diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpipv" "b/data_all_eng_slimpj/shuffled/split2/finalzzpipv" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpipv" @@ -0,0 +1,5 @@ +{"text":"\\section{Motion of a point charge in a Coulomb potential well}\n\nThe behavior of a point charge in a Coulomb potential well was studied well\n enough (see, e.g., \\cite{1}). By contrast, the behavior of {\\itshape\\bfseries a distributed} charge in a Coulomb potential well reveals some\n specific features. And it is the specific features arising in the behavior of {\\itshape\\bfseries a distributed} charge that we are going to address in the present paper. \n\nWe start by assuming a negative distributed charge with a density \n$\\rho(\\mathbf{r})$ to be placed into the field of a point positive charge $e$ of infinite mass at the origin of a coordinate system. For the sake of conveniency, we shall call this charge a nucleus. Our goal will be to reveal the specific features in the behavior of the distributed charge \n$\\rho(\\mathbf{r})$ in a Coulomb potential well.\nWe shall look for the solution of this problem in the framework of the approach that was employed, for instance, in monograph \\cite{1}, i.e., in the nonrelativistic approximation. This condition is reliably substantiated.\nFor instance, in an atom any element of charge located at a distance of the Bohr radius from the nucleus has a velocity on the order of $\\alpha c$, where $\\alpha$ is the fine structure constant, and $c$ is the velocity of light. Because in the nonrelativistic approximation the velocities of motion of individual charge elements are small, we will disregard in what follows the generation of magnetic field induced by charge motion. Said otherwise, we are going to neglect the vector potential $\\mathbf{A}(\\mathbf{r})$ induced by charge motion and assume the elements of a distributed charge to feel only the field of the scalar potential $\\Phi(\\mathbf{r})$.\n\nWe shall address in this paper the case where the density of the distributed charge is small. This licenses us to neglect the potential of the distributed charge $\\rho(\\mathbf{r})$ itself compared with that of the nucleus, and assume each element of the distributed charge to move only in the field of a point charge $e$ with the spherical potential $\\Phi(r)=e\/r$.\n\nNext we choose in the distributed charge a constant element of charge $dq$ with a mass $dm$ and, assuming this element to be a point charge, recall some features in the behavior of a point charge $dq$ in a Coulomb potential well, which we are going to use later on. (We shall denote this element sometimes by $dq$, and sometimes by $dm$, bearing in mind that in all cases we will deal with an element of charge $dq$ and mass $dm$). A comment is here, however, in order.\n\nAn element of charge $dq$ moving with acceleration must radiate energy. Our analysis of the motion of this element in the field of the nucleus will, however, be conducted under the assumption that the element does not radiate. Indeed, the element under consideration, rather than being a single isolated element of charge, has been selected by us out of the total distributed charge. As this will be demonstrated later on, there exist such states of a distributed charge which do not radiate. It is only such states that will be studied in this paper. Therefore in consideration of the motion of a charge element isolated from the total charge we also shall assume that this element does not radiate. Note that each given separate element of charge must radiate.\n\n\\medskip\n\nThe most convenient approach to identifying the significant details in the behavior of a point charge $dq$ in a Coulomb potential well is to resort to monograph \\cite{1}, \\$~15. It describes the behavior of a point particle in a field inversely proportional to $r^2$. Among such fields are Newton's gravitational and Coulomb's electrostatic fields.\nThe gravitational forces being weak, we are going to neglect them. The only force we will take account of is the Coulomb interaction of a negative element of charge $dq$ with the positively charged nucleus. Because the element $dq$ separated out from the total charge does not radiate (a point to be substantiated later on), one can invoke here all the conclusions drawn in \\cite{1}, \\$~15.\n\nConsider now the specific features in the behavior of an element of charge we shall treat in the discussion to follow.\n\nIn a Coulomb potential well, a constant element of charge \n$dq$ with a mass $dm$ can move in a circle or an ellipse, \nand the ellipse can degenerate into a straight line. The \nenergy of an element of charge depends on the semimajor axis \nof the ellipse (or on the radius of the circle). The actual \nshape of the ellipse (i.e., its semiminor axis) depends on \nthe angular momentum of the particle. Thus, all ellipses with \nthe same semimajor axis but different semiminor axes have the \nsame energies but different angular momenta, down to the zero \nmomentum (in which case the ellipse degenerates into a straight \nline). Said otherwise, elements of the same energy \ncan move in a Coulomb potential well along trajectories which \ndiffer in the value of the angular momentum.\n\n\nLet us analyze the various trajectories along which an \nelement of charge $dq$ with a mass $dm$ can move in the \ncase where the total energy of the element in each trajectory \nis the same. \n\n\\begin{figure}\n\\label{F1}\n\\begin{center}\n\\includegraphics[scale=1]{Figure1.eps}\n\\caption{}\n\\end{center}\n\\end{figure}\n\n\nFigure~1 for an element $dm$ located at some point\n$\\mathbf r$ illustrates several such orbits of all possible \nones: a circular orbit $a$, eight elliptical orbits \n$(b-k)$ with different eccentricities (and, hence, different \nangular momenta), and a linear orbit $l$ into which the \nellipse degenerates at an eccentricity of unity. This orbit \npasses through the nucleus. All the orbits are \ncharacterized by identical semimajor axes (if the energies \nof the elements are equal, the semimajor axes of the ellipses \nshould likewise be equal). All orbits lie in the same plane. \nThe elements of charge in all orbits rotate in the same \nsense.\n\nAll orbits focus at the same point. In this focus (in our \nfigure, this is the center $O$ of the circle) the nucleus \nis located. Using the focal properties of \nellipses, one can readily show that each elliptic trajectory \nintersects a circular orbit at the point where this ellipse \nintersects its semiminor axis. The dashed lines confine the \nregion of allowed trajectories along which an element $dm$\ncan move.\n\nSignificantly, the extended trajectories with an eccentricity close to unity pass not far from the nucleus. Therefore, within a small region in the vicinity of the nucleus the condition of the smallness of the element motion velocities does not hold. This region, however, is not large.\n\nBecause the energy of an element in each trajectory is the same, element $dm$ can move over {\\itshape\\bfseries any} of the trajectories specified. As follows from Fig.~1 and an analysis of the trajectories along which element $dm$ can propagate, all possible directions of motion of the element at a given point are strictly confined to the angle $\\pi$ (a few words on a semicircle located above orbit $l$---if the elements rotated in their orbits in the opposite direction, this semicircle would lie below orbit $l$).\n\nAn element residing in a Coulomb potential well moves in one plane. Note, however, that the number of planes which can be passed through the line connecting the element under consideration with the nucleus is infinite. Any of these planes can confine the trajectory of the given element, because the Coulomb field possesses spherical symmetry. In this case, all possible directions of the velocities of element motion at a given point lie inside the solid angle $2\\pi$ (a hemisphere).\nIn Fig.~1, this hemisphere extends above the plane passed through orbit $l$ perpendicular to the drawing. If elements in their orbits rotated in the opposite sense, the hemisphere would be located below this plane. Additional conditions related with the position of this plane will be specified below (in Sections 4 and 5).\n\nConsider now the magnitude of the velocities with which \nelements of charge having equal energies move in different \ndirections. The Reader can be conveniently referred to monograph \n\\cite{1}, \\$~14. The energy \n$d\\mathcal E$ of an element of charge $dq$ with a mass $dm$ \nmoving in a central field is preserved. Recalling the standard \nexpression for energy\n\n\\begin{equation}\n\\label{1}\nd\\mathcal E=d\\mathcal E_{Kin}+d\\mathcal E_{Pot}=\n\\frac{dm\\upsilon^2}{2}+d\\mathcal E_{Pot},\n\\end{equation}\nwhere $d\\mathcal E_{Kin}$ is the kinetic, and $d\\mathcal E_{Pot}$, \nthe potential energy of the element, and $\\upsilon$ is its \nvelocity, we obtain\n\n\\begin{equation}\n\\label{2}\n\\upsilon=\\sqrt{\\frac{2}{dm}(d\\mathcal E-d\\mathcal E_{Pot})}.\n\\end{equation}\n\nThis means that the magnitude of the velocity of an element \nwith a given energy $d\\mathcal E$ depends only on the potential \nenergy of this element $d\\mathcal E_{Pot}$. Equation (\\ref{2}) \ndoes not contain any other parameters, in particular, of \nparameters which would suggest the dependence of the magnitude \nof the velocity on its direction. As for the potential energy \nof this element, it depends only on the position of this element, \n$d\\mathcal E_{Pot}=edq\/r$. \nOne comes inevitably to the conclusion that the {\\itshape\\bfseries\n{magnitude of the velocity}} of an element at a given point \n{\\itshape\\bfseries{does not depend on \nthe direction}} of motion of this element.\n\nIt is a very significant statement, and, hence, it has to be \ncorroborated in more detail and in a more revealing way. \nRewrite Eq. (\\ref{1}) in the form\n\n\\begin{equation}\n\\label{3}\nd\\mathcal E=\\frac{dm}{2}(\\dot r^2+r^2\\dot\\zeta^2)+d\\mathcal E_{Pot}\n=\\frac{dm\\dot r^2}{2}+\\frac{dM^2}{2dmr^2}+d\\mathcal E_{Pot}.\n\\end{equation}\nHere $\\zeta$ is the angular coordinate in the plane in which \nthe element $dm$ rotates, and $dM$ is the angular momentum of \nthis element. Whence we come to \n\n\\begin{equation}\n\\label{4}\n\\dot r^2=\\frac{2}{dm}(d\\mathcal E-d\\mathcal E_{Pot})-\n\\frac{dM^2}{dm^2r^2}\n\\end{equation}\nThe term $r^2\\dot\\zeta^2$ can be derived from the expression \nfor the angular momentum $dM=dmr^2\\dot\\zeta$. Substituting it \ninto the expression for the velocity, we come to\n\n\\begin{equation}\n\\label{5}\n\\upsilon=\\sqrt{\\dot r^2+r^2\\dot\\zeta^2}=\\sqrt{\\frac{2}{dm}\n(d\\mathcal E-d\\mathcal E_{Pot})-\\frac{dM^2}{dm^2r^2}+\n\\frac{dM^2}{dm^2r^2}}.\n\\end{equation}\n\nAs seen from Eq.~(\\ref{5}), terms containing the angular \nmomentum cancel. Said otherwise, the magnitude of the \nvelocity does not depend on the angular momentum. Figure~1 \nshows, however, that it is different angular momenta of an \nelement of a given energy residing at a point that account \nfor the different directions of the elements velocity. \n\nThis brings us to the conclusion that the magnitude of the velocity \n{\\itshape\\bfseries at a given point} (i.e., at point $\\mathbf r$ where element $dm$ resides) does not depend on the trajectory on which element $dm$ moves. At a given point, the element $dm$ has the same velocity, no matter what trajectory is involved.\n\nOne more comment is here in order. A separate element of charge $dq$ rotating in a {\\itshape\\bfseries circular} trajectory radiates an ac field with the frequency of rotation. If, however, elements of the charge fill completely the circular trajectory, this charge distribution will not radiate ac fields, because this state is the steady-state which does not depend on time. The trajectory of such a state is closed and is actually an analog of a circular current. And circular current, as is well known, does not radiate ac fields while generating constant electric and magnetic fields.\n\n\n\\section{Motion of a distributed charge in a Coulomb potential well}\n\n\nNow recall that we are considering motion not of a point but ruther \nof a distributed charge.\n\nBecause in all trajectories which are shown in Fig.~1\nthe element of charge $dq$ has \nthe same energy, it can move \n{\\itshape\\bfseries along any} trajectory. Moreover, \nthis element of charge can move {\\itshape\\bfseries in all \ntrajectories at the same time}. This \ncan be visualized in the following way. Divide element \nof charge $dq$ in $k$ parts. Then one element of charge \n$dq'=dq\/k$ with mass $dm'=dm\/k$ can move along one elliptical \ntrajectory, another \ncharge $dq'$, along another trajectory, and so on. As $k$ \ntends to infinity, all the trajectories will criss-cross \nall of the allowed region containing trajectories of the \nelements of charge $dq'$ of the same energy but with \ndifferent angular momenta. Generally speaking, this \nprocess may be considered not as motion of elements of \ncharge along trajectories but rather as motion of a \ncontinuous medium, of a {\\itshape\\bfseries charge wave}. \n\nConsider now the velocity with which elements of a distributed charge move. As already shown, elements of a distributed charge can propagate from a given point along all trajectories simultaneously. In Section~1 it was demonstrated that elements of charge have at a specific point the same velocity in any trajectory. At this specific point, these trajectories are characterized by different directions (within a solid angle $2\\pi$, see Fig.~1). Hence, when a charge propagates over all trajectories simultaneously, the directions of velocities of different elements at a given point are confined within the solid angle $2\\pi$, while their magnitude is the same.\n\nThis motion {\\itshape\\bfseries of a charge wave} may be treated from two angles, to wit, either as motion of different elements of charge on all trajectories at the same time, or as motion of a wave. In the first case, the behavior of each element is described by equations of mechanics (allowing for the potentials in which these elements move; but for the description of the overall picture to be complete, one will have to take into account that the number of these elements is infinite). In the second case, one will have to resort to equations in partial derivatives. For description of the behavior of any continuous medium, and, in particular, of a distributed charge the second model appears to be simpler and, thus, preferable. For the present, however, we are going to adhere to the first approach -- it appears more graphic (while certainly more cumbersome). \n\nTurning now to Fig.~1, we see immediately that different trajectories (i.e., trajectories characterized by the same energy but different momenta) cross one another. Said otherwise, it turns out that the charges moving along all trajectories simultaneously {\\itshape\\bfseries can \"interpenetrate\"} one another without changing their trajectories.\n\nThis would seem at first glance to be in contradiction with the well known statement that like charges repel. Therefore the assumption that likely charged elements can penetrate into one another may sound surprising, to say the least. But point charges considered usually in science have an infinite density at the point of charge. Therefore like point charges cannot penetrate into one another and, moreover, cannot even approach one another to a short enough distance. The distributed charges considered by us have a specific finite charge density. We are going to show now that charges can penetrate into one another, depending on what external forces act on these charges and what relevant forces are generated by the charges themselves.\n\nThe force $d\\mathbf F=\\mathbf Edq$ acting on any element of charge \n$dq=\\rho dV$ is defined by the magnitude of the electric field \n$\\mathbf E$ at this point. The field $\\mathbf E(\\mathbf r)$ at a given point $\\mathbf r$ is a sum of the field $\\mathbf E_N(\\mathbf r)$ generated by the nucleus and the field $\\mathbf E_{\\rho}(\\mathbf r)$ created by all the charges surrounding the given charge:\n\n\\begin{equation}\n\\label{6}\n\\mathbf E(\\mathbf{r})=\\mathbf E_N(\\mathbf{r})+\\mathbf E_{\\rho}(\\mathbf{r})=\n\\frac{e\\mathbf r}{r^3}+\\int{\\frac{\\rho(\\mathbf r')(\\mathbf r-\\mathbf r')dV'}\n{|\\mathbf r-\\mathbf r'|^3}}.\n\\end{equation}\n\nThe presence of expression $|\\mathbf r-\\mathbf r'|^3$ in the denominator of the second term in equality (\\ref{6}) might cause an erroneous idea that the field $\\mathbf E_{\\rho}(\\mathbf{r})$ at point $\\mathbf r$ is large. In this case the element of charge located at point $\\mathbf r''$, near point\n $\\mathbf r$, will not be able to approach in its motion point \n$\\mathbf r$, because both these elements of charge have like signs.\n One can readily show that this is not so: indeed, the field \n$\\mathbf E_{\\rho}(\\mathbf{r})$ is finite at point $\\mathbf r$, and the difference between the fields at the two neighboring points $\\mathbf r$ and $\\mathbf r''$ tends to zero as the magnitude of \n$|\\mathbf r-\\mathbf r''|$ approaches zero. \n\nTo prove this, we use the approach employed in monograph \\cite{2}, $\\$44$. Circumscribe a sphere of radius $R_0$ around point $\\mathbf r$. The field generated by the charges outside the sphere $R_0$ is finite, because these charges are at a finite distance larger than $R_0$ from point $\\mathbf r$. We have now to verify that the field $\\hat\\mathbf E(\\mathbf r)$ generated by charges confined inside the sphere $R_0$, i.e., in the immediate vicinity of point $\\mathbf r$, is also finite. Denoting $|\\mathbf r-\\mathbf r'|=R$, and \n$(\\mathbf r-\\mathbf r')=\\mathbf R$, we can write for this part of the field:\n\n\\begin{equation}\n\\label{7}\n| \\hat \\mathbf E(\\mathbf r)|\\le\\int{\\frac{|\\rho(\\mathbf r')\\mathbf R|d\\hat V}\n{R^3}},\n\\end{equation}\nwhere integration of $d\\hat V$ is performed over the volume of the sphere \n$R_0$. But\n$$\n|\\rho(\\mathbf r')\\mathbf R|\\le |\\rho_{max}|R,\n$$ \nwhere $|\\rho_{max}|$ is the absolute value of the maximum density of charge inside sphere $R_0$.\n\nWe finally come to \n$$\n| \\hat \\mathbf E(\\mathbf r)|\\le|\\rho_{max}|\\int{\\frac{d\\hat V}\n{R^2}}.\n$$ \n\nIntroducing spherical coordinates $R, \\vartheta, \\varphi$ with the center at point $\\mathbf r$, with $d\\hat V=R^2sin\\vartheta d\\vartheta d\\varphi dR$,\nand integrating with respect to $R$ from $0$ to $R_0$, we obtain\n\n\\begin{equation}\n\\label{8}\n|\\hat \\mathbf E(\\mathbf r)|\\le 4\\pi|\\rho_{max}|R_0.\n\\end{equation}\n\nThus $\\hat \\mathbf E(\\mathbf r)$ is a finite quantity tending to zero with decreasing radius of the sphere $R_0$. Moreover, this immediately suggests a conclusion that the difference between the values of vector $\\mathbf E$ \nat two adjacent points, for instance, $\\mathbf r$ and $\\mathbf r''$, tends to zero with the distance between these points approaches zero too. Suppose that these two points are located inside the sphere $R_0$. The field created by charges outside the sphere $R_0$ is continuous, because these charges are at finite distances from points $\\mathbf r$ and $\\mathbf r''$.\n As for the field $\\hat \\mathbf E$ generated by charges confined inside the sphere $R_0$, the strength of this field in absolute magnitude, as proved above, cannot be larger than the value of $4\\pi|\\rho_{max}|R_0$. As the magnitude of $R_0$ is going to zero, we see that this part of the field also changes continuously and tends to zero as $R_0$ approaches zero.\n\nThus the electric field $\\mathbf E$ surrounding any element of the distributed charge $dq=\\rho dV$ is finite and varies continuously. And this is why the force $d\\mathbf F=\\mathbf Edq$ acting on this element of charge is finite and varies continuously. The element of distributed charge will be driven by this force to move in the direction of the total force acting at this point. \n\nIn actual fact, the statement that charges can pass through \none another does not carry anything supernatural in it. For \ninstance, electromagnetic fields can penetrate one into or \nthrough the other without at the same time affecting one \nanother---this is nothing but the standard principle of \nsuperposition. Two radar beams can cross without interaction; \nthis is just penetration of ac fields through one another. \nSuperposition of one dc field on another (the principle of \nsuperposition) may be regarded as penetration of one field \ninto another. Significantly, in this process the fields do \nnot act in any way on one another.\n\nAs for the charges, no statements concerning passage of one \ncharge through another without direct action on one another \n(interaction of charges is taken into account through the \nfields created by these charges) have thus far been made, \nalthough the principle of superposition is valid for charges \nas well. This statement should, however, be made. \n\nThis paper was intended to address the case of small density of the distributed charge. This means that we shall neglect the second term in expression (\\ref{6}) compared to the first one. In this case, any element of charge $dq$ with mass $dm$ moves only in the field of charge of the nucleus \n$e$, and for this element all the conditions specified in Section~1 are fully met, and the motion of this element will be subject to the laws described in the above monograph \\cite{1}, \\$~15.\n\nWe have considered earlier a set of trajectories, both elliptical and circular, which are characterized by the same energy. In a Coulomb potential well, however, elements moving along different, including circular, trajectories may have different energies. The potential energy of an element in a circular trajectory is constant and depends only on the distance of this element from the nucleus, $d\\mathcal E_{Pot}=edq\/r$. Each element in any circular trajectory can be identified by a set of elliptical trajectories with the same energy (see Fig.~1). This makes the total set of all trajectories extremely complex.\n\n\n\\section{Distributed charge has no spherical symmetry}\n\nConsider the shape which can have a charge in a Coulomb potential well.\n\nIn a spherical coordinate system $r, \\vartheta, \\varphi$ the angular part of any distribution of charges can be presented in the form of an expansion in spherical functions $Y_{lm}(\\vartheta, \\varphi)$. Let us see what spherical functions can be employed in description of a distributed charge in a Coulomb potential well.\n\nThe simplest spherical function is the spherically symmetric function \n$Y_{00}(\\vartheta, \\varphi)$. This function should be present in an expansion always (except for the cases where the total charge of the distribution is zero). This has the following natural explanation. The integral over all space of a charge expanded in spherical functions $Y_{lm}(\\vartheta, \\varphi)$ yields the total charge $Q$. But the only angular function whose integration yields a nonzero result is $Y_{00}(\\vartheta, \\varphi)$. Integration of all other functions will yield zero. \nTherefore only the function $Y_{00}(\\vartheta, \\varphi)$ can describe the presence itself of a charge in a volume. All the other angular functions participating in the expansion can only change the shape of the charge distribution, while not being capable of removing or adding a charge.\n\nOn the other hand, a distributed charge cannot be characterized with the use of one angular function $Y_{00}(\\vartheta, \\varphi)$ only. This becomes evident from the following consideration. A Coulomb potential well is spherically symmetric. Therefore, an elliptical trajectory of propagation of an element of charge may lie in any plane passing through the nucleus (i.e., through the origin of coordinates).\n\nAn analysis of all elements of a distributed charge reveals that their trajectories lie {\\itshape\\bfseries in all planes simultaneously}. Consider now different planes in which the elliptical trajectories of elements may lie. We choose for this purpose one of such planes, rotate it successively and follow several trajectories of motion of an element. The continuity of the distributed charge allows this operation.\n\n\\begin{figure}\n\\label{F2}\n\\begin{center}\n\\includegraphics[scale=1]{Figure2.eps}\n\\caption{}\n\\end{center}\n\\end{figure}\n\nThis situation is visualized in Fig.~2. It shows three planes which may confine the trajectories of motion $A$, $B$ and $C$. The nucleus is located at point $O$. A $K-K$ line passing through the nucleus is drawn in plane $A$. One can see three trajectories in plane $A$: a circular trajectory $a1$, an elliptical trajectory $a2$, and another elliptical trajectory $a3$, which is symmetric to $a2$ about the $K-K$ line.\nIt is shown that all elements lying in plane $A$ rotate in the same sense. We shall turn plane $A$ about the $K-K$ line in angle $\\eta$ starting from an initial direction. Because the Coulomb potential is spherically symmetric, and the charge continuous, it can be expected that on turning plane $A$ about the $K-K$ line through angle $\\pi$, we shall come to the state in which all trajectories will coincide with the ones that had been there before the turn of the plane.\n\nLet us perform this operation of the turn. Figure~2 shows two positions of the plane after a turn by $\\pi\/2$ and by an angle $\\pi$. By turning plane $A$ about the $K-K$ axis in angle $\\eta$ by an angle $\\pi\/2$, we come to plane \n$B$ with trajectories $b1$, $b2$, $b3$, and by angle $\\pi$, to plane $C$ with trajectories $c1$, $c2$, $c3$. \nSignificantly, trajectory $c1$ in plane $C$ will exactly coincide with trajectory $a1$ in plane $A$. Similarly, trajectory $a2$ will merge with trajectory $c3$, and trajectory $a3$, with $c2$. Thus for each elliptical trajectory in plane $A$ there will be always the corresponding trajectory in plane $C$. After the turn of the plane, these trajectories will coincide.\n\nIt will turn out, however, that motion along all these trajectories after completion of the turn of the plane (i.e., in position $C$) will occur in the sense opposite to that in which the elements moved in the plane before its turn (that is, in position $A$). The + sign on planes $A$, $B$, $C$ is put for the sake of convenience in following the effect of plane rotation.\n\nSince we are considering a {\\itshape\\bfseries distributed charge}, its elements should move simultaneously in all planes, including planes $A$ and $C$. As showed, however, our analysis, motion over the latter should occur in opposite directions. All elements just cannot rotate simultaneously in opposite directions (this would involve loss of energy).\n\nIt thus turns out that motion over these trajectories is impossible altogether. Hence, there should exist a direction in which trajectories for motion of charges do not exist at all. In the spherically symmetric potential well of the nucleus one cannot, however, isolate a specified direction (for instance, the direction from which we reckoned the angle $\\eta$ in Fig.~2). Hence, a distributed charge must have a specified direction, a direction in which the distributed charge does not exist. In this case, with no charge in this direction, one cannot expect existence of trajectories along which propagation could occur in opposite directions.\n\nIn other words, although the Coulomb potential is spherically symmetric, distributed charge loses spherical symmetry.\n\nThis appears only natural, because the angular momentums in plane $A$ (i.e., before the turn of the plane) do not coincide with those in plane $C$ (i.e., after the turn of the plane) and, moreover, have opposite orientation.\n\nTo sum up, in a spherically symmetric Coulomb potential well a spherically symmetric charge distribution just cannot exist; there must therefore exist a specified direction. An additional comment concerning a specified direction in which there is no distributed charge will be proposed in Section 5.\n\n\n\\section{Description of a charge with spherical functions}\n\nAs follows from previous considerations, description of a distributed charge has to be made with a spherically symmetric angular function \n$Y_{00}(\\vartheta, \\varphi)$ (this function specifies the presence itself of a charge in the given potential well). This function alone is not sufficient, however, because the $Y_{00}(\\vartheta, \\varphi)$ function does not have any specified direction (a specified direction appears as a result of the circular motion of the elements of charge around the nucleus). Therefore, description of a charge requires, even in the simplest case, invoking some other $Y_{lm}(\\vartheta, \\varphi)$ functions as well. Let us see what functions could be employed in description of a distributed charge.\n\nWe start by assuming that the specified direction discussed above coincides with the $\\vartheta=0$ direction of the spherical coordinate system. We shall call this direction the $Z$ axis. Then total rotation of the elements of charge will occur about the $Z$ axis, i.e., along the $\\varphi$ coordinate. We understand under total rotation here not the rotation of any one element but rather that of the totality of the elements, of the distributed charge as a whole. We discussed in Section 1 a plane with respect to which an element $dq$ (or $dm$) propagates as a wave in a solid angle $2\\pi$. In the present case this plane passes through the axis $Z$ and the position of the element at the given moment, i.e., perpendicular to the $\\varphi$ coordinate.\n\nIntroduce an additional condition; to wit, we are going to consider only stationary, i.e., time-independent, states of the charge. States of charge which do not depend on time, do not produce radiation of variable fields, while generating constant electric and magnetic fields. Because stationary states of a distributed charge do not vary with time, no need appears in proving that such states do not generate variable fields, i.e., fields depending on time. If total rotation of the charge occurs about the $Z$ axis, i.e., along the $\\varphi$ coordinate, absence of radiation can be described by using for description only functions which are axially symmetric about the $Z$ axis. This restricts the allowable set of functions to those of the $Y_{l0}(\\vartheta, \\varphi)$ group. Indeed, only \n$Y_{l0}(\\vartheta, \\varphi)$ functions do not depend on the coordinate \n$\\varphi$, i.e., have axial symmetry with respect to the $Z$ axis.\n\nIf we impose one more constraint, namely, that the distributed charge is symmetric with respect to the coordinate $\\vartheta=\\pi\/2$, i.e., about the equator (which is a more frequent situation), index $l$ of the spherical function $Y_{l0}$ can be only even.\n\nThere are no other proper functions for description of a charge which does not generate radiation. Indeed, $Y_{lm}(\\vartheta, \\varphi)$ functions contain a factor $exp(\\pm im\\varphi)$. Motion of such a charge along the coordinate $\\varphi$ will initiate dependence on time (a factor of the kind of $exp(\\pm im\\varphi-i\\omega t)$ will appear, where $\\omega$ is the frequency of the moving wave). The appearance of the dependence on time will inevitably give rise to radiation of variable fields. We disregard here such states involving radiation and focus our interest on stationary states only, which do not generate radiation.\n\nThe simplest function satisfying the above requirements is $Y_{20}$. We write therefore the angular part of the relation for a distributed charge in the form\n\n\\begin{equation}\n\\label{9}\nL=D(Y_{00}+D_{20}Y_{20}),\n\\end{equation}\n$$\n\\mbox{where }Y_{00}=\\frac{1}{\\sqrt{4\\pi}},\\quad Y_{20}\n=\\sqrt{\\frac{5}{4\\pi}}\\left(\\frac{3}{2}\\cos^2\\vartheta-\\frac{1}\n{2}\\right),\\quad \\mbox{(see, e.g., \\cite{3}).}\n$$\n\nThe coefficient $D_{20}$ can be determined from the condition that at \n$\\vartheta=0$ and $\\vartheta=\\pi$ (i.e., on the $Z$ axis) the function $L$ be zero. Coefficient $D$ can be found from the condition that at \n$\\vartheta=\\pi\/2$ (i.e., at the equator) function $L$ is unity. These conditions bring us to\n\n\n\n$$\nD=\\frac{2\\sqrt{4\\pi}}{3},\\qquad D_{20}=-\\frac{1}{\\sqrt{5}}.\n$$\nNow function L acquires the final form\n$$\nL=\\frac{2\\sqrt{4\\pi}}{3}\\left[\\frac{1}{\\sqrt{4\\pi}}-\\frac{1}\n{\\sqrt5}\\cdot\\sqrt{\\frac{5}{4\\pi}}\\left(\\frac{3}{2}\\cos^2\n\\vartheta-\\frac{1}{2}\\right)\\right].\n$$\nOne can readily verify that this function simply coincides with the function \n$\\sin^2\\vartheta$. This means that in this case the angular part of the density of distributed charge can be written in one of two ways:\n\n\\begin{equation}\n\\label{10}\nL=\\sin^2\\vartheta, \\quad \\mbox{or} \\quad L=D(Y_{00}+D_{20}Y_{20}),\n\\end{equation}\nand the density of distributed charge corresponding to this angular distribution will read\n\\begin{equation}\n\\label{11}\n\\rho(r,\\vartheta)=AR(r)\\sin^2\\vartheta, \\quad \\mbox{or} \\quad\n\\rho(r,\\vartheta)=AR(r)D(Y_{00}+D_{20}Y_{20}),\n\\end{equation}\nwhere $R(r)$ is the radial part of the distribution. The coefficient $A$ is derived from normalization of the distributed charge against the total charge $Q$.\n\nOne may choose any form that would seem appropriate in a given situation.\n\nThus, in describing a charge with spherical functions one obtains for the charge distribution in a Coulomb potential well in the simplest case a figure resembling a torus. All elliptical and circular trajectories of an elements of the charge (of any energy) should be confined to this torus. We note that at the $Z$ axis the charge is zero.\n\nA comment will be appropriate here. As shown earlier, at a given point the element $dm$ propagates in all directions (within a solid angle $2\\pi$) with the same velocity. It would seem that this is in direct contradiction with the statement that a distributed charge has a specified direction in which there is no charge. In particular, in the above example with a torus-shaped charge, it would seem that the trajectories lying on the equator must differ from those confined to a perpendicular plane, because these trajectories cross the $Z$ axis, where the charge is zero. \n\nThe following point may be in order here. The trajectories of the element $dm$ in Fig.~1 were considered by us under the tacit assumption that this element will continue to move along its original trajectory. (This assumption derives from the concept of the motion of a solid body.) This is, however, not necessarily so. An analysis of Fig.~1 shows that {\\itshape\\bfseries each point} is a center from which elements of charge propagate in all directions (within a solid angle $2\\pi$). Said otherwise, element $dm$ does not move along this trajectory all the time. At each point it breaks up into many elements $dm'$ which continue to move subsequently, but now along other trajectories.\n At the next point the situation repeats, with breakup into many elements, at the next point -- again into countless elements, and so on. Thus, the element $dm$, in starting its motion at a point on a trajectory, should not necessarily terminate it at the same trajectory. Actually, this is motion not of individual elements but rather that of a wave. This is why a mass (and a charge) may have different densities at different points in space.\n\nThis is an allowed process, because in each trajectory the corresponding element has the same energy. But {\\itshape\\bfseries at each given point} the velocity of the elements remains, as before, the same in all directions, irrespective of the density of charge or mass at the given point.\n\n \n\\section{Differences in the velocity of motion between a charge wave and individual elements}\n\nAs already pointed out, it appears more appropriate to consider the motion of elements of equal energy along different elliptical trajectories as that of a charge wave. It appears pertinent to compare now different parameters of motion of this wave with those of an individual element.\n\nThe total energy of all elements making up a charge wave is equal to the energy of one combined element lying at the point of crossing of all elliptical trajectories and moving along one of the circular trajectories. Considered in the context of equality of energies, the motion of a charge wave may be correlated with that of one point element, with the sum of the energies of all elements in this wave equal to the energy of one combined element moving in a circle. This energy can be readily determined.\n\nAn element moving along a circular trajectory retains both its total and the potential and kinetic energies. By the virial theorem (see, e.g. \\cite{1}, \n\\$~10), in this case the potential energy of an element is twice its total energy, and its total energy is equal to the kinetic energy taken with the opposite sign, \n$2d\\mathcal E=d\\mathcal E_{Pot}$, $d\\mathcal E=-d\\mathcal E_{Kin}$. No averaging is needed here, because the energies are constant. As for the potential energy of an element in a circular orbit, it can be derived simply from the radius $R$ of the circular orbit: $d\\mathcal E_{Pot}=edq\/R$. \n\nThe situation is different with the velocities of motion of a wave and of individual elements.\n \nAn analysis of Fig.~1 suggests that {\\itshape\\bfseries each point} of a charge is a center from which elements of charge propagate in all directions (within a solid angle $2\\pi$). One might say that the motion of a distributed charge at a given point represents a kind of \"a velocity fan\"\\ for all directions (within a solid angle $2\\pi$); note that, as shown in Section~1, in any direction the velocity has the same magnitude. \n\nThe velocity of an element being a vector, the velocity must \nretain its vector properties even in the case of the element \npropagating (in the form of a wave) within a solid angle of \n$2\\pi$. This, however, will be not the velocity of a single \nelement but rather that of motion of a wave, of propagation \nof a wave process. It turns out that the velocity of propagation \nof a wave process does not coincide with that of motion of \nelements of mass or charge. The momentum of an element $dm$ \npropagating as a wave likewise does not coincide with that \nof an element moving as a whole in one direction.\n\nDenote the velocity of motion of a charge wave by $\\mathbf \nv_w(\\mathbf r)$ (the subscript $w$ standing for wave), and \nthe magnitude of this velocity, by $\\upsilon_w(\\mathbf r)$, \nto discriminate this velocity from the velocity \n$\\mathbf v(\\mathbf r)$ and $\\upsilon(\\mathbf r)$ of motion \nof an element of charge. The momentum of an element $dm$ \npropagating as a wave will be denoted, accordingly, by \n$\\mathbf p_w(\\mathbf r)$.\n\nConsider this situation in more detail.\n\nIsolate an element of mass $dm$ at a point in space. If this \nelement moves as a whole with a velocity $\\mathbf{v}_l$ along \na trajectory $l$, the momentum $d\\mathbf{p}_l$ of this element \nwill be\n\\begin{equation}\n\\label{12}\nd\\mathbf{p}_l=dm\\mathbf{v}_l,\\qquad\\mbox{and the magnitude of \nthe momentum }\\qquad dp_l=dm\\upsilon.\n\\end{equation}\n\nTo describe the motion of this element of mass as that of a wave \nin the solid angle of $2\\pi$, divide the mass $dm$ into many parts \n$dm'$. Each part $dm'$ will propagate within a solid angle $d\\Omega$. \nIn this case, we can write $dm'=dm\\cdot\\frac{d\\Omega}{2\\pi}$. \nAs was shown in Section~1\nthe velocity of motion of each element $dm'$ is the same, equal \nin magnitude to the velocity $\\upsilon$ of motion of the whole \nelement $dm$. \n The direction of this velocity \n$\\mathbf{v'}$ is determined by the solid angle $d\\Omega$. \nFor the momentum of element $dm'$ in this case we can write\n\\begin{equation}\n\\label{13}\nd\\mathbf p'=dm'\\mathbf v'=dm\\frac{d\\Omega}{2\\pi}\n\\mathbf v',\\quad\\mbox{and for its magnitude}\\quad\ndp'=dm\\frac{d\\Omega}{2\\pi}\\upsilon.\n\\end{equation}\n\nIt might come up as a surprise that sometimes we discuss motion \nalong an elliptical trajectory (for instance, trajectory $l$) \nto stress that the element $dm$ moving in this trajectory obeys \nall laws of theoretical mechanics, while in other cases we prefer \nto identify the motion of the same element in a solid angle as \nthat of a wave.\n\nIn actual fact, we are speaking in these cases about different \nthings. When discussing the motion along a trajectory, we follow \nthe motion of {\\itshape\\bfseries{one}} specific element, be it \nelement $dm$ or $dm'$. It appears only natural that the motion \nof this element obeys all laws of theoretical mechanics.\n\nWhen, however, we discuss the motion of an element as a wave in \na solid angle, we have in mind rather the motion of \n{\\itshape\\bfseries{many}} elements crossing at a point \n$\\mathbf r$. This is shown in a revealing way in Fig.~1. \nThis set off elements could be formed of one element $dm$ as well. \nIt is for this purpose that we broke up element $dm$ into a set \nof elements $dm'$, which thereafter moved along {\\bfseries\\itshape\n{different}} trajectories.\n\n\\medskip\n\nWe chose to orient a spherical coordinate system such that the \ndirection $\\vartheta=0$ coincides with that of the angular momentum \nof the charge, and denoted this axis by $Z$. All elements of a \ndistributed charge rotate in the same sense (overall rotation \nis around the $Z$ axis). That all elements rotate in one sense only \nimplies that there are no velocity components along the negative \ndirection of the $\\varphi$ axis. In other words, on passing a \nplane through the $Z$ axis and the position of element $dm$, we end \nup with the following situation: element $dm$ propagates (as a wave) \ninto a solid angle of $2\\pi$, i.e., into the hemisphere located \non one side of this plane.\n\n\\begin{figure}\n\\label{F3}\n\\begin{center}\n\\includegraphics[scale=1]{Figure3.eps}\n\\caption{}\n\\end{center}\n\\end{figure}\n\nThis situation is illustrated by Fig.~3. In Fig.~3, the $Z$ axis is \ndirected at us (i.e., it is actually a top view). The nucleus \nis at point O. Dashed lines are lines of equal mass density. \nElement $dm$ propagates into a solid angle of $2\\pi$, i.e., into the \nhemisphere. We see a fanlike distribution of velocities $\\mathbf v'$ \nof the elements of mass $dm'$. The velocity of motion of element \n$dm'$ in any direction is the same. In Fig.~3, this is shown by all \nvelocity vectors $\\mathbf v'$ being of equal length.\n\nTo find the resultant momentum of an element $dm$ in the case where \nit propagates (as a wave) into a hemisphere, we have to sum all the \nmomentum vectors $d\\mathbf p'=dm'\\mathbf v'$. This can be done by \nintroducing a local spherical system of coordinates centered on the \nlocation of element $dm$, with angles $\\eta$ and $\\xi$. We orient \nthe coordinate system such that the $\\eta=0$ direction is \nperpendicular to the above-mentioned plane and denote this direction \nby $Z'$. In this particular case, $Z'$ coincides with the direction \nof the $\\varphi$ axis of the common coordinate system. Next we \nresolve the momentum into a component along the $Z'$ axis, and another,\nperpendicular to it. By virtue of the symmetry relative to \nthe $Z'$ axis, the perpendicular component will vanish after the \nsummation, leaving only the component along the $Z'$ axis, i.e., \nalong the $\\varphi$ axis. Thus, the resultant momentum of an element \nmoving in all directions simultaneously (into a solid angle of $2\\pi$, \ni.e., into the hemisphere) is aligned with the $\\varphi$ axis. \nCalculate now $dp_{w,\\varphi}$, i.e., the projection of the momentum \non the $\\varphi$ axis. To do this, we sum all $dp'_\\varphi$ components. \nUsing Eq. (\\ref{13}), we come to $dp'_\\varphi=dp'\\cos\\eta=\ndm'\\upsilon\\cos\\eta=dm\\frac{d\\Omega}{2\\pi}\\upsilon\\cos\\eta$, whence:\n\\begin{equation}\n\\label{14}\ndp_{w,\\varphi}=\\int{dm\\frac{d\\Omega}{2\\pi}\\upsilon\\cos\\eta}=\n\\frac{dm}{2\\pi}\\upsilon\\int\\limits_0^{\\pi\/2}\\cos\\eta\n\\sin\\eta d\\eta\\int\\limits_0^{2\\pi}d\\xi=\n\\frac{1}{2}dm\\upsilon=\\frac{1}{2}dp_l,\n\\end{equation}\nor\n\\begin{equation}\n\\label{15}\nd\\mathbf p_w=\\frac{1}{2}dm\\upsilon \\mathbf n_\\varphi ,\n\\end{equation}\nwhere $\\mathbf n_\\varphi$ is the unit vector along the $\\varphi$ axis.\n\nComparing now Eqs. (\\ref{14}) and (\\ref{15}) with relations (\\ref{12}), \nwe see that the momentum of an element propagating into a hemisphere \nis one half that in magnitude of an identical element moving in one \ndirection. Also, while the momentum of an element moving as a whole \nin one direction coincides in direction with its velocity, the \nmomentum of an element propagating into a hemisphere is directed \n{\\itshape\\bfseries {only along the}} $\\varphi$ {\\itshape\\bfseries \n{axis}}. {\\itshape\\bfseries {This momentum has no other components}}.\n\nThe above reasoning and the conclusions were conducted for a momentum. \nTo obtain a diverging wave, the element $dm$ was broken up \ninto smaller elements $dm'$, with all $dm'$ elements having the same \nvelocity $\\upsilon$, and each element of mass $dm'$ propagating in \nits solid angle $d\\Omega$. This is a rigorous treatment. It can be \nmade more convenient, however, by considering not the momentum but \nrather directly the velocity. To do this, we denote conventionally \n $d\\upsilon=\\upsilon d\\Omega$, understanding by $d\\upsilon$ a set \n of velocities with directions confined to the solid angle $d\\Omega$. \n Then in place of Eq. (\\ref{15}) and taking into account (\\ref{12})\n we come to\n\\begin{equation}\n\\label{16}\n\\mathbf v_w=\\frac{1}{2}\\upsilon\\mathbf n_\\varphi.\n\\end{equation}\n\nHere $\\mathbf v_w$ is no longer the velocity of a single element \n$dm$ moving in its trajectory; it is now the remaining vector part \nof the velocity of the element $dm$ propagating into the hemisphere, \nand $\\upsilon$ is, as before, the magnitude of the velocity of the \nelement moving in its trajectory. \n(Equation (\\ref{16}) can be also derived \ndirectly from Eq. (\\ref{15}) by canceling $dm$).\n\nThus, the velocity $\\mathbf v_w(\\mathbf r)$ is the velocity of \npropagation of a wave process, and the modulus of \n$\\mathbf v_w(\\mathbf r)$ is the magnitude of the velocity of the \nwave process $\\upsilon_w(\\mathbf r)$. The magnitude of the velocity \nof a wave process does not coincide with that of propagation of \nelements of charge. Indeed, taking the modulus of $\\mathbf v_w$ \n(using Eq. (\\ref{16}) and recalling that vector $\\mathbf v_w$ has \nonly one component, and it is directed along the $\\varphi$ axis), \nwe obtain only one half of the real velocity of an element $\\upsilon$. \nIndeed, some of the vector components vanish in propagation of \nthe element as a wave into the hemisphere. \nActually, these components do not disappear without trace, so that, \nfor instance, they have to be taken into account in calculation of \nthe energy, because in actual fact elements move along elliptical \ntrajectories with a velocity $\\upsilon$.\nSpecifically, for calculation of the kinetic energy one must use\nthe $\\upsilon$ quantity that is the real velocity of an element.\nFor calculation of the angular momentum (vector quantity) \nof the distributed charge one must use $\\mathbf v_w$ quantity.\n\n\n\\addcontentsline{toc}{chapter}{\u041b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0430}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\nGiven an integer $k > 0$ and a set $P$ of $n$ weighted points in the plane,\nour objective is to fit a $k$-step function to them so that the maximum weighted\nvertical distance of the points to the step function is minimized.\nWe call this problem the {\\em $k$-step function problem}.\nIt has applications in areas such as geographic information systems, \ndigital image analysis, data mining, facility locations, and\ndata representation (histogram), etc.\n\nIn the unweighted case, if the points are presorted,\nFournier and Vigneron \\cite{fournier2011} showed that\nthe problem can be solved in linear time using the results of \n\\cite{frederickson1991a,frederickson1984,gabow1984}.\nLater they showed that the weighted version of the problem can also be solved\nin $O(n\\log n)$ time~\\cite{fournier2013},\nusing Megiddo's parametric search technique \\cite{megiddo1983a}.\nPrior to these results, the problem had been discussed by several researchers\n\\cite{chen2009,diazbanez2001,liu2010,lopez2008,wang2002}. \n\nGuha and Shim \\cite{guha2007} considered this problem in the context of {\\em histogram construction}.\nIn database research, it is known as the {\\em maximum error histogram} problem.\nIn the weighted case,\n this problem is to partition the given points into $k$ buckets based on their $x$-coordinates,\nsuch that the maximum $y$-spread in each bucket is minimized.\nThis problem is of interest to the data mining community as well (see \\cite{guha2007} for references).\nGuha and Shim \\cite{guha2007} computed the optimum histogram of size $k$,\nminimizing the maximum error.\nThey present algorithms which run in linear time when the points are unweighted,\nand in $O(n\\log n + k^2\\log^6n)$ time and $O(n\\log n)$ space when the points are weighted.\n\nOur objective is to improve the above result to $O(n)$ time when $k$ is a constant.\nWe show that we can optimally fit a $k$-step function to unsorted weighted points in linear time.\nWe earlier suggested a possible approach to this problem at an OR workshop~\\cite{bhattacharya2013b}.\nHere we flesh it out, presenting a complete and rigorous algorithm and proofs.\nOur algorithm exploits the well-known properties of prune-and-search along the lines in \\cite{bhattacharya2007}.\n\n\nThis paper is organized as follows.\nSection \\ref{sec:prelim} introduces the notations used in the rest of this paper.\nIt also briefly discusses how the prune-and-search technique can be used\nto optimally fit a $1$-step function (one horizontal line) to a given set of weighted points.\nWe then consider in Section~\\ref{sec:cond2step} a variant of the 2-step function problem,\ncalled the anchored 2-step function problem.\nWe discuss a ``big partition'' in the context of the $k$-partition of a point set\ncorresponding to a $k$-step function in Section \\ref{sec:kstep}.\nSection \\ref{sec:algorithm} presents our algorithm for the optimal $k$-step function problem.\nSection \\ref{sec:conclusion} concludes the paper,\nmentioning some applications of our results.\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\\subsection{Model}\\label{sec:model}\nLet $P=\\{p_1,p_2,\\ldots, p_n\\}$ be a set of $n$ weighted points in the plane.\nFor $1\\leq i\\leq n$ let $p_i.x$ (resp. $p_i.y$) denote the $x$-coordinate (resp. $y$-coordinate)\nof point $p_i$, and let $w(p_i)$ denote its weight.\nThe points in $P$ are not sorted,\nexcept that $p_1.x\\leq p_i.x\\leq p_n.x$ holds for any $i=1, \\ldots, n$.\\footnote{For the sake\nof simplicity we assume that no two points have the same $x$ or $y$ coordinate.\nBut the results are valid if this assumption is removed.\n}\nLet $F_k(x)$ denote a generic $k$-step function,\nwhose $j^{th}$ segment (=step) is denoted by $s_j$.\nFor $1\\leq j \\leq k-1$, segment $s_j$ represents a half-open horizontal interval $[s_j^{(l)}, s_j^{(r)})$\nbetween two points $s_j^{(l)}$ and $s_j^{(r)}$.\nThe last segment $s_k$ represents a closed horizontal interval $[s_k^{(l)}, s_k^{(r)}]$.\nNote that $s_j^{(l)}.y= s_j^{(r)}.y$,\nwhich we denote by $s_j.y$.\nWe assume that for any $k$-step function $F_k(x)$,\n segments $s_1$ and $s_k$ satisfy $s_1^{(l)}.x = p_1.x$ and $s_k^{(r)}.x = p_n.x$,\nrespectively.\nSegment $s_j$ is said to {\\em span} a set of points $Q\\subseteq P$,\nif $s_j^{(l)}.x \\leq p.x U$; (b) $q$ can be ignored at $y>U$.\n}\n\\label{fig:2points1}\n\\end{figure}\nLet $y=U$ be the line at or above which at least 2\/3 of the upper bisectors lie,\nand at or below which at least 1\/3 of the upper bisectors lie.\nWe use $\\wp^U_{2\/3}$ and $\\wp^U_{1\/3}$ to name the subsets of\n $\\wp$ that have these two sets of bisectors, respectively.\n Note that $|\\wp^U_{2\/3}|\\geq n\/2\\times 2\/3 = n\/3$ and $|\\wp^U_{1\/3}|\\geq n\/2\\times 1\/3 = n\/6$.\nSimilarly, let $y=L$ be the line at or below which at least 2\/3 of the \nlower bisectors lie,\nand at or above which at least 1\/3 of the bisectors lie.\\footnote{{We define $U$ and $L$ this way,\nbecause many points could lie on them.}}\nWe use $\\wp^L_{2\/3}$ and $\\wp^L_{1\/3}$ to name the subsets of\n$\\wp$ that have these two sets of bisectors, respectively.\n Note that $|\\wp^L_{2\/3}|\\geq n\/2\\times 2\/3 = n\/3$ and $|\\wp^L_{1\/3}|\\geq n\/2\\times 1\/3 = n\/6$.\n\\begin{lemma} \\label{lem:one6th}\nWe can identify $n\/6$ points that can be removed without affecting the weighted 1-center\n for the values\nof their $y$-coordinates.\n\\end{lemma}\n\\begin{proof}\nConsider the following three possibilities.\n\\begin{enumerate}\n\\item[(i)]\nThe weighted 1-center lies above $U$.\n\\item[(ii)]\nThe weighted 1-center lies below $L$.\n\\item[(iii)]\nThe weighted 1-center lies between $U$ and $L$,\nincluding $U$ and $L$.\n\\end{enumerate}\n\nIn case (i), there are two subcases,\nwhich are shown in Fig.~\\ref{fig:2points1}(a) and (b), respectively.\nSince the center lies above $U$, \nwe are interested in the upper envelope of the costs in the \n$y$-region given by $y > U$.\nIn the case shown in Fig.~\\ref{fig:2points1}(a),\nthe costs of points $p$ and $q$ satisfy $d(y,p.y)w(p) < d(y,q.y)w(q)$ for $y > U$.\nThus we can ignore $p$.\nIn the case shown in Fig.~\\ref{fig:2points1}(b),\n the costs of points $p$ and $q$ satisfy\n$d(y,p.y)w(p) > d(y,q.y)w(q)$ for $y > U$.\nThus we can ignore $q$.\nSince $|\\wp^U_{1\/3}|\\geq n\/6$,\nin either case, one point from each such pair can be ignored,\ni.e., 1\/6 of the points in $P$ can be eliminated, because it cannot affect the weighted 1-center.\nIn case (ii) a symmetric argument proves that 1\/6 of the points in $P$ can be discarded.\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[]{\\includegraphics[height=1.5cm]{figs\/2points3.pdf}}\n\\hspace{2mm}\n\\subfigure[]{\\includegraphics[height=1.5cm]{figs\/2points4.pdf}}\n\\caption{2\/3 of upper bisectors are at $y> U$.}\n\\label{fig:2points3}\n\\end{figure}\n\nIn case (iii) see Fig.~\\ref{fig:2points3}.\nThe costs of each pair in $\\wp^U_{2\/3}$ (of the $2n\/3$ pairs) as functions of $y$ intersect at most once at $yL$.\nTherefore, $\\wp^U_{2\/3}\\cap \\wp^L_{2\/3}$ ($n\/3$ pairs must be common to both,)\ni.e., both intersections of each such pair occur outside of the $y$-interval $[L,U]$.\n$|\\wp^U_{2\/3}\\cap \\wp^L_{2\/3}| = n\/2 \\times 1\/3 = n\/6$.\nThis implies that their cost functions do not intersect within in $[L,U]$,\ni.e., one of each pair lies above that of the other in $[L,U]$,\nand can be discarded.\n\\end{proof}\n\n\n\\subsection{Optimal 1-step function}\\label{sec:1step}\nThis problem is equivalent to finding the weighted center for $n$ points on a line.\nWe pretend that all the points had the same $x$-coordinate.\nThen the problem becomes that of finding a weighted 1-center on a line,\ni.e., on the $y$-axis.\nThis can be solved in linear time using Megiddo's {\\em prune-and-search} method \\cite{bhattacharya2007,chen2015a,megiddo1983a}.\nIn \\cite{megiddo1983b} Megiddo presents a linear time algorithm in the case where the\npoints are unweighted. \nFor the weighted case we now present a more technical algorithm that we can apply\nlater to solve other related problems.\nThe following algorithm uses a parameter $c$ which is a small integer constant.\n\n\\begin{algorithm}{\\rm :} {\\tt 1-Step}$(P)$\n\\begin{enumerate}\n\\item\nPair up the points of $P$ arbitrarily.\n\\item\nFor each such pair $(p,q)$ determine their horizontal bisector lines. \n\\item\nDetermine a horizontal line, $y=U$ such that $|\\wp^U_{2\/3}|\\geq n\/3$\nand $|\\wp^U_{1\/3}|\\geq n\/6$ hold.\n\\item\nDetermine a horizontal line, $y=L$ such that $L$ $|\\wp^L_{2\/3}|\\geq n\/3$ and\n$|\\wp^L_{1\/3}|\\geq n\/6$ hold.\n\\item\nDetermine the critical points for $U$ and $L$.\n\\item\nIf there exist critical points for $U$ on both sides of (above and below) $U$, \nthen $y=U$ defines an optimal 1-step function, $F^*_1(x)$; Stop. \nOtherwise, let $s_U$ (higher or lower than $U$) be the side of $U$ on which the critical point\nlies.\n\\item\nIf there exist critical points for $L$ on both sides of $L$,\n$y=L$ defines $F^*_1(x)$; Stop.\nOtherwise, let $s_L$ (higher or lower than $L$) be the side of $L$ on which the critical point\nlies.\n\\item\nBased on $s_U$ and $s_L$, discard 1\/6 of the points from $P$,\nbased on Lemma~\\ref{lem:one6th}.\n\\item\nIf the size of the reduced set $P$ is greater than constant $c$, \nrepeat this algorithm from the beginning with the reduced set $P$.\nOtherwise, determine $F^*_1(x)$ using any known method\n(which runs in constant time).\n\\end{enumerate}\n\\end{algorithm}\n\n\\begin{lemma}\\label{lem:1step}\nAn optimal 1-step function $F^*_1(x)$ can be found in linear time.\n\\end{lemma}\n\\begin{proof}\nThe recurrence relation for the running time $T(n)$ of {\\tt 1-Step}$(P)$ for general $n$ is \n$T(n) \\leq T(n-n\/6) + O(n)$,\nwhich yields $T(n) = O(n)$.\n\\end{proof}\n\n\\section{Anchored $2$-step function problem}\\label{sec:cond2step}\nIn general, we denote an optimal $k$-step function by $F^*_k(x)$\nand its $i^{th}$ segment by $s^*_i$. \nLater, we need to constrain the first and\/or the last step of a step function to be\nat a specified height.\nA $k$-step function is said to be {\\em left-anchored} (resp. {\\em right-anchored}),\nif $s_1.y$ (resp. $s_k.y$) is assigned a specified value,\nand is denoted by $^{\\downarrow}\\!F_k(x)$ (resp. $F_k^{\\downarrow}(x)$).\nThe {\\em anchored $k$-step function} problem is defined as follows.\nGiven a set $P$ of points and two $y$-values $a$ and $b$,\ndetermine the optimal $k$-step function $^{\\downarrow}\\!F^*_k(x)$ (resp. $F_k^{\\downarrow *}(x)$)\nthat is left-anchored (resp. right-anchored) at \n$a$ (resp. $b$) such that cost $D(P, ^{\\downarrow}\\!\\!F^*_k(x))$ (resp. $D(P, F_k^{\\downarrow *}(x))$)\n is the smallest possible.\nIf a $k$-step function is both left- and right-anchored, \nit is said to be {\\em doubly anchored} and is\ndenoted by $^{\\downarrow}\\!F_k^{\\downarrow}(x)$.\n\n\n\\subsection{Doubly anchored 2-step function}\nSuppose that segment $s_1$ (resp. $s_2$) is anchored at $a$ (resp. $b$).\nSee Fig.~\\ref{fig:anchored2}(a).\n\\begin{figure}[ht]\n\\centering\n\\subfigure[]{\\includegraphics[height=3cm]{figs\/conditional2.pdf}}\n\\hspace{4mm}\n\\subfigure[]{\\includegraphics[height=3cm]{figs\/goftNhoft.pdf}}\n\\caption{(a) $s_1.y=a$ and $s_2.y=b$;\n(b) Monotone functions $g(x)$ (in blue) and $h(x)$ (in red).\n}\n\\label{fig:anchored2}\n\\end{figure}\nLet us define two functions $g(x)$ and $h(x)$ by\n\\begin{eqnarray}\ng(x) &=& \\max_{p.x\\leq x} \\{w(p)\\cdot|p.y - a|~\\mid p\\in P\\},\\label{eqn:g}\\\\\nh(x) &=&\\max_{p.x>x} \\{w(p)\\cdot |p.y - b|~\\mid p\\in P\\},\\label{eqn:h}\n\\end{eqnarray}\nwhere $g(x) =0$ for $x p_n.x$.\nIntuitively, if we divide the points of $P$ at $x$ into two partitions $P_1$ and $P_2$,\nthen $g(x)$ (resp. $h(x)$) gives the cost of partition $P_1$ (resp. $P_2$).\nSee Fig.~\\ref{fig:anchored2}(b).\nClearly the global cost for the entire $P$ is minimized for any $x$\nat the lowest point in the upper envelope of $g(x)$ and $h(x)$,\nwhich is named $\\overline{x}$.\nSince the points in $P$ are not sorted,\n$g(x)$ and $h(x)$ are not available explicitly,\nbut we can compute $\\overline{x}$ in linear time using the {\\em prune-and-search} method,\ntaking advantage of the fact that $\\max\\{g(x),h(x)\\}$ is unimodal.\n\n\\begin{algorithm}{\\rm :} {\\tt Doubly-Anch-2-Step}$(P,a,b)$\\label{alg:double}\n\\begin{enumerate}\n\\item\nInitialize $P'=P$.\n\\item\nFind the point in $P'$ that has the median $x$-coordinate, $x_m$.\n\\item\nEvaluate $g(x_m)$ (resp. $h(x_m)$) using (\\ref{eqn:g}) (resp. (\\ref{eqn:h})).\n\\item\nIf $g(x_m) = h(x_m)$ then $\\overline{x}=x_m$. Stop.\n\\item\nIf $g(x_m) < h(x_m)$ (resp. $g(x_m) > h(x_m)$), \ni.e., $\\overline{x}< x_m$ (resp. $\\overline{x}< x_m$),\nprune all the points $p$ with $p.x < x_m$ (resp. $p.x > x_m$),\n from $P'$,\nremembering just the maximum cost.\n\\item\nStop when $|P'|=2$, and find the lowest point $\\overline{x}$.\nOtherwise, go to Step~2.\n\\end{enumerate}\n\\end{algorithm}\n\nWe have the following lemma.\n\\begin{lemma}\\label{lem:doublyAnchored}\nAn optimal doubly anchored 2-step function\ncan be found in linear time.\n\\end{lemma}\n\\begin{proof}\nSteps~2 and 3 of Algorithm {\\tt Doubly-Anch-2-Step}$(P,a,b)$ can be carried out in linear time.\nSince Step~4 cuts the size of $P'$ in half every time, Step~2 is entered $O(\\log n)$ times.\nTherefore the total time is $O(n)$.\n\\end{proof}\n\n\\subsection{Left- or right-anchored 2-step function}\nWithout loss of generality, we discuss only a left-anchored 2-step function. \nGiven an anchor value $a$,\nwe want to determine the optimal 2-step function with the constraint\nthat $s^*_1.y=a$, denoted by $^{\\downarrow}\\!F^*_2(x)$.\nSee Fig.~\\ref{fig:anchored2}(a).\nIn this case, $b$ in (\\ref{eqn:h}) is not given; \nwe need to find the optimal value for it.\nBut assume for now that $b$ is also given,\nand execute {\\tt Doubly-Anch-2-Step}$(P,a,b)$.\nFrom the solution that it yields,\ncan we find the direction in which to move $b$ to find the optimal\nleft-anchored 2-step function?\n\\begin{lemma}\nLet $P_1$ (resp. $P_2$) be the left (resp right) partition of $P$ generated by {\\tt Doubly-Anch-2-Step}$(P,a,b)$\nsuch that $s_1.y=a$ (resp. $s_2.y=b$), where $a D(P_2, s_2)$)\nthen $P_1$ (resp. $P_2$) is the big partition.\n\\end{enumerate}\n\\end{procedure}\n\nIf $P_1$ is the big partition, we can eliminate all the points belonging to it,\nwithout affecting $^{\\downarrow}\\!F^*_2(x)$ that we will find.\nSee Step~4 of the Algorithm~\\ref{alg:cond2} given below.\nWe then repeat the process with the reduced set $P$.\nIf $P_2$ is the big partition, on the other hand, we need to do more work,\nsimilar to what we did to find an optimal 1-step function.\nNamely,\nwe determine values $U$ and $L$ for $P_2$ by executing Algorithm {\\tt 1-Step}$(P_2)$.\nWe then find a doubly anchored 2-step solution for $P$ with left anchor $a$\nand right anchor $U$.\n\n\n\\begin{algorithm}{\\rm :} {\\tt $l$-Anch-2-Step}$(P,a)$\\label{alg:cond2}\n\\begin{enumerate}\n\\item\nDivide $P$ into left partition $P_1$ and right partition $P_2$,\nwhose sizes differ by at most one.\\footnote{As before,\nwe assume that the points have different $y$-coordinates.\n}\n\\item\nLet $s_1$ be the segment with $s_1.y=a$ spanning $P_1$,\nand let $s_2$ be the 1-step (optimal) solution for $P_2$.\\footnote{\nSegment $s_2$ can be found in $O(|P_2|)$ time by Lemma~\\ref{lem:1step}.}\n\\item\nIf $D(P_1, s_1) = D(P_2, s_2)$ then\noutput $\\{s_1,s_2\\}$, which defines $^{\\downarrow}\\!F^*_2(x)$.\nStop.\n\\item\nIf $D(P_1, s_1) < D(P_2, s_2)$, remove from $P$ the points of $P_1$,\nexcept the critical point for $s_1$.\nGo to Step~6.\n\\item\nIf $D(P_1, s_1) > D(P_2, s_2)$ then carry out the following steps.\n\\begin{enumerate}\n\\item\nDetermine points $U$ and $L$ for $P_2$ as described in Algorithm {\\tt 1-Step}$(P)$.\n\\item\nExecute {\\tt Doubly-Anch-2-Step}$(P,a,U)$,\nand find the solution whose left partition is maximal.\nRepeat it with right anchor $L$.\n\\item\nEliminate 1\/6 of the points of $P_2$ from $P$, based on the two solutions\n(as in Steps 6--8 of Algorithm {\\tt 1-Step}$(P)$.)\n\\end{enumerate}\n\\item\nIf $|P| > c$ (a small constant),\nrepeat Steps~1 to 4.\nOtherwise, optimally solve the problem in constant time, using a known method.\n\\end{enumerate}\n\\end{algorithm}\nIn the example in Fig.~\\ref{fig:leftAnc}, \nassume that $b$ is not given,\nand $s_2$ is determined by Step~2.\nThen we have $D(P_1, s_1) > D(P_2, s_2)$,\nand Step~5 applies.\nAccording to Step~5(a), we determine $U$.\nWe then find the doubly anchored solution with the right anchor set to $b=U$.\n\n\n\\begin{lemma}\nAlgorithm {\\tt $l$-Anch-2-Step}$(P,a)$ computes $^{\\downarrow}\\!F^*_2(x)$ correctly,\nand runs in linear time.\n\\end{lemma}\n\\begin{proof}\nStep~3 is obviously correct.\nIf $D(P_1, s_1) < D(P_2, s_2)$ holds in Step~4,\nthen the first partition of $^{\\downarrow}\\!F^*_2(x)$ contains $P_1$.\nWe need to keep the critical point for $a$,\nbut all other points of $P_1$ can be ignored from now on\nbecause $P_1$ will expand.\nIf $D(P_1, s_1) > D(P_2, s_2)$ holds in Step~5,\nthen the first partition of $^{\\downarrow}\\!F^*_2(x)$ is contained in $P_1$.\n\nEach iteration of Steps 3 and 4 will eliminate at least $1\/2\\times1\/6=1\/12$ of the points of $P$.\nSuch an iteration takes linear time in the input size. \nThe total time needed for all the iterations is therefore linear.\n\\end{proof}\n\n\\section{$k$-step function}\\label{sec:kstep}\n\\subsection{Approach}\nTo design a recursive algorithm, assume that for any set of points $Q\\subset P$,\nwe can find the optimal $(j-1)$-step function and the optimal left- and right-anchored $j$-step function \nfor any $2\\leq j < k$ in $O(|Q|)$ time,\nwhere $k$ is a constant .\nWe have shown that this is true for $k=2$ in the previous two sections.\nSo the basis of induction holds.\n\nGiven an optimal $k$-step function $F^*_k(x)$, for each $i~(1\\leq i \\leq k)$,\nlet $P^*_i$ be the set of points vertically closest to segment $s^*_i$.\nBy definition, the partition\n$\\{P^*_i \\mid i = 1, 2, \\ldots, k\\}$ satisfies the contiguity condition.\nIt is easy to see that\nfor each segment $s^*_i$, there are (local) critical points with respect to $s^*_i$,\nlying on the opposite sides of $s^*_i$.\n\nIn finding an optimal $k$-step function,\nwe first identify a big partition that will be spanned by a segment in\nan optimal solution.\nBy Lemma~\\ref{lem:big},\nsuch a big partition always exists.\nOur objective is to eliminate a constant fraction of the points in a big partition.\nThis will guarantee that a constant fraction of the input set is eliminated when $k$ is a fixed constant.\nThe points in the big partition other than two critical points are ``useless''\nand can be eliminated from further considerations.\\footnote{Note that there may\nbe more than two critical points in which case all but two are ``useless.''}\nThis elimination process is repeated until the problem size gets small enough\nto be solved by an exhaustive method in constant time.\n\n\\subsection{Feasibility test}\\label{sec:feasibility}\nGiven a weighted distance (=cost) $D$,\na point set $P$ is said to be $D$-{\\em feasible} if there exists a $k$-step function\n$F_k(x)$ such that $D(P,F_k(x)) \\leq D$. \nTo test $D$-feasibility\nwe first try to identify the first segment $s_1$ of a possible $k$-step function $F_k(x)$.\nTo this end we compute the median $m$ of $\\{p_i.x\\mid i = 1, 2,\\ldots, n\\}$ in $O(n)$ time,\nand divide $P$ into two parts $P_1 = \\{p_i \\mid p_i.x \\leq m\\}$ and $P_2 = \\{p_i \\mid p_i.x > m\\}$,\nwhich also takes $O(n)$ time.\nNote that $|P_1| \\leq \\lceil |P|\/2\\rceil$ and $|P_2| \\leq \\lceil |P|\/2\\rceil$ hold.\nWe then find the intersection $I$ of the $y$-intervals in $\\{|p_i.y-y| \\leq D \\mid p_i \\in P_1\\}$.\nAssuming that $P$ is $D$-feasible,\nthen we have two cases.\n\nCase (a): [$|I|=\\emptyset$] $s_1$ ends at some point $p_j \\in P_1$.\nThrow away all the points in $P_2$ and look for the longest $s_1$ limited by cost $D$,\nconsidering only the points in $P_1$ from the left.\n\n\n\nCase (b): [$|I|\\not=\\emptyset$] $s_1$ may end at some point $p_j \\in P_2$.\nThrow away all the points in $P_1$\nand look for the longest $s_1$, using $I$ and the points in $P_2$ from the left.\n\nClearly,\nwe can find the longest $s_1$ in $O(n)$ time.\nRemove the points spanned by $s_1$ from $P$,\nand find $s_2$ in $O(n)$ time, and so on.\nSince we are done after finding $k$ steps $\\{s_1, \\ldots, s_k\\}$,\nit takes $O(kn)$ time.\n\n\\begin{lemma}\\label{lem:feasibility}\nWe can test $D$-feasibility in $O(kn)$ time.\n\\hfill$\\qed$\n\\end{lemma}\n\n\\subsection{Identifying a big partition}\\label{sec:findBig}\n\\begin{lemma}\\label{lem:big}\nLet ${\\cal P}=\\{P_i\\mid i=1,\\ldots,k\\}$ be any $k$-partition of $P$,\nsatisfying the contiguity condition,\nsuch that the sizes of the partitions differ by no more than 1,\nand let $\\{P^*_i\\mid i=1,\\ldots,k\\}$ be an optimal $k$-partition.\nThen there exists an index $j$ such that $P_j$ is a big partition spanned by $s^*_j$.\n\\end{lemma}\n\\begin{proof} \nLet $j$ be the smallest index such that $s^{(r)}_j.x \\leq s^{*(r)}_j.x$.\nSuch an index must exists, because if $s^{(r)}_j.x > s^{*(r)}_j.x$\nfor all $1\\leq j \\leq k-1$ then $s^{(r)}_k.x = s^{*(r)}_j.x$.\nWe clearly have $s_j\\subset s^*_j$,\nwhich implies that $s^*_j$ spans $P_j$.\n\\end{proof}\n\nGiven a point set $P$ in the $x$-$y$ plane,\nlet ${\\cal P}=\\{P_i\\mid i=1,\\ldots,k\\}$ be any $k$-partition of $P$,\nsatisfying the contiguity condition,\nsuch that the sizes of the partitions differ by no more than 1.\nThe following procedure returns a big partition $P_j$ spanned by $s^*_j$,\nwhose existence was proved by Lemma~\\ref{lem:big}.\nSince $P=\\cup\\{P_i \\mid P_i \\in {\\cal P}\\}$,\n$P$ is implicit in the input to the next procedure.\n\n\\begin{procedure}{\\rm :} {\\tt Big$({\\cal P}, k)$}\\label{proc:bigk}\n\n\\noindent\n\\begin{enumerate}\n\\item\nUsing Algorithm~{\\tt 1-Step}$(P)$, compute the optimal 1-step function for $P_1$\nand let $D_1$ be its cost for $P_1$.\nIf $P$ is not $D_1$-feasible (i.e., $D(P,F^*_{k}(x))>D_1$),\nthen return $P_1$ and stop.\\footnote{There exists an optimal solution for $P$\n in which $s^*_1$ spans $P_1$.}\n\\item\nUsing Algorithm~{\\tt 1-Step}$(P)$, compute the optimal 1-step function for $P_k$\nand let $D'_k$ be its cost for $P_k$.\nIf $P$ is not $D'_k$-feasible (i.e., $D(P,F^*_{k}(x))>D'_k$),\nthen return $P_k$ and stop.\n\\item\nFind an index $j~ (1 < j < k)$ such that for $D_{j-1}=D(\\cup_{i=1}^{j-1} P_i, F^*_{j-1}(x))$\n$P$ is $D_{j-1}$-feasible, \nand for $D_j=D(\\cup_{i=1}^{j} P_i, F^*_j(x))$ $P$ is not $D_{j}$-feasible.\\footnote{This means\nthat $D_{j-1}\\geq D^*$ and $D_{j}< D^*$,\nwhere $D^*$ is the cost of the optimal solution for $P$.\nUnless $P^*_i =P_i$ for all $i$, such an index $j$ always exists.\n[We should indicate why.]}\nReturn $P_j$ and stop.\n\\end{enumerate}\n\\end{procedure}\n\n\\begin{lemma}\\label{lem:Bigiscorrect}\nProcedure {\\tt Big$({\\cal P},k)$} is correct.\n\\end{lemma}\n\\begin{proof}\nIt is clear that Steps~1 and 2 are correct.\nTo show that Step~3 is also correct, \nwe {\\em stretch} a step $s$ of an optimal step function\nby making it as long as possible as follows.\nMove $s^{(l)}.x$ (resp. $s^{(r)}.x$) to the left (resp. right) as far as possible without changing the cost\nof the step function.\nThe step that has been stretched is called a {\\em stretched step.} \nLet us assume without loss of generality that $s^*_j$ corresponding to $P_j$ returned by Step~3 is stretched.\nSince $D_{j-1}\\geq D^*$,\nwe must have $s^{*(l)}_j.x \\leq s^{(l)}_j.x$.\n\nThe optimal solution $F^*_j(x)$ for $\\cup_{i=1}^j P_i$ has cost $D_j$,\nwhich is too small for $P$ to be $D_j$-feasible.\nRegarding the remaining points $\\cup_{i=j+1}^k P_i$,\nlet $G^*_j(x)$ denote the optimal $(k-j)$-step function for this point set.\nIf $D(\\cup_{i=j+1}^k P_i, G^*_j(x)) \\leq D_j$,\nthe $P$ would be $D_j$-feasible.\nSince it is not,\n $s^{(r)}_j.x$ would be stretched to the right under the optimal solution $F^*_k(x)$,\ni.e., $s^{*(r)}_j.x \\geq s^{(r)}_j.x$.\nTogether with $s^{*(l)}_j.x \\leq s^{(l)}_j.x$,\nit follows that $P_j$ is spanned by $s^*_j$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Bigislinear}\nProcedure {\\tt Big$({\\cal P},k)$} runs in linear time in $n$.\n\\end{lemma}\n\\begin{proof}\nIn Step~1, the optimal 1-step function for $P_1$ can be found in $O(|P_1|)$ time by Lemma~\\ref{lem:1step},\nand it takes $O(kn)$ time to test if $P$ is not $D_1$-feasible by Lemma~\\ref{lem:feasibility}.\nSimilarly, Step~2 can be carried out in $O(n)$ time.\nTo carry out Step~3,\nwe compute, using binary search, $\\lceil \\log n\\rceil$ values out of $\\{D_i\\mid 1\\leq i \\leq k-1\\}$,\nwhich takes $O(f(k)n)$ time for some function $f(k)$,\nunder the assumption that any $i$-step function problem, $i < k$,\nis solvable in time linear in the size of the input point set.\n\\end{proof}\n\n\n\\section{Algorithm}\\label{sec:algorithm}\n\\subsection{Optimal $k$-step function}\nIn this section we are assuming that we can solve any $(j-1)$-step and anchored\n$j$-step function problems for any $2\\le j 1$,\nwe need to find the left- and right-anchored solution for $U$ and $L$,\nand prune $1\/6$ of the points in $P_j$ using {\\tt Prune-Big$(k,P_j)$}, given below,\nwhich is very similar to Algorithm {\\tt 1-Step}$(P)$.\nLet $P_j$ be a big partition spanned by $s^*_j$,\nwhich is an input to the following procedure.\n\\begin{procedure}{\\rm :} {\\tt Prune-Big$(k,P_j)$}\n\n\\noindent\n{\\bf Output:} 1\/6 of points in $P_j$ removed.\n\n\n\\begin{enumerate}\n\\item\nDetermine $U$ and $L$ for $P_j$ as in Algorithm {\\tt 1-Step}$(P)$. \n\\item \nIf $j>1$, find two right-anchored $j$-step functions $F_j^{\\downarrow *}(x)$ for $\\cup_{i=1}^{j} P_i$,\none anchored by $L$ and the other anchored by $U$.\n\\item\nIf $j< k$, find two left-anchored $(k-j+1)$-step functions\n $^{\\downarrow}\\!F^*_{k-j+1}(x)$ for $\\cup_{i=j}^k P_i$,\none anchored by $L$ and the other anchored by $U$.\n\\item\nIdentify 1\/6 of the points in $P_j$ with respect to $L$ and $U$,\nwhich are ``useless''\\footnote{See Step~8 of Algorithm~{\\tt 1-Step}$(P)$.}\nbased on $F_j^{\\downarrow *}\\!(x)$\nand $^{\\downarrow}\\!F^*_{k-j+1}(x)$ found above,\nand remove them from $P$. \n\\end{enumerate}\n\\end{procedure}\n\n\n\n\\begin{lemma}\\label{lem:anchored}\n{\\tt Prune-Big$(k,P_j)$} runs in linear time when $k$ is a constant..\n\\hfill$\\qed$\n\\end{lemma}\n\nWe can now describe our algorithm formally as follows.\n\n\\begin{algorithm}{\\rm :} {\\tt $k$-Step}$(P)$. \n\n\\noindent\n{\\bf Output:} Optimal $k$-step function $F^*_k(x)$\n\\begin{enumerate}\n\\item \nDivide $P$ into partitions $\\{P_i \\mid i = 1, 2, \\ldots, k\\}$,\nsatisfying the contiguous condition,\nsuch that their sizes differ by no more than one.\n\\item \nExecute Procedure {\\tt Big$({\\cal P},k)$} to find a big partition $P_j$\nspanned by $s^*_j$. \n\\item\nExecute Procedure {\\tt Prune-Big$(k,P_j)$}.\n\\item\nIf $|P| > c$ for some fixed $c$, \nrepeat Steps~1 to 3 with the reduced $P$.\n\\end{enumerate}\n\\end{algorithm}\n\n\n\\subsection{Analysis of algorithm}\nTo carry out Step 1 of Algorithm {\\tt $k$-Step}$(P)$, \nwe first find the $(hn\/k)^{th}$ smallest among $\\{p_i.x \\mid 1\\leq i \\leq n\\}$,\nfor $h=1, 2, \\ldots, k-1$.\nWe then place each point in $P$ into $k$ partitions delineated by these\n$k-1$ values.\nIt is clear that this can be done in $O(kn)$ time.\\footnote{This could be done in $O(n\\log k)$ time.}\nAs for Step~2,\nwe showed in Sec.~\\ref{sec:findBig} that finding a big partition spanned by an optimal step\n$s_j^*$ takes $O(n)$ time, since $k$ is a constant.\nStep~3 also runs in $O(n)$ time by Lemma~\\ref{lem:anchored}.\nSince Steps 1 to 3 are repeated $O(\\log n)$ times,\neach time with a point set whose size is at most a constant fraction of the size of the previous set,\nthe total time is also $O(n)$, when $k$ is a constant.\nBy solving a recurrence relation for the running time of Algorithm~{\\tt $k$-Step}$(P)$,\nwe can show that it runs in $O(2^{2k\\log k}n)=O(k^{2k}n)$ time.\n\n\\begin{theorem}\nGiven a set of $n$ points in the plane $P=\\{p_1,p_2,\\ldots, p_n\\}$,\nwe can find the optimal $k$-step function that minimizes the maximum distance\nto the $n$ points in $O(k^{2k} n)$ time.\n\\hfill$\\qed$\n\\end{theorem}\nThus the algorithm is optimal for a fixed $k$.\n\n\n\\section{Conclusion and Discussion}\\label{sec:conclusion}\nWe have presented a linear time algorithm to solve the optimal $k$-step function problem,\nwhen $k$ a constant.\nMost of the effort is spent on identifying a ``big partition.''\nIt is desirable to reduce the constant of proportionality. \n\nThe {\\em size-$k$ histogram construction problem}~\\cite{guha2007},\nwhere the points are not weighted, \nis similar to the problem we addressed in this paper.\nIts generalized version,\nwhere the points are weighted,\nis equivalent to our problem, and thus can be solved in optimal linear time when $k$ is a constant.\nThe {\\em line-constrained $k$ center problem} is defined by:\nGiven a set $P$ of weighted points in the plane and a horizontal line $L$,\ndetermine $k$ centers on $L$\nsuch that the maximum weighted distance of the points to their closest centers is minimized.\nThis problem was solved in optimal $O(n\\log n)$ time for arbitrary $k$ even if the points\nare sorted~\\cite{karmakar2013,wang2014a}.\nOur algorithm presented here can be applied to solve this problem \nin $O(n)$ time if $k$ is a constant.\n\nA possible extension of our work reported here is to use a cost other than the weighted vertical distance.\nThere is a nice discussion in \\cite{guha2007} on the various measures one can use. \nOur complexity results are valid\nif the cost is more general than (\\ref{eqn:pointCost}),\nin particular, $D(p, F(x))\\triangleq d(p, F(x))^2 w(p)$,\nwhich is often used as an error measure.\n\\section*{Acknowledgement}\\label{sec:ack}\nThis work was supported in part by Discovery Grant \\#13883 from\nthe Natural Science and Engineering Research Council (NSERC) of Canada and in part by MITACS,\nboth awarded to Bhattacharya.\n\n\n\\section*{Reference}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\bf Introduction}\nOne of the most interesting new developments in hadron physics is\nthe application of the AdS\/CFT correspondence~\\cite{Maldacena} to\nnonperturbative QCD\nproblems~\\cite{Polchinski,Janik,Erlich,Karch,Brodsky1,Teramond1,Teramond2}.\nIt is well know that AdS\/CFT gives an important insight into the\nviscosity and other global properties of the hadronic system\nformed in heavy ion collisions~\\cite{Kovtun}. The essential ansatz\nfor the application of AdS\/CFT to hadron physics is the indication\nthat the QCD coupling $\\alpha_s(Q^2)$ becomes large. Therefore\nconformal symmetry can be applied to, for example solutions of the\nQCD Dyson Schwinger equations and phenomenological studies of QCD\ncouplings based on physical observables such as $\\tau$ decay and\nthe Bjorken sum rule show that the QCD $\\beta$ function vanishes\nand $\\alpha_s(Q^2)$ become constant at small virtuality; i.e.,\neffective charges develop an infrared fixed point. Fully\nexploiting the gauge\/gravity correspondence to produce a model for\nreal strong interaction physics- a method called \" holographic\nQCD\" or \"AdS\/QCD\"-may be attempted either through a top-dawn\napproach starting with a particular string theory and choosing a\nbackground that naturally produces QCD-like properties, or a\nbottom-up approach starting with real QCD properties and using\nthem to obtain constraints on viable dual gravity theories.\n\nThe first attempts have been made for constructing phenomenological\nholographic models of QCD~\\cite{Erlich}. Surprisingly simple models\nconsisting of gauge theory in an anti-de-Sitter space interval have\nturned out to provide a remarkably good description of the meson\nsector of QCD. Therefore, it will be interesting that the\ncalculation of the higher twist effects within holographic QCD in\nproton-proton collisions in the running coupling approach.\n\nThe large-order behavior of a perturbative expansion in gauge\ntheories is inevitably dominated by the factorial growth of\nrenormalon diagrams~\\cite{Hooft,Mueller,Zakharov,Beneke}. In the\ncase of quantum chromodynamics (QCD), the coefficients of\nperturbative expansions in the QCD coupling $\\alpha_{s}$ can\nincrease dramatically even at low orders. This fact, together with\nthe apparent freedom in the choice of renormalization scheme and\nrenormalization scales, limits the predictive power of perturbative\ncalculations, even in applications involving large momentum\ntransfer, where $\\alpha_{s}$ is effectively small.\n\nIn this work we apply the running coupling approach~\\cite{Agaev}\nin order to compute the effects of the infrared renormalons on the\npion production in proton-proton collisions within holographic\nQCD. This approach was also employed\npreviously~\\cite{Ahmadov3,Ahmadov4,Ahmadov5,Ahmadov6} to calculate\nthe inclusive meson production in proton-proton and photon-photon\ncollisions.\n\nFor the calculations of the higher-twist cross sections on the\ndependence of wave functions of pion, we used the holographic QCD\nprediction $\\Phi_{hol}(x)$~\\cite{Brodsky2,Brodsky3,Vega} and pion\nasymptotic wave functions $\\Phi_{asy}(x)$~\\cite{Lepage1} from the\nperturbative QCD evolution. Theoretically obtained predicted\nresults within holographic QCD were compared with results of the\nperturbative QCD which obtained by the running coupling and frozen\ncoupling constants approaches.\n\nThe frozen coupling constant approach in\nRefs.~\\cite{Bagger,Baier,Ahmadov1,Ahmadov2} was used for calculation\nof integrals, such as\n\\begin{equation}\nI\\sim \\int\\frac{\\alpha_{s}({Q}^2)\\Phi(x,{Q}^2)}{1-x}dx.\n\\end{equation}\nIt should be noted that, in pQCD calculations, the argument of the\nQCD coupling constant (or the renormalization and factorization\nscale) ${Q}^2$ should be taken equal to the square of the momentum\ntransfer of a hard gluon in a corresponding Feynman diagram. But\ndefinition of $\\alpha_{s}(\\hat{Q}^2)$ suffers from infrared\nsingularities. Therefore, in the soft regions as $x_{1}\\rightarrow\n0$, and $x_{2}\\rightarrow 0$, integrals (1.1) diverge and for their\ncalculation some regularization methods are needed for\n$\\alpha_{s}(Q^2)$ in these regions. Investigation of the infrared\nrenormalon effects in various inclusive and exclusive processes is\none of the most important and interesting problem in the\nperturbative QCD. It is known that infrared renormalons are\nresponsible for factorial growth of coefficients in perturbative\nseries for the physical quantities. But, these divergent series can\nbe resummed by means of the Borel transformation ~\\cite{Hooft} and\nthe principal value prescription ~\\cite{Contopanagos}. Studies of\nhigher-twist and renormalon effects also opened new prospects for\nevaluation of power suppressed corrections to processes\ncharacteristics.\n\nWe organize the paper as the follows. In Section \\ref{ht}, we\nprovide some formulas for the calculation of the contributions of\nthe higher twist and leading twist diagrams. In Section \\ref{ir},\nwe present the formulas and analysis of the higher-twist effects\non the dependence of the pion wave function by the running\ncoupling constant approach, and in Section \\ref{results}, the\nnumerical results for the cross section and discuss the dependence\nof the cross section on the pion wave functions are presented.\nFinally, some concluding remark are stated in Section \\ref{conc}.\n\n\\section{HIGHER TWIST AND LEADING TWIST CONTRIBUTIONS TO INCLUSIVE REACTIONS}\n\\label{ht} The higher-twist Feynman diagrams, which describe the\nsubprocess $q_1+\\bar{q}_{2} \\to \\pi^{+}(\\pi^{-})+\\gamma$ for the\npion production in the proton-proton collision are shown in Fig.1.\nThe amplitude for this subprocess can be found by means of the\nBrodsky-Lepage formula ~\\cite{Lepage2}\n\\begin{equation}\nM(\\hat s,\\hat\nt)=\\int_{0}^{1}{dx_1}\\int_{0}^{1}dx_2\\delta(1-x_1-x_2)\\Phi_{\\pi}(x_1,x_2,Q^2)T_{H}(\\hat\ns,\\hat t;x_1,x_2).\n\\end{equation}\nIn Eq.(2.1), $T_H$ is the sum of the graphs contributing to the\nhard-scattering part of the subprocess.\n\nThe Mandelstam invariant variables for subprocesses $q_1+\\bar{q}_{2}\n\\to \\pi^{+}(\\pi^{-})+\\gamma$ are defined as\n\\begin{equation}\n\\hat s=(p_1+p_2)^2,\\quad \\hat t=(p_1-p_{\\pi})^2,\\quad \\hat\nu=(p_1-p_{\\gamma})^2.\n\\end{equation}\nThe pion wave functions predicted by\nAdS\/QCD~\\cite{Brodsky2,Brodsky3,Vega} and the PQCD evolution\n~\\cite{Lepage1} has the form:\n$$\n\\Phi_{asy}^{hol}(x)=\\frac{4}{\\sqrt{3}\\pi}f_{\\pi}\\sqrt{x(1-x)},\n$$\n\\begin{equation}\n\\Phi_{VSBGL}^{hol}(x)=\\frac{A_1k_1}{2\\pi}\\sqrt{x(1-x)}exp\\left(-\\frac{m^2}{2k_{1}^2x(1-x)}\\right),\\quad\n\\Phi_{asy}^{p}(x)=\\sqrt{3}f_{\\pi}x(1-x)\n\\end{equation}\nwhere $f_{\\pi}$ is the pion decay constant.\n\nThe cross section for the higher-twist subprocess $q_1\\bar{q}_{2}\n\\to \\pi^{+}(\\pi^{-})\\gamma$ is given by the expression\n\\begin{equation}\n\\frac{d\\sigma}{d\\hat t}(\\hat s,\\hat t,\\hat u)=\\frac\n{8\\pi^2\\alpha_{E} C_F}{27}\\frac{\\left[D(\\hat t,\\hat\nu)\\right]^2}{{\\hat s}^3}\\left[\\frac{1}{{\\hat u}^2}+\\frac{1}{{\\hat\nt}^2}\\right]\n\\end{equation}\nwhere\n\\begin{equation}\nD(\\hat t,\\hat u)=e_1\\hat\nt\\int_{0}^{1}dx\\left[\\frac{\\alpha_{s}(Q_1^2)\\Phi_{\\pi}(x,Q_1^2)}{1-x}\\right]+e_2\\hat\nu\\int_{0}^{1}dx\\left[\\frac{\\alpha_{s}(Q_2^2)\\Phi_{\\pi}(x,Q_2^2)}{1-x}\\right].\n\\end{equation}\nIn the Eq.(2.5) $Q_{1}^2=(x-1)\\hat u \\,\\,\\,\\,$and $Q_{2}^2=-x\\hat\nt$ \\,\\, represent the momentum squared carried by the hard gluon\nin Fig.1, $e_1(e_2)$ is the charge of $q_1(\\overline{q}_2)$ and\n$C_F=\\frac{4}{3}$. The higher-twist contribution to the\nlarge-$p_{T}$ pion production cross section in the process\n$pp\\to\\pi^{+}(\\pi^{-})+\\gamma+X$ is ~\\cite{Owens,Greiner}\n\\begin{equation}\n\\Sigma_{M}^{HT}\\equiv E\\frac{d\\sigma}{d^3p}=\\int_{0}^{1}\\int_{0}^{1}\ndx_1 dx_2 G_{{q_{1}}\/{h_{1}}}(x_{1})\nG_{{q_{2}}\/{h_{2}}}(x_{2})\\frac{\\hat s}{\\pi} \\frac{d\\sigma}{d\\hat\nt}(q\\overline{q}\\to \\pi\\gamma)\\delta(\\hat s+\\hat t+\\hat u).\n\\end{equation}\nWe denote the higher-twist cross section obtained using the frozen\ncoupling constant approach by $(\\Sigma_{\\pi}^{HT})^0$.\n\nRegarding the higher-twist corrections to the pion production\ncross section, a comparison of our results with leading-twist\ncontributions is crucial. We take two leading-twist subprocesses\nfor the pion production:(1) quark-antiquark annihilation $q\\bar{q}\n\\to g\\gamma$, in which the $g \\to \\pi^{+}(\\pi^{-})$ and (2)\nquark-gluon fusion, $qg \\to q\\gamma $, with subsequent\nfragmentation of the final quark into a meson, $q \\to\n\\pi^{+}(\\pi^{-})$ ~\\cite{Ahmadov3,Ahmadov5}.\n\n\\section{THE HIGHER TWIST MECHANISM IN HOLOGRAPHIC QCD AND INFRARED RENORMALONS}\\label{ir}\n\nThe main problem in our investigation is the calculation of integral\nin (2.5) by the running coupling constant approach within\nholographic QCD and also discussion of the problem of normalization\nof the higher twist process cross section in the context of the same\napproach. Therefore, it is worth noting that, the renormalization\nscale (argument of $\\alpha_s$) according to Fig.1 should be chosen\nequal to $Q_{1}^2=(x-1)\\hat u$, $Q_{2}^2=-x\\hat t$. The integral in\nEq.(2.5) in the framework of the running coupling approach takes the\nform\n\\begin{equation}\nI(\\mu_{R_{0}}^2)=\\int_{0}^{1}\\frac{\\alpha_{s}(\\lambda\n\\mu_{R_0}^2)\\Phi_{M}(x,\\mu_{F}^2)dx}{1-x}.\n\\end{equation}\nThe $\\alpha_{s}(\\lambda \\mu_{R_0}^2)$ has the infrared singularity\nat $x\\rightarrow1$, for $\\lambda=1-x$ or $x\\rightarrow0$, for\n$\\lambda=x$ and so the integral $(3.1)$ diverges. For the\nregularization of the integral, we express the running coupling at\nscaling variable $\\alpha_{s}(\\lambda \\mu_{R_0}^2)$ with the aid of\nthe renormalization group equation in terms of the fixed one\n$\\alpha_{s}(Q^2)$. The solution of renormalization group equation\nfor the running coupling $\\alpha\\equiv\\alpha_{s}\/\\pi$ has the form\n~\\cite{Contopanagos}\n\\begin{equation}\n\\frac{\\alpha(\\lambda)}{\\alpha}=\\left[1+\\alpha\n\\frac{\\beta_{0}}{4}\\ln{\\lambda}\\right]^{-1}.\n\\end{equation}\nThen, for $\\alpha_{s}(\\lambda Q^2)$, we get\n\\begin{equation}\n\\alpha(\\lambda Q^2)=\\frac{\\alpha_{s}}{1+\\ln{\\lambda\/t}}\n\\end{equation}\nwhere $t=4\\pi\/\\alpha_{s}(Q^2)\\beta_{0}=4\/\\alpha\\beta_{0}$.\n\nHaving inserted Eq.(3.3) into Eq.(2.5) we obtain\n$$\nD(\\hat t,\\hat u)=e_{1}\\hat t\\int_{0}^{1}dx\\frac{\\alpha_{s}(\\lambda\n\\mu_{R_0}^2)\\Phi_{M}(x,Q_{1}^2)}{1-x}+ e_{2}\\hat\nu\\int_{0}^{1}dx\\frac{\\alpha_{s}(\\lambda\n\\mu_{R_0}^2)\\Phi_{M}(x,Q_{2}^2)}{1-x}\n$$\n\\begin{equation}\n=e_{1}\\hat t\\alpha_{s}(-\\hat u)t_{1}\\int_{0}^{1}dx\n\\frac{\\Phi_{M}(x,Q_{1}^2)}{(1-x)(t_{1}+\\ln\\lambda)} + e_{2}\\hat\nu\\alpha_{s}(-\\hat t)t_{2}\\int_{0}^{1}dx\n\\frac{\\Phi_{M}(x,Q_{2}^2)}{(1-x)(t_{2}+\\ln\\lambda)}\n\\end{equation}\nwhere $t_1=4\\pi\/\\alpha_{s}(-\\hat u)\\beta_{0}$ and\n$t_2=4\\pi\/\\alpha_{s}(-\\hat t)\\beta_{0}$.\n\nAlthough the integral (3.4) is still divergent, it is recast into a\nsuitable form for calculation. Making the change of variable as\n$z=\\ln\\lambda$, we obtain\n\\begin{equation}\nD(\\hat t,\\hat u)=e_{1}\\hat t \\alpha_{s}(-\\hat u) t_1\\int_{0}^{1}dx\n\\frac{\\Phi_{M}(x,Q_{1}^2)}{(1-x)(t_1+z)}+ e_{2}\\hat u\n\\alpha_{s}(-\\hat t) t_2 \\int_{0}^{1} dx\n\\frac{\\Phi_{M}(x,Q_{2}^2)}{(1-x)(t_2+z)}\n\\end{equation}\nIn order to calculate (3.5) we will apply the integral\nrepresentation of $1\/(t+z)$ ~\\cite{Zinn-Justin,Erdelyi}.\n\\begin{equation}\n\\frac{1}{(t+z)}=\\int_{0}^{\\infty}e^{-(t+z)u}du,\n\\end{equation}\ngives\n\\begin{equation}\nD(\\hat t,\\hat u)=e_{1} \\hat{t} \\alpha_{s}(-\\hat u) t_1 \\int_{0}^{1}\n\\int_{0}^{\\infty} \\frac{\\Phi_{\\pi}(x,Q_{1}^2)e^{-(t_1+z)u}du\ndx}{(1-x)}+ e_{2} \\hat{u} \\alpha_{s}(-\\hat t) t_2 \\int_{0}^{1}\n\\int_{0}^{\\infty} \\frac{\\Phi_{\\pi}(x,Q_{2}^2)e^{-(t_2+z)u}du\ndx}{(1-x)}\n\\end{equation}\nIn the case $\\Phi_{asy}^{hol}(x)$ for the $D(\\hat t,\\hat u)$ it is\nwritten as\n\\begin{equation}\nD(\\hat t,\\hat u)=\\frac{16 f_{\\pi} e_{1} \\hat t}{\\sqrt{3}\\beta_{0}}\n\\int_{0}^{\\infty} du\ne^{-t_{1}u}B\\left(\\frac{3}{2},\\frac{1}{2}-u\\right)+ \\frac{16\nf_{\\pi} e_{2} \\hat u}{\\sqrt{3}\\beta_{0}} \\int_{0}^{\\infty} du\ne^{-t_{2}u}B\\left(\\frac{3}{2},\\frac{1}{2}-u\\right)\n\\end{equation}\nand for $\\Phi_{asy}^{p}(x)$ wave function\n\\begin{equation}\nD(\\hat t,\\hat u)=\\frac{4\\sqrt{3}\\pi f_{\\pi}e_{1}\\hat t}{\\beta_{0}}\n\\int_{0}^{\\infty}du e^{-t_{1}u}\n\\left[\\frac{1}{1-u}-\\frac{1}{2-u}\\right] +\\frac{4\\sqrt{3}\\pi\nf_{\\pi}e_{2}\\hat u}{\\beta_{0}} \\int_{0}^{\\infty}du e^{-t_{2}u}\n\\left[\\frac{1}{1-u}-\\frac{1}{2-u}\\right].\n\\end{equation}\nwhere $B(\\alpha,\\beta)$ is Beta function. The structure of the\ninfrared renormalon poles in Eq.(3.8) and Eq.(3.9) strongly depend\non the wave functions of the pion. To remove them from Eq.(3.8) and\nEq.(3.9) we adopt the principal value prescription. We denote the\nhigher-twist cross section obtained using the running coupling\nconstant approach by $(\\Sigma_{\\pi}^{HT})^{res}$.\n\n\\section{NUMERICAL RESULTS AND DISCUSSION}\\label{results}\n\nIn this section, we discuss the higher-twist contributions\ncalculated in the context of the running and frozen coupling\nconstant approaches on the dependence of the chosen pion wave\nfunctions in the process $pp \\to \\pi^{+}(or\\,\\, \\pi^{-})\\gamma+X$.\nIn numerical calculations for the quark distribution function\ninside the proton, the MSTW distribution function ~\\cite{Martin},\nand the gluon and quark fragmentation ~\\cite{Albino} functions\ninto a pion have been used. The results of our numerical\ncalculations are displayed in Figs.2-14. Firstly, it is very\ninteresting comparing the higher-twist cross sections obtained\nwithin holographic QCD with the ones obtained within perturbative\nQCD. In Fig.2 and Fig.3 we show the dependence of higher-twist\ncross sections $(\\Sigma_{\\pi^{+}}^{HT})^{0}$,\n$(\\Sigma_{\\pi^{+}}^{HT})^{res}$ calculated in the context of the\nfrozen and running coupling constant approaches as a function of\nthe pion transverse momentum $p_{T}$ for different pion wave\nfunctions at $y=0$. It is seen from Fig.2 and Fig.3 that the\nhigher-twist cross section is monotonically decreasing with an\nincrease in the transverse momentum of the pion. In Fig.4-Fig.7,\nwe show the dependence of the ratios\n$(\\Sigma_{HT}^{hol})$\/$(\\Sigma_{HT}^p)$,\n$(\\Sigma_{\\pi}^{HT})^{res}$\/$(\\Sigma_{\\pi^{+}}^{HT})^{0}$,\n$(\\Sigma_{\\pi^{+}}^{HT})^{0}$\/$(\\Sigma_{\\pi^{+}}^{LT})$ and\n$(\\Sigma_{\\pi^{+}}^{HT})^{res}$\/$(\\Sigma_{\\pi^{+}}^{LT})$ as a\nfunction of the pion transverse momentum $p_{T}$ for\n$\\Phi_{\\pi}^{hol}(x)$, $\\Phi_{\\pi}^{p}(x)$ and\n$\\Phi_{VSBGL}^{hol}(x)$ pion wave functions. Here\n$\\Sigma_{\\pi^{+}}^{LT}$ is the leading-twist cross section,\nrespectively. As shown in Fig.4, in the region\n$2\\,\\,GeV\/c