin{aligned}\ne^{-\\Delta S_2}\n\\propto&\\exp\\left\\{-\\frac{\\beta}{2}\\sum_{n,j}\\left[T_{\\alpha\\beta}T'_{\\alpha\\beta}-A_{\\alpha\\beta}T_{\\beta\\alpha}-B^T_{\\alpha\\beta}T'_{\\beta\\alpha}\\right]\\right\\}, \\label{Delta S_2}\n\\end{aligned}\n\\end{equation}\nwith the saddle point values $T_{\\alpha\\beta}=\\left\\langle B^T\\right\\rangle_{\\beta\\alpha}=-\\left\\langle V^+_{n+0,j}P^-_0\\right\\rangle_{\\alpha\\beta}$ and $T'_{\\alpha\\beta}=\\left\\langle A\\right\\rangle_{\\beta\\alpha}=\\left\\langle V^-_{n,j}P^+_0\\right\\rangle_{\\beta\\alpha}$.\n\nNext we decompose $\\left\\langle V^+_{n,j}\\right\\rangle$ and $\\left\\langle V^-_{n,j}\\right\\rangle$ into spinor components as follows:\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\langle V_{n,j}^+ \\rangle & \\equiv v_s^+ + i \\gamma_5 v_p^+ + \\sum_\\mu \\gamma_\\mu v_{v \\mu}^+ + \\sum_\\mu i \\gamma_5 \\gamma_\\mu v_{a \\mu}^+,\\\\\n\\langle V_{n,j}^- \\rangle & \\equiv v_s^- + i \\gamma_5 v_p^- + \\sum_\\mu \\gamma_\\mu v_{v \\mu}^- + \\sum_\\mu i \\gamma_5 \\gamma_\\mu v_{a \\mu}^-,\n\\end{aligned}\n\\right. \\label{V^+-V^-}\n\\end{equation}\nwhere the first, second, third and fourth terms are the components of scalar, pseudoscalar, vector and pseudovector (axial vector) mode, respectively.\nThe terms $\\left\\langle V^+_{n,j}\\right\\rangle$ and $\\left\\langle V^-_{n,j}\\right\\rangle$ are equivalent to the propagator from a point to another point.\nOnly the scalar and vector modes appear when parity is not broken, and the pseudoscalar and pseudovector modes may also appear when parity is broken.\nTherefore these four modes should be considered in Eq. (\\ref{V^+-V^-}).\n\nAfter the calculation in the appexdix, we obtain the $\\mathcal{O}(\\beta)$ contribution to the action as\n\\begin{widetext}\n\\begin{equation}\n\\begin{aligned}\n\\Delta S=&\\beta\\sum_{n,j}\\left[(1-r^2) v_s^- v_s^+ +(1+r^2) v_p^- v_p^+ +(1-r^2) v_{v0}^- v_{v0}^+ - (1+r^2) \\sum_l v_{vl}^- v_{vl}^+ -(1+r^2) v_{a0}^- v_{a0}^+ + (1-r^2) \\sum_l v_{al}^- v_{al}^+ \\right]\\\\\n&+\\sum_{n,j} \\left[ \\bar{\\psi}_n \\mathcal{A}_- \\psi_{n+\\hat{j}} + \\bar{\\psi}_{n+\\hat{j}} \\mathcal{A}_+ \\psi_n \\right], \\label{Delta S}\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\langle \\mathcal{A}_- \\rangle &=\\frac {\\beta}{4}\\left[\n-(1-r^2) v_s^- + (1+r^2) i \\gamma_5 v_p^- - (1-r^2)\\gamma_0 v_{v0}^- \n+ (1+r^2)\\sum_l \\gamma_l  v_{vl}^- +  (1+r^2)i \\gamma_5 \\gamma_0 v_{a0}^- - (1-r^2)\\sum_l i \\gamma_5 \\gamma_l v_{al}^-\\right]^T,\\\\\n\\langle \\mathcal{A}_+ \\rangle &=\\frac {\\beta}{4}\\left[\n-(1-r^2) v_s^+ + (1+r^2) i \\gamma_5 v_p^+ - (1-r^2)\\gamma_0 v_{v0}^+ \n+ (1+r^2)\\sum_l \\gamma_l  v_{vl}^+ +  (1+r^2)i \\gamma_5 \\gamma_0 v_{a0}^+ - (1-r^2)\\sum_l i \\gamma_5 \\gamma_l v_{al}^+\\right]^T.\n\\end{aligned}\n\\end{equation}\nThen doing the Fourier transform and combining Eqs. (\\ref{S_eff_SCL}) and (\\ref{Delta S}), we get the effective action up to $\\mathcal{O}(\\beta)$ with auxiliary fields:\n\\begin{equation}\n\\begin{aligned}\nS_{\\mathrm{eff}}=S_{\\mathrm{eff}}^{\\mathrm{aux}}\\left(\\phi_\\sigma,\\phi_\\pi, v^\\pm_s, v^\\pm_p, v^\\pm_{v\\mu}, v^\\pm_{a\\mu}\\right)+\\sum_k \\bar{\\psi}_k\\mathcal{M}\\left(\\bm{k};\\phi_\\sigma,\\phi_\\pi, v^\\pm_s, v^\\pm_p, v^\\pm_{v\\mu}, v^\\pm_{a\\mu}\\right)\\psi_k.\n\\end{aligned}\n\\end{equation}\nFor the explicit forms of $S_{\\mathrm{eff}}^{\\mathrm{aux}}$ and $\\mathcal{M}$, see the appendix.\n\nFinally, after eliminating the auxiliary fields $v$'s by the stationary conditions, we arrive at the effective potential up to $\\mathcal{O}(\\beta)$ given by\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{F}_{\\mathrm{eff}}\\left(\\phi_\\sigma,\\phi_\\pi\\right)=&(1-r^2)\\phi_\\sigma^2+(1+r^2)\\phi_\\pi^2-2\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\log X_0\n-\\frac{\\beta}{3}(1+r^2)\\sum_j\\left[\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\frac{\\sin^2 k_j}{X_0}\\right]^2\\\\\n&-\\frac{\\beta}{3}(1+r^2)\\left[\\frac{1+r^2}{2}\\phi_\\pi\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\frac{\\sum_j\\cos k_j}{X_0}\\right]^2\n-\\frac{\\beta}{3}(1-r^2)\\left[\\int_{-\\pi}^{\\pi}\\frac{d^3k}{(2\\pi)^3}\\frac{\\tilde{m}(\\bm{k})+r}{X_0}\\sum_j\\cos k_j\\right]^2+\\mathcal{O}(\\beta^2), \\label{F_eff-final}\n\\end{aligned}\n\\end{equation}\nwhere we have defined $X_0= {\\sum}_j\\sin^2k_j+\\left[\\tilde{m}(\\bm{k})+r\\right]^2+\\frac{(1+r^2)^2}{4}\\phi_\\pi^2$.\n\\end{widetext}\n\\section{Numerical Results}\nAt first, we found that the value of $\\phi_\\pi$ is zero at the stationary point for any set of $(r,m_0)$.\nHence in the following, we set $\\phi_\\sigma=-\\sigma$ and $\\phi_\\pi=0$ in Eq. (\\ref{Feff-SCL}) to calculate the value of the chiral condensate $\\sigma$.\nThe term $i\\bar{\\psi}\\gamma_5\\psi$ is odd under both time-reversal and inversion.\nTherefore, this means that the phase with spontaneously broken time-reversal and inversion symmetries does not arise in the strong coupling (electron correlation) limit.\nA mean-field study of Wilson fermions with the short-range interaction from the weak coupling\\cite{Sekine2012}  and a lattice strong coupling expansion study of the Kane-Mele model on a honeycomb lattice\\cite{Araki2013} suggest the existence of this phase.\nSuch a phase, \"Aoki phase\" (where parity and flavor symmetry are spotaneously broken) has also confirmed in the lattice QCD with Wilson fermions\\cite{Aoki1984,Aoki1986,Sharpe1998}.\nWe mention the main difference between this analysis and lattice QCD\nexcept for the gauge group as follows.\nOur effective model has only temporal (timelike) link variablies in contrast with lattice QCD.\nSpatial link variables are absent, like in the case of free fermions. \nParity-flavor symmetry is not spontaneously broken in free fermions. \nThis is one of the reasons why the parity broken phase does not appear in this analysis.\n\nThe $m_0$-dependence of the chiral condensate $\\sigma$ is shown in Fig. \\ref{fig1}(a).\nThe value of $\\sigma$ is expected to be quantitatively correct, based on the fact that the result of a strong coupling expansion study in graphene\\cite{Araki2010,Araki2011} is in good agreement with that of lattice Monte Carlo studies\\cite{Hands2008,Drut2009,Drut2009a,Armour2010}.\nAs mentioned above, in the noninteracting limit (i.e. at $\\beta=\\infty$), the system with $0>m_0>-2r$ ($m_0>0$) is identified as a topological (normal) insulator.\nThe chiral condensate is equivalent to a correction to the bare mass.\nHence it is natural to define the effective mass in Eq. (\\ref{m_eff}):\n\\begin{equation}\n\\begin{aligned}\nm_{\\rm{eff}}=m_0+(1-r^2)\\sigma\/2.\n\\end{aligned}\n\\end{equation}\nThe phase diagram with $r=0.5$ in the strong coupling limit calculated by the $Z_2$ invariant (Eq. (\\ref{Z2invariant})) is shown in Fig \\ref{fig1}(b).\nIn the strong coupling limit, the system with $0>m_{\\rm eff}>-2r$ ($m_{\\rm eff}>0$) is identified as a topological (normal) insulator.\nFrom this phase diagram, we see that the effect of the long-range Coulomb interaction is to shift the region of the topological insulator phase.\nThis result doesn't contradict that of a mean-field analysis from the weak coupling\\cite{Sekine2012}.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=\\columnwidth,clip]{fig1.eps}\n\\caption{(Color online) (a) $m_0$-dependence of the chiral condensate $\\sigma$ in the strong coupling limit ($\\beta=0$).\n(b) Phase diagram with $r=0.5$ in the strong coupling limit. The phase boundaries are determined by the condition $m_{\\rm eff}=0$ or $m_{\\rm eff}=-2r$.}\\label{fig1}\n\\end{center}\n\\end{figure}\n\nThe $\\beta$-dependence of the chiral condensate $\\sigma$ is shown in Fig. \\ref{fig2}.\nWe see that $\\sigma$ is a monotonically decreasing function of the coupling strength $\\beta$.\nThis behavior is consistent with a mean-field analysis from the weak coupling\\cite{Sekine2012}.\nOur result shows that the mass gap remains finite, in contrast to the mean-field analysis in which the mass gap becomes infinity in the strong coupling limit.\nWe see also that as $r$ becomes smaller, the rate of decrease of $\\sigma$ becomes notable.\nNamely, as the original mass of doublers becomes smaller, the energy gap of the system becomes smaller, as is understood intuitively.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.83\\columnwidth,clip]{fig2.eps}\n\\caption{(Color online) $\\beta$-dependence of the chiral condensate $\\sigma$ at $m_0=-r$.}\\label{fig2}\n\\end{center}\n\\end{figure}\nFrom Fig \\ref{fig2}, it is concluded that the gapped phases (normal or topological insulator phases) are stable in the strong coupling region.\nThis contrasts with the result of the strong coupling expansion in graphene\\cite{Araki2010,Araki2011}.\nIn graphene, the rate of decrease of $\\sigma$ from $\\beta=0$ to $\\beta=0.5$ is about 60\\%\\cite{Araki2011}, whereas that of our model is about 3\\% at $r=0.2$.\nNamely, in our model, the topological insulator phase survives in the strong coupling limit, although graphene undergoes the semimetal-insulator transition in the strong coupling region.\n\n\n\\section{Discussion and Summary}\nSo far we have obtained the value of the effective mass up to of the order of the coupling strength $\\beta$.\nWe can connect the phase boundary in the noninteracting limit and that in the strong coupling region.\nA possible phase diagram of the Wilson fermions interacting via the long-range Coulomb interaction is shown in Fig. \\ref{fig3}.\nA similar behavior of the phase boundary between the topological insulator phase and the normal insulator phase have been obtained in a mean-field analysis of the Wilson fermions with the short-range interaction\\cite{Sekine2012}.\nOne might wonder why the topological insulator phase survives at infinite coupling.\nIf the interaction is short-range, i.e., Hubbard-like, the antiferromagnetic phase is considered to be dominant.\nHowever, in the present case, the interaction is pure $1\/r$ Coulomb interaction.\nThis difference may affect the phase structure.\nA lattice strong coupling expansion study of the Kane-Mele model on a honeycomb lattice shows a similar result that when spin-orbit coupling is sufficiently strong, the topological insulator phase survives in the strong coupling limit.\n\nTo summarize, we have studied the strong electron correlation effect in a 3D topological insulator which effective Hamiltonian can be described by the Wilson fermions.\nBased on the U(1) lattice gauge theory, we have performed the strong coupling expansion.\nIt was found that the effect of long-range Coulomb interaction corresponds to the renormalization of the bare mass.\nThe values of the chiral condensate, which is regarded as a correction to the bare mass in the strong coupling limit, are expected to be correct quantitatively.\nThe behavior of the chiral condensate in our model is similar to that of the lattice QCD with Wilson fermions.\nThe phase where time-reversal and inversion symmetries are spontaneously broken (\"Aoki phase\") was not found in the strong coupling region, in contrast to the case of lattice QCD.\nIt was also found that the gapped phase is stable in the strong coupling region.\nThis suggests that the topological insulator phase survives in the strong coupling limit.\nIn this study, the bulk property of a 3D topological insulator was examined.\nIt will be interesting to examine the strong correlation effect in the surface Dirac fermions.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=0.90\\columnwidth,clip]{fig3.eps}\n\\caption{(Color online) A possible phase diagram of the Wilson fermions interacting via the long-range Coulomb interaction ($r=0.5$).}\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\\begin{acknowledgments}\nT. Z. N. is thankful to H. Iida, D. Satow, and S. Gongyo for fruitful discussions.\nY. A. is thankful to T. Kimura for valuable discussions.\nThis work was supported by the Grants-in-Aid for Scientific Research (No. 24740211 and No. 10J03314) from the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT).\nA. S. is supported by the global COE program \"Weaving Science Web beyond Particle-Matter Hierarchy\" from MEXT.\nT. Z. N. is supported by the Fellowship for Young Scientists (No. 22-3314) from Japan Society for the Promotion of Science (JSPS) and the global COE program \"The Next Generation of Physics, Spun from Universality and Emergence\" from MEXT.\nY. A. is supported by JSPS Postdoctoral Fellowship for Research Abroad (No.25.56).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{The Appelquist-Terning One Family Walking TC Model (Model A) }\n\nWe take this model as a typical example of the improved TC models\nwithout assisted by topcolor. To reduce the value of the oblique correction \nparameter $S$, this model is designed such that the techniquark ($Q$) sector \nrespects the custodial $SU(2)$ symmetry, while the technilepton ($L$) sector \nis custodial $SU(2)$ violating, and the vacuum expectation values (VEV's)\nof $\\bar{Q}Q$ and $\\bar{L}L$ are further designed to be $~F_Q\\gg F_L$ \\cite{AT}. \nThe color-singlet would-be Goldstone bosons eaten by $W$ and $Z$\nare mainly composed of techiquarks. There are 36 PGBs in this model \\cite{AT},\nin which the color-singlet PGBs are mainly composed of technileptons which \nare irrelevant to the $s$-channel $t\\bar{t}$ production. At the hadron \ncolliders, the color-octet PGBs $\\Pi ^{0a}~(a=1,...,8)$ composed of \ntechniquarks can contribute to the $s$-channel $t\\bar{t}$ productions via the\ntechniquark and top quark triangle loops [cf. Fig. 1]. This is the main \ndifference between the present case and the $\\gamma\\gamma\\to t\\bar{t}$ case in \nRef.\\cite{gamgamtt}. The decay constant of the color-octet PGBs is\n$F_\\Pi=123$ GeV \\cite{AT}.  The masses of $\\Pi^{0a}$ are model-dependent.\nFollowing Ref.\\cite{AT}, we take $M_{\\Pi^{0a}}$ in the range \n$400~{\\rm GeV}\\alt M_{\\Pi^{0a}} \\alt 500$ GeV.\n\nSince the techniquark $Q$ is very heavy, the triangle loop in Fig. 1(a) can be\nsimply evaluated by the Adler-Bell-Jackiw anomaly \\cite{anomaly}, and the\ngeneral form of which is \\cite{Lubicz,DE}\n\\begin{eqnarray}                              \n\\frac{S_{\\Pi^aB_1B_2}}{4\\pi^2 F_{\\pi}} \\epsilon_{\\mu\\nu\\lambda\\rho}\nk_1^{\\lambda}k_2^{\\rho}\\,,\n\\label{ABJ}\n\\end{eqnarray}\nwhere $B_1$ and $B_2$ denote the two gauge fields which, in our case, are the\ntwo gluons $g_b$ and $g_c$ with the color indices $b$ and $c$, respectively.\nThe factor $S_{\\Pi^a g_bg_c}$ can be easily obtained from the formulae\nin Ref.\\cite{Lubicz,DE,hadtt}, and it is\\footnote{Here, and in (\\ref{t-loop}),\n(\\ref{J-Pia}) and (\\ref{S(B)}), we have corrected some typos in \nRef.\\cite{hadtt}.}\n\\begin{eqnarray}                              \nS_{\\Pi^{0a}g_bg_c} =\\frac{1}{\\sqrt{2}}g_s^2 N_{TC}d_{abc}\\,,\n\\label{S_Piagg}\n\\end{eqnarray}\nwhere $d_{abc}$ is the symmetric tensor in the color $SU(3)_c$ group.\n\nThe evaluation of the triangle loop in Fig. 1(b) needs more consideration. The \ntop quark is not heavy enough for the validity of simply using the \nAdler-Bell-Jackiw anomaly. Correction of the $m_t$ effect has to be\ntaken into account. This has been calculated in Ref.\\cite{J}, and the\nresult is\n\\begin{eqnarray}\n-i\\frac{C_t g_s^2}{8\\pi^2 F_{\\pi}}\\frac{d_{abc}}{2} J(R_{\\hat{s}})\n\\epsilon_{\\mu \\nu \\lambda \\rho} k_1^{\\lambda} k_2^{\\rho}\\,,             \n\\label{t-loop}\n\\end{eqnarray}\nwhere $C_t$ is a model-dependent coupling constant which is expected to be\nof order 1 \\cite{TC2-2,Lubicz,RS}, $\\hat{s}$ is the center-of mass energy of \nthe $t\\bar t$ system and $J(R_{\\hat{s}})$ is defined as \\cite{J}\n\\begin{eqnarray}\nJ(R_{\\hat{s}})&\\equiv& -\\frac{1}{R^2_{\\hat{s}}}\n\\int_0^1 \\frac{dx}{x(1-x)}\\nonumber\\\\\n&&\\times \\ln[1 - R^2_{\\hat{s}} x(1-x) ]\\,,          \n\\label{J-Pia}\n\\end{eqnarray}\nwith $R_{\\hat{s}} \\equiv \\sqrt{\\hat s}\/m_t$.\n\nCombining (\\ref{ABJ}), (\\ref{S_Piagg}) and (\\ref{t-loop}), we obtain the \nproduction amplitude for Fig. 1\n\\begin{eqnarray}                                                     \n\\displaystyle\n {\\cal M}^{(A)}_{\\Pi^{0a}}&=& \\frac{C_tm_tg_s^2[N_{TC}+\\frac{1}{2\\sqrt{2}}\n C_tJ(R_{\\hat s})]d_{abc}}{4\\sqrt{2}\\pi^2F^2_{\\Pi}[\\hat{s}-M^2_{\\Pi^{0a}}\n+iM_{\\Pi{0a}}\\Gamma_{\\Pi^{0a}}]}\n\\nonumber\\\\\n&&\\times(\\bar{t}\\gamma_5\\frac{\\lambda_a}{2}t)\\epsilon_{\\mu\\nu\\lambda\\rho}\n\\varepsilon_1^{\\mu}\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\n\\end{eqnarray}\nwhere $\\Gamma_{\\Pi^{0a}}$ is the total width of $\\Pi^{0a}$ which\nhas been given in Ref.\\cite{hadtt}. The total production amplitude is then\n\\begin{eqnarray}                                                    \n{\\cal M}^{(A)}={\\cal M}^{SM}_{tree}+ {\\cal M}^{(A)}_{\\Pi^{0a}}\\,.\n\\label{ATamplitude}\n\\end{eqnarray}\n\n\\subsection{The Original Topcolor-Assisted Technicolor Model (Model B)}\n\nSince the original topcolor-assisted technicolor model (Model B) was\nproposed \\cite{TC2-1}, there have been refinements of the model to make\nit more realistic \\cite{TC2-1'}. In the present study, we are only\ninterested in the characteristic PGB effects of this kind of model in \n$t\\bar t$ productions which do not concern the subtleties of the\nrefinements, so that we simply take the original Model B as a typical\nexample of this kind of model in our calculation.\nIn this model, the TC sector is taken to be the standard extended\ntechnicolor model in which there are 60 TC PGBs with the decay\nconstant $F_\\Pi\\approx 120$ GeV\\footnote{This is slightly smaller than\nthe usual value $F_\\Pi=123$ GeV in the extended technicolor model since, in \nthe topcolor-assisted technicolor model, the total vacuum expectation value is \nalso contributed by the topcolor sector.}, and both the \nthe color-octet PGB $\\Pi^{0a}$ and the color-singlet PGB $\\Pi^0$ contribute to \nthe $t\\bar t$ production. As in Model A, we take \n$400\\alt M_{\\Pi^{0a}}\\alt 500$ GeV. The mass of $\\Pi^0$\nis lighter, say around $150$ GeV \\cite{ETC}. The coupling of $\\Pi^0$ to\ngluons via the techniquark and top quark triangle loops is\ndecscribed by \\cite{Lubicz,hadtt}\n\\begin{eqnarray}                                         \n\\displaystyle\nS^{(B)}_{\\Pi^0 g_b g_c}&=&\\frac{1}{2\\sqrt{3}}g^2_s\\delta_{bc}N_{TC}\\nonumber\\\\\n&&+\\frac{1}{\\sqrt{2}}g^2_sJ(R_{\\hat s})\\delta_{bc}\\,,\n\\label{S(B)}\n\\end{eqnarray}\nwhere the first term is from the techniquark loop and the second term\nis from the top quark loop.\n\nThere is a topcolor sector in this model responsible for causing the main part \nof the top quark mass. In the topcolor sector, there is a PGB called top-pion \n$\\Pi_t$ with decay constant $F_{\\Pi_t}=50$ GeV. The mass $M_{\\Pi_t}$\nwas first estimated as around $200$ GeV in the original paper \\cite{TC2-1}. \nHowever, recent phenomenological analyses up to one-loop calculations show \nthat the LEP\/SLD precision data of $R_b$ give severe constraint on the value \nof $M_{\\Pi_t}$ due to the large negative contribution to $R_b$ from the \ncorrections related to $\\Pi_t$, and it requires $M_{\\Pi_t}$ to be of the order \nof 1 TeV \\cite{BK,LT}. However, as is pointed out in Ref.\\cite{BK}, such a \nconstraint can only be regarded as a rough estimate since higher order\ncorrections related to $\\Pi_t$ may be substantial due to the large \n$\\Pi_t-t-\\bar b$ coupling. Furthermore, the extended technicolor gauge boson \ncontribution to $R_b$, which has been shown to be positive \\cite{R_b}, is not \ntaken into account in the analyses in Refs.\\cite{BK,LT}, and the actual \nconstraint on $M_{\\Pi_t}$ may be relaxed when such a positive contribution is \ntaken into account.\nTherefore, to see the $M_{\\Pi_t}$-dependence of the cross \nsection, we take $M_{\\Pi_t}$ to vary in the range $~500~{\\rm GeV}\\alt\nM_{\\Pi_t}\\alt 1~{\\rm TeV}~$ in our calculation. \n\nThe top quark mass $m_t$ comes from two sources in this model. The TC\nsector gives rise to a small portion of it, and we call this portion \n$m^\\prime_t$. The value of $m^\\prime_t$ is model-dependent.\nLow energy data, especially the $b\\to s\\gamma$ experiment, give constraints on \n$m^\\prime_t$, and the reasonable range of $m^\\prime_t$ is about \n$5~{\\rm GeV}\\alt m^\\prime_t\\alt 20~{\\rm GeV}$ \\cite{TC2-1,Balaji}. The rest \npart of $m_t$, say $m_t-m^\\prime_t$ comes from the topcolor sector. Thus the \ncouplings of the technipions to the top quark can be written as \n\\cite{EL}\\cite{Lubicz}\n\\begin{eqnarray}\n\\displaystyle\n\\frac{C_t m_t^\\prime}{\\sqrt{2} F_\\Pi} \\Pi^0 (\\bar{q} \\gamma^5 q)   \n\\end{eqnarray}\n\\begin{eqnarray}                                               \n\\frac{C_t m_t^{\\prime}}{F_\\Pi} \\Pi^{0a} (\\bar{q} \\gamma^5 \\frac{\\lambda^a}{2} \nq)\n\\end{eqnarray}\nwhere $\\lambda^a$ is the Gell-Mann matrix of the color group. \nThe interactions of the top-pions with the top quark is \\cite{TC2-1,TC2-1'}\n\\begin{eqnarray}\n\\displaystyle\n\\frac{m_t - m_t^{\\prime}}{\\sqrt{2} F_{\\Pi_t}} [\\bar{t} \\gamma_5 t \\Pi_t^0 +\n\\frac{i}{\\sqrt{2}} \\bar{t} (1 - \\gamma_5) b \\Pi_t^+ \\nonumber\\\\\n+ \\frac{i}{\\sqrt{2}}\\bar{b} (1 + \\gamma_5) t \\Pi_t^-]\\,.              \n\\end{eqnarray}\n\nWith these couplings, the PGB contributed production amplitudes in this\nmodel described in Fig. 1 are\n\\begin{eqnarray}                                                     \n\\displaystyle\n {\\cal M}^{(B)}_{\\Pi^{0a}}&=& \\frac{C_tm_t^\\prime g_s^2[N_{TC}+\\frac{1}\n{2\\sqrt{2}}C_t J(R_{\\hat s})]\nd_{abc}}{4\\sqrt{2}\\pi^2F^2_{\\Pi}[\\hat{s}-M^2_{\\Pi^{0a}}\n+iM_{\\Pi{0a}}\\Gamma_{\\Pi^{0a}}]}\n\\nonumber\\\\\n&&\\times(\\bar{t}\\gamma_5\\frac{\\lambda_a}{2}t)\\epsilon_{\\mu\\nu\\lambda\\rho}\n\\varepsilon_1^{\\mu}\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\\label{MPi0a}\n\\end{eqnarray}\n\\begin{eqnarray}                                                \n\\displaystyle\n{\\cal M}^{(B)}_{\\Pi^0}&=&\\frac{C_tm_t^{\\prime}g_s^2[N_{TC}+\\frac{\\sqrt{6}}{2}\nC_t J(R_{\\hat s})]\\delta_{bc}}\n{8\\sqrt{6}\\pi^2F^2_{\\Pi}[\\hat{s}-M^2_{\\Pi^0}+iM_{\\Pi^0}\\Gamma_{\\Pi^0}]}\n\\nonumber\\\\\n&&\\times (\\bar{t}\\gamma_5t)\\epsilon_{\\mu\\nu\\lambda\\rho}\\varepsilon_1^\n{\\mu}\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\\label{MPi0}\n\\end{eqnarray}\n\\begin{eqnarray}                                               \n\\displaystyle\n{\\cal M}^{(B)}_{\\Pi_t}&=&\\frac{(m_t-m_t^{\\prime})g_s^2J(R_{\\hat s})\\delta_{bc}}\n{8\\sqrt{6}\\pi^2F^2_{\\Pi_t}[\\hat{s}-M^2_{\\Pi_t^0}+iM_{\\Pi_t^0}\\Gamma_{\\Pi_t^0}]}\n\\nonumber\\\\\n&&\\times (\\bar{t}\\gamma_5t)\\epsilon_{\\mu\\nu\\lambda\\rho}\\varepsilon_1^{\\mu}\n\\varepsilon_2^{\\nu}k_1^{\\lambda}k_2^{\\rho}\\,,\\label{M^B_Pi_t}\n\\end{eqnarray}\nwhere $\\Gamma_{\\Pi^0}$ and $\\Gamma_{\\Pi_t}$ are, respectively, the total \nwidths of $\\Pi^0$ and $\\Pi_t$ given in Ref.\\cite{hadtt}.\nThe total production amplitude in this model is then\n\\begin{eqnarray}                                            \n{\\cal M}^{(B)}={\\cal M}^{SM}_{tree}+{\\cal M}^{(B)}_{\\Pi^{0a}}\n+{\\cal M}^{(B)}_{\\Pi^0}+{\\cal M}^{(B)}_{\\Pi_t}\\,.\n\\end{eqnarray}\nCompared with ${\\cal M}^{(A)}$, the amplitude ${\\cal M}^{(B)}$ contains two \nextra terms ${\\cal M}^{(B)}_{\\Pi^0}$ and ${\\cal M}^{(B)}_{\\Pi_t}$. As we shall \nsee later that this makes Model A and Model B experimentally distinguishable at\nthe LHC.\n\n\\subsection{The Topcolor-Assisted Multiscale Technicolor Model (Model C)}\n\nThe topcolor-assisted multiscale technicolor model (Model C) \n\\cite{TC2-2,EL,hadtt} is different from Model B by its extended technicolor \nsector which is taken to be the multiscale technicolor model \\cite{MTC}.\nIn this model, the value of the decay constant $F_\\Pi$ is $F_\\Pi=40$ GeV \nrather than $120$ GeV, and the technipion $\\Pi^0$ is almost \ncomposed of pure techniquarks (ideal mixing) \\cite{TC2-2} which leads to\n\\begin{eqnarray}                                           \n\\displaystyle\nS^{(C)}_{\\Pi^0 g_b g_c}=\\frac{1}{\\sqrt{3}}N_{TC}\\delta_{bc}\\,.\n\\label{S(C)}\n\\end{eqnarray}\nThen the production amplitudes in Model C is\n\\begin{eqnarray}                                           \n{\\cal M}^{(C)}={\\cal M}^{SM}_{tree}+{\\cal M}^{(C)}_{\\Pi^{0a}}\n+{\\cal M}^{(C)}_{\\Pi^0}+{\\cal M}^{(C)}_{\\Pi_t}\\,,\n\\end{eqnarray}\nwith\n\\begin{itemize}\n\\item ${\\cal M}^{(C)}_{\\Pi_t}={\\cal M}^{(B)}_{\\Pi_t}$;\n\\item the formula for ${\\cal M}^{(C)}_{\\Pi^0}$ differs from that for\n${\\cal M}^{(B)}_{\\Pi^0}$ by a factor of $2$;\n\\item the value of $F_\\Pi$ in ${\\cal M}^{(C)}_{\\Pi^{0a}}$ and \n${\\cal M}^{(C)}_{\\Pi^0}$ is $F_\\Pi=40$ GeV rather than $123$ GeV.\n\\end{itemize}\nThe smallness of the value of $F_\\Pi$ and the ideal mixing of $\\Pi^0$\nin model C enhance the technicolor PGB contributions in $t\\bar t$ production\nrelative to the top-pion contribution. As we shall see later that this\nmakes Model C experimentally distinguishable from Model B and Model A.\n\n\\null\\vspace{0.4cm}\n\\begin{center}\n{\\bf III. CROSS SECTIONS AND NUMERICAL RESULTS}\n\\end{center}\n\nWe take the method in Ref.\\cite{helicity} to do the numerical calculation. \nOnce the elementary cross section $\\hat \\sigma$ is calculated at the \nparton-level, the total cross section $\\sigma$ can be obtained by folding \n$\\hat \\sigma$ with the parton distribution functions $f^{p(\\bar p)}_i(x_i,Q)$ \n\\cite{EHLQ}\n\\begin{eqnarray}                                              \n\\sigma(pp(\\bar{p})\\to t\\bar{t})&=&\\sum\\limits_{ij}\\int dx_idx_jf_i^{(p)}(x_i,Q)\nf_j^{(p(\\bar{p}))}(x_j,Q)\\nonumber\\\\\n&&\\times\\hat{\\sigma}(ij\\to t\\bar{t})\n\\end{eqnarray}\nwhere $i$ and $j$ stand for the partons $g$, $q$ and $\\bar{q}$; $x_i$ is the \nfraction of longitudinal momentum of the proton (antiproton) carried by the \n$i$th parton; $Q^2\\approx \\hat s$; and $f_i^{(p(\\bar{p}))}$ is the parton \ndistribution function in the proton (antiproton). In this paper, we take the \nMRS setA$^\\prime$ parton distribution for $f_i^{p(\\bar{p})}$ \\cite{MRS}. To take \naccount of the QCD corrections, we shall multiply the obtained cross section \nby a factor of 1.6 \\cite{LSN} as what was done in Ref.\\cite{hadtt}. The\nvalues of the tree-level SM cross section $\\sigma_0$ at the $\\sqrt{s}=2$ TeV\nTevatron and the $\\sqrt{s}=14$ TeV LHC are, respectively\n\\begin{eqnarray}                                            \n&&{\\rm Tevatron}:~~~~~~~~~~~~~~\\sigma_0=8.02~{\\rm pb}\\,,\\nonumber\\\\\n&&{\\rm LHC}:~~~~~~~~~~~~~~~~~~~~\\sigma_0=826~{\\rm pb}\\,.\n\\label{sigma_0}\n\\end{eqnarray}\nIn the numerical calculations, we take $\\alpha_s(\\sqrt{\\hat s})$ the same as\nthat in the MRS set A$^\\prime$ parton distributions, $m_t=174$ GeV,\nand we simply take the technicolor model parameter $C_t=1$. In the\nfollowing analysis, we consider the one-year-run integrated luminosities\nfor the Tevatron Run II and the LHC\n\\begin{eqnarray}                                      \n{\\rm Tevatron}:~~~~~~~~\\int {\\cal L}dt&=&2~{\\rm fb}^{-1}\\,,\\nonumber\\\\\n{\\rm LHC}:~~~~~~~~~~~~~\\int {\\cal L}dt&=&100~{\\rm fb}^{-1}\\,,\n\\label{luminosity}\n\\end{eqnarray}\nand assume a $10\\%$ detecting efficiency. \n\nThe obtained total production\ncross sections can be compared with the recently measured $t\\bar t$ \nproduction cross sections by the CDF Collaboration and the D0 Collaboration \n\\cite{Heinson}\n\\begin{eqnarray}                                       \n{\\rm CDF}:~~~~~~~~~~\\sigma(p\\bar p\\to t\\bar t)&=&10.1\\pm 1.9^{+4.1}_{-3.1}\n~{\\rm pb}\\,,\\nonumber\\\\\n{\\rm D0}:~~~~~~~~~~~\\sigma(p\\bar p\\to t\\bar t)&=&7.1\\pm 2.8\\pm 1.5~{\\rm pb}\\,.\n\\label{sigmatt}\n\\end{eqnarray}\nThe data in (\\ref{sigmatt}) can serve as a constraint on the parameters in the \nTC models.\n\n\\subsection*{A. Results of Model A}\n\nIn Table I, we list the results of the cross sections at the Tevatron Run II \nand the LHC in Model A with $M_{\\Pi^{0a}}$ varying from 400 GeV to 500 GeV. We \nsee from Table I that the values of $\\sigma^{(A)}_{t\\bar t}$ for the Tevatron \nare consistent with the recent CDF and D0 measurements (\\ref{sigmatt}). The \nrelative technicolor corrections to the SM tree-level cross section $\\sigma_0$\nare $\\Delta\\sigma^{(A)}\/\\sigma_0\\approx (10-36)\\%$ for the Tevatron\n\n\\null\\vspace{0.4cm}\n\\noindent\n{\\small Table I.  Cross sections in Model A at the $\\sqrt{s}=2$ TeV Tevatron\nand the $\\sqrt{s}=14$ TeV LHC with $M_{\\Pi^{0a}}$ varying from 400 GeV to 500 \nGeV. $\\sigma_0$ denotes the SM tree-level cross section,\n$\\Delta\\sigma^{(A)}$ denotes the correction to $\\sigma_0$,\nand $\\sigma^{(A)}_{t\\bar{t}}=\\sigma_0+\\Delta\\sigma^{(A)}$ is the total cross \nsection. All masses are in GeV.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 0.5pt\n\\begin{tabular}{c cc cc}\n\\hline\\hline\n& $~~~~~~~~~~$Tevatron& &$~~~~~~~$LHC& \\\\\n\\hline\n$M_{\\Pi^{0a}}$&$\\Delta\\sigma^{(A)}(pb)$&$\\sigma^{(A)}_{t\\bar{t}}(pb)$ &\n$~~~~\\Delta\\sigma^{(A)}(nb)$ & $~~\\sigma^{(A)}_{t\\bar{t}}(nb)~~$\\\\\n\\hline\n400 & 2.92 & 10.94 ~~& 1.36 & 2.19 \\\\\n450 & 1.54 & 9.56 ~~& 1.04 & 1.87 \\\\\n500 & 0.84 & 8.86 ~~& 0.81 & 1.63 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{0.4cm}\n\n\\noindent\nand $\\Delta\\sigma^{(A)}\/\\sigma_0\\approx (98-165)\\%$ for the LHC, which are \nquite large due to the $\\Pi^{0a}$ resonance effects. The relative corrections \nare much larger than those in the $\\gamma\\gamma\\to t\\bar t$ process given in \nRef.\\cite{gamgamtt} because of the existence of the $\\Pi^{0a}$ contribution\nat the hadron colliders.\nWith the integrated luminosities in (\\ref{luminosity}) and assuming a $10\\%$ \ndetecting efficiency, we see from Table I that Model A predicts around\n2000 $t\\bar t$ events at the Tevatron and around $2\\times 10^7~ t\\bar t$ events\nat the LHC. The statistical uncertainty at the $95\\%$ C.L. in the case of the \nTevatron is then around $4\\%$ which is about the same level as the expected \nsystematic error of the $t\\bar t$ cross section measurement ($\\sim 5\\%$ \n\\cite{Liss}), and the statistical uncertainty in the case of the LHC is \naround $4\\times 10^{-4}$ which is much smaller than the expected systematic \nerror ($\\sim {\\rm few}\\%$ \\cite{Liss}). The relative corrections \n$\\Delta\\sigma^{(A)}\/\\sigma_0$ from Table I are all larger than the above\nuncertainties and thus {\\it these events are all experimentally\ndetectable at both the Tevatron and the LHC}. To illustrate the resonances, \nwe further plot the $t\\bar t$ invariant mass distributions for\n$M_{\\Pi{0a}}=400$ GeV at the Tevatron and the LHC in Fig. 2(a) and Fig. 2(b), \nrespectively. The resonance effects at $M_{\\Pi^{0a}}$ can be clearly\nseen. Comparing Fig. 2(a) with the new vector resonances (with\nthe width about $20\\%$ of the mass) shown in Ref.\\cite{HP}, we see that the \n$\\Pi^{0a}$ resonance is sharper.\n\n\\subsection*{B. Results of Model B}\n\nThe results of the cross sections in Model B at the Tevatron are listed in \nTable II. Since $M_{\\Pi^0}$ is much lower than the $t\\bar t$ threshold, there \nis almost no $\\Pi^0$ resonance effect, so that we simply take a typical value \n$M_{\\Pi^0}=150$ GeV in the calculation. To see the resonance effects of \n$\\Pi^{0a}$ and $\\Pi_t$ with various values of $M_{\\Pi^{0a}}$ and $M_{\\Pi_t}$, \nwe take their masses varying in the ranges $~400~{\\rm GeV}\\alt M_{\\Pi^{0a}}\n\\alt 500~{\\rm GeV}~$ and $~500~{\\rm GeV}\\alt M_{\\Pi_t}\\alt 1~{\\rm TeV}$, \nrespectively. For the parameter $m^\\prime_t$, we take two typical values \n$m^\\prime_t=5$ GeV (denoted by the superscript $i=1$) and $m^\\prime_t=15$ GeV \n(denoted by the superscript $i=2$), and the cross sections with these\ntwo values of $m^\\prime_t$ are denoted by $\\sigma^{(B1)}_{t\\bar t}$ \n\n\\null\\vspace{0.4cm}\n\\noindent\n{\\small Table II. Cross sections in Model B at the $\\sqrt{s}=2$ TeV Tevatron. \n$\\Delta\\sigma^{(Bi)}$ denotes the correction to the SM tree-level cross\nsection $\\sigma_0$, and $\\sigma^{(Bi)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Bi)}$ is the total cross section. The superscript $i$\ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$ GeV\n($i=2$). All masses are in GeV, and all cross sections are in pb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\\hline\\hline\n$M_{\\Pi_t^0}$  & $M_{\\Pi^{0a}}$  & $\\Delta\\sigma^{(B1)}$ &\n$\\sigma^{(B1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{(B2)}$ & $\\sigma^{(B2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.13 & 8.15 & 0.47 & 8.49 \\\\\n500 & 450 & 0.11 & 8.13 & 0.29 & 8.31 \\\\\n500 & 500 & 0.09 & 8.11 & 0.18 & 8.20 \\\\\n600 & 400 & 0.10 & 8.12 & 0.44 & 8.46 \\\\\n600 & 450 & 0.08 & 8.09 & 0.26 & 8.28 \\\\\n600 & 500 & 0.06 & 8.08 & 0.15 & 8.17 \\\\ \n700 & 400 & 0.08 & 8.10 & 0.42 & 8.44 \\\\\n700 & 450 & 0.06 & 8.08 & 0.24 & 8.26 \\\\\n700 & 500 & 0.05 & 8.06 & 0.14 & 8.15 \\\\\n800 & 400 & 0.07 & 8.09 & 0.42 & 8.43 \\\\\n800 & 450 & 0.05 & 8.07 & 0.24 & 8.25 \\\\\n800 & 500 & 0.04 & 8.06 & 0.13 & 8.15 \\\\\n900 & 400 & 0.07 & 8.08 & 0.41 & 8.43 \\\\\n900 & 450 & 0.05 & 8.06 & 0.23 & 8.25 \\\\\n900 & 500 & 0.03 & 8.05 & 0.13 & 8.14 \\\\\n1000 & 400 & 0.06 & 8.08 & 0.41 & 8.42 \\\\\n1000 & 450 & 0.04 & 8.06 & 0.23 & 8.24 \\\\\n1000 & 500 & 0.03 & 8.05 & 0.12 & 8.14 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{0.4cm}\n\\noindent\nand $\\sigma^{(B2)}_{t\\bar t}$, respectively. From the values of \n$\\sigma^{(B1)}_{t\\bar t}$ and $\\sigma^{(B2)}_{t\\bar t}$ in Table II, we\nsee that they are consistent with the recent CDF and D0 measurements \n(\\ref{sigmatt}). We know that the width of a heavy $\\Pi_t$ is \nrather large due to the largeness of $(m_t-m^\\prime_t)\/F_{\\Pi_t}$ (the\nsmallness of $F_{\\Pi_t}$), thus  \nthe cross sections depend more sensitively on $M_{\\Pi^{0a}}$ than on \n$M_{\\Pi_t}$ as we see in Table II. Moreover, the $\\Pi^{0a}$ couplings are \nproportional to $m^\\prime_t$, while the $\\Pi_t$ couplings are\nproportional to $m_t-m^\\prime_t$. The former is much sensitive to\n$m^\\prime_t$ than the latter does since $m_t\\gg m^\\prime_t$. Thus the cross \nsections with $m^\\prime_t=15$ GeV are all larger than those with \n$m^\\prime_t=5$ GeV in Table II. From Table II we see that all relative \ncorrections $\\Delta\\sigma^{(B1)}\/\\sigma_0$ and $\\Delta\\sigma^{(B2)}\/\\sigma_0$ \nare at most $6\\%$ which is of the same order as the expected systematic error \n($\\sim 5\\%$). Hence {\\it Model B can hardly be detected at the Tevatron}.\n\nThe obtained cross sections in Model B at the LHC are listed in Table III.\nNow the relative corrections $|\\Delta\\sigma^{(B1)}|\/\\sigma_0$\nin Table III are around $(3\\--7)\\%$ depending on the value of $\\Pi_t$.\nSince the statistical uncertainty is of the order of $10^{-4}$ as can\nbe seen from Table III and\n\\newpage\n{\\small Table III. Cross sections in Model B at the $\\sqrt{s}=14$ TeV LHC. \n$\\Delta\\sigma^{(Bi)}$ denotes the correction to the SM tree level cross \nsections $\\sigma_0$, and $\\sigma^{(Bi)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Bi)}$ is the total cross section. The superscript $i$ \ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$ GeV\n($i=2$). All masses are in GeV, and all cross sections are in nb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$M_{\\Pi_t^0}$  & $M_{\\Pi^{0a}}$  & $\\Delta\\sigma^{(B1)}$ &\n$\\sigma^{(B1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{(B2)}$ & $\\sigma^{(B2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.06 & 0.89 & 0.23 & 1.05 \\\\\n500 & 450 & 0.06 & 0.88 & 0.19 & 1.02 \\\\\n500 & 500 & 0.05 & 0.88 & 0.15 & 0.98 \\\\\n600 & 400 & 0.05 & 0.87 & 0.21 & 1.04 \\\\\n600 & 450 & 0.05 & 0.87 & 0.18 & 1.01 \\\\\n600 & 500 & 0.04 & 0.87 & 0.14 & 0.97 \\\\ \n700 & 400 & 0.04 & 0.87 & 0.20 & 1.03 \\\\\n700 & 450 & 0.04 & 0.86 & 0.17 & 1.00 \\\\\n700 & 500 & 0.03 & 0.86 & 0.13 & 0.96 \\\\\n800 & 400 & 0.04 & 0.86 & 0.20 & 1.03 \\\\\n800 & 450 & 0.03 & 0.86 & 0.17 & 0.99 \\\\\n800 & 500 & 0.03 & 0.86 & 0.13 & 0.95 \\\\\n900 & 400 & 0.03 & 0.86 & 0.20 & 1.02 \\\\\n900 & 450 & 0.03 & 0.86 & 0.16 & 0.99 \\\\\n900 & 500 & 0.03 & 0.85 & 0.12 & 0.95 \\\\\n1000 & 400 & 0.03 & 0.86 & 0.19 & 1.02 \\\\\n1000 & 450 & 0.03 & 0.85 & 0.16 & 0.99 \\\\\n1000 & 500 & 0.02 & 0.85 & 0.12 & 0.95 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{0.4cm}\n\\noindent\neq.(\\ref{sigmatt}), {\\it the PGB effects for\n$m^\\prime_t=5$ GeV in Model B can be marginally detected at the LHC (at\nleast for $M_{\\Pi_t}\\alt 800$ GeV and $M_{\\Pi^{0a}}\\alt 450$ GeV)}.\nFor $m^\\prime_t=15$ GeV, the relative corrections\n$\\Delta\\sigma^{(B2)}\/\\sigma_0$ are in the range of $(15\\--27)\\%$\nwhich are larger than the systematic error and the statistical uncertainty.\nThus {\\it the PGB effects for $m^\\prime_t=15$ GeV in Model B can be\nclearly detected at the LHC}. Comparing the cross sections in Table I and \nTable III, we see that the relative differences between Model A and Model B \nat the LHC are $R^{(1)}_{AB}\\equiv (\\sigma^{(A)}_{t\\bar t}\n-\\sigma^{(B1)}_{t\\bar t})\/\\sigma^{(A)}_{t\\bar t}\\approx (46\\--61)\\%$ and\n$R^{(2)}_{AB}\\equiv (\\sigma^{(A)}_{t\\bar t}-\\sigma^{(B2)}_{t\\bar t})\/\n\\sigma^{(A)}_{t\\bar t}\\approx (40\\--53)\\%$. These are all much larger\nthan the systematic error and the statistical uncertainty, so that\n{\\it Model A and Model B can be clearly distinguished at the LHC}.\n\nAs an illustration, the $t\\bar t$ invariant mass distributions in Model B for\n$M_{\\Pi^{0a}}=400$ GeV and $M_{\\Pi_t}=500$ GeV at the Tevatron and\nthe LHC are shown in Fig. 3. Since the width of $\\Pi^{0a}$ in Model B depends \non $m^\\prime_t\/F_\\Pi$ rather than on $m_t\/F_\\Pi$, {\\it the resonance of \n$~\\Pi^{0a}$ in Model B is much sharper than that in Model A}. This is a clear \ndistinction between Model B and Model A. The width of $\\Pi_t$ is very wide due \nto the largeness of $(m_t-m^\\prime_t)\/F_{\\Pi_t}$ (the smallness of \n$F_{\\Pi_t}$). Because of the large width of $\\Pi_t$, no resonance peak\nof $\\Pi_t$ can be seen, and the contribution of $\\Pi_t$ is just a\nslight enhancement of the $M_{t\\bar t}$ distribution in a certain region. In \nFig. 3(b), the solid curve and the dotted curve denote the $M_{t\\bar{t}}$ \ndistribution with and without the $\\Pi_t$ contribution, respectively. From the\ndifference of these two curves, we can see the effect of the $\\Pi_t$ \ncontribution.\nWe see that both the $\\Pi^{0a}$ and the $\\Pi_t$ contributions look very\ndifferent from those of the new heavy vector resonances (with the \nwidth about $20\\%$ of the mass) shown in Ref.\\cite{HP}.\n\n\n\\subsection*{C. Results of Model C}\n\nThe obtained cross sections in Model C at the Tevatron and the LHC are\nlisted in Table IV and Table V, respectively. The cross sections\nin Table IV are consistent with the CDF and D0 data. In Model C, the decay\nconstant $F_\\Pi$ is much smaller than that in Model B, so that the\n$\\Pi^{0a}$ and $\\Pi^0$ contributions are enhanced\\footnote{In this\npaper, we have considered the effect of ideal mixing of $\\Pi^0$ in\nmodel C, while this effect is not considered in Ref.\\cite{hadtt}.}, and\nthus the cross sections in Tables IV and V are larger than those in Tables\nII and III. \n\n\\null\\vspace{0.4cm}\n{\\small Table IV. Cross sections in Model C at the $\\sqrt{s}=2$ TeV Tevatron. \n$\\Delta\\sigma^{(Ci)}$ denotes the correction to the SM tree-level cross\nsection $\\sigma_0$, and $\\sigma^{(Ci)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Ci)}$ is the total cross section. The superscript $i$\ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$\nGeV ($i=2$). All masses are in GeV, and all cross sections are in pb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$M_{\\Pi_t^0}$ & $M_{\\Pi^{0a}}$  & $\\Delta\\sigma^{(C1)}$ &\n$\\sigma^{(C1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{C(2)}$ & $\\sigma^{(C2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.50 & 8.52 & 3.46 & 11.48 \\\\\n500 & 450 & 0.33 & 8.35 & 1.91 & 9.93 \\\\\n500 & 500 & 0.21 & 8.23 & 0.99 & 9.00 \\\\\n600 & 400 & 0.48 & 8.49 & 3.43 & 11.45 \\\\\n600 & 450 & 0.30 & 8.32 & 1.88 & 9.90 \\\\\n600 & 500 & 0.18 & 8.20 & 0.96 & 8.98 \\\\\n700 & 400 & 0.46 & 8.47 & 3.42 & 11.44 \\\\\n700 & 450 & 0.28 & 8.30 & 1.87 & 9.89 \\\\\n700 & 500 & 0.16 & 8.18 & 0.95 & 8.96 \\\\\n800 & 400 & 0.45 & 8.47 & 3.42 & 11.43 \\\\\n800 & 450 & 0.28 & 8.29 & 1.86 & 9.88 \\\\\n800 & 500 & 0.16 & 8.17 & 0.94 & 8.96 \\\\\n900 & 400 & 0.44 & 8.46 & 3.41 & 11.43 \\\\\n900 & 450 & 0.27 & 8.29 & 1.86 & 9.38 \\\\\n900 & 500 & 0.15 & 8.17 & 0.94 & 8.95 \\\\\n1000 & 400 & 0.44 & 8.46 & 3.41 & 11.43 \\\\\n1000 & 450 & 0.27 & 8.29 & 1.86 & 9.87 \\\\\n1000 & 500 & 0.15 & 8.16 & 0.93 & 8.95 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\nFrom Table IV we see that, at the tevatron, the relative correction\n$\\Delta\\sigma^{(C1)}\/\\sigma_0$ is about $(2\\--6)\\%~$ which is at most\nof the same order as the expected systematic error, and \n$\\Delta\\sigma^{(C2)}\/\\sigma_0$ is around $~(12\\--43)\\%~$ which is larger \nthan the systematic error and the statistical uncertainty. So that, at the \nTevatron, {\\it the PGB effects in Model C ~for $m^\\prime_t=5$ GeV can hardly \nbe detected, while those for $m^\\prime_t=15$ GeV can be clearly detected}. \nThe relative differences \n$R^{(2)}_{CB}\\equiv (\\sigma^{(C2)}_{t\\bar t}-\\sigma^{(B2)}_{t\\bar t})\/\n\\sigma^{(C2)}_{t\\bar t}\\approx (9\\--26)\\%$, so that, for\n$m^\\prime_t=15$ GeV, {\\it Model C can be distinguished from Model B at the \nTevatron}. However, the relative difference \n$R^{(2)}_{CA}\\equiv (\\sigma^{(C2)}_{t\\bar t}-\\sigma^{(A)}_{t\\bar t})\/\n\\sigma^{(C2)}_{t\\bar t}$ is at most $5\\%$, therefore, {\\it even for \n$m^\\prime_t=15$ GeV, Model C can hardly be distinguished from Model A at the \nTevatron}.\n\n\n\\vspace{0.4cm}\n{\\small Table V. Cross sections in Model C at the $\\sqrt{s}=14$ TeV LHC. \n$\\Delta\\sigma^{(Ci)}$ denotes the correction to the SM tree level cross \nsections $\\sigma_0$, and $\\sigma^{(Ci)}_{t\\bar{t}}=\\sigma_0\n+\\Delta\\sigma^{(Ci)}$ is the total cross section. The superscript $i$ \ndenotes the two cases of $m_t^ {\\prime}=5$ GeV ($i=1$) and $m_t^\\prime=15$ GeV\n($i=2$). All masses are in GeV, and all cross sections are in nb.}\n\\begin{center}\n\\doublerulesep 0.5pt\n\\tabcolsep 5pt\n\\begin{tabular}{cccccc}\n\\hline\\hline\n$M_{\\Pi_t^0}$ & $M_{\\Pi^{0a}}$ & $\\Delta\\sigma^{(C1)}$ &\n$\\sigma^{(C1)}_{t\\bar{t}}$ & $\\Delta\\sigma^{(C2)}$ & $\\sigma^{(C2)}\n_{t\\bar{t}}$ \\\\\n\\hline\n500 & 400 & 0.22 & 1.04 & 1.64 & 2.47 \\\\\n500 & 450 & 0.20 & 1.02 & 1.35 & 2.18 \\\\\n500 & 500 & 0.16 & 0.99 & 1.02 & 1.85 \\\\\n600 & 400 & 0.20 & 1.03 & 1.63 & 2.45 \\\\\n600 & 450 & 0.19 & 1.01 & 1.34 & 2.17 \\\\\n600 & 500 & 0.15 & 0.98 & 1.01 & 1.84 \\\\\n700 & 400 & 0.19 & 1.02 & 1.62 & 2.45 \\\\\n700 & 450 & 0.18 & 1.00 & 1.33 & 2.16 \\\\\n700 & 500 & 0.14 & 0.97 & 1.00 & 1.83 \\\\\n800 & 400 & 0.19 & 1.02 & 1.62 & 2.44 \\\\\n800 & 450 & 0.17 & 1.00 & 1.33 & 2.15 \\\\\n800 & 500 & 0.14 & 0.96 & 1.00 & 1.82 \\\\\n900 & 400 & 0.19 & 1.01 & 1.61 & 2.44 \\\\\n900 & 450 & 0.17 & 1.00 & 1.33 & 2.15 \\\\\n900 & 500 & 0.13 & 0.96 & 1.00 & 1.82 \\\\\n1000 & 400 & 0.18 & 1.01 & 1.61 & 2.44 \\\\\n1000 & 450 & 0.17 & 0.99 & 1.32 & 2.15 \\\\\n1000 & 500 & 0.13 & 0.96 & 0.99 & 1.82 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\n\\null\\vspace{0.4cm}\n\nFrom Table V we see that, at the LHC, the relative corrections \n$\\Delta\\sigma^{(C1)}\/\\sigma_0\\approx (16\\--26)\\%$,\n$\\Delta\\sigma^{(C2)}\/\\sigma_0\\approx (120\\--200)\\%$. These are all much\nlarger than the systematic error and the statistical uncertainty. So that \n{\\it the PGB effects in Model C, for both $m^\\prime_t=5$ Gev and \n$m^\\prime_t=15$ GeV, can be clearly detected at the LHC}.\nComparing the cross sections in Table V with those in Table I and Table III, we\nsee that the relative differences are~~ $R^{(1)}_{AC}\\equiv \n(\\sigma^(A)_{t\\bar t}-\\sigma^{(C1)}{t\\bar t})\/\n\\sigma^{(A)}_{t\\bar t}\\approx (40\\--54)\\%$,~~  $R^{(1)}_{CB}\\equiv\n(\\sigma^{(C1)}_{t\\bar t}-\\sigma^{(B1)}_{t\\bar t})\/\\sigma^{(C1)}_{t\\bar t}\n\\approx (11\\--15)\\%$,~~  $R^{(2)}_{CA}\\equiv (\\sigma^{(C2)}_{t\\bar t}\n-\\sigma^{(A)}_{t\\bar t})\/\\sigma^{(C2)}_{t\\bar t}\\approx (11\\--16)\\%$, \n~~ $R^{(2)}_{CB}\\equiv (\\sigma^{(C2)}_{t\\bar t}-\\sigma^{(B2)}_{t\\bar\nt})\/\\sigma^{(C2)}_{t\\bar t}\\approx (47\\--58)\\%$. These are all much\nlarger than the systematic error and the statistical uncertainty. So\nthat for both $m^\\prime_t=5$ GeV and $m^\\prime_t=15$ GeV, {\\it Model C\ncan be clearly distinguished from Model A and Model B at the LHC}.\n\nFor comparison with Fig. 2 and Fig. 3, the corresponding $t\\bar t$ invariant \nmass distributions at the Tevatron and the LHC in Model C are illustrated in \nFig. 4. We see that {\\it the resonances of $\\Pi^{0a}$ are significantly wider \nthan those in Model B, and clearly narrower than those in Model A} \nbecause the width of $\\Pi^{0a}$ depends on $m^\\prime_t\/F_\\Pi$, and the\nvalues of $F_\\Pi$ are very different in Model B and Model C. This character \nshows the clear distinction of the three kinds of TC models. Here we\nsee again that the $\\Pi_t$ contribution does not show up as a resonance\npeak, and its effect can be seen from the difference between the curve\nwith its contribution (the solid curve) and the curve without its\ncontribution (the dotted curve). The shapes of the $\\Pi^{0a}$ and $\\Pi_t$\ncontributions in Model C all look very different from those of the new\nheavy vector reaonaces shown in Ref.\\cite{HP}.\n\n\\null\\vspace{0.4cm}\n\\begin{center}\n{\\bf IV. CONCLUSIONS}\n\\end{center}\n\nIn this paper, we have studied the pseudo-Goldstone boson contributions\nto the $t\\bar t$ production cross sections at the Fermilab Tevatron Run\nII and the CERN LHC in various technicolor models, and have examined the\npossiblity of testing and distinguishing different technicolor models in the \nexperiments. We take the Appilquist-Terning one-family walking\ntechnicolor model (Model A), the original topcolor-assisted technicolor\nmodel (Model B), and the topcolor-assisted multiscale technicolor model\n(Model C) as three typical examples of technicolor models with and\nwithout topcolor. At the hadron colliders, the $s$-channel\npseudo-Goldstone boson contrbutions described in Fig. 1 dominate. In the\ncalculation, the MRS set A$^\\prime$ parton distribution functions are used to\nobtain the $p(\\bar p)\\to t\\bar t$ cross sections, and the\npseudo-Goldstone boson masses $M_{\\Pi^{0a}}$ and $M_{\\Pi_t}$ are taken\nto vary in certain ranges (as discussed in Sec. II) to see the dependence of \nthe cross sections on them. The obtained results are compared with the recent \nCDF and D0 data on the $t\\bar t$ production cross sections at the Tevatron \n[cf. eq.(\\ref{sigmatt})]. It is shown that all the obtained cross\nsections at the Tevatron are consistent with the CDF and D0 data.\n\nThe results of the calculated cross sections are listed in Table I to\nTable V. Considering the expected systematic error at the Tevatron and\nthe LHC, and assuming a $10\\%$ detecting efficiency, we have the following \nconclusions:\n\n\\begin{enumerate}\n\\item Model A can be clearly detected both at the Tevatron and the LHC.\n\\item In Model B and Model C, the $\\Pi^{0a}$ couplings are proportional\nto $m^\\prime_t$ ($m^\\prime_t\\ll m_t$) rather than to $m_t$ as in Model A.\nTherefore the $\\Pi^{0a}$ contributions in Model B and Model C are\nsignificantly reduced relative to Model A. This causes the fact that,\nconsidering the expected systematic error and the statistic uncertainty,\nModel B can hardly be detected at the Tevatron, and model C can be\ndetected at the Tevatron only for large $m^\\prime_t$, say $m^\\prime_t=15$ GeV. \nThe situation is much better for the LHC. Model B with $m^\\prime_t=15$ GeV and \nModel C (with $m^\\prime_t=5~{\\rm GeV~and}~m^\\prime_t=15$ GeV) can all \nbe clearly detected, and Model B with\n$m^\\prime_t=5$ GeV can be marginally detected at the LHC.\n\\item Due to the smallness of $F_{\\Pi_t}$ (the largeness of \n$(m_t-m^\\prime_t)\/F_{\\Pi_t}$), the width of the $\\Pi_t$ resonance is\nvery large which causes the fact that the cross sections are not so sensitive \nto the variation of $M_{\\Pi_t}$.\n\\item For the detectable cases, all the three kinds of models can be\nexperimentally distinguished by the significant differences of their\ncross sections. Furthermore, the $\\Pi^{0a}$ resonance peaks in the invariant \nmass $M_{t\\bar t}$ distributions for the three kinds of models are also very \ndifferent. \nThe width of the $\\Pi^{0a}$ resonance in the three models are: \n$\\Gamma^{(A)}_{\\Pi^{0a}}>\\Gamma^{(C)}_{\\Pi^{0a}}>\\Gamma^{(B)}_{\\Pi^{0a}}$.\nThis can serve as a clear distinction between the three kinds of \nmodels.\n\\item Comparing the present results with the heavy vector \nresonances (with the width about $20\\%$ of the mass) shown in Ref.\\cite{HP}, \nwe see that the $\\Pi^{0a}$ resonances are much sharper and the $\\Pi_t$ \ncontributions do not show up as resonances. The behavior of the \npresent resonances are very different from those heavy vector \nresonances studied in Ref\\cite{HP}. \n \n\n\\end{enumerate}\n\nIn summary, the PGB effects in $t\\bar t$ productions at the LHC provide \nfeasible tests of technicolor models including distinguishing different \ntypical models. It is complementary to other tests such as the tests studied in\nRefs.\\cite{HP,TC2-1,DE,RS,test}.\n\n\\vspace{0.4cm}\n\n\\begin{center}\n{\\bf Acknowledgment}\n\\end{center}\n\nThis work is supported by the National Natural Science Foundation of China,\nthe Fundamental Research Foundation of Tsinghua University, and a\nspecial grant from the Ministry of Education of China.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of rough path analysis has been developed from the initial paper\nby Lyons \\ \\cite{L}. The aim of this theory is to analyze dynamical\nsystems $dx_{t}=f(x_{t})dy_{t}$, where the control function $y$ is not\ndifferentiable but has finite $p$-variation for some $p> 1$.\nThere is a wide literature on rough path analysis (see, for instance,\nLyons and Qian \\cite{L-Q},  Friz and Victoir  \\cite{FV2}, Lejay \\cite{Le}, Lyons   \\cite{LCL} or  Gubinelli  \\cite{Gu}).  \n\n\n A path-wise approach to classical stochastic\ncalculus has been one of the motivations to build rough path analysis theory. A nice application of the rough path\nanalysis is the stochastic calculus with respect to the fractional Brownian\nmotion with Hurst parameter $H\\in (0,1)$. We\nrefer, for instance to Coutin and Lejay \\cite{CA}, Friz and Victoir \\cite{Fr}, Friz \\cite{FV} and Ledoux {\\it et al.} \\cite{LQZ} for some\napplications of   rough path analysis to the stochastic calculus.\n\n Nualart and R\\u{a}\\c{s}canu in  \\cite{NR} developed an alternative approach to the study of dynamical\nsystems   $dx_{t}=f(x_{t})dy_{t}$, where the control function $y$ is H\\\"{o}lder\ncontinuous of order $\\beta >\\frac{1}{2}$.\nIn this case the Riemann-Stieltjes integral $\\int_{0}^{t}f(x_s)dy_s$     can be expressed as a Lebesgue integral\nusing fractional derivatives  following the ideas of Z\\\"{a}hle \\cite{Z}.\nLater, Hu and Nualart \\cite{H-N} extended this approach  to the case  $\\beta \\in (\\frac{1}{3},\\frac{1}{2})$ . In this work they give an  explicit expression for  the integral $\\int_{0}^{t}f(x_s)dy_s$\nthat depends on the functions $x$, $y$ and a quadratic multiplicative functional \n$x\\otimes y$.  Using this formula,  the authors have established the existence and uniqueness of a solution for the dynamical system $dx_{t}=f(x_{t})dy_{t}$ driven by a H\\\"{o}lder\ncontinuous function $y$ of order $\\beta \\in (\\frac{1}{3},\\frac{1}{2})$. Finally,  using the same approach, Besal\\'u and Nualart \\cite{B-N} got estimates\nfor the supremum norm of the solution.\n\n\n The purpose of this paper is to study a differential delay equation with non-negativity constraints \n driven by a H\\\"older continuous function $y$ of order \n\\beta\\in\\left(\\frac{1}{3},\\frac{1}{2}\\right)$ using the methodology introduced in \\cite{H-N}.\n We will consider the problems of existence, uniqueness and boundedness of the solutions.  As an\napplication we will study a\nstochastic delay  differential equations with non-negativity constraints  driven by a fractional Brownian motion\nwith Hurst parameter $H\\in\\left(\\frac{1}{3},\\frac{1}{2}\\right)$. \nThese results extend the work by Besal\\'u and Rovira \\cite{B-R}, where is considered the case $H>\\frac 12$.\n\nMore precisely, we consider a  delay differential equation with positivity constraints on $\\R^d$ of the form:\n\\begin{eqnarray}\nx(t)&=&\\eta(0)+\\int_0^t b(s,x)ds+\\int_0^t \\si(x(s-r))dy_s+z(t),\\quad t\\in(0,T],\\nonumber\\\\\nx(t)&=& \\eta(t),\\qquad t\\in[-r,0], \\nonumber\n\\end{eqnarray} \nwhere $r$ denotes a strictly positive time delay, $y$ is a $m$-dimensional $\\beta$-H\\\"older continuous function  with  $\\frac13<\\beta<\\frac{1}{2}$, $b(s,x)$ the hereditary term, depends on the path $\\left\\{x(u),-r\\leq u\\leq s\\right\\}$, while $\\eta:[-r,0]\\rightarrow\\R^d_+$ is a non negative smooth function, with $\\R^d_+=\\left\\{u\\in\\R^d;\\,u_i\\geq 0\\;\\mathrm{for}\\;i=1,\\ldots,d\\right\\}$ and $z$ is a vector-valued non-decreasing process which ensures that the non-negativity constraints on $x$ are enforced.\n\n\n\n\n\n\n\\vskip 7pt\n\nThen, we  will apply pathwise  our deterministic result to  a stochastic delay differential equation with positivity constraints on $\\R^d$ of the form:\n\\begin{eqnarray}\nX(t)&=&\\eta(0)+\\int_0^t b(s,X)ds+\\int_0^t \\si(X(s-r))dW_s^H+Z(t),\\quad t\\in(0,T],\\nonumber\\\\\nX(t)&=& \\eta(t),\\qquad t\\in[-r,0], \\nonumber\n\\end{eqnarray} \nwhere  $W^H=\\left\\{W^{H,j},\\,j=1,\\ldots,m\\right\\}$ are independent fractional Brownian motions with Hurst parameter $\\frac13<H<\\frac{1}{2}$ defined in a complete probability space $(\\Omega,\\cf,\\G)$,  while $\\eta:[-r,0]\\rightarrow\\R^d_+$ is a deterministic  non negative smooth function and $Z$ is a vector-valued non-decreasing process which ensures that the non-negativity constraints on X are enforced.\n\n\\vskip 4pt\n\nAs far as we know, stochastic delay differential equations with constraints and driven by a fractional Brownian motion has only been considered when $H>\\frac12$ (\\cite{B-R}). Furthermore, \nthe literature about stochastic delay differential equations  driven by a fractional Brownian\nmotion is scarce. For the  case $H > \\frac12$ has been studied the existence and uniqueness of solution (\\cite{FR}, \\cite{LT}),\nthe existence and regularity of the density (\\cite{LT}) and the  convergence when the delay goes to zero (\\cite{F-R}). For $H<\\frac12$ we can find the\nresults about the existence and uniqueness of solution  (\\cite{NNT}, \\cite{TT}). Actually, in \\cite{NNT}  the authors consider a similar equation to our case  but without reflection. Moreover, they use another approach in order to define the stochastic integral based on L\\'evy area. In any case,\nwe will use some results on fractional Brownian motion taken from this paper.\n\n\n\n\\vskip 4pt\n\nAnyway, as it has been described in this paper of Kinnally and Williams \\cite{K-W} there are some models afected by some type of noise where the dynamics are related to propagation delay and some of them are naturally non-negative quantities. So, it is natural to continue the study of the  stochastic delay differential equations and non-negativity constraints driven by a fractional Brownian motion.\n\n\\vskip 7pt\n\nIn our work, we will make use of the techniques introduced by Hu and Nualart \\cite{H-N}  with some ideas borrowed from Besal\\'u and Rovira \\cite{B-R}. In this framework, let us point out again that one novelty of our paper is the non-negative constraints dealing with equations driven by a H\\\"older continuous function of order $\\beta \\in (\\frac13,\\frac12)$. We have used the\nSkorohod's  mapping. Let us recall now the Skorokhod problem.\nSet  $$\\cc_+(\\R_+,\\R^d):=\\left\\{x\\in\\cc(\\R_+,\\R^d): x(0)\\in\\R^d_+\\right\\}.$$\n\\vskip 5pt\n\\noindent\n \n\n\\begin{definition}\nGiven a path $z\\in\\cc_+(\\R_+,\\R^d)$, we say that a pair $(x,y)$ of functions in $\\cc_+(\\R_+,\\R^d)$ solves the Skorokhod problem for $z$ with reflection if\n\\begin{enumerate}\n\\item $x(t)=z(t)+y(t)$ for all $t\\geq 0$ and $x(t)\\in\\R_+^d$ for each $t\\geq 0$,\n\\item for each $i=1,\\ldots, d$, $y^i(0)=0$ and $y^i$ is nondecreasing,\n\\item for each $i=1,\\ldots, d$, ${\\displaystyle \\int_0^t x^i(s)dy^i_s=0}$ for all $t\\geq 0$, so $y^i$ can increase only when $x^i$ is at zero.\n\\end{enumerate}\n\\end{definition}\n\\noindent\nIt is known that we have an explicit formula for $y$ in terms of $z$: for each $i=1,\\ldots,d$\n\\begin{equation*}\ny^i(t)=\\max_{s\\in[0,t]} \\left(z^i(s)\\right)^-.\n\\end{equation*}\n\nThe path $z$ is called the reflector of $x$ and the path $y$ is called the regulator of $x$. We use the Skorokhod mapping for constraining a continuous real-valued function to be non-negative by means of reflection at the origin. \n\n\\vskip 7pt\n\n\n\nThe structure of the paper is as follows:\nin the next section we give some preliminaries, our hypothesis and we state the main results of our paper.\nIn Section 3, we give some basic facts about fractionals integrals. \nSection 4  is devoted to prove our main result: the existence and\nuniqueness  for the solution for deterministic equations, while Section 5 deals with the problem of the boundedness.\nIn  Section 6 we  apply the deterministic results to the stochastic case. Finally, Section 7\nis devoted to give some technical results, as a fixed point theorem, and some properties related to the Skorohod problem.\n\n\n\\renewcommand{\\theequation}{2.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Main results}\n\\vskip 5pt\n\\noindent\nFix a time interval $[0,T]$. For any function $x:[0,T]\\rightarrow \\mathbb{R\n{}^{n}$, the $\\gamma $-H\\\"{o}lder norm of $x$ on the interval $[s,t] \\subset \\lbrack 0,T]$, where $0< \\gamma \\le 1$, will be denoted by \n\\begin{equation*}\n\\left\\| x\\right\\| _{\\gamma(s,t) }=\\sup_{s< u<v< t}\\frac{|x(v)-x(u)|}\n(v-u)^{\\gamma }}.\n\\end{equation*\nIf $\\Delta_T :=\\left\\{ (s,t):0\\leq s<  t\\leq T\\right\\} $, for any $(s,t)\\in\n\\Delta_T $ and for any $g:\\Delta_T \\rightarrow {\\mathbb{R}}^{n}$ we set \n\\begin{equation*}\n\\left\\| g\\right\\| _{\\gamma(s,t)}=\\sup_{s< u<v< t}\\frac{|g(u,v)|}\n(v-u)^{\\gamma }}.\n\\end{equation*\nWe will also set $\\left\\| x\\right\\| _{\\gamma}=\\left\\| x\\right\\|\n_{\\gamma(0,T)}$ and $\\left\\| x\\right\\| _{\\gamma(r)}=\\left\\| x\\right\\|\n_{\\gamma(-r,T)}$. Moreover, $\\left\\| \\cdot \\right\\| _{\\infty(s,t)}$ will\ndenote the supremum norm in the interval $(s,t)$, and for simplicity $\\left\\| x\\right\\| _{\\infty}=\\left\\| x\\right\\|\n_{\\infty(0,T)}$ and $\\left\\| x\\right\\| _{\\infty(r)}=\\left\\| x\\right\\|\n_{\\infty(-r,T)}$.\n\\vskip 5pt\n\\noindent\nFix $0<\\beta \\le 1$. As in \\cite{L} we introduce the\nfollowing definition.\n\n\\begin{definition}\\label{d2.1}\nWe will say that $(x,y,x\\otimes y)$ is an {$(d,m)$-dimensional $\\beta $-\n\\\"{o}lder continuous multiplicative functional} if:\n\n\\begin{enumerate}\n\\item $x:[0,T]\\rightarrow \\mathbb{R}{}^{d}$ and $y:[0,T]\\rightarrow \\mathbb{\n}{}^{m}$ are $\\beta $-H\\\"{o}lder continuous functions,\n\n\\item $x\\otimes y:\\Delta_T \\rightarrow \\mathbb{R}{}^{d}\\otimes \\mathbb{R}{}^{m}\n$ is a continuous function satisfying the following properties:\n\n\\begin{enumerate}\n\\item {\\upshape(Multiplicative property)} For all $s\\leq u\\leq t$ we have \n\\begin{equation*}\n(x\\otimes y)_{s,u}+(x\\otimes y)_{u,t}+(x(u)-x(s))\\otimes\n(y(t)-y(u))=(x\\otimes y)_{s,t}.\n\\end{equation*}\n\n\\item For all $(s,t)\\in \\Delta_T $ \n\\begin{equation*}\n\\left| (x\\otimes y)_{s,t}\\right| \\leq c|t-s|^{2\\beta }.\n\\end{equation*}\n\\end{enumerate}\n\\end{enumerate}\n\\end{definition}\nWe will denote by $M_{d,m}^{\\beta\n}(0,T)$ the space of $(d,m)$-dimensional $\\beta $-H\\\"{o}lder continuous\nmultiplicative functionals. \nFurthermore, we will denote by $M_{d,m}^{\\beta\n}(a,b)$ the obvious extension of the definition $M_{d,m}^{\\beta\n}(0,T)$ to a general interval $(a,b)$. \nWe refer the reader to \\cite{L} and  \\cite{H-N} for a more detailed presentation on $\\beta $-H\\\"{o}lder continuous\nmultiplicative functionals.\n\\vskip 5pt\n\n We have now the tools to give our results.  Set $\\beta\\in(\\frac13,\\frac12)$. \n\nConsider   the deterministic stochastic differential equation on $\\R^d$\n\\begin{eqnarray}\nx(t)&=&\\eta(0)+\\int_0^t b(s,x)ds+\\int_0^t\\sigma(x(s-r))dy_s+ z(t),\\quad t\\in (0,T],\\nonumber\\\\\nx(t) &=& \\eta(t),\\quad t\\in[-r,0],\\label{det}\n\\end{eqnarray} \n{where for each} $i=1,\\ldots,d$,\n\\begin{equation*}\n{\\mathit z^i(t)=\\max_{s\\in[0,t]}\\left(\\xi^i(s)\\right)^-,\\qquad t\\in[0,T],}\n\\end{equation*}\n{and}\n\\begin{equation*}\n{\\mathit \\xi(t)= \\eta(0)+\\int_0^t b(s,x)ds+\\int_0^t \\si(x(s-r))dy_s,\\qquad t\\in[0,T].}\n\\end{equation*}\nLet us consider the following hypothesis:\n\\begin{itemize}\n\\item[\\bfseries(H1)] $\\sigma:\\R^d\\rightarrow \\R^d\\times\\R^m$ is a continuously differentiable function such that $\\sigma'$ is locally $\\gamma$- H\\\"older continuous for $\\gamma>\\frac{1}{\\beta}-2$.\n\\item[\\bfseries(H2)] $b:[0,T]\\times C(-r,T;\\R^d)\\rightarrow \\R^d$ is a measurable function such that for every $t>0$ and $f\\in C(-r,T;\\R^d),\\, b(t,f)$ depends only on $\\left\\{f(s);-r\\leq s\\leq t\\right\\}$. Moreover, there exists $b_0\\in L^\\rho(0,t;\\R^d)$ with $\\rho\\geq 2$ and $\\forall N\\geq 0$ there exists $L_N>0$ such that:\n\\begin{itemize}\n\\item[{\\bf (1)}] $\\left|b(t,x)-b(t,y)\\right|\\leq L_N \\left\\|x-y\\right\\|_{\\infty(-r,t)},\\; \\forall x,y\\;\\mathrm{such\\,that} \\left\\|x\\right\\|_{\\infty(r)}\\leq N,$ \n$\\left\\|y\\right\\|_{\\infty(r)}\\leq N,\\;\\forall t\\in[0,T]$,\n\\item[{\\bf (2)}] $\\left|b(t,x)\\right|\\leq L_0 \\left\\|x\\right\\|_{\\infty(-r,t)}+b_0(t),\\quad \\forall t\\in[0,T]$.\n\\end{itemize}\n\\end{itemize}\n\n\\vskip 5pt\n\\noindent\nThe result of existence and uniqueness states as follows:\n\n\\begin{theorem}\\label{tt1}\nAssume that $\\sigma$ and $b$ satisfy the hypothesis {\\upshape \\bfseries(H1)} and  {\\upshape \\bfseries(H2)} respectively with $\\rho\\geq \\frac{1}{1-\\beta}$. Assume also  that $\\eta\\geq 0$, $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,r)$ and $(y_{\\cdot-r},y,y_{\\cdot-r}\\otimes y)\\in M^\\beta_{m,m}(r,T)$. Then the equation (\\ref{det}) has a unique solution $x\\in \\cc(-r,T;\\R^d_+)$.\n\\end{theorem}\n\n\\vskip 5pt\n\n\\noindent {\\bf Remark}. If we assume that $\\eta\\geq 0$ is a differentiable continuous function with positive derivative, then the assumptions on $\\eta$ of this theorem are satisfied.\n\n\\medskip\n\n\\noindent\nIn order to study the boundedness of the solutions we need to stronger our hyphotesis. Consider now:\n\\begin{itemize}\n\\item[\\bfseries(H3)] $b$ and $\\sigma'$ are bounded function.\n\\end{itemize}\nThen, the result is as follows:\n\\begin{theorem}\\label{tt2}\nAssume that $\\sigma$ and $b$ satisfy the hypothesis {\\upshape \\bfseries(H1)}, {\\upshape \\bfseries(H2)} and  {\\upshape \\bfseries(H3)}. Also assume that $\\eta\\geq 0$  satisfies $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,r)$ and finally that $(y_{\\cdot-r},y,y_{\\cdot-r}\\otimes y)\\in M^\\beta_{m,m}(r,T)$. Set\n$$\\mu=\\left\\|b\\right\\|_{\\infty}+\\left\\|\\sigma\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\gamma}.$$\nThen, the solution of (\\ref{det}) is bounded as follows\n\\begin{equation}\n\\|x\\|_{\\infty}\\le 2+ \\eta(0) +  T \\left\\{K\\left(\\left\\|\\eta\\right\\|_{\\beta}\n+\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n+\\mu (d^\\frac12 +1)  \\left[\\left\\|y\\right\\|_{\\beta}\n+\\left\\|y\\right\\|_{\\beta}^2+ \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\right]\\right)\\right\\}^\\frac{1}{\\beta},\\label{e5.1}\\end{equation}\nwhere $K$ is a universal constant depending only on $\\beta$ and  $\\gamma$,\nand\n\\begin{eqnarray*}\n\\left\\|\\eta\\right\\|_{\\beta}&:=&\\left\\|\\eta\\right\\|_{\\beta(-r,0)},\\\\\n\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n&:=&\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta(0,r)},\\\\\n\\left\\|y\\right\\|_{\\beta}&:=&\\left\\|y\\right\\|_{\\beta(0,T)},\\\\\n\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n&:=&\n\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta(r,T)}.\n\\end{eqnarray*}\n\\end{theorem}\n\n\\vskip 5pt\n\\noindent\nOur last result is an application of the above theorems to stochastic delay differential equations. More precisely, let us consider a\nstochastic delay differential equation with positivity constraints on $\\R^d$ of the form:\n\\begin{eqnarray}\nX(t)&=&\\eta(0)+\\int_0^t b(s,X)ds+\\int_0^t \\si(X(s-r))dW_s^H+Z(t),\\quad t\\in(0,T],\\nonumber\\\\\nX(t)&=& \\eta(t),\\qquad t\\in[-r,0], \\label{eqstoc}\n\\end{eqnarray} \nwhere  $W^H=\\left\\{W^{H,j},\\,j=1,\\ldots,m\\right\\}$ are independent fractional Brownian motions with Hurst parameter $\\frac13<H<\\frac{1}{2}$ defined in a complete probability space $(\\Omega,\\cf,\\G)$, \n {and for each} $i=1,\\ldots,d$\n\\begin{equation*}\n{\\mathit Z^i(t)=\\max_{s\\in[0,t]}\\left(\\Xi^i(s)\\right)^-,\\qquad t\\in[0,T],}\n\\end{equation*}\n{and}\n\\begin{equation*}\n{\\mathit \\Xi(t)= \\eta(0)+\\int_0^t b(s,X)ds+\\int_0^t \\si(s,X(s-r))dW^H_s,\\qquad t\\in[0,T].}\n\\end{equation*}\n\n\\noindent\nThen, our result for the stochastic equation, reads as follows:\n\n\\begin{theorem}\\label{tt3} Fix $\\beta \\in (\\frac13,H)$.\nAssume that $\\sigma$ and $b$ satisfy the hypothesis {\\upshape \\bfseries(H1)} and  {\\upshape \\bfseries(H2)} respectively with $\\rho\\geq \\frac{1}{1-\\beta}$. Assume also  that $\\eta$ is a non-negative bounded function such that $(\\eta_{\\cdot-r},W^H,\\eta_{\\cdot-r}\\otimes W^H)\\in M_{d,m}^{\\beta}(0,r)$ almost surely. Then the equation (\\ref{eqstoc}) has a unique solution\n \\[X\\in L^0(\\Omega,\\cf,\\G; \\cc(-r,T;\\R^d_+)).\\]\n\n\\noindent\nFurthermore, if  {\\upshape \\bfseries(H3)} is satisfied and $E(\\left\\|\\eta_{\\cdot-r}\\otimes W^H\\right\\|_{2\\beta(0,r)}^p)  < \\infty, \n\\forall p \\ge 1$, then $E ( \\Vert X \\Vert^p_{\\infty} ) < \\infty,\n\\forall p \\ge 1.$\n\\end{theorem}\n\n\n\\renewcommand{\\theequation}{3.\\arabic{equation}} \\setcounter{equation}{0}\n\n\\section{Fractional integrals and derivatives}\n\nIn this section we recall some definitions and results on fractional integrals. We refer the reader to \\cite{H-N} for a more detailed presentation.\n\\vskip 5pt\n\\noindent\nLet $a,b\\in  \\mathbb{R}$ with $a<b$. Let $f\\in L^{1}(a,b)$ and $\\alpha >0$. The\nleft-sided and right-sided fractional Riemann-Liouville integrals of $f$ of\norder $\\alpha $ are defined for almost all $t\\in (a,b)$ by \n\\begin{equation*}\nI_{a+}^{\\alpha }f(t)=\\frac{1}{\\Gamma (\\alpha )}\\int_{a}^{t}(t-s)^{\\alpha\n-1}f(s)ds\n\\end{equation*\nand \n\\begin{equation*}\nI_{b-}^{\\alpha }f(t)=\\frac{(-1)^{-\\alpha }}{\\Gamma (\\alpha )\n\\int_{t}^{b}(s-t)^{\\alpha -1}f(s)ds,\n\\end{equation*\nrespectively, where $(-1)^{-\\alpha }=e^{-i\\pi \\alpha }$ and $\\Gamma (\\alpha\n)=\\int_{0}^{\\infty }r^{\\alpha -1}e^{-r}dr$ is the Euler gamma function. For any $p\\ge 1$, let \nI_{a+}^{\\alpha }(L^{p})$ (resp. $I_{b-}^{\\alpha }(L^{p})$) be the image of \nL^{p}(a,b)$ by the operator $I_{a+}^{\\alpha }$ (resp. $I_{b^{-}}^{\\alpha }\n). If $f\\in I_{a+}^{\\alpha }(L^{p})$ (resp. $f\\in I_{b-}^{\\alpha }(L^{p})$)\nand $0<\\alpha <1$, then the Weyl derivatives are defined as \n\\begin{eqnarray}\nD_{a+}^{\\alpha }f(t) &=&\\frac{1}{\\Gamma (1-\\alpha )}\\left( \\frac{f(t)}\n(t-a)^{\\alpha }}+\\alpha \\int_{a}^{t}\\frac{f(t)-f(s)}{(t-s)^{\\alpha +1}\nds\\right) ,  \\nonumber \\\\\nD_{b-}^{\\alpha }f(t) &=&\\frac{(-1)^{\\alpha }}{\\Gamma (1-\\alpha )}\\left( \n\\frac{f(t)}{(b-t)^{\\alpha }}+\\alpha \\int_{t}^{b}\\frac{f(t)-f(s)}\n(s-t)^{\\alpha +1}}ds\\right) , \\nonumbe\n\\end{eqnarray\nwhere $a\\leq t\\leq b$ (the convergence of the integrals at the singularity \ns=t$ holds point-wise for almost all $t\\in (a,b)$ if $p=1$ and moreover in\nthe $L^{p}$-sense if $1<p<\\infty $).\\newline\n\\noindent If $f\\in C^{\\lambda }(a,b)$ and $g\\in C^{\\mu }(a,b)$ with $\\lambda\n+\\mu >1$, it is proved in \\cite{Z} that the Riemman-Stieltjes integral \n\\int_{a}^{b}fdg$ exists. The following proposition provides an explicit\nexpression for the integral $\\int_{a}^{b}fdg$ in terms of fractional\nderivatives (see \\cite{Z}).\n\n\\begin{proposition}\nSuppose that $f\\in C^{\\lambda }(a,b)$ and $g\\in C^{\\mu }(a,b)$ with $\\lambda\n+\\mu >1$. Let $1-\\mu <\\alpha <\\lambda $. Then the Riemann-Stieltjes integral \n$\\int_{a}^{b}fdg$ exists and it can be expressed as \n\\begin{equation}\n\\int_{a}^{b}fdg=(-1)^{\\alpha }\\int_{a}^{b}D_{a+}^{\\alpha\n}f(t)D_{b-}^{1-\\alpha }g_{b-}(t)dt,  \\label{forpart}\n\\end{equation\nwhere $g_{b-}(t)=g(t)-g(b).$\n\\end{proposition}\n\\noindent\nBut if $x,y \\in C^{\\beta} (a,b)$ with $\\beta \\in (\\frac13,\\frac12)$ we can not use Equation (\\ref{forpart}) to define the integral $\\int_{a}^{b}f(x(t))dy_{t}$, so\nwe need to recall the construction of the integral $\\int_{a}^{b}f(x(t))dy_{t}$\ngiven by Hu and Nualart in \\cite{H-N} using fractional derivatives.\n\\begin{definition}\n\\label{defhn} Let $(x,y,x\\otimes y)\\in M_{d,m}^{\\beta }(0,T)$. Let $f\n\\mathbb{R}{}^{d}\\rightarrow \\mathbb{R}{}^{m}\\otimes \\mathbb{R}{}^{d}$ be a\ncontinuously differentiable function such that $f^{\\prime }$ is locally \n\\lambda $-H\\\"{o}lder continuous, where $\\lambda >\\frac{1}{\\beta }-2$. Fix \n\\alpha >0$ such that $1-\\beta <\\alpha <2\\beta $, and $\\alpha <\\frac{\\lambda\n\\beta +1}{2}$. Then, for any $0\\leq a<b\\leq T$ we define \n\\begin{eqnarray*}\n\\int_{a}^{b}f(x(r))dy_{r}&=&(-1)^{\\alpha\n}\\sum_{j=1}^{m}\\int_{a}^{b}\\widehat{D}_{a+}^{\\alpha }f_{j}(x)(r)D_{b-}^{1-\\alpha\n}y_{b-}^{j}(r)dr \\\\\n&& -(-1)^{2\\alpha\n-1}\\sum_{i=1}^{d}\\sum_{j=1}^{m}\\int_{a}^{b}D_{a+}^{2\\alpha -1}\\partial\n_{i}f_{j}(x)(r)D_{b-}^{1-\\alpha }\\mathcal{D}_{b-}^{1-\\alpha }(x\\otimes\ny)^{i,j}(r)dr,\n\\end{eqnarray*}\n\\end{definition}\nwhere for $r\\in \\left( a,b\\right)$ \n\\begin{equation*}\n\\begin{array}{l}\n\\displaystyle\\widehat{D}_{a+}^{\\alpha }f(x)(r)=\\frac{1}{\\Gamma (1-\\alpha )\n\\left( \\frac{f(x(r))}{(r-a)^{\\alpha }}\n +\\alpha \\int_{a}^{r}\\frac\nf(x(r))-f(x(\\theta ))-\\sum_{i=1}^{m}\\partial _{i}f(x(\\theta))(x^{i}(r)-x^{i}(\\theta))}{(r-\\theta )^{\\alpha +1}}d\\theta \\right)\n\\end{array\n\\end{equation*}\nis  the \\textit{compensated fractional\nderivative}  \nand  \n\\begin{equation*}\n\\mathcal{D}_{b-}^{1-\\alpha }(x\\otimes y)(r)=\\frac{(-1)^{1-\\alpha }}{\\Gamma\n(\\alpha )}\\left( \\frac{(x\\otimes y)_{r,b}}{(b-r)^{1-\\alpha }}+(1-\\alpha\n)\\int_{r}^{b}\\frac{(x\\otimes y)_{r,s}}{(s-r)^{2-\\alpha }}ds\\right)\n\\end{equation*}\nis the extension of the fractional derivative of $x\\otimes y$.\n\n\\vskip 5pt\n\\noindent\nLet us finish this section recalling two propositions from \\cite{H-N}. In the sequel, $k$ denotes a generic constant that may depend on the\nparameters $\\beta, \\alpha$ and $\\gamma$.\n\n\\begin{proposition}\\label{prop23}\nLet $(x,y,x \\otimes y)$ be in $M_{d,m}^\\beta (0,T)$. Assume that $f: \\R^d \\longrightarrow \\R^m$ is a continuous\ndifferentiable function such that $f'$ is bounded and $\\gamma$-H\\\"older continuous, where $\\gamma > \\frac{1}{\\beta} -2.$ Then for any\n$0 \\le a < b \\le T$ we have\n\\begin{eqnarray*}\n\\Vert \\int f(x(r)) dy_r \\Vert_{\\beta(a,b)} &\\le & k \\vert f(x(a)) \\vert \\Vert y \\Vert_{\\beta(a,b)} + k \\Phi_{a,b,\\beta} (x,y)\n\\\\\n&&  \\times \\left( \\Vert f' \\Vert_\\infty + \\Vert f' \\Vert_\\gamma \\Vert x \\Vert^\\gamma_{\\beta(a,b)} (b-a)^{\\gamma \\beta} \\right) (b-a)^\\beta,\n\\end{eqnarray*}\nwhere\n$$ \\Phi_{a,b,\\beta} (x,y)= \\Vert x \\otimes y \\Vert_{2\\beta(a,b)} + \\Vert x \\Vert_{\\beta(a,b)} \\Vert y \\Vert_{\\beta(a,b)}.$$\n\\end{proposition}\n\n\\begin{proposition}\\label{prop24}\nSuppose that  $(x,y,x \\otimes y)$ and   $(y,z,y \\otimes z)$   belong to  $M_{d,m}^\\beta (0,T)$. Let $f: \\R^d \\longrightarrow \\R^m$ be a continuously\ndifferentiable function such that $f'$ is $\\gamma$-H\\\"older continuous and bounded, where $\\gamma > \\frac{1}{\\beta} -2.$ \nFix $\\alpha >0$ such that $1 - \\beta < \\alpha <2 \\beta, \\alpha < \\frac{\\gamma \\lambda +1}{2}.$ Then the following estimate holds:\n\\begin{eqnarray*}\n\\vert \\int_a^b f(x(r)) d (y\\otimes z)_{\\cdot,b}(r) \\vert & \\le & k \\vert f(x(a)) \\vert\\Phi_{a,b,\\beta}(y,z) (b-a)^{2\\beta}\n\\\\\n&& + k \\left( \\Vert f' \\Vert_\\infty + \\Vert f' \\Vert_\\gamma \\Vert x \\Vert^\\gamma_{\\beta(a,b)} (b-a)^{\\gamma \\beta} \\right) \\Phi_{a,b,\\beta}(x,y,z) (b-a)^{3\\beta},\n\\end{eqnarray*}\nwhere\n$$ \\Phi_{a,b,\\beta} (x,y,z)= \\Vert x \\Vert_{\\beta(a,b)} \\Vert y \\Vert_{\\beta(a,b)} \\Vert z \\Vert_{\\beta(a,b)}  + \\Vert z \\Vert_{\\beta(a,b)}\\Vert x \\otimes y \\Vert_{2\\beta(a,b)} + \\Vert x \\Vert_{\\beta(a,b)} \\Vert y \\otimes z  \\Vert_{2\\beta(a,b)}.$$\n\\end{proposition}\n\n\\renewcommand{\\theequation}{4.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Existence and uniqueness for deterministic integral equations}\n\\vskip 5pt\n\\noindent\nThe aim of this section is the proof of Theorem \\ref{tt1}. For simplicity let us assume $T=Mr$.\n\\vskip 5pt\n\\noindent\n{\\bf Proof of Theorem \\ref{tt1}:}\nIn order to prove that equation (\\ref{det}) admits a unique continuous solution on $[-r,T]$, we will use an induction argument.\nWe shall prove that if the equation (\\ref{det}) admits a unique solution $x^{(n)}$  on $[-r,nr]$ we can  prove that there is a unique solution $x^{(n+1)}$ on $[-r,(n+1)r]$. More precisely, \nour induction hypothesis is the following:\\\\\n$\\bf{(H_n)}$ The equation\n\\begin{eqnarray*}\nx^{(n)}(t)&=&\\eta(0)+\\int_0^t b(s,x^{(n)})ds+\\int_0^t\\sigma(x^{(n-1)}(s-r))dy_s+ z^{(n)}(t),\\quad t\\in (0,nr],\\nonumber\\\\\nx^{(n)}(t) &=& \\eta(t),\\quad t\\in[-r,0],\\label{sn}\n\\end{eqnarray*}\nwhere for $i=1,\\dots,d$, $(z^{(n)})^{i}(t)=\\max_{s\\in[0,t]}((\\xi^{(n)})^{i}(s))^-$ with\n\\[\\xi^{(n)}(t)=\\eta(0)+\\int_0^t b(s,x^{(n)})ds+\\int_0^t\\sigma(x^{(n-1)}(s-r))dy_s,\\]\nhas a unique solution $x^{(n)}\\in \\cc(-r,nr,\\R_+^d)$ and moreover $(x^{(n)}_{\\cdot-r},y,x^{(n)}_{\\cdot-r}\\otimes y)\\in M^\\beta_{d,m}(0,(n+1)r)$.\n\\vskip 5pt\n\\noindent\nActually, when we want to check  $\\bf{(H_{n+1})}$ assuming $\\bf{(H_{n})}$, we can write the equation of $\\bf{(H_{n+1})}$\n as\n\\begin{eqnarray}\nx^{(n+1)}(t)&=&\\eta(0)+\\int_0^t b(s,x^{(n+1)})ds+\\int_0^t\\sigma(x^{(n)}(s-r))dy_s+ z^{(n+1)}(t),\\quad t\\in (0,(n+1)r],\\nonumber\\\\\nx^{(n+1)}(t) &=& \\eta(t),\\quad t\\in[-r,0].\\label{sn1}\n\\end{eqnarray}\nSince $(x^{(n)}_{\\cdot-r},y,x^{(n)}_{\\cdot-r}\\otimes y)\\in M^\\beta_{d,m}(0,(n+1)r)$ we know that we can use  Definition {\\ref{defhn}}  to define the integral $\\int_0^t\\sigma(x^{(n)}(s-r))dy_s$ appearing in  equation (\\ref{sn1}).\nThen, \nthe proof will consist in checking the following  steps:\n\\begin{enumerate}\n\\item Existence of a solution of the equation (\\ref{sn1}) in the space $\\cc(-r,(n+1)r;\\R_+^d)$.\n\\item Uniqueness of a solution of the equation (\\ref{sn1}) in the space $\\cc(-r,(n+1)r;\\R_+^d)$.\n\\item The solution $x^{(n+1)}$ satifies that  $(x^{n+1}_{\\cdot-r},y,x^{(n+1)}_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,(n+2)r)$.\n\\end{enumerate}\n\nActually, we will only proof the first case, that is  $\\bf{(H_{1})}$. Notice that \nthe induction step, that is the proof of $\\bf{(H_{n+1})}$ assuming that $\\bf{(H_{n})}$ is true, can be done repeating  the computations of this initial case.\n\\vskip 5pt\n\\noindent So, let us check   $\\bf{(H_{1})}$. We will deal with the equation\n\\begin{eqnarray}\nx^{(1)}(t)&=&\\eta(0)+\\int_0^t b(s,x^{(1)})ds+\\int_0^t\\sigma(\\eta(s-r))dy_s+ z^{(1)}(t),\\quad t\\in (0,r],\\nonumber\\\\\nx^{(1)}(t) &=& \\eta(t),\\quad t\\in[-r,0],\\label{s1}\n\\end{eqnarray}\nwhere for $i=1,\\;\\ldots,\\;d$,  $(z^{(1)})^i(t)=\\max_{s\\in[0,t]}((\\xi^{(1)})^i(s))^-$ and\n\\[\\xi^{(1)}(t)=\\eta(0)+\\int_0^t b(s,x^{(1)})ds+\\int_0^t\\sigma(\\eta(s-r))dy_s.\\]\nNote that since $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,r)$ we can use Definition \\ref{defhn} in order to define the  integral $\\int_0^t\\sigma(\\eta(s-r))dy_s$ appearing in (\\ref{s1}). The proof of this initial case will be divided en 3 steps:\n\\begin{enumerate}\n\\item Existence of a solution in the space $\\cc(-r,r;\\R^d_+)$.\n\\item Uniqueness of a solution in the space $\\cc(-r,r;\\R^d_+)$.\n\\item The solution $x^{(1)}$ satisfies that $(x^{(1)}_{\\cdot-r},y,x^{(1)}_{\\cdot-r}\\otimes y)\\in M_{d,m}^{\\beta}(0,2r)$\n\\end{enumerate}\n\\noindent To simplify the proof we will assume $d=m=1$.\n\n\\vskip 5pt\n\\noindent\n{\\underline{Step 1:}} In order to prove the existence of solution we will use Lemma \\ref{puntfix}, a fixed point  argument on $\\cc(-r,r,\\R_+)$.\n\\vskip 5pt\n\\noindent\nLet us consider the operator \n\\[\\cl: \\cc(-r,r;\\R_+)\\rightarrow \\cc(-r,r;\\R_+)\\]\nsuch that\n\\begin{eqnarray*}\n\\cl(u)(t)&=&\\eta(0)+\\int_0^t b(s,u)ds+\\int_0^t\\si(\\eta(s-r))dy_s+z(t),\\qquad t\\in[0,r],\\\\\n\\cl(u)(t)&=&\\eta(t),\\qquad t\\in[-r,0].\n\\end{eqnarray*}\nwhere setting \n\\[\\xi(t)=\\eta(0)+\\int_0^t b(s,u)ds+ \\int_0^t \\si(\\eta(s-r))dy_s,\\]\nthen ${\\displaystyle z(t)=\\max_{s\\in[0,t]}(\\xi(s))^-}$.\n\\vskip 5pt\n\\noindent \nClearly\n$\\cl$ is well defined. Let us use the notation $u^*=\\cl(u)$.\\\\\nNow, we need to introduce a family of norms in the space $\\cc(-r,r;\\R_+)$. That is, for any $\\la\\geq 1$, let us consider\n\\[\\left\\|f\\right\\|_{\\infty,\\la(-r,r)}:= \\sup_{t\\in[-r,r]} e^{-\\la t}|f(t)|.\\]\nIt is easy to check that these norms are equivalent to $\\left\\|f\\right\\|_{\\infty(-r,r)}$.\n\\vskip 5pt\n\\noindent\nUsing standard arguments (see for instance \\cite{B-R} for similar computations) we obtain that\n\\begin{eqnarray}\n\\left\\|u^*\\right\\|_{\\infty,\\lambda(-r,r)}&\\leq&\\left\\|\\eta\\right\\|_{\\infty,\\la(-r,0)}+2|\\eta(0)|\n+2 \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\int_0^t b(s,u)ds\\right|\\nonumber\\\\\n&&+2 \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\int_0^t \\si(\\eta(s-r))dy_s\\right|.\\label{cotaE1}\n\\end{eqnarray}\nWe obtain easily (see again  \\cite{B-R}) that\n{\\begin{eqnarray}\n\\sup_{t\\in[0,r]} e^{-\\la t}\\left|\\int_0^t b(s,u)ds\\right|&\\leq& \\frac{L_0}{\\la} \\left\\|u\\right\\|_{\\infty,\\la(-r,r)}+\\frac{C_\\rho}{\\la^{1-\\rho}}\\left\\|b_0\\right\\|_{L^{\\rho}}. \\label{c1}\n\\end{eqnarray}}\n\\vskip 5pt\n\\noindent \nIt only remains the study of the term with the fractional integral. Using the bound appearing on the proof of  Proposition \\ref{prop23}, we get for any $\\lambda \\ge 1$,\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle\\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\int_0^t\\sigma(\\eta({s-r}))dy_s\\right|\\\\\n\\qquad\\quad\\displaystyle\\leq k|\\sigma(\\eta({-r}))|\\left\\|y\\right\\|_{\\beta(0,r)}\\sup_{t\\in[0,r]}e^{-\\la t}t^\\beta+\\\\\\qquad\\qquad\n\\displaystyle+k\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}\\sup_{t\\in[0,r]}e^{-\\la t}t^{2\\beta}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\sup_{t\\in[0,r]}e^{-\\la t}t^{(\\gamma+2)\\beta}\\right)\\\\\n\\qquad\\quad\\displaystyle\\leq k|\\sigma(\\eta({-r}))|\\left\\|y\\right\\|_{\\beta(0,r)} \\left(\\frac{\\beta}{\\lambda}\\right)^\\beta e^{-\\beta}+\\\\\n\\qquad\\qquad\\displaystyle k\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}\\left(\\frac{2\\beta}{\\lambda}\\right)^{2\\beta} e^{-2\\beta}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\left(\\frac{(\\gamma+2)\\beta}{\\lambda}\\right)^{(\\gamma+2)\\beta} e^{(\\gamma+2)\\beta}\\right)\\\\\n\\qquad\\quad\\displaystyle\\leq C_{\\beta,\\gamma}\\frac{1}{\\la^{\\beta}}\\left(|\\sigma(\\eta({-r}))|\\left\\|y\\right\\|_{\\beta(0,r)}+\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\right)\\right),\n\\end{array} \\label{a2} \n\\end{equation}\nwhere in the last inequality we have used that\n\\[\\sup_{t\\in[0,r]} t^\\mu e^{-\\la t} \\leq \\left(\\frac{\\mu}{\\la}\\right)^\\mu e^{-\\mu}\\]\nand $C_{\\beta,\\gamma}$ is a constant depending on $\\beta$ and $\\gamma$.\n\\vskip 5pt\n\\noindent\nSo putting together (\\ref{cotaE1}), (\\ref{c1}) and (\\ref{a2}) we have\n\\[\\left\\|u^*\\right\\|_{\\infty,\\lambda(-r,r)}\\leq M_1(\\lambda)+M_2(\\lambda)\\left\\|u\\right\\|_{\\infty,\\lambda(-r,r)},\\]\nwhere\n\\begin{eqnarray*}\nM_1(\\lambda)&=&\\left\\|\\eta\\right\\|_{\\infty,\\la(-r,0)}+2|\\eta(0)|+\\frac{2C_\\rho}{\\la^{1-\\rho}}\\left\\|b_0\\right\\|_{L^{\\rho}}\\\\\n&&+C_{\\beta,\\gamma}\\frac{2}{\\la^{\\beta}}\\left(|\\sigma(\\eta(-r))|\\left\\|y\\right\\|_{\\beta(0,r)}+\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)\\left(\\left\\|\\sigma'\\right\\|_{\\infty}+\\left\\|\\sigma'\\right\\|_{\\gamma}\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}\\right)\\right),\\\\\nM_2(\\la)&=&2L_0\\frac{1}{\\la}.\n\\end{eqnarray*}\nNow, we can choose $\\la=\\la_0$ large enough such that $M_2(\\la_0) \\leq \\frac{1}{2}$, Then,  ${\\displaystyle\\left\\|u\\right\\|_{\\infty,\\la_0(-r,r)}\\leq 2M_1(\\la_0)}$ yields that $${\\displaystyle\\left\\|u^*\\right\\|_{\\infty,\\la_0(-r,r)}\\leq 2M_1(\\la_0)}$$ and  ${\\displaystyle \\cl(B_0)\\subseteq B_0}$ for\n\\[B_0=\\left\\{u\\in \\cc(-r,r;\\R^d_+); \\left\\|u\\right\\|_{\\infty,\\la_0(-r,r)}\\leq 2M_1(\\la_0)\\right\\}.\\]\nThe first hypothesis in Lemma \\ref{puntfix} is now satisfied with the metric $\\rho_0$ associated to the norm $\\left\\|\\cdot\\right\\|_{\\infty,\\la_0(-r,r)}$.\nTo finish the proof it suffices  to find a metric $\\rho_1$ satisfying the second hypothesis in Lemma \\ref{puntfix}.\n\\vskip 5pt \\noindent\n Notice first that if $u\\in B_0$ then $\\left\\|u\\right\\|_{\\infty(-r,r)}\\leq 2e^{\\la_0r}M_1(\\la_0):=N_0$.\nConsider $u,u'\\in B_0$ and $\\la\\geq 1$. Then\n\\begin{equation}\n \\left\\|\\cl(u)-\\cl(u')\\right\\|_{\\infty,\\la(-r,r)}\\leq \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\xi(t)-\\xi'(t)\\right|\n +\\sup_{t\\in[0,r]}e^{-\\la t}\\left|z(t)-z'(t)\\right|.\\label{laa}\n\\end{equation} \nFrom Lemma \\ref{le2} notice that given $t \\in [0,r]$ there exists $t_2 \\le t$ such that\n$$ \n\\left|z(t)-z'(t)\\right| \\le K_l \\left|\\xi(t_2)-\\xi'(t_2)\\right| .$$\nSo\n$$ \ne^{-\\lambda t}\\left|z(t)-z'(t)\\right|  \\le K_l e^{-\\lambda t_2} \\left|\\xi(t_2)-\\xi'(t_2)\\right|$$\nand it follows easily that\n\\begin{equation}\n\\sup_{t\\in[0,r]}e^{-\\la t}\\left|z(t)-z'(t)\\right| \\le\nK_l \\sup_{t\\in[0,r]}e^{-\\la t}\\left|\\xi(t)-\\xi'(t)\\right|.\\label{lab}\\end{equation}\nFrom (\\ref{laa}) and (\\ref{lab}) we can write\n\\begin{eqnarray*}\n\\left\\|\\cl(u)-\\cl(u')\\right\\|_{\\infty,\\la(-r,r)}&\\leq &(1+K_l)  \\sup_{t\\in[0,r]}e^{-\\la t}\\int_0^t\\left|b(s,u)-b(s,u')\\right|ds\\\\\n&\\leq & L_{N_0} (1+K_l) \\sup_{t\\in[0,r]}e^{-\\la t}\\int_0^t\\sup_{0\\leq v\\leq s}\\left|u(v)-u'(v)\\right|ds\\\\\n&\\leq & L_{N_0}(1+K_l)\\sup_{t\\in[0,r]}\\int_0^t e^{-\\la( t-s)} e^{-\\la s} \\sup_{-r\\leq v\\leq s}\\left|u(v)-u'(v)\\right|ds\\\\\n&\\leq & L_{N_0}(1+K_l )\\frac{1}{\\la}\\left\\|u-u'\\right\\|_{\\infty,\\la(-r,r)} .\n\\end{eqnarray*}\n\\vskip 6pt \\noindent\nSo,  choosing $\\la=\\la_1$ such that ${\\displaystyle \\frac{L_{N_0}(1+K_l)}{\\la_1}\\leq \\frac{1}{2}}$, the second hypothesis is satisfied for the metric $\\rho_1$ associated with the norm $\\left\\|\\cdot\\right\\|_{\\infty,\\la_1(-r,r)}$ and $a=\\frac12$.\n\\vskip 7pt\n\\noindent\n{\\underline{Step 2:}} We deal now with the uniqueness problem. \\\\\nLet $x$ and $x'$ be two solutions of (\\ref{s1}) in the space $\\cc(-r,r;\\R_+)$ and choose $N$ large enough such that $\\left\\|x\\right\\|_{\\infty(-r,r)}\\leq N$ and $\\left\\|x'\\right\\|_{\\infty(-r,r)}\\leq N$.\n\\vskip 5pt\n\\noindent\nFor any $t\\in[0,r]$,\n$$\n\\sup_{s\\in[0,t]} \\left|x(s)-x'(s)\\right| \\le \\sup_{s\\in[0,t]}\\left|\\xi(s)-\\xi'(s)\\right|+ \\sup_{s\\in[0,t]} \\left|z(s)-z'(s)\\right|.$$\nMoreover, using Lemma \\ref{le2} we have\n$$\n\\sup_{s\\in[0,t]} \\left|z(s)-z'(s)\\right|\\leq K_l \\sup_{s\\in[0,t]} \\left|\\xi(t)-\\xi'(t)\\right|.$$\nSo, putting together the last two inequalities we get that\n\\begin{eqnarray*}\n\\sup_{s\\in[0,t]} \\left|x(s)-x'(s)\\right|&\\le & (1+K_l)  \\sup_{s\\in[0,t]} \\left|\\xi(s)-\\xi'(s)\\right| \\\\&\\leq&\n(1+K_l)  \\sup_{s\\in[0,t]}\n\\left|\\int_0^s \\left(b(\\tau,x)-b(\\tau,x')\\right)d\\tau\\right|\\\\&\\leq&\n(1+K_l) L_N\\sup_{s\\in[0,t]}\\left|\\int_0^s\\sup_{0\\leq v\\leq \\tau}|x(v)-x'(v)|d\\tau\\right|\\\\&\\leq& L_N(1+K_l)\\int_0^t\\sup_{v\\in[0, \\tau]}|x(v)-x'(v)|d\\tau.\n\\end{eqnarray*}\nApplying now Gronwall's inequality, we have that for all $t\\in[0,r]$\n\\[\\sup_{s\\in[0,t]}|x(s)-x'(s)|= 0.\\]\nSo\n\\[\\left\\|x-x'\\right\\|_{\\infty(-r,r)}= 0 \\]\nand the uniqueness has been proved.\n\\vskip 5pt\n\\noindent\n{\\underline{Step 3:}} \nWe have to prove that  $(x^{(1)}_{\\cdot-r},y,x^{(1)}_{\\cdot-r}\\otimes y)\\in M_{1,1}^{\\beta}(0,2r)$.\\\\\n\nWe have to check the three conditions appearing in Definition \\ref{d2.1}:\n\\begin{enumerate}\n\\item $y:[0,2r]\\rightarrow\\R$ is $\\beta$-H\\\"older continuous. This condition is one of the hypothesis of our theorem.\n\\item $x^{(1)}_{\\cdot-r}:[0,2r]\\rightarrow \\R$ is $\\beta$-H\\\"older continuous.\\\\\nWe can write that\n\\begin{eqnarray*}\n\\left\\|x^{(1)}_{\\cdot-r}\\right\\|_{\\beta(0,2r)}&=&\\left\\|x^{(1)}\\right\\|_{\\beta(-r,r)}=\\sup_{-r\\leq v\\leq w\\leq r} \\frac{|x^{(1)}(w)-x^{(1)}(v)|}{(w-v)^\\beta}\\\\\n&\\leq& \\sup_{-r\\leq v\\leq w<0} \\frac{|\\eta(w)-\\eta(v)|}{(w-v)^\\beta}+\\sup_{\\substack{-r\\leq v\\leq 0\\\\ 0\\leq w\\leq r}}\\frac{|x^{(1)}(w)-\\eta(v)|}{(w-v)^\\beta}\\\\\n&&+ \\sup_{0\\leq v\\leq w\\leq r} \\frac{|x^{(1)}(w)-x^{(1)}(v)|}{(w-v)^\\beta}.\n\\end{eqnarray*}\nNote that \n\\[\\frac{|x^{(1)}(w)-\\eta(v)|}{(w-v)^\\beta}\\leq\\frac{|x^{(1)}(w)-\\eta(0)|}{(w-0)^\\beta}+\\frac{|\\eta(0)-\\eta(v)|}{(0-v)^\\beta}. \\]\nSo\n\\begin{equation}\n\\left\\|x^{(1)}_{\\cdot-r}\\right\\|_{\\beta(0,2r)}\\leq 2\\left\\|\\eta\\right\\|_{\\beta(-r,0)}+2\\left\\|x^{(1)}\\right\\|_{\\beta(0,r)}. \\label{a3}\n\\end{equation}\n\n\nMoreover\n\\begin{equation}\n\\left\\|x^{(1)}\\right\\|_{\\beta(0,r)} \\leq \\left\\|\\int_0^\\cdot b(s,x^{(1)})ds\\right\\|_{\\beta(0,r)}+\\left\\|\\int_0^\\cdot \\sigma(\\eta(s-r))dy_s\\right\\|_{\\beta(0,r)}+\\left\\|z^{(1)}\\right\\|_{\\beta(0,r)}. \\label{a4}\n\\end{equation}\nUsing Lemma \\ref{le2} we also get that\n\\begin{equation}\n\\left\\|z^{(1)}\\right\\|_{\\beta(0,r)}\\leq \\left\\|\\xi^{(1)}\\right\\|_{\\beta(0,r)}. \\label{a5}\n\\end{equation}\n\nFurthermore,  putting together (\\ref{a3}), (\\ref{a4}) and (\\ref{a5}) and using again Proposition  \\ref{prop23}  we obtain that\n\\begin{eqnarray*}\n\\left\\|x^{(1)}\\right\\|_{\\beta(-r,r)}&\\leq&4\\left\\|\\eta\\right\\|_{\\beta(-r,0)}+4\\left\\|\\int_0^\\cdot b(s,x^{(1)} )ds\\right\\|_{\\beta(0,r)}\n+4\\left\\|\\int_0^\\cdot \\sigma(\\eta (s-r))dy_s\\right\\|_{\\beta(0,r)}\\\\\n&\\leq&4\\left\\|\\eta\\right\\|_{\\beta(-r,0)}+4 C\\left(1+\\left\\|x^{(1)}\\right\\|_{\\infty(-r,r)}\\right)\n+4(k|\\sigma(\\eta (-r))|\\left\\|y\\right\\|_{\\beta(0,r)}\\\\\n&&+\\Phi_{0,r,\\beta}(\\eta_{\\cdot-r},y)(\\left\\|\\sigma'\\right\\|_\\infty+\\left\\|\\sigma'\\right\\|_\\gamma\\left\\|\\eta_{\\cdot-r}\\right\\|^\\gamma_{\\beta(0,r)}r^{\\gamma\\beta}))r^\\beta.\n\\end{eqnarray*}\nSo we can conclude that $x^{(1)}_{\\cdot-r}$ is $\\beta$-H\\\"older continuous.\n\n\\item Let us define $(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}$ for $s,t\\in \\Delta_{2r}$. For completeness, we will give this definition for any dimensions $d$ and $m$, unless we will still  consider  $d=m=1$ in the proofs. For any $k \\in \\{1,\\cdots,d\\}$ and $l \\in \\{1,\\cdots,m\\}$, set:\n\\begin{itemize}\n\\item\nif $s,\\,t\\in[0,r]$, \n\\[(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}^{k,l}=(\\eta_{\\cdot-r}\\otimes y)_{s,t}^{k,l},\\]\n\\item if $s,\\,t\\in [r,2r]$, set\n\\begin{eqnarray*}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}^{k,l}&=&\\int_s^t (y^l(t)-y^l(v))b^k(v-r,x^{(1)})dv+\\sum_{j=1}^m \\int_{s}^{t} \\si^k_j(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot, t}^{j,l}(v)\\\\\n&&+\\int_s^t (y^l(t)-y^l(v))d(z^{(1)})_{v-r}^k,\n\\end{eqnarray*}\n\\item if $s\\in[0,r]$ and $t\\in[r,2r]$, \n\\[(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}^{k,l}=(\\eta_{\\cdot-r}\\otimes y)_{s,r}^{k,l}+(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}^{k,l}+(\\eta^k(0)-\\eta^k(s-r))\\otimes (y^l(t)-y^l(r)).\\]\n\\end{itemize}\n\\vskip 5pt\n\\noindent\nLet us check that the multiplicative property (let us recall that we consider again $d=m=1$ for simplicity) is satisfied, that is, \nfor any $ 0 \\le s \\le u \\le t \\le 2r$ it holds that\n\\begin{equation}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}+(x^{(1)}(u-r)-x^{(1)}(s-r))\\otimes (y(t)-y(u))=(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}. \\label{s2}\n\\end{equation}\n\nWe have to distinguish several cases:\n\\begin{itemize}\n\\item[a)] Case $0\\leq s\\leq u\\leq t\\leq r$.\n \nSince on $\\Delta_r$ it holds that\n$$(x^{(1)}_{\\cdot-r},y,x^{(1)}_{\\cdot-r}\\otimes y)=(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y),$$ \nthe multiplicative property follows from the fact that  we are assuming  that $(\\eta_{\\cdot-r},y,\\eta_{\\cdot-r}\\otimes y)$  \n is a  $\\beta-$ H\\\"older continuous functional.\n\n\\item[b)] Case  $r\\leq s\\leq u\\leq t\\leq 2r$. \n\nNotice first that,\n\\begin{equation*}\n\\begin{array}{l}\n\\displaystyle (x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}=\\int_s^u (y(u)-y(v))b(v-r,x^{(1)})dv\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_{s}^{u} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)+\\int_s^u(y(u)-y(v))dz^{(1)}_{v-r}\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_u^t (y(t)-y(v))b({v-r},x^{(1)})dv+\\int_{u}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_u^t(y(t)-y(v))dz^{(1)}_{v-r}\\\\[4mm]\n\\displaystyle \\qquad =\\int_s^t (y(t)-y(v))b({v-r},x^{(1)})dv+\\int_{s}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\\\[4mm]\n\\displaystyle \\qquad\\quad +\\int_s^t(y(t)-y(v))dz^{(1)}_{v-r}\\\\[4mm]\n\\displaystyle \\qquad\\quad+(y(u)-y(t))\\left(\\int_s^u b({v-r},x^{(1)})dv+z^{(1)}(u-r)-z^{(1)}(s-r)\\right)\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_{s}^{u} \\si(\\eta(v-2r))(d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)-d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)).\n\\end{array}\n\\end{equation*}\nSo\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle (x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}\\\\[4mm]\n\\displaystyle \\qquad =(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}+(y(u)-y(t))\\left(\\int_s^u b({v-r},x^{(1)})dv+z^{(1)}(u-r)-z^{(1)}(s-r)\\right)\n\\\\[4mm]\n\\displaystyle \\qquad\\quad+\\int_{s}^{u} \\si(\\eta(v-2r))(d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)-d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v))\n\\end{array}\\label{xxa}\n\\end{equation}\nOn the other hand, from Definition \\ref{d2.1} we obtain that\n\\begin{equation}\n\\int_{s}^{u} \\si(\\eta(v-2r))(d(y_{\\cdot-r}\\otimes y)_{\\cdot,u}(v)-d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v))=(y(u)-y(t))\\int_{s}^{u} \\si(\\eta(v-2r))dy_{v-r}.\n\\label{xxb}\n\\end{equation}\nFinally, using that\n\\begin{eqnarray*}\n\\int_s^u b({v-r},x^{(1)})dv&=&\\int_{s-r}^{u-r} b({v},x^{(1)})dv,\\\\\n\\int_{s}^{u} \\si(\\eta(v-2r))dy_{v-r}&=&\\int_{s-r}^{u-r} \\si(\\eta(v-r))dy_{v},\n\\end{eqnarray*}\nand putting together (\\ref{xxa}) and (\\ref{xxb}) we get the multiplicative property (\\ref{s2}).\n\n\\item[c)] Case  $0\\leq s\\leq r$ and $r\\leq u\\leq t\\leq 2r$.\\\\\nNotice first that from the definition of $(x^{(1)}_{\\cdot-r}\\otimes y)$  it follows that\n\\begin{equation}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}=(\\eta_{\\cdot-r}\\otimes y)_{s,r}+(x^{(1)}_{\\cdot-r}\\otimes y)_{r,u}+(\\eta(0)-\\eta(s-r))\\otimes (y(u)-y(r)).\n\\label{bba}\n\\end{equation}\nOn the other hand, we have seen in the case b) (choosing $s=r$) that\n\\begin{equation}\n(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}=(x^{(1)}_{\\cdot-r}\\otimes y)_{r,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}+(x^{(1)}(u-r)-\\eta(0))\\otimes (y(t)-y(u)). \n\\label{bbb}\n\\end{equation}\nSo, putting together (\\ref{bba}) and (\\ref{bbb}) we can write\n\\begin{equation*}\n\\begin{array}{l}\n\\displaystyle (x^{(1)}_{\\cdot-r}\\otimes y)_{s,u}+(x^{(1)}_{\\cdot-r}\\otimes y)_{u,t}+(x^{(1)}(u-r)-\\eta(s-r))\\otimes (y(t)-y(u))\\\\[4mm]\n\\displaystyle \\qquad =(\\eta_{\\cdot-r}\\otimes y)_{s,r}+(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}+(\\eta_0-\\eta_{s-r})\\otimes (y_t-y_r)\\\\[4mm]\n\\displaystyle \\qquad=(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t},\n\\end{array}\n\\end{equation*}\nwhere the last equality follows for the definition of $(x^{(1)}_{\\cdot-r}\\otimes y)$. The proof of this case is now finished.\n\\item[d)] Case $0\\leq s\\leq u\\leq r$ and $r\\leq t\\leq 2r$.\n\nThis case can be done following the same ideas that the case c).\n\\end{itemize}\n\n\\item Now only remains to prove that $|(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}|\\leq C|t-s|^{2\\beta}$. We will distinguish again three cases:\\\\\n\n\\begin{enumerate}\n\n\\item Assume that $s,t\\in[r,2r]$. Then\n\\begin{eqnarray*}\n|(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}|&\\leq& \\left|\\int_s^t(y(t)-y(v))b({v-r},x^{(1)})dv\\right|+\\left|\\int_{s}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\right|\\\\ &&+\\left|\\int_s^t(y(t)-y(v))dz^{(1)}_{v-r}\\right|.\n\\end{eqnarray*}\nSince $y$ is $\\beta-$H\\\"older continuous function, we have that\n\\[\\left|\\int_s^t (y(t)-y(v))dz^{(1)}_{v-r}\\right|\\leq K|t-s|^\\beta|z^{(1)}(t-r)-z^{(1)}(s-r)|,\\]\nfor a constant $K$. \nThen, using Lemma \\ref{le2} we get\n\\begin{equation}\n\\left|\\int_s^t (y(t)-y(v))dz^{(1)}_{v-r}\\right|\\leq K|t-s|^{2\\beta}. \\label{s3}\n\\end{equation}\nOn the other hand, using the hypothesis on $b$ we have\n\\begin{eqnarray}\n& &\\left|\\int_s^t(y(t)-y(v))b(v-r,x^{(1)})dv\\right| \\leq K|t-s|^\\beta\\left|\\int_s^t(L_0\\sup_{-r\\leq u\\leq v-r}|x^{(1)}(u)|+b_0(v))dv\\right|\\nonumber\\\\\n& & \\qquad  \\leq K|t-s|^{\\beta+1}\\left\\|x^{(1)}\\right\\|_{\\infty(-r,r)}+|t-s|^{\\beta+1-\\frac{1}{\\rho}}\\left\\|b_0\\right\\|_{L^\\rho}. \\label{s4}\n\\end{eqnarray}\nFinnally using Proposition  \\ref{prop24} we get\n\\begin{equation}\n\\begin{array}{l}\n\\displaystyle \n\\left|\\int_{s}^{t} \\sigma(\\eta(v-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(v)\\right|\\leq k|\\sigma(\\eta(s-2r))|\\Phi_{s,t,\\beta}(y_{\\cdot-r},y)(t-s)^{2\\beta}\\\\[4mm]\n\\displaystyle \\qquad\\quad+k\\left(\\left\\|\\sigma'\\right\\|_\\infty+\\left\\|\\sigma'\\right\\|_\\gamma\\left\\|\\eta_{.-2r}\\right\\|^\\gamma_{\\beta(s,t)}(t-s)^{\\gamma\\beta}\\right)\\Phi_{s,t,\\beta}(\\eta_{\\cdot-2r},y_{\\cdot-r},y)(t-s)^{3\\beta},\n\\end{array} \\label{s5}\n\\end{equation}\nwhere \n\\begin{eqnarray*}\n\\Phi_{a,b,\\beta}(x,y,z)&=&\\left\\|y\\right\\|_{\\beta(a,b)}\\left\\|z\\right\\|_{\\beta(a,b)} \\left\\|x\\right\\|_{\\beta(a,b)} \\\\\n&& +\\left\\|z\\right\\|_{\\beta(a,b)}\\left\\|x\\otimes y\\right\\|_{\n2\\beta(a,b)}+\\left\\|x\\right\\|_{\\beta(a,b)}\\left\\|y\\otimes z\\right\\|_{2\\beta(a,b)}.\n\\end{eqnarray*}\n\\vskip 5pt\n\\noindent\nNow putting together (\\ref{s3}), (\\ref{s4}) and (\\ref{s5}) we finish the proof.\n\\vskip 5pt\n\\noindent\n\\item If $s\\in[0,r]$ and $t\\in[r,2r]$,\n\\begin{eqnarray*}\n|(x^{(1)}_{\\cdot-r}\\otimes y)_{s,t}|&\\leq&|(\\eta_{\\cdot-r}\\otimes y)_{s,r}|+|(x^{(1)}_{\\cdot-r}\\otimes y)_{r,t}|+|(\\eta(0)-\\eta(s-r))\\otimes (y(t)-y(r))|\\\\\n&\\leq& K|r-s|^{2\\beta}+K|t-r|^{2\\beta}+K|s-r|^\\beta|t-r|^\\beta\\leq K|t-s|^{2\\beta}.\n\\end{eqnarray*}\n\\item  If $s,\\,t\\in[0,r]$ then $x^{(1)}_{\\cdot-r}=\\eta_{\\cdot-r}$ and the result is already true.\n\\end{enumerate}\n\\end{enumerate}\n\\hfill $\\Box$\n\n\\renewcommand{\\theequation}{5.\\arabic{equation}} \\setcounter{equation}{0}\n\n\\section{Boundedness for deterministic integral equations}\n\nThe aim of this section is the proof of Theorem \\ref{tt2}. For simplicity let us assume $T=Mr$.\n\n\n\\vskip 5pt\n\\noindent\n{\\bf Proof of Theorem \\ref{tt2}:}\nThe proof will be done in several steps.\n\\vskip 5pt\n\\noindent\n{\\underline{Step 1:}}  Assuming that $(x,y,x\\otimes y)\\in M^\\beta_{d,m}(0,T)$, let us  define \n$\\left(x_{\\cdot-r}\\otimes y_{\\cdot-r}\\right)_{s,t}$. Set\n\\begin{equation}\\label{e5.1biss}\\left(x_{\\cdot-r}\\otimes y_{\\cdot-r}\\right)_{s,t}:=\\left(x\\otimes y\\right)_{s-r,t-r}.\\end{equation}\nIt clearly t belongs to $M^\\beta_{d,m}(r,T)$. Notice that the functions $x_{\\cdot-r}$ and $y_{\\cdot-r}$ are $\\beta$-H\\\"older\ncontinuous and $x_{\\cdot-r}\\otimes y_{\\cdot-r}$ is a continuous functions satisfying the multiplicative property. Indeed, we have, for \n$s\\leq u\\leq t$, \n\\begin{equation*}\\begin{array}{l}\n\\displaystyle (x_{\\cdot-r}\\otimes y_{\\cdot-r})_{s,u}+(x_{\\cdot-r}\\otimes y_{\\cdot-r})_{u,t}+(x(u-r)-x(s-r))\\otimes\n(y(t-r)-y(u-r))\\\\\n\\qquad \\displaystyle \n=(x\\otimes y)_{s-r,u-r}+(x\\otimes y)_{u-r,t-r}+(x(u-r)-x(s-r))\\otimes\n(y(t-r)-y(u-r))\\\\\n\\qquad \\displaystyle =(x\\otimes y)_{s-r,t-r} =(x_{\\cdot-r}\\otimes y_{\\cdot-r})_{s,t}.\n\\end{array}\\end{equation*}\nFinally, we also have that,\nfor all $(s,t)\\in \\{(s,t): r\\le s<t\\le T\\}$, \n\\begin{equation*}\n\\left| (x_{\\cdot-r}\\otimes y_{\\cdot-r})_{s,t}\\right|= \\left| (x\\otimes y)_{s-r,t-r}\\right|\n\\leq c|t-s|^{2\\beta }.\n\\end{equation*}\n\\vskip 5pt\n\\noindent\n{\\underline{Step 2:}}  Set, for any $s, t \\in [nr,(n+1)r]$, $n=0,\\dots, (M-1)$,\n\\begin{eqnarray*}\nJ_1^n(x,y,x\\otimes y) (t)&=&\\eta(0)+\\int_0^t b(s,x)ds+\\int_0^t\\sigma(x(s-r))dy_s+ z(t),\\\\\nJ_2^n(x,y,x\\otimes y )(s,t)&=&\n\\left\\{\\begin{array}{ll}\n(\\eta_{\\cdot-r}\\otimes y)_{s,t}, & {\\rm if}\\ s,t\\in [0,r] \\, {\\rm and} \\, n=1,\\\\\n(x_{\\cdot-r}\\otimes y)_{s,t}, & {\\rm otherwise}, \n\\end{array}\\right.\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n(x_{\\cdot-r}\\otimes y)_{s,t}&=&\\int_s^t (y(t)-y(u)) b(u-r,x)du+\\int_{s}^{t}\\sigma(x(u-2r))d(y_{\\cdot-r}\\otimes y)_{\\cdot,t}(u)\\\\\n&&\n+ \\int_s^t(y(t)-y(u))dz_{u-r},\\end{eqnarray*}\nwith $z$ given in (\\ref{det}). \n\n\\noindent Set\n \\begin{displaymath}\n \\Delta_y= \\left(\\left\\|\\eta\\right\\|_{\\beta}\n+\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n+(d^\\frac12 +1)(1+3k) \\mu \\left[\\left\\|y\\right\\|_{\\beta}\n+\\left\\|y\\right\\|_{\\beta}^2+ \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\right]\\right)^{-\\frac{1}{\\beta}},\n  \\end{displaymath}\n  with $k$ depending only on $\\beta$ and $\\gamma$.\nIf we consider $s,t$ such that\n$$0<t-s\\le \\Delta_y \\wedge 1,$$\nthen we will have that\n\\begin{eqnarray}\n(t-s)^\\beta&\\le &\\Delta_y^\\beta\\le \\frac{1}{\\left\\|\\eta\\right\\|_{\\beta}},\\label{e5.7}\\\\[2mm]\n(t-s)^\\beta&\\le &\\Delta_y^\\beta\\le \\frac{1}{\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}},\\label{e5.8}\\\\[2mm]\n(t-s)^\\beta&\\le &\\Delta_y^\\beta\\le \\frac{1}{(d^\\frac12 +1)(1+3k)\\ \\mu\\left(1 +\\left\\|y\\right\\|_{\\beta}\n+\\left\\|y\\right\\|_{\\beta}^2+ \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\right)}.\\label{e5.9}\n\\end{eqnarray}\n\\vskip 5pt\n\\noindent We will use an induction argument  to prove that for any $n$ and for all \n$s,t$  such that $nr \\le s < t \\le (n+1)r$ with $0<t-s\\le \\Delta_y \\wedge 1,$ it hols that\n$$\\Vert J_1^n \\Vert_{\\beta(s,t)} = \\Vert x \\Vert_{\\beta(s,t)} \\le \\mu (d^\\frac12 +1)\\left[1 +2k+ 3k \\left\\|y\\right\\|_{\\beta}\\right].$$\n\n\n\\vskip 5pt\n\\noindent\nAssuming this last inequality, the proof of (\\ref{e5.1}) is standard. Indeed, notice that\nit follows easily, that  for any  $0 \\le s<t \\le T$ with  $t-s\\le \\Delta_y$, \n$$ \\Vert x \\Vert_{\\beta(s,t)} \\le 2 \\mu (d^\\frac12 +1) \\left[1+2k+3k \\left\\|y\\right\\|_{\\beta}\\right].$$\nThen,  if $t-s\\le \\Delta_y$, $0\\le s<t$, we have\n\\begin{equation}\\label{e5.16}\n\\sup_{r\\in[s,t]} \\left|x(r)\\right|\\le \\left|x(s)\\right| + (t-s)^\\beta \\left\\|x\\right\\|_{\\beta(s,t)}\\le  \\left|x(s)\\right| +2,\n\\end{equation}\nand, particularly,  \n$$\\sup_{r\\in[0,\\Delta_y]} \\left|x(r)\\right| \\le  \\eta(0) +2.$$\nNow we divide the interval $[0,T]$ into $n=\\left[T\\Delta_y^{-1}\\right]+1$ intervals of length $\\Delta_y$, where $[a]$ denotes the largest integer bounded by $a$. Then, applying (\\ref{e5.16}) on the intervals $\\left[\\Delta_y,2\\Delta_y\\right], \\dots, \\left[(n-1)\\Delta_y,n\\Delta_y\\right]$ and the previous inequality, we obtain\n\\begin{displaymath}\n\\sup_{r\\in[0,T]} \\left|x(r)\\right|\\le \\eta(0) +2n \\le 2+\\eta(0) + 2\\left[T\\Delta_y^{-1}\\right],\n\\end{displaymath}\nand we can conclude that the estimate (\\ref{e5.1}) is true.\n\n\n\\vskip 5pt\n\\noindent Let us come back to check our induction argument to finish the proof.\n\n\n\n\n\n\n\n\n\n\n\n\\vskip 5pt\n\\noindent {\\underline{Step 2.1:}}\nAssume  $s,t \\in [0,r]$. \nOn the one hand, \n\\begin{equation*\n\\left\\|J_2^0\\right\\|_{2\\beta (s,t)}=\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}.\n\\end{equation*}\nOn the other hand, using Lemma \\ref{le2} and  Proposition  \\ref{prop23}, we obtain that\n\\begin{eqnarray*}\n\\left\\|J_1^0\\right\\|_{\\beta (s,t)}&\\le & (d^\\frac12 +1)\\bigg[\\|b\\|_\\infty + k\\Big(\\left\\|\\sigma\\right\\|_{\\infty} \\left\\|y\\right\\|_\\beta+ \\left[\\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}+\\|\\eta_{\\cdot-r}\\|_{\\beta(s,t)}\\left\\|y\\right\\|_\\beta\\right]\\\\\n&&\\times\\left[\\left\\| \\sigma' \\right\\|_{\\infty}+\n\\left\\| \\sigma' \\right\\|_{\\gamma} \\left\\| \\eta_{\\cdot-r} \\right\\|_{\\beta(s,t)}^\\gamma(t-s)^{\\gamma \\beta}\\right](t-s)^\\beta\\Big)\\bigg].\n\\end{eqnarray*}\nBy means of  (\\ref{e5.7}) and (\\ref{e5.8}), we have\n\\begin{eqnarray*}\n\\left\\| \\eta_{\\cdot-r} \\right\\|_{\\beta(s,t)}(t-s)^\\beta&\\le &1,\\\\\n \\left\\|\\eta_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)} (t-s)^\\beta&\\le &1.\\end{eqnarray*}\nThen we conclude that\n\\begin{equation}\\label{e5.11}\n\\left\\|J_1^0\\right\\|_{\\beta (s,t)}\\le  \\mu (d^\\frac12 +1)\\left[1 +2k+ 3k \\left\\|y\\right\\|_{\\beta}\\right].\n\\end{equation}\n\\vskip 5pt\n\\noindent\n{\\underline{Step 2.2:}} Since the first step does not follow the general case, in order to prove our induction we have also to consider the next interval.\n\n\\noindent So, assume now $s,t\\in [r,2r]$. \nWe first need to deal with $\\left\\|J_2^1\\right\\|_{2\\beta (s,t)}=\\left\\|x_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}$.\nApplying Proposition  \\ref{prop24} and dealing with the integrals with $b$ and $z$, we get\n\\begin{eqnarray*}\n\\left\\|J_2^1\\right\\|_{2\\beta (s,t)}&\\le & \\left\\|y\\right\\|_\\beta \\left\\|z\\right\\|_{\\beta(s-r,t-r)}\n+\\mu \\bigg[ \\left\\|y\\right\\|_\\beta  +k\\Big(\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n+\\left\\|y\\right\\|_{\\beta}^2\\\\&&+\\left[1+ \\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}^\\gamma(t-s)^{\\gamma \\beta}\\right]\n \\Big[\\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}\\left\\|y\\right\\|_{\\beta}^2\\\\\n &&+\\left\\|x_{.-2r}\\otimes y_{\\cdot-r}\\right\\|_{2\\beta (s,t)}\\left\\|y\\right\\|_{\\beta}+\\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}\n\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\Big](t-s)^\\beta\\Big)\\bigg].\n\\end{eqnarray*}\nThe same computations given to bound $J_1^0$ in (\\ref{e5.11}) and Lemma \\ref{le2} yields that\n\\begin{displaymath}\n\\left\\|z\\right\\|_{\\beta(s-r,t-r)}\\le d^\\frac12  \\mu \\left[1 +2k+3k \\left\\|y\\right\\|_{\\beta}\\right].\n\\end{displaymath}\nAs before, the estimate (\\ref{e5.7}) gives\n\\begin{displaymath}\n\\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}(t-s)^\\beta=\\left\\| \\eta_{.-2r} \\right\\|_{\\beta(s,t)}(t-s)^\\beta\\le 1,\n\\end{displaymath}\nand (\\ref{e5.1biss}) and (\\ref{e5.8}) imply\n$$\n\\left\\| x_{.-2r}\\otimes y_{\\cdot-r} \\right\\|_{2\\beta(s,t)}(t-s)^\\beta = \\left\\| \\eta_{\\cdot-r}\\otimes y  \\right\\|_{2\\beta(s-r,t-r)}(t-s)^\\beta\n \\le 1.\n$$\nSo, using all these inequalities, we can bound $J_2^1$ as follows:\n\\begin{eqnarray}\\label{e5.12}\n\\left\\|J_2^1\\right\\|_{2\\beta (s,t)}&\\le & (d^\\frac12 +1)\n(1+2k) \\mu \\left\\|y\\right\\|_{\\beta}+(d^\\frac12 +1) 3 k \\mu \\left\\|y\\right\\|_{\\beta}^2+\n3 k\\mu \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\nonumber\\\\\n&\\le & \n (d^\\frac12 +1)\n(1+3k) \\mu \\left(\\left\\|y\\right\\|_{\\beta}+\n\\left\\|y\\right\\|_{\\beta}^2+ \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\right).\n\\end{eqnarray}\nProposition  \\ref{prop23}  again yields\n\\begin{eqnarray*}\n\\left\\|J_1^1\\right\\|_{\\beta (s,t)}&\\le &  (d^\\frac12 +1)\\bigg[\\|b\\|_\\infty + k\\Big(\\left\\|\\sigma\\right\\|_{\\infty} \\left\\|y\\right\\|_\\beta+ \\left[\\left\\|x_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}+\\left\\|x_{\\cdot-r}\\right\\|_{\\beta(s,t)}\\left\\|y\\right\\|_\\beta\\right]\\\\\n&&\\times\\left[\\left\\| \\sigma' \\right\\|_{\\infty}+\n\\left\\| \\sigma' \\right\\|_{\\gamma} \\left\\| x_{\\cdot-r} \\right\\|_{\\beta(s,t)}^\\gamma(t-s)^{\\gamma \\beta}\\right](t-s)^\\beta\\Big)\\bigg].\n\\end{eqnarray*}\n Now, using the computations that we have done in the first step\n\\begin{displaymath}\\left\\| x_{\\cdot-r} \\right\\|_{\\beta(s,t)}=\\left\\| J_1^0 \\right\\|_{\\beta(s-r,t-r)}\\le \n\\mu (d^\\frac12 +1)  \\left[1+ 2k+ 3k \\left\\|y\\right\\|_{\\beta}\\right],\\end{displaymath}\nand (\\ref{e5.9}) yields that\n\\begin{displaymath}\\left\\|x_{\\cdot-r}\\right\\|_{\\beta(s,t)}(t-s)^\\beta\\le 1.\\end{displaymath}\nMoreover, also thanks to (\\ref{e5.12}) and (\\ref{e5.9}) \n\\begin{displaymath}\\left\\|x_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}(t-s)^\\beta\n=\\left\\|J^1_2\\right\\|_{2\\beta (s,t)}(t-s)^\\beta\n\\le 1.\\end{displaymath}\nCombining these two last bounds allow us to show that\n\\begin{equation}\\label{e5.13}\n\\left\\|J_1^1\\right\\|_{\\beta (s,t)}\\le \\mu  (d^\\frac12 +1) \\left[1+2k +3k\\left\\|y\\right\\|_{\\beta}\\right].\n\\end{equation}\n\\vskip 5pt\n\\noindent\n{\\underline{Step 2.3:}}  We can  write now our hypothesis of induction. Set  $(H_l)$: for any $s,t \\in [lr, (l+1)r]$, the following hypothesis are satisfied \n\\begin{displaymath}\n(H_l)\\qquad \\qquad \\left\\{\\begin{array}{l}\n\\displaystyle \\left\\|J_1^l\\right\\|_{\\beta (s,t)}\\le \\mu  (d^\\frac12 +1) \\left[1+2k+3k \\left\\|y\\right\\|_{\\beta}\\right],\\\\[2mm]\n\\displaystyle \\left\\|J_2^l\\right\\|_{2\\beta (s,t)}\\le \\mu  (d^\\frac12 +1) (1+3k)\n\\left[ \\left\\|y\\right\\|_{\\beta}+ \\left\\|y\\right\\|_{\\beta}^2+\n \\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\right].\n\\end{array}\\right.\n\\end{displaymath}\nWe have checked  ($H_1$). Let us suppose that ($H_l$) is satisfied for any $l=1,\\dots, n$. Our goal is to proved that $(H_{n+1})$ is also satisfied.\nFor $s,t \\in [(n+1)r,(n+2)r],$, the first two inequalties can be proved as in the previous case:\n\\begin{eqnarray*}\n\\left\\|J_1^{n+1}\\right\\|_{\\beta (s,t)}&\\le &  (d^\\frac12 +1) \\mu \\bigg[1 + k\\Big( \\left\\|y\\right\\|_\\beta+ \\left[\\left\\|x_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}+\\left\\|x_{\\cdot-r}\\right\\|_{\\beta(s,t)}\\left\\|y\\right\\|_\\beta\\right]\\\\\n&&\\times\\left[1+\n \\left\\| x_{\\cdot-r} \\right\\|_{\\beta(s,t)}^\\gamma(t-s)^{\\gamma \\beta}\\right](t-s)^\\beta\\Big)\\bigg],\\\\\n\\left\\|J_2^{n+1}\\right\\|_{2\\beta (s,t)}&=&\\left\\|x_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s,t)}\\\\\n&\\le & \\left\\|y\\right\\|_\\beta \\left\\|z\\right\\|_{\\beta(s-r,t-r)}\n+\\mu \\bigg[ \\left\\|y\\right\\|_\\beta  +k\\Big(\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\n+\\left\\|y\\right\\|_{\\beta}^2\\\\&&+\\left[1+ \\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}^\\gamma(t-s)^{\\gamma \\beta}\\right]\\\\\n&&\\times \n \\Big[\\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}\\left\\|y\\right\\|_{\\beta}^2\n +\\left\\|x_{.-2r}\\otimes y_{\\cdot-r}\\right\\|_{2\\beta (s,t)}\\left\\|y\\right\\|_{\\beta}\\\\\n &&+\\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}\n\\left\\|y_{\\cdot-r}\\otimes y\\right\\|_{2\\beta}\\Big](t-s)^\\beta\\Big)\\bigg].\n\\end{eqnarray*}\nIn order to study $J_1^{n+1}$ and $J_2^{n+1}$,  we will use that\n\\begin{eqnarray*}\n\\left\\|z\\right\\|_{\\beta(s-r,t-r)}&\\le & d^\\frac12 \\mu\\left[1+2k+3k\\left\\|y\\right\\|_{\\beta}\\right],\\\\\n\\left\\| x_{.-2r} \\right\\|_{\\beta(s,t)}(t-s)^\\beta&=& \\left\\|J_1^{n-1}\\right\\|_{\\beta(s-2r,t-2r)}(t-s)^\\beta\\le 1,\\\\\n\\left\\|x_{.-2r}\\otimes y_{\\cdot-r}\\right\\|_{2\\beta (s,t)}(t-s)^\\beta&=& \n\\left\\|x_{\\cdot-r}\\otimes y\\right\\|_{2\\beta (s-r,t-r)}(t-s)^\\beta\\\\\n&=&\n\\left\\|J_2^{n}\\right\\|_{2\\beta(s-r,t-r)}(t-s)^\\beta\\le 1,\\\\\n\\left\\|x_{\\cdot-r}\\right\\|_{\\beta(s,t)}(t-s)^\\beta&=& \\left\\|J_1^{n}\\right\\|_{\\beta(s-r,t-r)}(t-s)^\\beta\\le 1.\n\\end{eqnarray*}\nThen, it is not difficult to check that\n\\begin{eqnarray*}\n\\left\\|J_2^{n+1}\\right\\|_{2\\beta (s,t)}& \\le &\\mu (d^\\frac12 +1) \n(1+3k)\n\\left[ \\left\\|y\\right\\|_{\\beta}+  \\left\\|y\\right\\|_{\\beta}^2+\n \\left\\|y\\otimes y\\right\\|_{2\\beta}\\right],\\\\\n\\left\\|J_1^{n+1}\\right\\|_{\\beta (s,t)}&\\le &\\mu (d^\\frac12 +1) \\left[1+2k+3k \\left\\|y\\right\\|_{\\beta}\\right],\n\\end{eqnarray*}\nwhere the bound of $J_2^{n+1}$ is used to prove the last inequality.\n\\vskip 5pt\n\\noindent\n\\hfill $\\Box$\n\n\\renewcommand{\\theequation}{5.\\arabic{equation}} \\setcounter{equation}{0}\n\n\\section{Stochastic case}\n\nFix a parameter $H \\in(\\frac13, \\frac{1}{2})$. Set $W^H=\\left\\{W^H(t),t\\in \\lbrack 0,T]\\right\\} $ a $m$-dimensional fractional Brownian\nmotion of Hurst parameter $H$. The components $W^{H,1},\\ldots,W^{H,m}$ are independent  centered Gaussian processes\n with the covariance\nfunction\n$$ \\ R\\left( t,s\\right)\n=\\frac{1}{2}\\left( s^{2H}+t^{2H}-|t-s|^{2H}\\right).\n$$\n\n\n\n\nLet us consider now a Stratonovich type integral with respect to\n$W^H$.\nFollowing the approach by Russo and Vallois \\cite{RV}, we have:\n\n\\begin{definition}\n\\label{def1}Let $u=\\{u(t),t\\in \\lbrack 0,T]\\}$ be a stochastic\nprocess with integrable trajectories. The Stratonovich integral of\n$u$ with respect to $W^{H,i}$ is defined as the limit in probability as\n$\\varepsilon $ tends to zero of\n\\begin{equation*}\n(2\\varepsilon )^{-1}\\int_{0}^{T}u(s)(W^{H,i}_{s+\\varepsilon}\n-W^{H,i}_{s-\\varepsilon})ds,\n\\end{equation*}\nprovided this limit exists. When the limit exists, it is denoted by\n$\\int_{0}^{T}u(t) \\circ dW^{H,i}_t$.\n\\end{definition}\n\\vskip 5pt\n\\noindent\nThen, we can consider the tensor product defined by\n$$\n( W^{H,i}_{\\cdot-r} \\otimes W^{H,j} )_{s,t} = \\int_s^t (W^{H,i}_{v-r} - W^{H,i}_{s-r} ) \\circ dW_v^{H,j},$$\nfor $ 0 \\le s < t \\le T$ and $i,j=1,\\ldots,m.$ Notice that when $i \\not= j$, the Stratotovich intergal coincides with the Skorohod integral.\n\nIt is proved in Proposition 5.2 in \\cite{NNT}  that\n\\begin{equation}\\label{fin1}\nE (\\vert ( W^{H,i}_{\\cdot-r} \\otimes W^{H,j} )_{s,t} \\vert^p ) \\le C_p \\vert t- s\\vert^{2pH},\n\\end{equation}\nfor all $i,j \\in \\{1,\\ldots,m\\}$ and $p \\ge 1$. Furthermore, they also have checked that there is a version of $( W^{H}_{\\cdot-r} \\otimes W^{H} )$ such that, for almost\nall sample paths of $W^H$, satisfy that $ ( W^{H,i}_{\\cdot-r} \\otimes W^{H,j} )_{s,t} \\in \\cc_2^{2\\beta}  (\\R^{m\\times m})$ for any  $\\beta \\in (\\frac13, H)$ and $i,j \\in \\{1,\\ldots,m\\},$  where $\\cc_2^{2\\beta} $ denotes a space of $2 \\beta$-H\\\"older continuous functions of two variables.\n\nSince the multiplicative property can be checked easily using the definition and the properties of the Stratonovich integral, it follows  that $(W^H_{\\cdot-r}, W^H, W^H_{\\cdot-r}\\otimes W^H)$ is a $\\beta$-H\\\"older continuous multiplicative functional for a fixed $\\beta \\in (\\frac13,H)$.\n\nNotice also that (\\ref{fin1}) implies that\n\\begin{equation*}\nE (\\Vert  W^{H,i}_{\\cdot-r} \\otimes W^{H,j} )\\Vert_{2\\beta(r,T)}^p )\\le C_p \\vert t- s\\vert^{2p(H-\\beta)},\n\\end{equation*}\nfor any $\\beta \\in (\\frac13,H)$ and for all $p \\ge 1$.\n\nThen Theorem \\ref{tt3} follows easily from  Theorems \\ref{tt1}  and Theorem \\ref{tt2}, when we apply its results pathwise.\n\n\n\n\\renewcommand{\\theequation}{5.\\arabic{equation}}\n\\setcounter{equation}{0}\n\\section{Appendix}\n\nLet us recall a fixed point Theorem from \\cite{B-R}.\n\n\\begin{lemma} \\label{puntfix}\nLet $(X,\\rho)$ be a complete metric space, and $\\rho_0$ and $\\rho_1$ two metrics on X equivalents to $\\rho$. If $\\cl: X\\rightarrow X$ satisfies:\n\\begin{enumerate}\n\\item There exists $r_0>0$, $x_0\\in X$ such that if $B_0=\\left\\{x\\in X; \\rho_0(x_0,x)\\leq r_0\\right\\}$ then $\\cl(B_0)\\subseteq B_0$,\n\\item There exists $a\\in (0,1)$ such that $\\rho_1\\left(\\cl(x),\\cl(y)\\right)\\leq a\\rho_1(x,y)$ for all $x,y\\in B_0$.\n\\end{enumerate}\nThen there exists $x^*\\in\\cl(B_0)\\subseteq X$ such that $x^*=\\cl(x^*)$.\n\\end{lemma}\n\nWe also need   a result with some properties of the solution of Skorohod's problem.\n\n\\begin{lemma} \\label{le2}\nFor each path $\\xi\\in \\cc(\\R_+,\\R^d)$, there exists a unique solution $(x,z)$ to the Skorokhod problem for $\\xi$. Thus there exists a pair of functions  ${\\displaystyle (\\phi,\\varphi): \\cc_+(\\R_+,\\R^d)\\rightarrow\\cc_+(\\R_+,\\R^{2d})}$ defined by $\\left(\\phi(\\xi),\\varphi(\\xi)\\right)=(x,z)$. The pair $\\left(\\phi,\\varphi\\right)$ satisfies the following:\n\\vskip 5pt\nThere exists a constant $K_l>0$ such that for any $\\xi_1,\\xi_2\\in\\cc_+(\\R_+,\\R^d)$ we have for each $t\\geq 0$,\n\\begin{eqnarray*}\n\\left\\|\\phi(\\xi_1)-\\phi(\\xi_2)\\right\\|_{\\infty(0,t)}&\\leq& K_l\\left\\|\\xi_1-\\xi_2\\right\\|_{\\infty(0,t)},\\\\\n\\left\\|\\varphi(\\xi_1)-\\varphi(\\xi_2)\\right\\|_{\\infty(0,t)}&\\leq& K_l\\left\\|\\xi_1-\\xi_2\\right\\|_{\\infty(0,t)}.\n\\end{eqnarray*}\nMoreover  for each $0\\leq s<t\\leq T$\n\\[\\left\\|\\varphi(\\xi)\\right\\|_{\\beta(s,t)}\\leq d^\\frac12 \\left\\|\\xi\\right\\|_{\\beta(s,t)},\\]\n\\end{lemma}\n\\vskip 6pt\n\\noindent\n{\\bf Proof:} We refer the reader to Proposition A.0.1 in \\cite{K-W} for the proof of the first part of the Lemma.\n\\vskip 5pt\n\\noindent\nSet $z=\\varphi(\\xi)$. Take $u,v$ such that $ s \\le u <v \\le t$. Fixed a component $i$, we wish to study\n$$ \\frac{\\vert z^i(v) - z^i(u) \\vert}{(v-u)^\\beta}.$$\nWhen $z^i(v)=z^i(u)$, this is clearly zero.\nOn the other hand, when $z^i(v)>z^i(u)$, let us define\n\\begin{eqnarray*}\nu^*&:=&\\sup\\{ u'\\ge u; z^i(u)=z^i(u') \\},\\\\\nv^*&:=&\\inf\\{ v' \\le v; z^i(v)=z^i(v') \\}.\n\\end{eqnarray*}\nThen, $u \\le u^* < v^* \\le v$ and $z^i(u)=z^i(u^*), z^i(v)=z^i(v^*)$. So\n$$ \\frac{\\vert z^i(v)-z^i(u) \\vert}{(v-u)^\\beta} \\le \\frac{\\vert  z^i(v^*)-z^i(u^*) \\vert}{(v^*-u^*)^\\beta} =\n\\frac{\\vert \\xi^i(v^*)-\\xi^i(u^*) \\vert}{(v^*-u^*)^\\beta} $$\nwhere the last equality follows from the fact that\n$\\xi^i$ and $z^i$ coincides whenever $z^i$ is not constant.\n\nThen, note that\n$$\n\\sup_{s < u <v< t} \\frac{\\vert  z^i(v)-z^i(u) \\vert}{(v-u)^\\beta}   \\le\n\\sup_{s < u^* <v^*< t} \\frac{\\vert \\xi^i(v^*)-\\xi^i(u^*) \\vert}{(v^*-u^*)^\\beta} \\leq\n \\Vert \\xi \\Vert_{\\beta(s,t)}.$$\nFinally, we get that\n$$\n\\Vert z \\Vert_{\\beta(s,t)} \\leq \\Big( \\sum_{i=1}^d \\big( \\sup_{s < u <v < t} \\frac{\\vert z^i(v)-z^i(u) \\vert}{(v-u)^\\beta} \\big)^2 \\Big)^\\frac12   \\le d^\\frac12\n \\Vert \\xi \\Vert_{\\beta(s,t)}.$$\n\n\\hfill $\\Box$\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nNeural style transfer aims at rendering a content image with style patterns from a reference image. The pioneering style transfer algorithm is proposed by Gatys \\etal \\cite{o1,o2}, which iteratively optimizes an image with perceptual losses. To accelerate the stylization process, a bunch of feed-forward network based approaches \\cite{f1,f2,f3,f4,f5,f6,f7,f8,f9} flourished. These methods render the content image with a single forward propagation but cannot generalize well to unseen styles. Arbitrary style transfer methods \\cite{a1, a2, a3, a5, a6} further extend the feed-forward network to arbitrary styles. These flexible yet efficient methods have received widespread attention from academics and industrial. \n\n\nRecently, style attentional network (SANet)\\cite{att1} and its following works \\cite{att2, att3} have achieved state-of-the-art performance. With a learnable kernel, these methods compute pair-wise similarities to generate the attention map, which serves as a per-point feature transformation for fine-grained stylization. Still, the attentional style transfer methods are imperfect: they tend to stylize the semantic regions inconsistently. As demonstrated in the first row of \\cref{fig:fig1}, the sky region is rendered by highly different style patterns from various semantic regions, causing visual artifacts. According to \\cite{mani1}, each semantic region corresponds to a feature manifold, and the feature follows a multi-manifold distribution. To render a content region consistently, features from it should only be stylized by those from the most related style manifold. However, since the per-point transformation failed to capture the manifold distributions, features of the sky manifold are stylized independently, leading to chaotic results. \n\n\nTo address this problem, we want the attention module to learn a manifold-aware measurement, which consistently matches features between related content and style manifolds. Unfortunately, since the content semantic and style semantic are inherently dissimilar, their manifold distributions are heterogeneous. It is difficult for the attention module to learn the desired measurement. Therefore, we adopt the manifold alignment to reveal the cross manifold correspondence. We align each content manifold to its most related style manifold, thus increasing the structural similarity between them. Afterward, features from the corresponding content and style manifolds are close in space, making the attention module match them consistently. The existing manifold alignment style transfer method \\cite{mani1} employs a global channel transformation for stylization. Instead of aligning the related content and style manifolds individually, this method aligns the multi-manifold distributions as a whole. As a result, it is not suitable to work with the attention module. \n\n\nIn this paper, we proposed the progressive attentional manifold alignment (PAMA) framework, which performs attention operations and space-aware interpolations multiple times. The proposed PAMA dynamically aligns each of the content manifolds to their most related style manifolds, enabling the consistent attention mechanism between regions. Firstly, the attention operation is employed to rearrange the style features according to the spatial structure of content features. By matching content and style features, the related content manifolds and style manifolds correspond on the feature map. Afterward, the space-aware interpolation fuses the corresponding manifolds with adaptive weights. The interpolation can increase the structural similarity of the corresponding manifolds dynamically, making the attention module easier to match features between them. However, a single alignment cannot build a strong enough correspondence between manifolds for the attention module. Therefore we repeat the manifold alignment process multiple times and employ multistage loss functions. An image reconstruction loss is also used to maintain the shared space for manifold alignment. Our contributions can be summarized as:\n\n\n\\begin{itemize}\n\\item[$\\bullet$]We proposed a new arbitrary style transfer framework named PAMA, which gradually aligns content manifolds to style manifolds with the attention mechanism for consistent stylization between semantic regions. \n\\item[$\\bullet$]A multistage loss function is designed to enable progressive manifold alignment while preserving the regional consistency. We also adopt an image reconstruction loss to maintain the shared space for manifold alignment.\n\\item[$\\bullet$]Experiments show that the proposed framework can generate fine-grained stylization result in real-time (101 fps for 512px images on Tesla V100 GPU).  \n\\end{itemize}\n\n\\section{Related Works}\n\\label{sec:rel}\n\\subsection{Arbitrary Style Transfer}\nThe goal of arbitrary style transfer is to generate stylization results in real-time with arbitrary content-style pairs. The mainstream arbitrary style transfer algorithms can be divided into two groups: the global transformation based and local patch based. The global transformation based method utilizes the global statistic for feature transformation. One of the representative methods is AdaIN\\cite{a1}, which forces the mean and variance of content features to be the same as the style features for stylization. The WCT\\cite{a2} changes the covariance matrix of content features with whitening and coloring transformation. For Linear\\cite{a4}, it learns the feature transformation matrix explicitly with a subnet.  Although effective, the global transformation is vulnerable to local content distortions. \n\nFor the local patch based methods, they manipulate the feature patches for stylization. The style swap proposed by Chen \\etal \\cite{patch1} is the earliest patch based method, which swaps the content patches with the most similar style patches. The DFR\\cite{patch2} and AAMS\\cite{patch3} further extend this method with the global statistics. These local patch based methods can preserve the content structure fairly well but tend to trigger visual artifacts. Recently, the SANet\\cite{att1} is proposed to match content and patches with a learnable similarity kernel dynamically. Then MANet\\cite{att2} disentangles the content and style representation for attention mechanism. AdaAttN\\cite{att3} enhances the attention mechanism with multi-layer features to reduce the content distortion. But as the attentional methods render the feature points independently, they fail to capture the multi-manifold distribution of features, producing inconsistent stylization results.\n\\subsection{Manifold Alignment}\nManifold alignment aims at revealing the relationship of two datasets from different manifold distributions. These algorithms learn to project the two datasets to a shared subspace, where the correspondence of two datasets is established, and their manifold structures are preserved. Existing manifold alignment methods can be divided into semi-supervised methods\\cite{m1,m2,m3} and unsupervised methods\\cite{m4,m5,m6}. Huo \\etal introduce manifold alignment methods to the style transfer community\\cite{mani1}, which makes an assumption that the image features follow a multi-manifold distribution. This method aligns the content manifolds to style manifolds with a global channel transformation. However, It cannot increase the similarity of related manifolds in a targeted manner, and thus not suitable for consistent attention mechanism between semantic regions.\n\n\n\\section{Method}\n\\subsection{Pre-Analysis}\nIn this section, we will explain the inconsistent stylization phenomenon from a manifold perspective. According to \\cite{mani1}, the image features follow a multi-manifold distribution that each semantic region corresponds to a manifold. If there were $m$ semantic regions in the content image, we can divide the content features $F_c$ into $m$ subsets corresponding to the manifolds:\n\\begin{equation}\n\\label{eq:eq1}\nF_c = \\cup_{i=1}^m F_{c,i} \\ , F_c \\subseteq \\mathbb{R}^{C \\times H_cW_c}, F_{c,i} \\subseteq \\mathbb{R}^{C\\times M_i}\n\\end{equation}\nwhere $F_{c,i}$ denotes the $i$-th subset (corresponding to the $i$-th manifold) which consists of $M_i$ features. The style features $F_s$ follows the same definition: \n\\begin{equation}\n\\label{eq:eq2}\nF_s = \\cup_{i=1}^n F_{s,i} \\ , F_s \\subseteq \\mathbb{R}^{C \\times H_sW_s}, F_{s,i} \\subseteq \\mathbb{R}^{C\\times N_i}\n\\end{equation}\nhere we have defined the content features and style features in a manifold perspective.\n\nThe inconsistency happens because the per-point rendering of the attention mechanism neglects the multi-manifold distribution. Suppose the $p$-th content manifold is most related to the $q$-th style manifold, which means that features from $F_{c,p}$ should only match with features from $F_{s,q}$ in attention. For any content feature $f_c^x$ from $F_c$, if it belongs to $F_{c,p}$, we can write the attention operation as:  \n\\begin{equation}\n\\label{eq:eq3}\n\\begin{aligned}\n\\hat f_s^x & = \\sum_{i=1}^{H_sW_s} sim(f_c^x, f_s^i)f_s^i \\\\\n& = \\sum_{i=1}^{N_q} sim(f_c^x, f_{s,q}^i)f_{s,q}^i+\\eta(f_c^x, F_s)\n\\end{aligned}\n\\end{equation}\nwhere $f_s^i$ denotes the $i$-th feature from $F_s$, and the $f_{s,q}^i$ refers to the $i$-th feature from the $q$-th subset of $F_s$. The learned similarity kernel of attention mechanism abbreviates as $sim(\\cdot, \\cdot)$. The $\\eta(\\cdot, \\cdot)$ is the mismatch term measuring the inconsistency. This equation suggests that features from a content manifold should only be stylized by features from the most related style manifold, and any out-of-manifold rendering is considered as mismatch. If the mismatch term $\\eta$ is relatively high, the stylization result tend to be inconsistent. To reduce the mismatch term and enable consistent attention module, we proposed the progressive attentional manifold alignment (PAMA) to align the content manifolds to their most related style manifolds.\n\n\n\n\n\n\n\n\\subsection{Overall Framework}\n\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure2.pdf}\n\n   \\caption{The architecture of our network. The content manifolds is gradually aligned to the style manifolds with three independent attentional manifold alignment (AMA) blocks. The dash lines are only forwarded during training to generate the intermediate results for loss calculation. }\n   \\label{fig:fig2}\n\\end{figure}\n\n\n\\cref{fig:fig2} shows the proposed progressive attentional manifold alignment (PAMA) framework. Our method uses a pre-trained VGG \\cite{vgg} network to encode the content image $I_c$ and style image $I_s$ to obtain the $ReLU4\\_1$ features $F_c$ and $F_s$. The attentional manifold alignment (AMA) consists of an attention module and a space-aware interpolation module. Passed through three AMA blocks, the content feature $F_c$ gradually fuses style information, during which the content manifolds are aligned to the style manifolds. In this way, the attention mechanism can capture the manifold distributions and render the semantic regions consistently. Finally, the aligned content feature will be fed into the decoder to generate the stylized image. The structure of the decoder is symmetric to the encoder. Notice that the intermediate stylized features are only decoded during training to calculate the multistage perceptual loss (dash lines in \\cref{fig:fig2}). \n\n\n\n\n\n\n\n\\subsection{Loss Function}\n\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=0.8\\linewidth]{figure\/figure3.pdf}\n\n   \\caption{Attention mechanism will change the original VGG\\cite{vgg} space. The first row is generated by the proposed PAMA, the second row shows the result of SANet\\cite{att1}.}\n   \\label{fig:fig3}\n\\end{figure}\n\n\nThe attentional transformation in existing methods like \\cite{att1, att2, att3} will change the original VGG\\cite{vgg} space of style feature. \\cref{fig:fig3} demonstrates that the decoder of SANet failed to parse the original content and style feature in the VGG space. This characteristic is devastating for our progressive alignment that the attention module needs to learn a cross space similarity measurement, making the inconsistency problem even worse. To constraint all of the features in the VGG space, we employed an image reconstruction loss: \n\\begin{equation}\n\\label{eq:eq4}\n\\begin{aligned}\n& L_{rec} = \\lambda(\\|(I_{rc}-I_c)\\|_2 + \\|(I_{rs}-I_s)\\|_2) + \\\\\n& \\sum_{i}(\\|\\phi_i(I_{rc})-\\phi_i(I_c)\\|_2 + \\|\\phi_i(I_{rs})-\\phi_i(I_s)\\|_2)\n \\end{aligned}\n\\end{equation}\nwhere $I_{rc}$ and $I_{rs}$ are the content and style image reconstructed from VGG features, and the $\\lambda$ is a constant weight. The $\\phi_i(I)$ refers to the $ReLU\\_i\\_1$ layer VGG feature of image $I$. Since this loss forces the decoder to reconstruct VGG features, all features between the encoder and decoder are restricted in the VGG space. In the manifold alignment perspective, this loss function maintains the shared space for the alignment between content and style manifolds.\n\nFor the progressive manifold alignment, the overall loss function consists of multiple stages. In each stage, the loss is a weighted summation:\n\\begin{equation}\n\\label{eq:eq5}\nL = \\sum_{i=1}(\\lambda_{ss}^i L_{ss} + \\lambda_r^i L_r + \\lambda_m^i L_m + \\lambda_{h}^i L_{h})+L_{rec}\n\\end{equation}\nwhere the $\\lambda_x^i$ denotes the weight for $L_x$ in the $i$-th stage. The $L_{ss}$ is the content loss, while the $L_r$, $L_m$, and $L_h$ serve as the style losses. The initial value of the content weight $\\lambda_{ss}^1$ is relatively high to preserve the content manifold structure. In the next stages, the content weight decreases gradually to encourage more vivid style patterns.\n\n\nOur content loss is based on the structure self-similarity descriptor\\cite{remd1} between the content feature $F_c$ and the VGG feature of stylized image $F_{cs}$:\n\\begin{equation}\n \\label{eq:eq6}\nL_{ss} = \\frac{1}{H_c W_c} \\sum_{i,j} |\\frac{D_{ij}^c}{\\sum_i D_{ij}^c}-\\frac{D_{ij}^{cs}}{\\sum_j D_{ij}^{cs}}|\n\\end{equation}\nwhere $D_{ij}^c$ and $D_{ij}^{cs}$ are the pairwise cosine distance matrices of $F_c$ and $F_{cs}$ respectively. This loss function is effective for manifold structure preservation, because it regulates the feature correlation within manifolds to be invariant. \n\nFollowing the setting of \\cite{remd1, remd2} we adapts the relaxed earth mover distance (REMD):\n\\begin{equation}\n\\label{eq:eq7}\nL_{r} = \\max (\\frac{1}{H_sW_s}\\sum_i \\min_j C_{ij}, \\frac{1}{H_cW_c}\\sum_j \\min_i C_{ij})\n\\end{equation}\nwhere the $C_{ij}$ denotes the pair-wise cosine distance matrix between $F_{cs}$ and $F_s$. As suggested in \\cite{remd2}, this loss function optimizes along the manifold surface of style features, which works well with our manifold alignment process. We also added the moment matching loss to regularize the magnitude of features:\n\\begin{equation}\n\\label{eq:eq8}\nL_m = \\|\\mu_{cs} - \\mu_s\\|_1 + \\|\\Sigma_{cs} - \\Sigma_s\\|_1\n\\end{equation}\nwhere the $\\mu$ and $\\Sigma$ denotes the mean and covariance matrix of feature vectors.\n\nAlthough our proposed PAMA generates high-quality stylized images, it sometimes outputs color-mixed images. The self-similarity loss causes this limitation, for it forces the attention mechanism rendering a region overly uniform, making it mix the style patterns. To fix it, we referred to the differentiable color histogram loss proposed in HistoGAN\\cite{hist}:\n\\begin{equation}\n\\label{eq:eq9}\nL_{h} = \\frac{1}{\\sqrt 2} \\| H_s^{1\/2} - H_{cs}^{1\/2} \\|_2\n\\end{equation}\nwhere the $H$ refers the color histogram feature, the $H^{1\/2}$ denotes the element-wise square root.  At the expense of a little consistency, this loss function can reduce the color mixing problem. \n\n\n\n\n\n\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure4.pdf}\n\n   \\caption{The attentional manifold alignment (AMA) block. The attention module generates the attention map $A_{cs}$ to rearrange the style feature. The space-aware interpolation dense the channel information to obtain the adaptive weight $W$, which is applied to interpolate between the content feature $F_c$ and the rearranged style feature $\\hat F_s$.}\n   \\label{fig:fig4}\n\\end{figure}\n\n\\subsection{Attentional Manifold Alignment Block}\n\\cref{fig:fig4} demonstrates the attention manifold alignment block, which consists of an attention module and a space-aware interpolation module. \n\n\\textbf{Attention Module}. In the attention module, the content and style features are firstly normalized and embedded to compute the attention map:\n\\begin{equation}\n\\label{eq:eq10}\nA_{cs} = softmax(f(Norm(F_c))^T \\otimes g(Norm(F_s)))\n\\end{equation}\nwhere the $f(\\cdot)$ and $g(\\cdot)$ denote 1x1 convolution blocks for feature embedding, the $Norm(\\cdot)$ refers to the mean-variance normalization, and the $\\otimes$ is the matrix multiplication. The attention map contains the pair-wise similarities between the content features and style features. Then the attention map serves as an affine transformation to spatially rearrange the style features: \n\\begin{equation}\n\\label{eq:eq11}\n\\hat F_s = \\theta(h(F_s)^T \\otimes A_{cs})\n\\end{equation}\nagain the $g(\\cdot)$ and $\\theta(\\cdot)$ are the 1x1 convolutions for feature embedding.\n\nThe attention module is exactly the same as the one in \\cite{att1, att2}, which produces inconsistent results. In the attention module of PAMA, we aim at finding the correspondence between related content and style semantic regions while minimizing the mismatch term $\\eta$ in \\cref{eq:eq3}. Therefore, we adopt a high initial value of self-similarity loss (\\cref{eq:eq6}) and decrease it gradually during the multi-stage manifold alignment. In the first stage, the high self-similarity loss encourages the manifold structure of the rearranged style features $\\hat F_s$ to be the same as the content feature $F_c$, thus forcing the attention module to match the features consistently. \\cref{fig:fig5} (a) shows that the style features $F_s$ are rearranged according to the spatial structure of content features $F_c$, and the related semantic regions correspond on the feature map. With this correspondence, the space-aware interpolation module can increase the structural similarity between the related manifolds. In the next stages, even if we decrease the self-similarity loss for more vivid style patterns, the attention module will reveal the relationship between manifolds and rendering the features consistently. We have verified this in the experiment part with \\cref{fig:fig8}.\n\n\n\\textbf{Space-aware Interpolation Module.} This module adaptively interpolates between $F_c$ and $\\hat F_s$ with regional information. Initially, the channel dense operation applies convolution kernels of different scale on the concatenated feature to summarize multi-scale regional information:\n\\begin{equation}\n \\label{eq:eq12}\nW = \\frac{1}{n}\\sum_{i=1}^n \\psi_i ([F_c, \\hat F_s]) \n\\end{equation}\nwhere $\\psi_i (\\cdot)$ represent the $i$-th convolution kernel, and the $[\\cdot, \\cdot]$ denotes the channel concatenation operation. The concatenated feature can help us to identify the differences between the corresponding content and style manifolds, figuring out the local inconsistencies triggered by the attention module. This learnable channel dense operation outputs the scalar spatial weights $W \\in \\mathbb{R}^{H \\times W}$, which is used for interpolation:\n\\begin{equation}\n \\label{eq:eq13}\nF_{cs} = W \\odot F_c +(1-W) \\odot \\hat F_s\n\\end{equation}\nwhere the $\\odot$ refers to the dot production. Different from the ordinary feature fusion in attentional methods like \\cite{att1, att2, att3}, the space-aware interpolation fuses features in the same space. Therefore the interpolated content feature will not suffer from degradation (see \\cref{fig:fig3}). As you can see in \\cref{fig:fig5} (b), the interpolation fully solved the local distortion. Moreover, the affinity between the corresponding manifolds are increased, making it easier for the attention module to render semantic regions consistently. Then the stylized feature $F_{cs}$ will be feed into the next attentional alignment block for further refinement.\n\n\n \n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure5.pdf}\n\n   \\caption{The manifold alignment process in the first stage. (a) The attention operation to find the correspondence between manifolds; (b) The interpolation to increase the structural similarity between corresponding manifolds.}\n   \\label{fig:fig5}\n\\end{figure}\n\n\n\n\n\n\n\\section{Experiment}\n\\subsection{Implementation Details}\nOur proposed PAMA is trained with content images from COCO\\cite{coco} and style images from wikiart\\cite{wiki}. Features from $ReLU\\_3\\_1$, $ReLU\\_4\\_1$, and $ReLU\\_5\\_1$ are used to compute the self-similarity loss $L_{ss}$, the REMD loss $L_r$, and the moment matching loss $L_m$. For self-similarity loss, we set its weights $\\lambda_{ss}^1$, $\\lambda_{ss}^2$, $\\lambda_{ss}^3$ to 12, 9, 7 respectively. All weights of the style loss terms $L_r$ and $L_m$ are set to 2. These terms are normalized with the summation of their weights (\\eg $\\frac{12}{12+2+2}$ for $\\lambda_{ss}^1$). The weights of color histogram loss $\\lambda_h^1$,  $\\lambda_h^2$, $\\lambda_h^3$ are set to 0.25, 0.5, and 1. The $\\lambda$ for the reconstruction loss in \\cref{eq:eq5} is 50.  We use the Adam\\cite{adam} as our optimizer with learning rate of 0.0001 and momentum of 0.9. During training, we take 8 content-style image pairs as a batch. The smaller dimension of content and style images are rescaled to 512, and then we randomly crop a 256x256 patch for efficient training. In the testing phase, our fully convolutional network can tackle images with any size.\n\n\\begin{figure*}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure6.pdf}\n   \\caption{Qualitative Comparison.}\n   \\label{fig:fig6}\n\\end{figure*}\n\n\\subsection{Comparison with Prior Arts}\nTo evaluate the propose approach, we compared it with other arbitrary style transfer methods, including the pioneering method AdaIN\\cite{a1}, the first manifold-alignment-based method MAST \\cite{mani1}, and the attention-based methods \\cite{att1, att2, att3}. \n\n\\textbf{Qualitative Comparison}\nIn \\cref{fig:fig6} we show the comparison results. AdaIN\\cite{a1} adjusts the mean and variance of the content feature globally for stylization. For its global transformation neglects the local details, AdaIN tends to bring distortions (1st, 2nd, 3rd, 5th rows) and visual artifacts(2nd, 3rd rows). MAST\\cite{mani1} is an arbitrary style transfer algorithm based on manifold alignment, which can stylize the semantic regions consistently. Rather than treating different manifolds dynamically, MAST aligns the multi-manifold distribution as a whole with a global channel transformation. This makes the MAST suffers from style degradation (2nd, 4th, 5th rows) and blurry (2nd, 5th rows). SANet\\cite{att1} and MANet\\cite{att2} use the attention mechanism to transform the deep features point-wise. They produce fine-grained results with vivid style patterns but are still not free from the distortion problem (2nd, 4th rows) and the visual artifacts (3rd, 5th rows). Because these attentional methods render the feature points independently without considering the manifold distribution, their semantic regions are not consistent (1st, 2nd, 3rd rows). The AdaAttN\\cite{att3} extends the attention mechanism with shallow features and achieves better content preservation. However, as you can see in the 3rd and 4th rows, increasing content protection capabilities sacrifices style patterns: the color distribution and brushstrokes of AdaAttN are quite different from the style image.  The AdaAttN is also affected by inconsistency (1st, 2nd rows) and visual artifacts (5th row). In the 1st and 2nd row, because the PAMA has aligned the content and style manifolds for the attention module, it can produce much more consistent results. In the 3rd and 5th rows, you can see that our methods also generate significantly fewer visual artifacts. The proposed PAMA achieves a better balance between content and style while increasing the consistency in semantic regions.\n\n\\textbf{User Study.}\nWe use 25 content images and 25 style images to generate 625 stylization results in total for each method. Then 20 samples are randomly drawn from the 625 results and distributed to users. For each sample, users are asked to choose their favorite stylization results among all methods. We used 4 evaluation indicators: content preservation, style quality, regional consistency, and overall performance. We collect votes from 100 users and plot \\cref{fig:fig7}. The result demonstrates that our method produces results with significantly higher consistency and overall performance. \n\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure7.pdf}\n   \\caption{User preference result of AdaIN\\cite{a1}, MAST\\cite{mani1}, SANet\\cite{att1}, MANet\\cite{att2}, and AdaAttN\\cite{att3}. }\n   \\label{fig:fig7}\n\\end{figure}\n\n\\textbf{Efficiency.}\nIn \\cref{tab:tab1}, we compare the running time of our method with other baselines at 256px and 512px. All of the methods are evaluated on a server with an Intel Xeon Gold 5118 CPU @ 2.30GHz and an NVIDIA V100 PCIe 32G GPU. The results are the average running time of 100 image pairs. Thanks to the lightweight design of the attentional manifold alignment block, even if we align the manifold multiple times, our method can still generate artworks in real-time (101 fps at 512px).\n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{p{2cm} p{2cm} p{2cm}}\n    \\toprule\n    Method & Time (256px) & Time (512px) \\\\\n    \\midrule\n    AdaIN & 2.813ms  &  2.912ms \\\\\n    MAST & 1488ms & 2743ms \\\\\n    SANet & 4.792ms  &  6.351ms \\\\\n    MANet & 8.202ms  &  8.992ms \\\\\n    AdaAttN & 19.76ms  &  22.52ms\\\\\n    Ours & 8.527ms &  9.871ms\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Running time comparison.}\n  \\label{tab:tab1}\n\\end{table}\n\n\n\n\n\n\n\\subsection{Verify the Effectiveness}\n\n\\textbf{Attention Operation.}\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure8.pdf}\n   \\caption{Multi-stage attention results. Images in the first row are stylization results. Images from the second row are the decoded results of rearrange style feature $\\hat F_s$. (a) Results from the first stage; (b) Results from the second stage; (c) Results from the third stage.}\n   \\label{fig:fig8}\n\\end{figure}\nTo analyze the behavior of attention operation during the progressive manifold alignment process, we visualize the rearranged style feature $\\hat F_s$ in different stages. For the first stage, a relatively high self-similar loss is used for training, which regulates the manifold structure of the content image to be invariant. As we can see in \\cref{fig:fig8} (a) that the rearranged style feature is well aligned with the content structure. Although the result is consistent, the attention module fails to capture the manifold distributions, rendering all of the regions uniformly. Here, the consistency is provided by the self-similarity loss rather than the attention module. In the second stage, for the content manifold are fused with related style information in the first stage, the attention module can distinguish different content manifolds and stylize them with features from various style manifolds. In the final stage, as shown in \\cref{fig:fig8}, the content manifolds are well aligned with the most similar style manifolds, and the content semantic regions are rendered consistently by the style manifolds. The above proves the effectiveness of the manifold alignment method for the consistent attention mechanism.\n\n\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure9.pdf}\n   \\caption{Space-aware Interpolation results. The first row is the visualization of the adaptive weight $W$, where the yellow pixels denote higher stylization, and the blue pixels denote higher content preservation. The second row is the interpolation results of the first row respectively. (a) Interpolation of the first stage; (b) Interpolation of the second stage; (c) Interpolation of the third stage. }\n   \\label{fig:fig9}\n\\end{figure}\n\\begin{figure*}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure10.pdf}\n   \\caption{The effectiveness of the self-similarity loss and color histogram loss. (a) Original PAMA; (b) High self-similarity loss; (c) Low self-similarity loss; (d) High histogram loss; (e) low histogram loss.}\n   \\label{fig:fig10}\n\\end{figure*}\n\\begin{figure}[t]\n  \\centering\n   \\includegraphics[width=\\linewidth]{figure\/figure11.pdf}\n   \\caption{The effectiveness of multi-stage manifold alignment. (a) Single-stage manifold alignment; (b) Two-stage manifold alignment; (c) Three-stage manifold alignment. }\n   \\label{fig:fig11}\n\\end{figure}\n\n\\textbf{Space-aware Interpolation.}\nThis part verifies the effectiveness of the space-aware interpolation of all three manifold alignment stages. \\cref{fig:fig9} demonstrates that the space-aware interpolations are sensitive to edge information and tend to preserve content structures around the edges. This can help the network to remove the local inconsistency (or distortions) in salient areas. Also, the interpolation module has learned to detect the uniform regions and render them with higher strength. In this way, we can add style patterns without hurting the manifold structure significantly. For the reason that we decrease the self-similarity content loss gradually during manifold alignment, the interpolation module fuses more style information in the latter stages (more yellow pixels in \\cref{fig:fig9}), producing results with vivid style patterns.\n\n\\textbf{Loss Analysis.}\nIn Figure 10 we explore the effect of different loss weights, including the self-similarity loss $\\lambda_{ss}$ and the color histogram loss $\\lambda_h$. \\cref{fig:fig10} (a) is the original result of the proposed PAMA. The sky part has been rendered with red and yellow patterns chaotically. This is due to the color histogram loss forcing the color distribution of the result to be the same as the style image. \\cref{fig:fig10} (b) shows the result with high self-similarity loss, its weights are $\\lambda_{ss}^{1}=20$, $\\lambda_{ss}^2=15$, $\\lambda_{ss}^3=10$. After zooming in, you can find that the sky part is rendered uniformly, and the slings of the suspension bridge are well preserved. The self-similarity loss protects the structure of content regions. For \\cref{fig:fig10} (c), its $\\lambda_{ss}^1=10$, $\\lambda_{ss}^2=6$, $\\lambda_{ss}^3=4$. Although contains more vivid colors, the high style loss has introduced distortions and inconsistencies. In terms of the color histogram loss, \\cref{fig:fig10} (d) uses $L_h^1=0.5$, $L_h^2=1$, $L_h^3=2$. The high color histogram loss leads to more appropriate color rendering but also brings inconsistent color blocks in the image. $\\cref{fig:fig10}$ (e) is the result with color histogram weights of $L_h^1=0.125$, $L_h^2=0.25$, $L_h^3=0.5$. This example has the highest consistency in that there is no fluctuation in color in most of the parts. However, the overly uniform leads to the color mixing problem, the color of the sky is mixed into a pinkish color. In conclusion, the balance of self-similarity loss and color histogram loss controls the balance of consistency and color distribution. \n\n\\textbf{Multistage Manifold Alignment.}\nTo investigate the behavior of multistage manifold alignment, we trained another two networks with single-stage manifold alignment and two-stage manifold alignment, respectively. For the single-stage manifold alignment network, we trains it with content loss weight $L_{ss}^1=7$, REMD loss weight $L_r^1=2$, moment loss weight $L_m^1=2$, and histogram loss weight $L_h^1=1$. It is exactly the weight of the last stage of PAMA. \\cref{fig:fig11} (a) demonstrates that the grassland is stylized by green and purple patterns inconsistently. Single-stage manifold alignment is not enough to reveal the manifold structure. The two-stage manifold alignment network is trained with the same loss function and weights in the last two stages of PAMA. The consistency of the grassland improves, but local inconsistency still exists. \\cref{fig:fig11} (c) is the result generated by the proposed PAMA, and you can see a monotonous increase of consistency. \n\n\n\\section{Conclusion}\nIn this paper, we proposed the progressive attention manifold alignment framework (PAMA) to alleviate the inconsistency problem and improve the quality of stylization. PAMA uses the attention mechanism to reveal the correspondence between the most related content and style manifolds and then applies the space-aware interpolation to fuse the related manifolds adaptively. By performing this process multiple times, the structural similarity between the most related content and style manifolds increases, making it easier for the attention mechanism to match features consistently between them. The attention mechanism can capture the multi-manifold distribution of features to generate high-quality consistent results.\n\\clearpage\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s1}\n\nIn this note we are interested in non-trivial integer solutions, that is,\nsolutions with $k\\ge 2$, of the equation\n\\begin{equation}\n\\label{EME}\n1^k+2^k+\\dots+(m-2)^k+(m-1)^k=m^k.\n\\end{equation}\nConjecturally such solutions do not exist. For $k=1$ one has\nclearly the solution $1+2=3$ (and no further ones). From now on\nwe will assume that $k\\ge 2$.\nMoser~\\cite{Moser} showed in 1953 that if $(m,k)$ is a solution of \\eqref{EME}, then\n$m>10^{10^6}$ and $k$ is even. His result has since then been improved on.\nButske et al.~\\cite{graphs} have shown by computing, rather\nthan estimating, certain quantities in Moser's original proof that $m>1.485\\cdot 10^{\\,9\\,321\\,155}$.\nBy proceeding along these lines this bound cannot be improved on substantially.\nButske et al.~\\cite[p. 411]{graphs} expressed\nthe hope that new insights will eventually make it possible to reach the more natural benchmark\n$10^{10^7}$.\n\nUsing that $\\Sigma_k(m)=1^k+2^k+\\dots+m^k\\le\\int_1^m t^k \\d t$\nand $\\Sigma_k(m+1)>\\int_0^m t^k\\d t$ we obtain that $k+1<m<2(k+1)$.\nThis shows that the ratio $k\/m$ is bounded.\nBy a more elaborate reasoning along these lines Krzysztofek~\\cite{K}\nobtained that $k+2<m<\\frac32(k+1)$. This\nimplies that $k\\ge 4$ and hence\n\\begin{equation}\n\\label{ineq}\nk+2<m<2k.\n\\end{equation}\nDividing both sides of \\eqref{EME} by $m^k$ one sees that for every integer $m\\ge 2$, \\eqref{EME}\nhas precisely one \\emph{real} solution~$k$. It is known that\n$\\lim_{m\\to \\infty}k\/m=\\log 2$ and we show here that in fact the behaviour of $k$ as a function\nof $m$ can be determined in a much more explicit way (Theorem~\\ref{th:As} and Section~\\ref{sec:As}).\n\nMoree et al.~\\cite{MRU}, using properties of the\nBernoulli numbers and polynomials (an approach initiated in\nUrbanowicz \\cite{Urbi}), showed that\n$N_1=\\operatorname{lcm}(1,2,\\dots,200)\\mid k$. Kellner \\cite{Kellner} in 2002\nshowed that also all primes $200<p<1000$ have to divide~$k$. Actually, Moree et al.~\\cite[p.~814]{MRU} proved\na slightly stronger result and on combining this with Kellner's, one obtains that $N_2\\mid k$ with\n\\begin{align*}\nN_2\n=2^8\\cdot 3^5\\cdot 5^4\\cdot 7^3\\cdot 11^2\\cdot 13^2\\cdot 17^2\\cdot 19^2\\cdot\\prod_{23\\le p\\le 997}p\n>5.7462\\cdot 10^{427}.\n\\end{align*}\nFor some further references and info on the Erd\\H{o}s--Moser equation we refer to the book\nby Guy~\\cite[D7]{Guy}.\n\nIn this note we attack \\eqref{EME} using the theory of\ncontinued fractions. This approach was first explored in~1976 by Best and\nte Riele \\cite{Best} in their attempt to solve the related conjecture of\nErd\\H{o}s~\\cite{E} that there are infinitely many pairs $(m,k)$ such that\n$\\Sigma_k(m)\\ge m^k$ and $2(m-1)^k<m^k$. In this\ncontext they also gave the following variant of one of their results (without\nproof), namely, \\eqref{eq:2} with $O(m^{-2})$ replaced with $o(m^{-1})$.\nThe proof we give here uses the\nsame circle of ideas as used by Best and te Riele. It seems that after their work continued\nfractions in the Erd\\H{o}s--Moser context have been completely ignored.\nWe hope the present paper makes clear that this is unjustified.\n\n\\begin{Thm}\n\\label{th:As}\nFor integer $m>0$ and \\emph{real} $k>0$ satisfying equation~\\eqref{EME},\nwe have the asymptotic expansion\n\\begin{equation}\n\\label{eq:2}\nk=\\log2\\biggl(m-\\frac32-\\frac{c_1}m+O\\biggl(\\frac 1{m^2}\\biggr)\\biggr)\n\\quad\\text{as}\\; m\\to\\infty,\n\\end{equation}\nwith $c_1=\\frac{25}{12}-3\\log 2\\approx 0.00389\\dots$\\,.\nMoreover, if $m>10^9$ then\n\\begin{equation}\n\\frac km=\\log2\\biggl(1-\\frac3{2m}-\\frac {C_m}{m^2}\\biggr),\n\\qquad\\text{where}\\quad 0<C_m<0.004.\n\\label{E01}\n\\end{equation}\n\\end{Thm}\n\n\\begin{cor}\n\\label{cor:1}\nIf $(m,k)$ is a solution of {\\rm\\eqref{EME}} with $k\\ge 2$, then\n$2k\/(2m-3)$ is a convergent $p_j\/q_j$ of $\\log 2$ with $j$ even.\n\\end{cor}\n\\begin{cor}\n\\label{cor:2}\nThe number of solutions $m\\le x$ of \\eqref{EME}, as $x$ tends to infinity, is at most $O(\\log x)$.\n\\end{cor}\nThe equation \\eqref{EME} seems to be a sole example of an exponential Diophantine\nequation in just two unknowns for which even the finiteness of solutions is not yet established. The best\nresult in this direction is given by Corollary \\ref{cor:2}, which is an immediate consequence of the exponential\ngrowth of $p_j$ as a function of $j$ and Corollary \\ref{cor:1}.\n\nCorollary \\ref{cor:1} is not the only result which relates convergents to solutions of\nDiophantine equations. For example, if $(x_0,y_0)$ is a positive solution to Pell's\nequation $x^2-dy^2=\\pm 1$, with $d$ a positive square-free integer, then $x_0\/y_0$ is\na convergent of the continued fraction expansion of $\\sqrt{d}$.\nOn the other hand, in our situation the number in question, $\\log2$, is transcendental and\nits continued fraction expansion is expected to be sufficiently `generic'\n(unlike that of quadratic irrationals).\n\nCorollary \\ref{cor:1} naturally leads us to investigate common factors of $k$ and $2m-3$. This can\nbe done using the method of Moser, but is not in the literature, as before there was no special\nreason for considering $2m-3$.\n\nA key role in this arithmetic study is played by the congruence\n\\begin{equation}\n\\label{staudt}\n\\frac{\\sum_{j=1}^{l-1}j^r}y=\\begin{cases}\n0\\pmod{\\frac12} & \\text{if $r>1$ is odd}; \\\\\n-\\sum_{p\\mid l, \\, p-1\\mid r}\\frac1p\\pmod1 &\\text{otherwise}.\n\\end{cases}\n\\end{equation}\nThis identity can be proved using the Von Staudt--Clausen theorem; for\nalternative proofs see, e.g., Carlitz \\cite{Carlitz} or Moree \\cite{Canada}.\nIts relevance for the study of \\eqref{EME} was first pointed out by Moree~\\cite{Oz}.\n\nGiven $N\\ge 1$, put\n$$\n{\\mathcal{P}}(N)=\\{p:p-1\\mid N\\}\\cup\\{p:\\text{$3$ is a primitive root modulo $p$}\\}.\n$$\nBy a classical result of Hooley \\cite{Hooley} it follows, assuming the\nGeneralized Riemann Hypothesis (GRH), that ${\\mathcal{P}}(N)$ has\na natural density $A$, with $A=0.3739558136\\dots$ the Artin constant, in the set of primes.\nIf $2k\/(2m-3)=p_j\/q_j$ is a convergent of $\\log 2$ arising in Corollary~\\ref{cor:1}, then it can be shown that\n$(q_j,6)=1$ and,\nif $p\\in {\\mathcal{P}}(N_2)$ and $p$ divides $q_j$, then\n$$\n\\nu_p(q_j)=\\nu_p(3^{p-1}-1)+1\\ge 2,\n$$\nwhere we write $\\nu_p(n)=a$ if $p^a\\mid n$ and $p^{a+1}\\nmid n$.\nAll primes $p\\le 2017$ are in ${\\mathcal{P}}(N_2)$. For $p\\ne 3$ we have $\\nu_p(3^{p-1}-1)=1$ unless\n$3^{p-1}\\equiv 1\\pmod{p^2}$, that is, $p$~is a Mirimanoff prime. (It is known that the only\nMirimanoff primes $p<10^{14}$ are $11$ and $1006003$.)\n\nThe main idea of this paper is, in essence, to make use of the fact that the convergents\n$p_j\/q_j$ of $\\log 2$ have no reason to also satisfy $N_2\\mid p_j$. The first piece of information\ncomes from asymptotic analysis and the latter piece from arithmetic. Analysis and\narithmetic give rise to conditions on the solutions that `do not feel\neach other' and this is exploited in our main result:\n\n\\begin{Thm}\n\\label{main}\nLet $N\\ge 1$ be an arbitrary integer. Let\n$$\\frac{\\log 2}{2N} = [a_0,a_1,a_2,\\dots] = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cdots}}$$\nbe the (regular) continued fraction of\n$({\\log 2})\/(2N)$, with $p_i\/q_i = [a_0,a_1,\\dots,a_i]$ its $i$-th partial convergent.\n\nSuppose that the integer pair $(m,k)$ with $k\\ge 2$ satisfies \\eqref{EME}\nwith $N\\mid k$.\nLet $j=j(N)$ be the smallest integer such that:\n\\begin{itemspec}\n\n\\itemindent18pt\n\\item\n\\label{ca}\n$j$ is even;\n\n\\item\n\\label{cb}\n$a_{j+1}\\ge 180N-2$;\n\n\\item\n\\label{cc}\n$(q_j,6)=1$; and\n\n\\item\n\\label{cd}\n$\\nu_p(q_j)=\\nu_p(3^{p-1}-1)+\\nu_p(N)+1$\nfor all primes $p\\in {\\mathcal{P}}(N)$ dividing $q_j$.\n\\end{itemspec}\nThen $m>q_j\/2$.\n\\end{Thm}\n\nComputing many partial quotients (that is, continued fraction digits)\nof $\\log 2$ is closely related to computing $\\log 2$ with\nmany digits of accuracy. Indeed, it is a well-known result of Lochs that for a generic number knowing\nit accurately up to $n$ decimal digits implies that we can compute about $0.97n$\n(where $0.97\\approx6(\\log 2)(\\log 10)\/\\pi^2$)\ncontinued fraction digits accurately. For example, knowing 1000 decimal digits of~$\\pi$ allows one\nto compute 968 continued fraction digits.\n\nIt seems a hopeless problem to prove anything about ${\\mathsf E}(\\log q_{j(N)})$, the\nexpected value of $\\log q_{j(N)}$ produced by the result. However, metric theory\nof continued fractions offers some hope of proving a non-trivial lower bound for\n${\\mathsf E}(\\log q_{j(N)}(\\xi))$,\nwhere we require conditions \\ref{ca}, \\ref{cb}, \\ref{cc} and~\\ref{cd} to be satisfied but replace\n$(\\log 2)\/(2N)$ by a `generic' $\\xi\\in [0,1]\\setminus \\mathbb Q$.\nIn this context recall the result of L\\'evy~\\cite{L} that, for such a $\\xi$,\n\\begin{equation}\n\\lim_{j\\to\\infty}\\frac{\\log q_j(\\xi)}j=\\frac{\\pi^2}{12\\log 2}\\approx1.18.\n\\label{Levy}\n\\end{equation}\nThe Gauss--Kuz'min statistics asserts that,\nfor a generic~$\\xi$, the probability that a given term\nin its continued fraction expansion is at least~$b$,\nequals $\\log_2(1+1\/b)$. This allows one to deal with the case where we only have\ncondition~\\ref{cb}. Likewise a result of Moeckel~\\cite{Moeckel},\nreproved in a very different way a few years later by Jager and\nLiardet \\cite{JL}, allows one to deal with the case\nwhere we only focus on condition~\\ref{cc}. Their result says\nthat for a generic $\\xi\\in[0,1]\\setminus \\mathbb Q$ we have\n$$\n\\lim_{n\\to\\infty}\\frac{\\{1\\le m\\le n:q_m(\\xi)\\equiv a\\pmod{d}\\}}{n}\n=\\frac d{J(d)}\\,\\frac{\\varphi((a,d))}{(a,d)},\n$$\nwhere $\\varphi$ denotes Euler's totient function, $J(m)=m^2\\prod_{p\\mid m}(1-1\/p^2)$ Jordan's\ntotient and $(a,m)$ the greatest common divisor of $a$ and~$m$.\nThis result shows that $(q_j,6)=1$ with probability $1\/2$\n(note that a natural number is coprime to~6 with probability $1\/3$).\nP.~Liardet communicated\nto us that methods of his paper \\cite{Liardet} can be used to take into account\nboth conditions \\ref{ca} and~\\ref{cc};\nalso the authors of \\cite{Harman} claim that this can be done.\nWe expect that there is a positive constant $c_1$ such that for a generic\n$\\xi$ satisfying conditions\n\\ref{ca}, \\ref{cb} and \\ref{cc}, we have ${\\mathsf E}(\\log q_{j(N)}(\\xi))\\sim c_1N$ as $N$ tends to\ninfinity. Furthermore, we expect that for a generic $\\xi$ satisfying conditions\n\\ref{ca}, \\ref{cb}, \\ref{cc} and \\ref{cd},\n${\\mathsf E}(\\log q_{j(N)}(\\xi))\\sim c_2 N\\log^{\\beta} N$\nfor some positive constants $c_2$ and $\\beta$; condition~\\ref{cb}\nis responsible for $N$, condition~\\ref{cd} for $\\log^{\\beta}N$, while conditions \\ref{ca} and~\\ref{cc}\naffect $c_2$. We are definitely not experts in metric aspects of number theory,\nthus leave this problem to the interested reader acquainted with the subject.\nIndeed, we even expect that going beyond computing the expected value of $\\log q_{j(N)}(\\xi)$ is\npossible, and a probability distribution for $\\log q_{j(N)}(\\xi)$ can be obtained.\n\nUsing the above results from metric theory of continued fractions and some heuristics we are led\nto believe that roughly speaking we can get\n$$\nm>10^{257N}\n$$\nfrom Theorem~\\ref{main}. Being able to compute the convergents of $(\\log 2)\/(2N)$ arbitrarily far,\nwe would expect (taking $N=N_2$) to show that $m>10^{10^{400}}$.\nWith the current computer technology computing sufficiently many convergents is the bottleneck.\nTaking this into consideration we would expect to get\n$$\nm>10^{0.515 r},\n$$\nfrom Theorem~\\ref{main}, where $r$ is the number of convergents we can compute accurately\nand $0.515$ is the base~10 logarithm of L\\'evy's constant~\\eqref{Levy}.\nNote that the fact that $N_2$ has many divisors gives us some flexibility and increases\nthe likelihood of the heuristics to be applicable. Indeed, our numerical experimenting agrees well\nwith our heuristic considerations (see Section~\\ref{s4}). Early 2009, A.~Yee and R.~Chan \\cite{AJY}\nreached $r>31\\cdot10^9$ for~$\\log 2$. On the other hand,\nY.~Kanada and his team~\\cite{Kanada} computed $\\pi$ to over 1.24 trillion decimal\ndigits already in 2002, using formulae of the same complexity as those used\nfor the computation of $\\log2$ (see \\cite[Chapter~3]{BB} for details).\nThus, given the present computer (im)possibilities,\none could hope to show (with a lot of effort!) that $m>10^{10^{12}}$.\n\nApplying Theorem~\\ref{main} with $N=2^8\\cdot 3^5\\cdot 5^3$\nor $N=2^8\\cdot 3^5\\cdot 5^4$, and invoking the result of\nMoree et al.~\\cite{MRU} that $N\\mid k$, we obtain the following\n\n\\begin{Thm}\n\\label{YVES}\nIf an integer pair $(m,k)$ with $k\\ge 2$ satisfies \\eqref{EME}, then\n$$m > 2.7139 \\cdot 10^{\\,1\\,667\\,658\\,416}.$$\n\\end{Thm}\n\nAs an application we can show that $\\omega(m-1)\\ge 33$,\nthis improves on the result of Brenton and Vasiliu \\cite{BV},\nwho have shown that $\\omega(m-1)\\ge 26$, where $\\omega$\ndenotes the number of distinct prime divisors; see Section~\\ref{s5.1} for further details.\n\nThe fact $N_2\\mid k$ naively implies that $k$~is of size $10^{427}$ (at least),\nwhich is much smaller than Moser's $10^{10^6}$. However, in this\npaper we show that the fact actually yields that $k>10^{10^9}$ (and likely even\n$k>10^{10^{400}}$)\\,---\\,a modestly {\\small small} number dividing~$k$\nleads to a huge lower bound for~$k$.\nThus, on revisiting \\cite{MRU} after 16~years, its main result is seen\nto be far more powerful than the second author thought at that time.\n\nIn the three following sections we prove Theorems~\\ref{th:As},\n\\ref{main} and \\ref{YVES}, respectively.\nOur final Section~\\ref{s5} is devoted to discussing some problems\nrelated to the Erd\\H os--Moser equation.\n\n\\section{Asymptotic dependence of $k$ in terms of $m$}\n\\label{sec:As}\n\nOur proof of Theorem~\\ref{th:As} makes use of the following lemma.\n\\begin{Lem}\n\\label{lem:As}\nFor any real $k>0$, we have\n\\begin{equation}\n(1-y)^k=e^{-ky}\\biggl(1-\\frac k2y^2-\\frac k3y^3\n+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5+O(y^6)\\biggr)\n\\quad \\text{as}\\; y\\to0.\n\\label{E02}\n\\end{equation}\nMoreover, for $k>8$ and $0<y<1$, the inequality\n\\begin{align}\n&\ne^{-ky}\\biggl(1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5-\\frac{k^3}6y^6\\biggr)\n\\nonumber\\\\ &\\quad\n<(1-y)^k\n<e^{-ky}\\biggl(1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k^2}2y^5\\biggr)\n\\label{E03}\n\\end{align}\nholds.\n\\end{Lem}\n\n\\begin{proof}\nAs for the asymptotic relation in~\\eqref{E02}, we simply develop\nthe Taylor expansion of $(1-y)^ke^{ky}$ up to~$y^5$.\nUnfortunately, estimates coming from the classical forms\nfor the remainder are not sufficient to derive a sharp\ndependence on $k$ as in~\\eqref{E03} for the last term. Therefore,\nwe need more drastic methods to quantify the asymptotics in~\\eqref{E02}\nwhen $0<y<1$.\n\nFirst note that\n\\begin{align*}\n(1-y)e^y\n&=1-\\sum_{n=2}^\\infty\\frac{n-1}{n!}y^n\n\\\\\n&=1-\\frac{y^2}2-\\frac{y^3}3-\\frac{y^4}8-\\frac{y^5}{30}-\\dotsb,\n\\qquad 0<y<1.\n\\end{align*}\nSince all coefficients, starting from $n=2$, in this power series are\nnegative and their sum is exactly $-1$, for these values of~$y$ we have the inequality\n\\begin{equation}\n1-\\frac{y^2}2-\\frac{y^3}3-\\frac{y^4}8-\\frac{y^5}{30}-\\frac{y^6}{120}\n<(1-y)e^y\n<1-\\frac{y^2}2-\\frac{y^3}3-\\frac{y^4}8.\n\\label{E04}\n\\end{equation}\nThe quantities\n\\begin{equation}\nx_1=\\frac{y^2}2+\\frac{y^3}3+\\frac{y^4}8\n\\quad\\text{and}\\quad\nx_2=\\frac{y^2}2+\\frac{y^3}3+\\frac{y^4}8+\\frac{y^5}{30}+\\frac{y^6}{120},\n\\label{E05}\n\\end{equation}\nwhich appear in~\\eqref{E04}, lie between $0$ and~$1$ for $0<y<1$.\n\nOur next ingredient is Gerber's generalization of the Bernoulli inequality~\\cite{Gerber} (see\nalso Alzer \\cite{Alzer}). It states that the remainder after $k$ terms of the (possibly divergent) binomial\nseries for $(1+x)^a$ ($a,x$ real with $-1<x$) has the same sign as the first neglected term.\nIn particular we have for real $k>2$ and $0<x<1$,\n\\begin{equation}\n(1-x)^k<1-kx+\\frac{k(k-1)}2x^2,\n\\label{E06}\n\\end{equation}\nand for real $k>3$ and $0<x<1$,\n\\begin{equation}\n(1-x)^k>1-kx+\\frac{k(k-1)}2x^2-\\frac{k(k-1)(k-2)}6x^3.\n\\label{E07}\n\\end{equation}\n\nUsing the right inequality in~\\eqref{E04} and taking $x=x_1$ in~\\eqref{E06} we obtain, for $k>2$,\n\\begin{align}\n(1-y)^ke^{ky}\n&<1-k\\biggl(\\frac{y^2}2+\\frac{y^3}3+\\frac{y^4}8\\biggr)\n+\\frac{k(k-1)}2\\biggl(\\frac{y^2}2+\\frac{y^3}3+\\frac{y^4}8\\biggr)^2\n\\nonumber\\\\\n&=1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4\n\\nonumber\\\\ &\\qquad\n+k(k-1)y^5\\biggl(\\frac16+\\frac{17}{144}y+\\frac1{24}y^2+\\frac1{128}y^3\\biggr)\n\\nonumber\\\\\n&<1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4\n+\\frac{385}{1152}k(k-1)y^5\n\\label{E08}\n\\end{align}\nimplying the upper estimate in~\\eqref{E03}.\nIn the same vein, the application of the left identity in~\\eqref{E04} and of~\\eqref{E07}\nwith $x=x_2$ results, for $k>3$, in\n\\begin{align}\n(1-y)^ke^{ky}\n&>1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5\n\\nonumber\\\\ &\\qquad\n-ky^6\\sum_{n=0}^{12}(a_nk^2+b_nk+c_n)y^n,\n\\label{E09}\n\\end{align}\nwhere the polynomials $p_n(k)=a_nk^2+b_nk+c_n$, $n=0,1,\\dots,12$, all have\npositive leading coefficients $a_n$; moreover, $p_n(k)>0$ for $k>3$ and\n$n=2,3,\\dots,12$, \\ $p_1(k)=\\frac1{24}k^2-\\frac{11}{60}k+\\frac{17}{120}>0$\nfor $k>4$, and $p_0(k)=\\frac1{48}k^2-\\frac{13}{72}k+\\frac{121}{720}>0$\nfor $k>8$. Using this positivity of the polynomials we can continue\nthe inequality in~\\eqref{E09} for $k>8$ as follows:\n\\begin{align}\n(1-y)^ke^{ky}\n&>1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5\n\\nonumber\\\\ &\\qquad\n-ky^6\\sum_{n=0}^{12}(a_nk^2+b_nk+c_n)\n\\nonumber\\\\\n&=1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\frac{k(5k-6)}{30}y^5\n\\nonumber\\\\ &\\qquad\n-ky^6\\biggl(\\frac16k^2-\\frac{17}{24}k+\\frac{11}{20}\\biggr),\n\\label{E10}\n\\end{align}\nfrom which we deduce the left inequality in~\\eqref{E03}, and the lemma\nfollows.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~{\\rm\\ref{th:As}}]\nThe original equation~\\eqref{EME} is equivalent to\n\\begin{equation}\n\\label{eq:1}\n1=\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k.\n\\end{equation}\nApplying to each term on the right-hand side the inequality\nfrom~\\eqref{E03} we obtain\n\\begin{align}\n&\nS_0-\\frac k{2m^2}S_2-\\frac k{3m^3}S_3+\\frac{k(k-2)}{8m^4}S_4+\\frac{k(5k-6)}{30m^5}S_5-\\frac{k^3}{6m^6}S_6\n\\nonumber\\\\ &\\qquad\n<\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k\n<S_0-\\frac k{2m^2}S_2-\\frac k{3m^3}S_3+\\frac{k(k-2)}{8m^4}S_4+\\frac{k^2}{2m^5}S_5,\n\\label{E11}\n\\end{align}\nwith the notation\n$$\nS_n=\\sum_{j=1}^{m-1}j^ne^{-kj\/m}\n=\\sum_{j=1}^{m-1}j^nz^j\\bigg|_{z=e^{-k\/m}}.\n$$\n\nBy (\\ref{ineq}) we have $e^{-1}<z<e^{-1\/2}$, where $z=e^{-k\/m}$, and\nhence $1\/(1-z)<1\/(1-e^{-1\/2})<3$, and in the closed-form expression of the sum\n$$\nS_0=\\sum_{j=1}^{m-1}z^j=\\frac z{1-z}-\\frac{z^m}{1-z},\n$$\nthe second term as well as its $z\\frac{\\d}{\\d z}$-derivatives are bounded:\n\\begin{gather*}\n0<\\frac{z^m}{1-z}\\bigg|_{z=e^{-k\/m}}<3e^{-k}\n\\quad\\text{and}\n\\\\\n0<\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac{z^m}{1-z}\\biggr)\\bigg|_{z=e^{-k\/m}}<3^{n+1}m^ne^{-k},\n\\quad\\text{for $n=1,2,\\dots$}\\,.\n\\end{gather*}\nTherefore, we can write the inequality in~\\eqref{E11} as\n\\begin{align*}\n&\nS_0'-\\frac k{2m^2}S_2'-\\frac k{3m^3}S_3'+\\frac{k(k-2)}{8m^4}S_4'+\\frac{k(5k-6)}{30m^5}S_5'-\\frac{k^3}{6m^6}S_6'\n-\\biggl(\\frac{3^3k}2+\\frac{3^4k}3+\\frac{3^7k^3}6\\biggr)e^{-k}\n\\\\ &\\qquad\n<\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k\n\\\\ &\\qquad\n<S_0'-\\frac k{2m^2}S_2'-\\frac k{3m^3}S_3'+\\frac{k(k-2)}{8m^4}S_4'+\\frac{k^2}{2m^5}S_5'\n+\\biggl(3+\\frac{3^5k(k-2)}8+\\frac{3^6k^2}2\\biggr)e^{-k}\n\\end{align*}\nimplying\n\\begin{align}\n&\nS_0'-\\frac k{2m^2}S_2'-\\frac k{3m^3}S_3'+\\frac{k(k-2)}{8m^4}S_4'+\\frac{k(5k-6)}{30m^5}S_5'-\\frac{k^3}{6m^6}S_6'\n-500k^3e^{-k}\n\\nonumber\\\\ &\\qquad\n<\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k\n<S_0'-\\frac k{2m^2}S_2'-\\frac k{3m^3}S_3'+\\frac{k(k-2)}{8m^4}S_4'+\\frac{k^2}{2m^5}S_5'\n+500k^2e^{-k},\n\\label{E12}\n\\end{align}\nwhere\n\\begin{align*}\nS_n'\n&=\\sum_{j=1}^\\infty j^nz^j\\bigg|_{z=e^{-k\/m}}\n=\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac z{1-z}\\biggr)\\bigg|_{z=e^{-k\/m}}\n\\\\\n&=(-1)^n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac1{z-1}\\biggr)\\bigg|_{z=e^{k\/m}}\n\\quad\\text{for $n=0,1,\\dots$};\n\\end{align*}\nin particular,\n\\begin{gather*}\nS_0'=\\frac1{z-1},\n\\quad\nS_2'=\\frac{z+z^2}{(z-1)^3},\n\\quad\nS_3'=\\frac{z+4z^2+z^3}{(z-1)^4},\n\\quad\nS_4'=\\frac{z+11z^2+11z^3+z^4}{(z-1)^5},\n\\\\\nS_5'=\\frac{z+26z^2+66z^3+26z^4+z^5}{(z-1)^6},\n\\quad\nS_6'=\\frac{z+57z^2+302z^3+302z^4+57z^5+z^6}{(z-1)^7}\n\\end{gather*}\nwith $z=e^{k\/m}$. Since\n$500k^3e^{-k}<(2k)^{-3}<m^{-3}$\nfor $k>m\/2>30$, using our equation \\eqref{eq:1} we can write\nthe estimates~\\eqref{E12} as\n\\begin{align}\n&\n\\frac{k(5k-6)}{30m^5}S_5'-\\frac{k^3}{6m^6}S_6'-\\frac1{m^3}\n\\nonumber\\\\ &\\qquad\n<1-S_0'+\\frac k{2m^2}S_2'+\\frac k{3m^3}S_3'-\\frac{k(k-2)}{8m^4}S_4'\n<\\frac{k^2}{2m^5}S_5'+\\frac1{m^3}.\n\\label{E13}\n\\end{align}\nNoting that $e^{1\/2}<z=e^{k\/m}<e$, we find\n\\begin{align*}\n0<S_5'&<\\frac{e+26e^2+66e^3+26e^4+e^5}{(e^{1\/2}-1)^6}\n<41438,\n\\\\\n0<S_6'&<\\frac{e+57e^2+302e^3+302e^4+57e^5+e^6}{(e^{1\/2}-1)^7}\n<658544,\n\\end{align*}\nwe continue~\\eqref{E13} as follows:\n\\begin{multline}\n\\biggl|1-\\frac1{z-1}+\\frac k{2m^2}\\frac{z+z^2}{(z-1)^3}\n+\\frac k{3m^3}\\frac{z+4z^2+z^3}{(z-1)^4}\n\\\\\n-\\frac{k(k-2)}{8m^4}\\frac{z+11z^2+11z^3+z^4}{(z-1)^5}\\biggr|\n<\\frac{110000}{m^3},\n\\label{E14}\n\\end{multline}\nwhere $z=e^{k\/m}$.\n\nWe already know that $k\/m$ is bounded as $m\\to\\infty$;\nmaking the ansatz $k\/m=c+O(1\/m)$, hence $z=e^{k\/m}=e^c+O(1\/m)$,\nwe find from~\\eqref{E14} that\n$$\n1-\\frac1{e^c-1}=O\\biggl(\\frac 1m\\biggr) \\quad\\text{as}\\; m\\to\\infty,\n$$\nhence $e^c=2$ and $c=\\log2$. Now we take\n$$\n\\frac km=\\log2+\\frac am+\\frac b{m^2}+O\\biggl(\\frac 1{m^3}\\biggr) \\quad\\text{as}\\; m\\to\\infty,\n$$\nhence\n$$\nz=e^{k\/m}=2+\\frac{2a}m+\\frac{a^2+2b}{m^2}+O\\biggl(\\frac 1{m^3}\\biggr) \\quad\\text{as}\\; m\\to\\infty.\n$$\nSubstituting these formulas into~\\eqref{E14} results in\n\\begin{align*}\nO\\biggl(\\frac 1{m^{3}}\\biggr)\n&=1-\\frac1{1+2a\/m+(a^2+2b)\/m^2+O(m^{-3})}\n\\\\ &\\qquad\n+\\frac{\\log2+a\/m+O(m^{-2})}{2m}\\,\\frac{6+10a\/m+O(m^{-2})}{1+6a\/m+O(m^{-2})}\n\\\\ &\\qquad\n+\\frac{\\log2+O(m^{-1})}{3m^2}\\,\\frac{26+O(m^{-1})}{1+O(m^{-1})}\n\\\\ &\\qquad\n-\\frac{\\log^22+O(m^{-1})}{8m^2}\\,\\frac{150+O(m^{-1})}{1+O(m^{-1})}\n+O\\biggl(\\frac 1{m^3}\\biggr)\n\\\\\n&=\\frac{2a+3\\log2}m\n-\\frac{3a^2-3a+13a\\log2-2b+\\frac{75}4\\log^22-\\frac{26}3\\log2}{m^2}\n+O\\biggl(\\frac 1{m^3}\\biggr),\n\\end{align*}\nhence $a=-\\frac32\\log2$, $b=(3\\log2-\\frac{25}{12})\\log2$ and, finally,\nwe get the asymptotic formula~\\eqref{eq:2}.\n\nTo quantify this asymptotic expansion, we introduce the function\n\\begin{align*}\nf_m(C)\n&=\\biggl(1-\\frac1{z-1}\n+\\frac\\lambda{2m}\\,\\frac{z+z^2}{(z-1)^3}\n+\\frac\\lambda{3m^2}\\,\\frac{z+4z^2+z^3}{(z-1)^4}\n\\\\ &\\qquad\n-\\frac{\\lambda(\\lambda-2\/m)}{8m^2}\\,\n\\frac{z+11z^2+11z^3+z^4}{(z-1)^5}\\biggr)\\bigg|_{z=e^\\lambda},\n\\end{align*}\nwhere\n$$\n\\lambda=\\lambda(C)\n=\\log2\\biggl(1-\\frac3{2m}-\\frac C{m^2}\\biggr)\n$$\nagrees with our $k\/m$ up to $O(m^{-2})$.\nDirect computation then shows that\n$$\nf_m(0)>0.005m^{-2}-100m^{-3}\n\\quad\\text{and}\\quad\nf_m(0.004)<-0.00015m^{-2}+100m^{-3}\n$$\nfor $m\\ge100$. Therefore,\n$f_m(0)>110000\/m^3$ for $m>2202\\cdot10^4$ and\n$f_m(0.004)<-110000\/m^3$ for $m>734\\cdot10^6$,\nso that $|f_m(C)|<110000\/m^3$ is possible only\nif $0<C<0.004$. Comparing this result with~\\eqref{E14} we conclude\nthat, for $k$ and $m>10^9$ satisfying \\eqref{eq:1}, we necessarily\nhave\n$$\n\\frac km=\\log2\\biggl(1-\\frac3{2m}-\\frac{C_m}{m^2}\\biggr)\n$$\nwith $0<C_m<0.004$.\n\\end{proof}\n\nClearly, the strategy to deduce further terms in the expansion~\\eqref{eq:2} remains\nthe same, but in order to achieve precision $O(m^{-n})$ for an integer $n\\ge2$\nwe have to use the Taylor expansion of $(1-y)^ke^{ky}$ up to $y^{2n+1}$ (each new term\nin~\\eqref{eq:2} requires two extra terms in the expansion of $(1-y)^ke^{ky}$).\nIn this way we get\n\\begin{align}\nk\n&=cm-\\frac{3}{2}c-\\biggl(\\frac{25}{12}c-3c^2\\biggr)m^{-1}\n+\\biggl(-\\frac{73}{8}c+\\frac{61}2c^2-25c^3\\biggr)m^{-2}\n\\nonumber\\\\ &\\qquad\n+\\biggl(-\\frac{41299}{720}c+\\frac{657}2c^2-598c^3+\\frac{1405}4c^4\\biggr)m^{-3}+O(m^{-4})\\biggr)\n\\nonumber\\\\\n&\\approx 0.69314718m-1.03972077-0.00269758m^{-1}+0.00323260m^{-2}\n\\nonumber\\\\ &\\qquad\n+0.00217182m^{-3}+O(m^{-4}),\n\\label{km}\n\\end{align}\nwhere $c=\\log2$. However, we do not possess any clear general strategy\nto quantify such expansions. Already proving a sharp dependence\non $k$ for the remainder of the $n$-th truncation of the Taylor expansion of\n$(1-y)^ke^{ky}$ (like we do for $n=4$ in Lemma~\\ref{lem:As}) seems\nto be a difficult task. We discuss related problems in Section~\\ref{s5}.\n\n\\begin{proof}[Proof of Corollary~{\\rm\\ref{cor:1}}]\nLet $(m,k)$ be a non-trivial integer solution of~\\eqref{EME}. By Moser's result we know that\n$m>10^9$. It follows from Theorem~\\ref{th:As} that\n\\begin{equation}\n\\label{drie}\n0<\\log 2-\\frac{2k}{2m-3} < \\frac{0.0111}{(2m-3)^2}.\n\\end{equation}\nBy Legendre's theorem, $|\\log2-p\/q|<1\/(2q^2)$ implies that $p\/q$ is a convergent\nof~$\\log2$, while $\\log2>p\/q$ insures that the index of the convergent is even.\nThus, $2k\/(2m-3)$ is a convergent $p_j\/q_j$ of the continued fraction of $\\log 2$\nwith $j$~even.\n\\end{proof}\n\n\\section{The proof of the main theorem}\n\\label{s3}\n\nIn this section we prove Theorem~\\ref{main}. The restrictions on the prime factorization\nof~$q_j$ in that result are established using an argument in the style of Moser given\nin the proof of the following lemma.\n\n\\begin{Lem}\n\\label{twee}\nLet $(m,k)$ be a solution of {\\rm\\eqref{EME}} with $k\\ge 2$. Let $p$ be a prime\ndivisor of $2m-3$. If $p-1\\mid k$, then\n$$\n\\nu_p(2m-3)=\\nu_p(3^{p-1}-1)+\\nu_p(k)+1\\ge 2.\n$$\nIf $3$~is a primitive root modulo $p$, then $p-1\\mid k$.\n\\end{Lem}\n\n\\begin{proof}\nUsing that $k$ must be even, we find that\n\\begin{align*}\n\\sum_{j=1}^{2m-4}j^k\n& \\equiv \\sum_{j=1}^{m-1}j^k+\\sum_{j=1}^{m-3}(2m-3-j)^k\n\\equiv \\sum_{j=1}^{m-1}j^k+\\sum_{j=1}^{m-3}j^k\\pmod{2m-3}\n\\\\\n& \\equiv m^k+m^k-(m-1)^k-(m-2)^k\n\\equiv 2(3^k-1)(m-1)^k\\pmod{2m-3},\n\\end{align*}\nwhere we used that $m^k\\equiv (2m-3+m)^k\\equiv 3^k(m-1)^k\\pmod{2m-3}$ and\n$(m-2)^k\\equiv (2m-3-m+1)^k\\equiv (m-1)^k\\pmod{2m-3}$.\nOn applying \\eqref{staudt} with $l=2m-3$ and $r=k$ we then obtain that\n\\begin{equation}\n\\label{sta}\n\\frac{2(3^k-1)(m-1)^k}{2m-3}\\equiv -\\sum_{\\substack{p\\mid 2m-3\\\\p-1\\mid k}}\\frac1p\\pmod1.\n\\end{equation}\nIf $p\\mid 2m-3$ and $p-1\\mid k$, the $p$-order of the right-hand side is~$-1$. The $p$-order of\nthe left-hand side must also be $-1$, that is, we must have\n$$\n\\nu_p(2m-3)=\\nu_p(3^k-1)+k\\nu_p(m-1)+1=\\nu_p(3^{p-1}-1)+\\nu_p(k)+1,\n$$\nwhere we used that $m-1$ and $2m-3$ are coprime.\nNow suppose that $p\\mid 2m-3$ and $3$ is a primitive root modulo $p$ (thus $p\\mid 3^k-1$ implies\n$p-1\\mid k$). If $p-1\\nmid k$, the $p$-order of the left-hand side is $\\le -1$ and $>-1$ on the\nright-hand side. Thus, we infer that $p-1\\mid k$.\n\\end{proof}\n\nThis completes the required ingredients needed in order to prove the main result.\n\n\\begin{proof}[Proof of Theorem {\\rm\\ref{main}}]\nSince by assumption $N\\mid k$, we can write $k=Nk_1$ and thus rewrite \\eqref{drie} as\n\\begin{equation}\n\\label{E}\n0<\\frac{\\log 2}{2N}-\\frac{k_1}{2m-3}<\\frac{0.0111}{2N(2m-3)^2}.\n\\end{equation}\nWe infer that $k_1\/(2m-3)=p_j\/q_j$ is a convergent to\n$(\\log 2)\/(2N)$ with $j$~even.\nSince $p\\mid m$ implies $p-1\\nmid k$ (see, e.g., Moree \\cite[Proposition 9]{Oz}),\nwe have $(6,q_j)=1$.\nWe rewrite~\\eqref{E} as\n$$\n0<\\frac{\\log 2}{2N}-\\frac{p_j}{q_j}<\\frac{0.0111}{2Nd^2q_j^2},\n$$\nwith $d$ the greatest common divisor of $k_1$ and $2m-3$.\nOn the other hand,\n$$\n\\frac{\\log 2}{2N}-\\frac{p_j}{q_j}>\\frac1{(a_{j+1}+2)q_j^2},\n$$\nhence $(a_{j+1}+2)^{-1}<0.0111\/(2Nd^2)$, from which the\nresult follows on also noting that $2m-3\\ge q_j$ and invoking Lemma~\\ref{twee}\n(note that if $\\nu_p(q_j)\\ge 1$, then $\\nu_p(q_j)=\\nu_p(2m-3)-\\nu_p(k_1)$).\n\\end{proof}\n\nTo prove that $p\\mid m$ implies $p-1\\nmid k$ one uses that $k$ must be even\nand takes $l=m$ in~\\eqref{staudt}, showing that $\\sum_{p\\mid m,\\,p-1\\mid k}\\frac1p$ must\nbe an integer. Since a sum of reciprocals of distinct primes can never be an integer,\nthe result follows.\n\n\\section{Computation of the continued fractions}\n\\label{s4}\n\nWe make use of conditions \\ref{ca}, \\ref{cb}, \\ref{cc} of Theorem~\\ref{main}. We recall that\nwe expect ${\\mathsf E}(\\log q_{j(N)}(\\xi))\\sim c_1N$ for a generic $\\xi\\in [0,1]$ satisfying\nthese conditions.\nIndeed, on the basis of theoretical results, heuristics and numerical experiments, we conjecture\nthat $c_1=60\\pi^2$.\n\n\\begin{table}[ht\n\\begin{center}\n\\begin{tabular}{|c|r|r|l|c|c|}\n\\hline\n\\multicolumn{1}{|c|}{\\T $N$} & \\multicolumn{1}{|c|}{$j=j(N)$} & \\multicolumn{1}{|c|}{$a_{j+1}$} &\n\\multicolumn{1}{|c|}{$q_j$ \\ (rounded down)} & \\multicolumn{1}{|c|}{\\small $q_j \\bmod 6$} & \\multicolumn{1}{|c|}{$p = p(q_j)$} \\\\\n\\hline\n\\T $1$ & $642$ & $764$ & $2.383153 \\cdot 10^{\\,330}$ & $-1$ & $149$ \\\\\n\\hline\n\\T $2$ & $664$ & $1\\,529$ & $2.383153 \\cdot 10^{\\,330}$ & $-1$ & $149$ \\\\\n\\hline\n\\T $2^2$ & $1\\,254$ & $21\\,966$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^3$ & $1\\,264$ & $43\\,933$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^4$ & $1\\,280$ & $87\\,866$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^5$ & $1\\,294$ & $175\\,733$ & $1.132014 \\cdot 10^{\\,638}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $2^6$ & $8\\,950$ & $26\\,416$ & $3.458446 \\cdot 10^{\\,4\\,589}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^7$ & $8\\,926$ & $52\\,834$ & $3.458446 \\cdot 10^{\\,4\\,589}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8$ & $119\\,476$ & $122\\,799$ & $1.374540 \\cdot 10^{\\,61\\,317}$ & $+1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3$ & $119\\,008$ & $368\\,398$ & $1.374540 \\cdot 10^{\\,61\\,317}$ & $+1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^2$ & $139\\,532$ & $782\\,152$ & $9.351282 \\cdot 10^{\\,71\\,882}$ & $+1$ & $56\\,131$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^3$ & $6\\,168\\,634$ & $1\\,540\\,283$ & $8.220719 \\cdot 10^{\\,3\\,177\\,670}$ & $+1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^4$ & $22\\,383\\,618$ & $5\\,167\\,079$ & $5.128265 \\cdot 10^{\\,11\\,538\\,265}$ & $+1$ & $17$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5$ & $155\\,830\\,946$ & $31\\,664\\,035$ & $2.257099 \\cdot 10^{\\,80\\,303\\,211}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5$ & $351\\,661\\,538$ & $85\\,898\\,211$ & $9.729739 \\cdot 10^{\\,181\\,214\\,202}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5^2$ & $1\\,738\\,154\\,976$ & $1\\,433\\,700\\,727$ & $1.594940 \\cdot 10^{\\,895\\,721\\,905}$ & $+1$ & $5$ \\\\\n\\hline\n\\T $$ & $1\\,977\\,626\\,256$ & $853\\,324\\,651$ & $1.196828 \\cdot 10^{\\,1\\,019\\,133\\,881}$ & $-1$ & $$ \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5^3$ & $2\\,015\\,279\\,170$ & $4\\,388\\,327\\,617$ & $5.565196 \\cdot 10^{\\,1\\,038\\,523\\,018}$ & $-1$ & $19$ \\\\\n\\hline\n\\T $$ & $3\\,236\\,170\\,820$ & $2\\,307\\,115\\,390$ & $5.427815 \\cdot 10^{\\,1\\,667\\,658\\,416}$ & $+1$ & \\\\\n\\hline\n\\T $2^8 \\cdot 3^5 \\cdot 5^4$ & $2\\,015\\,385\\,392$ & $21\\,941\\,638\\,090$ & $5.565196 \\cdot 10^{\\,1\\,038\\,523\\,018}$ & $-1$ & $19$ \\\\\n\\hline\n\\T $$ & $3\\,236\\,257\\,942$ & $11\\,535\\,576\\,954$ & $5.427815 \\cdot 10^{\\,1\\,667\\,658\\,416}$ & $+1$ & $$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Smallest integers $j$ satisfying conditions \\ref{ca}, \\ref{cb} and \\ref{cc} of Theorem~\\ref{main}}\n\\label{tab:j}\n\\end{table}\n\nThe computation of $(\\log 2)\/(2N)$ is done in two steps. First, we generate $d$~digits\nof~$\\log2$. For this we use the $\\gamma$-cruncher~\\cite{AJY}. With\nthis program, A.~Yee and R.~Chan computed 31~billion decimal digits\nof~$\\log2$ in about 24~hours. Second, we set a rational approximation\nof $(\\log 2)\/(2N)$ with a relative error bounded by~$10^{-d}$. Then\npartial quotients of the continued fraction of $(\\log 2)\/(2N)$ are\ncomputed: about $0.97 d$ of them can be evaluated, with safe error\ncontrol~\\cite{BPR96} (cf.\\ the result of Lochs mentioned in Section~\\ref{s1}).\nWe maintain a floating point approximation\nof numbers~$q_j$ (rounded down) and residues of $q_j \\pmod 6$\nby the formula $q_{i+1} = a_{i+1} q_i + q_{i-1}$ for $i \\geq 0$,\nwhere $q_0 = 1$ and $q_{-1} = 0$.\n\nTable~\\ref{tab:j} was created with the `basic method' of~\\cite{BPR96}\nfor $N \\leq 2^8 \\cdot 3^4$. It was fast enough to reach the benchmark\n$m > 10^{10^7}$ in four days with $50\\cdot 10^6$ digits of~$\\log2$.\nBit-complexity of this algorithm (or of the indirect or direct\nmethods~\\cite{BPR96}) is quadratic and reaching the $m > 10^{10^{10}}$ milestone\nwould take centuries.\n\nSome subquadratic GCD algorithms were discovered that have\nasymptotic running time $O(n(\\log n)^2\\log\\log n)$~\\cite{MOL08}.\nA faster version of the program was written: this time a recursive\nHGCD method is applied. It is adapted for computing a continued\nfraction by using Lemma~3 of~\\cite{BPR96} (which is similar to\nAlgorithm~1.3.13 of~\\cite{Cohen93}) for error control. With it\nthe program leaps over $10^{10^8}$ in just about one hour.\nFinally, the new benchmark $m > 10^{10^9}$ is established in no more\nthan 10~hours with $3\\cdot 10^9$ digits of~$\\log2$, $N = 1555200$\nand condition~\\ref{cd}: the first found solution fits\nconditions \\ref{ca}--\\ref{cc}, but not \\ref{cd}. With $N = 7776000$, $m > 10^{10^9}$\nis achieved for the smallest $j$. See Table~\\ref{tab:j}: in the\nlast column, $p$~is a prime such that $p \\in \\mathcal{P}(N)$ and\n$\\nu_p(q_j) = 1$, that is, such that condition~\\ref{cd} of Theorem~\\ref{main}\nis violated.\n\nNow, computation time is not a problem to achieve the $m > 10^{10^{10}}$\nmilestone, a few days will be sufficient on a computer with a large\namount of memory. We remark that the complexity and\nhardware requirement for computation of the digits of~$\\log2$, respectively\nfor computation of its continued fraction expansion, are similar.\n\n\\section{Miscellaneous}\n\\label{s5}\n\n\\subsection{{\\tt The number of distinct prime factors of $m-1$}}\n\\label{s5.1}\n\nThere is a different application of Theorem~\\ref{YVES} suggested\nby the work of Brenton and Vasiliu~\\cite{BV}, to factorization\nproperties of the number $m-1$ coming from a non-trivial solution $(m,k)$ of~\\eqref{EME}.\nA result of Moser~\\cite{Moser} (which can also be deduced from the key identity~\\eqref{staudt},\ncf.\\ the proof of Lemma~\\ref{twee} above) asserts that\n\\begin{equation}\n\\sum_{p\\mid m-1}\\frac1p+\\frac1{m-1}\\in\\mathbb Z;\n\\label{pm}\n\\end{equation}\nin particular, the number $m-1$ is square-free. Since the sum of reciprocals\nof the first 58~primes is less than~2, we conclude that either\n$\\omega(m-1)\\ge 58$ or the integer\nin~\\eqref{pm} is equal to~1. In the latter case,\nwe can apply Curtiss' bound~\\cite{Cur} for positive integer solutions\nof Kellogg's equation\n$$\n\\sum_{i=1}^n\\frac1{x_i}=1,\n$$\nnamely, $\\max_i\\{x_i\\}\\le A_n-1$, where the Sylvester sequence\n$\\{A_n\\}_{n\\ge1}=\\{2,3,7,43,\\dots\\}$ is defined by the recurrence\n$A_n=1+\\prod_{i=1}^{n-1}A_i$ (for some further info, see e.g. Odoni \\cite{Odoni}).\n{}From this result and the estimate\n$A_n<(1.066\\cdot10^{13})^{2^{n-7}}$, we infer\n$$\nm<(1.066\\cdot10^{13})^{2^{\\omega(m-1)-6}},\n$$\nwhich together with the lower bound on~$m$ from Theorem~\\ref{YVES} yields\n$\\omega(m-1)\\ge 33$.\nA similar estimate on the basis of another \\eqref{pm}-like identity of Moser implies\nthat $\\omega(m+1)\\ge 32$.\n\n\\subsection{{\\tt Generalized EM equation}}\n\\label{s5.2}\n\nThe method we use in Section~\\ref{sec:As} for deriving the asymptotics of $k$ in terms of~$m$\nworks for the more general equation\n\\begin{equation}\n\\label{EMEt}\n1^k+2^k+\\dots+(m-1)^k=tm^k,\n\\end{equation}\nwith $t\\in\\mathbb N$ fixed, as well. Indeed, the coefficients in the Taylor series expansion\n\\begin{equation}\n\\label{TS}\n(1-y)^ke^{ky}\n=1-\\frac k2y^2-\\frac k3y^3+\\frac{k(k-2)}8y^4+\\dotsb\n=\\sum_{n=0}^\\infty g_n(k)y^n\n\\end{equation}\nare polynomials satisfying\n\\begin{equation}\n\\label{TSa}\ng_0(k)=1, \\quad g_1(k)=0,\n\\qquad\\text{and}\\qquad\n\\deg_kg_n(k)=\\biggl[\\frac n2\\biggr], \\quad\ng_n(0)=0 \\quad\\text{for $n\\ge2$};\n\\end{equation}\nthe latter follows from raising the series $(1-y)e^y=1-y^2\/2-y^3\/3-\\dotsb$\nto the power~$k$. In these settings, equation~\\eqref{EMEt} becomes\n\\begin{align*}\nt\n&=\\sum_{j=1}^{m-1}\\biggl(1-\\frac jm\\biggr)^k\n=\\sum_{j=1}^{m-1}e^{-kj\/m}\\sum_{n=0}^\\infty g_n(k)\\biggl(\\frac jm\\biggr)^n\n\\\\\n&=\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}\\sum_{j=1}^{m-1}j^ne^{-jk\/m}\n\\\\ \\intertext{(since $\\sum_{j=m}^\\infty j^ne^{-jk\/m}=O(m^ne^{-k})$)}\n&\\sim\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}\\sum_{j=1}^\\infty j^ne^{-jk\/m}\n=\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}\n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac z{1-z}\\biggr)\\bigg|_{z=e^{-k\/m}}\n\\\\\n&=\\sum_{n=0}^\\infty\\frac{g_n(k)}{m^n}(-1)^n\n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac1{z-1}\\biggr)\\bigg|_{z=e^{k\/m}},\n\\end{align*}\nhence in the notation $\\lambda=k\/m$ and $x=1\/m$ we have\n\\begin{equation}\n\\label{ast}\nt=\\sum_{n=0}^\\infty g_n\\biggl(\\frac\\lambda x\\biggr)(-x)^n\n\\biggl(\\biggl(z\\frac{\\d}{\\d z}\\biggr)^n\\frac1{z-1}\\biggr)\\bigg|_{z=e^\\lambda}.\n\\end{equation}\nSearching $\\lambda$ in the form $\\lambda=c_0+c_1x+c_2x^2+\\dotsb$,\nwe find successively\n\\begin{gather*}\nc_0=c(t)=\\log\\biggl(1+\\frac1t\\biggr)=\\log\\frac{t+1}t,\n\\qquad c_1=-\\biggl(t+\\frac12\\biggr)c,\n\\\\\nc_2=\\biggl(t+\\frac12\\biggr)^3c^2-\\biggl(t+\\frac12\\biggr)^2c\n-\\frac14\\biggl(t+\\frac12\\biggr)c^2+\\frac c6,\n\\end{gather*}\nand so on. Note that $c_n(-(t+1))=(-1)^{n+1}c_n(t)$ for $n=0,1,2,\\dots$;\nthis reflects the equivalence of equation \\eqref{EMEt} and\n\\begin{equation}\n\\label{EMEtdual}\n1^k+2^k+\\dots+(m-1)^k+m^k=(t+1)m^k.\n\\end{equation}\n\n{}From this asymptotics we see that\n\\begin{align}\n\\frac{2k}{2m-t_1}\n&=c+\\frac{{t_1^3}c^2-2{t_1^2}c-t_1c^2+4c\/3}{2(2m-t_1)^2}+O\\biggl(\\frac1{(2m-t_1)^3}\\biggr),\n\\label{CFt}\n\\end{align}\nwhere $t_1=2t+1$ and $c=\\log(1+1\/t)$.\nIt can be checked that for all positive integers $t$ we have the inequality\n$$\n-0.22<{t_1^3}c^2-2{t_1^2}c-t_1c^2+\\frac{4c}3<0,\n$$\nand hence $2k\/(2m-2t-1)$ is a convergent (with even index) of this logarithm\n$c=\\log(1+1\/t)$ for $m$~large enough.\n\n\\subsection{{\\tt Saddle-point method}}\n\\label{s5.3}\n\nA different approach to treat the asymptotic behaviour of $k$ in terms of $m$\nfor $k$ and $m$ satisfying \\eqref{EME} (or, more generally, \\eqref{EMEt})\nis based on the integral representation\n$$\n1^k+2^k+\\dots+(m-1)^k\n=\\frac{\\Gamma(k)}{2\\pi i}\\int_{C-i\\infty}^{C+i\\infty}\n\\frac{e^{mz}}{(e^z-1)z^{k+1}}\\,\\d z,\n$$\nwhere $C$~is an arbitrary positive real number (cf.~\\cite[p.~273]{Delange}).\nOn noting that\n$$\n\\frac{e^{mz}}{e^z-1}\n=\\frac{e^{(m-1)z}}{1-e^{-C}}\\biggl(1+\\frac{1-e^{z-C}}{e^z-1}\\biggr)\n$$\none obtains, on taking $C=(k+1)\/(m-1)$ and after invoking some rather\ntrivial estimates, that\n\\begin{equation}\n\\label{hubert}\n1^k+2^k+\\dots+(m-1)^k=\\frac{(m-1)^k}{1-e^{-(k+1)\/(m-1)}}\\bigl(1+\\rho_k(m)\\bigr),\n\\end{equation}\nwith\n$$\n|\\rho_k(m)|<\\frac{\\sqrt{2(k+1)}C}{\\sqrt\\pi(k-1)(e^C-1)}.\n$$\n(This part of the argument is due to Delange; for more details see\n\\cite[pp.~273--274]{Delange}.) By~\\eqref{ineq},\n$C$~is bounded and we infer that $|\\rho_k(m)|=O(k^{-1\/2})=O(m^{-1\/2})$.\nOn putting $m^k$ on the left-hand side of~\\eqref{hubert}\nand using $(1-1\/m)^m=\\exp(-1+O(m^{-1}))$, we immediately conclude\nthat, as $m\\to\\infty$,\n$$\n\\frac km=\\log2+O\\biggl(\\frac1{\\sqrt{m}}\\biggr),\n$$\nwhere the implied constant is absolute. A more elaborate analysis,\nusing the saddle-point method, will very likely allow one as many terms\nin the latter expansion as required.\n\n\\subsection{{\\tt Experimental asymptotics}}\n\\label{s5.4}\n\nIt is worth mentioning a fast experimental approach of doing asymptotics\nlike~\\eqref{km}. Given numerically a few hundred terms of a sequence\n$s=\\{s_n\\}_{n\\ge1}$ that one believes has an asymptotic expansion\nin inverse powers of $n$, one can try to apply the $\\mathtt{asymp}_k$\ntrick, a simple but often powerful method to numerically determine the\ncoefficients in the ansatz\n$$\ns_n\\sim c_0+\\frac{c_1}{n}+\\frac{c_2}{n^2}+\\dotsb.\n$$\nAs a second step one tries to identify the so-found coefficients\nwith (linear combinations of) known constants. Thus, one arrives\nat a conjecture that hopefully can be turned into a proof.\nFor more details and some `victories' achieved by\nthe $\\mathtt{asymp}_k$ method, see Gr\\\"unberg and Moree \\cite{GM}.\n\nD.~Zagier has applied this trick to the sequence of $k=k(m)$ obtained from\n\\eqref{EME} on letting $m$ run through the first thousand values.\nExcellent agreement with our theoretical results was obtained in this way.\n\n\\medskip\n{\\tt Acknowledgements}. The second author is very indebted to Jerzy Urbanowicz\nfor involving him in the early nineties in his EM research (with \\cite{MRU} as\nvisible outcome). The second and third author would like to thank D.~Zagier\nfor verifying some of our results using the $\\mathtt{asymp}_k$ trick and for some\ninformative discussions regarding the saddle-point method (reflected in the final section). H.~te Riele provided us\nwith the unpublished report\n\\cite{Best}, which became the `initial spark' for the current project. C.~Baxa pointed out the\nrelevance of \\cite{Harman} to us. Further thanks are due to T.~Agoh and I.~Shparlinski.\n\nThis research was carried out whilst the third author was visiting in the Max Planck Institute for Mathematics\n(MPIM) and the Hausdorff Center for Mathematics (HCM) financially supported by these institutions.\nHe and the second author thank the MPIM and HCM for providing such a nice research environment.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}