552 research outputs found

    Quantity Constrained General Equilibrium

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    In a standard general equilibrium model it is assumed that there are no price restrictions and that prices adjust infinitely fast to their equilibrium values.In case of price restrictions a general equilibrium may not exist and rationing on net demands or supplies is needed to clear the markets.In the mid 1970s it was shown that in case of upper and lower bound restrictions on the prices there exists a quantity constrained equilibrium at which not both demand and supply of a good are rationed simultaneously and there is rationing on the net supply or net demand of a good only if the price of that good is on its lower or upper bound, respectively.For an arbitrary set of admissible prices it was recently proposed to let the rationing schemes be determined by the components of a vector being a direction in which the prices are restricted to move.When the set of restricted prices is convex and compact, it was shown that there exists a connected set of such quantity constrained equilibria, containing two trivial no-trade equilibria without trade opportunities.In this paper we refine the concept of quantity constrained equilibrium and propose a specific quantity constrained equilibrium which may serve as a general equilibrium in case of price restrictions.At this equilibrium demand rationing and supply rationing are in balance with each other, so that trade opportunities are maximal and therefore trivial no-trade and other equilibria with less trade opportunities are excluded.Moreover, in equilibrium only relative prices matter. Any homogenous transformation or normalization of the set of admissible prices yields the same set of quantity constrained general equilibria up to scaling of the price vectors.exchange economy;price restrictions;general equilibrium;rationing

    Tune optimization for maximum dynamic acceptance; 1, formulation

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    In order to combine the acceptance limitation due to a mechanical obstacle at radius rmech with that due to magnetic imperfections present in the lattice, a quantity eda to be called ``dynamic accepta nce'' is introduced. Using lowest order theory (with transfer matrices and no Hamiltionian) perturbed linear betatron motion is calculated and used to derive the dependence eda(rmech). Being in analyt ic form, this acceptance reduction provides a figure of merit that can be used to optimize the lattice tunes (thereby refining the prescription ``stay away from low order resonances''). Apart from its definition as an acceptance rather than an aperture, what distinguishes eda(rmech )from the commonly employed ``dynamic aperture'' is its dependence on rmech and the importance of this distinction fad es as rmech becomes large. In this Part~I the method is formulated and, to demonstrate the method, optimal fractional tunes are found with only random errors present-the loss of acceptance is dominate d by sextupole errors. But the intended application is for field errors that are systematic over sections of the lattice, but not necessarily over the whole lattice. Such field errors are unavoidable and are especially important in a high tune accelerator like the LHC

    The Microwave Undulator

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    Tune Optimization for Maximum Dynamic Acceptance 2: Qx = 65, Qy = 58

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    By minimizing the "figure of merit" FOM (defined in part I to be the fractional reduction in dynamic acceptance) in the presence of nonlinear elements in the LHC, optimal lattice parameters can be determined. Emphasis here is placed on determining the optimal integer tunes in the ranges 59 #log# DistList LAforwarding PSforwarding RNCAN3 batch done files files.tar helpND.html images images.tar insert_new.sh insert_new.sh~ log modifying pending running waiting Qx #log# DistList LAforwarding PSforwarding RNCAN3 batch done files files.tar helpND.html images images.tar insert_new.sh insert_new.sh~ log modifying pending running waiting 66, 56 #log# DistList LAforwarding PSforwarding RNCAN3 batch done files files.tar helpND.html images images.tar insert_new.sh insert_new.sh~ log modifying pending running waiting Qy #log# DistList LAforwarding PSforwarding RNCAN3 batch done files files.tar helpND.html images images.tar insert_new.sh insert_new.sh~ log modifying pending running waiting 63 in the presence of systematic errors, constant or time-dependent. The fractional tunes (always taken to be 0.28 and 0.31) have been intentionally chosen to avoid low order resonances. Other than chromaticity sextupoles (that keep the chromaticities near zero) the only nonlinear field errors treated are systematic sextupole, octupole, and decapole, all both erect and skew, and only in the main bending elements. Unlike the fractional tunes, for which random sextupole effects are dominant (see part I 1 ) systematic octupoles, either erect or skew, in conjunction with the chromaticity sextupoles, drive the choice of integer tunes. Making assumptions that appear to be reasonable concerning field errors to be expected, the optimal tunes have been found to be Qx = 65; Qy = 58. Since the optimum is partly based on compensation over single arcs the same choice should also be good for systematic-per-arc errors

    Summary of the Working Group on Errors

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    The Storage Ring as Radiation Source

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    Equilibrium ion distribution in the presence of clearing electrodes and its influence on electron dynamics

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    Here we compute the ion distribution produced by an electron beam when ion-clearing electrodes are installed. This ion density is established as an equilibrium between gas ionization and ion clearing. The transverse ion distributions are shown to strongly peak in the beam's center, producing very nonlinear forces on the electron beam. We will analyze perturbations to the beam properties by these nonlinear fields. To obtain reasonable simulation speeds, we develop fast algorithms that take advantage of adiabatic invariants and scaling properties of Maxwell's equations and the Lorentz force. Our results are very relevant for high current Energy Recovery Linacs, where ions are produced relatively quickly, and where clearing gaps in the electron beam cannot easily be used for ion elimination. The examples in this paper therefore use parameters of the Cornell Energy Recovery Linac project. For simplicity we only consider the case of a circular electron beam of changing diameter. However, we parameterize this model to approximate non-round beams well. We find suitable places for clearing electrodes and compute the equilibrium ion density and its effect on electron-emittance growth and halo development. We find that it is not sufficient to place clearing electrodes only at the minimum of the electron beam potential where ions are accumulated

    An Explicit Formula for Undulator Radiation

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    Theorem on Magnet Fringe Field

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    Transverse particle motion in particle accelerators is governed almost totally by non-solenoidal magnets for which the body magnetic field can be expressed as a series expansion of the normal (b{sub n}) and skew (a{sub n}) multipoles, B{sub y} + iB{sub x} = {summation}(b{sub n} + ia{sub n})(x + iy){sup n}, where x, y, and z denote horizontal, vertical, and longitudinal (along the magnet) coordinates. Since the magnet length L is necessarily finite, deflections are actually proportional to ``field integrals`` such as {bar B}L {equivalent_to} {integral} B(x,y,z)dz where the integration range starts well before the magnet and ends well after it. For {bar a}{sub n}, {bar b}{sub n}, {bar B}{sub x}, and {bar B}{sub y} defined this way, the same expansion Eq. 1 is valid and the ``standard`` approximation is to neglect any deflections not described by this expansion, in spite of the fact that Maxwell`s equations demand the presence of longitudinal field components at the magnet ends. The purpose of this note is to provide a semi-quantitative estimate of the importance of {vert_bar}{Delta}p{sub {proportional_to}}{vert_bar}, the transverse deflection produced by the ion-gitudinal component of the fringe field at one magnet end relative to {vert_bar}{Delta}p{sub 0}{vert_bar}, the total deflection produced by passage through the whole magnet. To emphasize the generality and simplicity of the result it is given in the form of a theorem. The essence of the proof is an evaluation of the contribution of the longitudinal field B{sub x} from the vicinity of one magnet end since, along a path parallel to the magnet axis such as path BC

    Equilibrium adjustment of disequilibrium prices

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    We consider an exchange economy in which price rigidities are present. In the short run the non-numeraire commodities have a exible price level with respect to the numeraire commodity but their relative prices are mutually fixed. In the long run prices are assumed to be completely exible. For a given price level and fixed relative prices, markets can be equilibrated by means of quantity rationing on demand and supply. Keeping markets in equilibrium through rationing, we provide an adjustment process in prices and quantities converging from a trivial equilibrium with complete demand rationing on all non-numeraire markets to a Walrasian equilibrium. Along the path initially all relative prices are kept fixed and the price level is increased. Rationing schemes are adjusted to keep markets in equilibrium. Doing so the process reaches a short run equilibrium with only demand rationing and no rationing on the numeraire and at least one of the other commodities. The process allows for a downward price adjustment of non-rationed non-numeraire commodities and reaches a Walrasian equilibrium in the long run.Equilibrium Theory;market economy;Prices;Disequilibrium Theory;Rationing;economic theory
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