article_id
stringlengths 9
15
| section_id
stringclasses 5
values | document
stringlengths 503
67.8k
| summary
stringlengths 205
16.2k
| abstract
stringlengths 293
26.7k
|
|---|---|---|---|---|
math0503267
|
i
|
in recent years , the problem of finding index formulas for elliptic pseudodifferential operators on singular manifolds has been studied in numerous papers ( e.g. , see plamenevsky - rozenblum @xcite , melrose - nistor @xcite , rozenblum @xcite , schulze - sternin - shatalov @xcite , nazaikinskii - sternin @xcite ) . despite considerable progress in this direction , the situation is far from being clear yet . indeed , some of the formulas known in the literature fail to express the index via the principal symbol treated as an element of an appropriate calkin algebra ( e.g. , @xcite ) , in other formulas , separate terms are not homotopy invariant ( e.g. , see fedosov - schulze - tarkhanov @xcite ) , and the few formulas that combine both desirable properties ( e.g. , see @xcite ) are valid only for an important but rather narrow class of operators satisfying certain symmetry conditions . this situation is primarily caused by the complicated symbol structure for pseudodifferential operators on singular manifolds . these symbols consist of components ( in general , operator - valued ) corresponding to the strata of the manifold and satisfying certain matching conditions for the adjacent strata see , e.g. schulze @xcite . in a number of existing results , the index of the operator is expressed as a sum of contributions from these symbol components . these contributions are usually noninteger and lack homotopy invariance , which is not at all surprising in the presence of the matching conditions . it is also difficult to assign to them any straightforward topological or algebraic meaning . one can ensure their homotopy invariance only by severely restricting the class of operators to be considered ( say , by imposing some symmetry conditions ) . a detailed analysis of symmetry conditions and their role in obtaining invariant index formulas can be found in savin - schulze - sternin @xcite and in @xcite , and we do not dwell on the topic here . in the present paper , we propose another approach to the construction of index formulas on manifolds with singularities . this approach is based on @xmath0-theory of algebras and cyclic cohomology . we take a slightly different viewpoint as to what a `` good '' index formula must be . instead of trying to use topological invariants of separate components of the symbol ( which is hopeless due to results of @xcite ) , we consider the symbol as a whole , that is , as an element of an appropriate symbol algebra @xmath2 . moreover , instead of topological objects , one naturally deals with algebraic objects like the chern connes character viewed as an element of the cyclic cohomology group of the symbol algebra . in the abstract framework , the scheme is well known . if the algebra @xmath2 is separable , then every quantization of @xmath2 ( in particular , the pseudodifferential quantization , which is of interest to us ) is equivalent ( e.g. , see blackadar @xcite ) . ] to some ( generalized ) toeplitz quantization @xmath3 in the hilbert space @xmath4 defined as the range of an orthogonal projection @xmath5 in some hilbert @xmath2-module @xmath6 ; here @xmath5 is assumed to almost commute with the action of @xmath2 . under the additional condition that the commutators of elements of the algebra with the projection are not only compact but also belong to an appropriate von neumann schatten class , the toeplitz representation permits one to write out analytic index formulas ( see connes @xcite ) . namely , if @xmath7 is a dense local subalgebra of @xmath2 such that the restriction of the toeplitz quantization to @xmath8 is @xmath9-summable,$ ] belongs to the @xmath9th von neumann schatten class @xmath10 for every @xmath11 . ] then for an arbitrary elliptic element @xmath12 one has the index formula @xmath13\otimes\operatorname{tr}\bigr ) \bigl(\underbrace{\mathfrak{a}^{-1},\mathfrak{a},\mathfrak{a}^{-1 } , \mathfrak{a } , \ldots,\mathfrak{a}^{-1 } , \mathfrak{a}}_{\text{$n+1 $ arguments}}\bigr),\ ] ] where @xmath14 is odd , @xmath15 is the matrix trace , and the _ chern connes character _ @xmath16 $ ] of the quantization @xmath17 is given by the formula @xmath18(a_0,a_1,\ldots , a_n)\\=\sqrt{2i } ( -1)^{n(n-1)/2}\gamma{\biggl}(\frac n2 + 1{\biggr})^{-1}\operatorname{tr}\{a_0[p , a_1][p , a_2]\dotsm [ p , a_n]\}.\end{gathered}\ ] ] ( here @xmath19 is the operator trace in @xmath6 . ) to apply this formalism to manifolds with singularities , we do the following . for the case of a smooth compact closed manifold @xmath20 , there is a well - known toeplitz quantization @xmath21 of the symbol algebra @xmath22 with the help of the szeg - caldern projection @xmath23 in the space @xmath24 of square integrable functions on the cosphere bundle @xmath25 , see guillemin , @xcite , boutet de monvel @xcite , boutet de monvel - guillemin @xcite . this construction was explained in @xcite as showing that pseudodifferential operators are toeplitz operators in disguise . in the present paper , we generalize this quantization and construct a toeplitz quantization equivalent to the standard pseudodifferential quantization for manifolds with conical singularities . thus pseudodifferential operators on a manifold with conical singularities are toeplitz operators in disguise , too . ( this toeplitz quantization acts in a more complicated space , which , however , turns into @xmath24 if there are no singularities . ) in the smooth case , the equivalence ( and even the almost isomorphism ) of the toeplitz quantization to the pseudodifferential quantization as a homogeneous function of degree zero in the fibers is assumed . ] @xmath26 is given by the so - called guillemin transform @xmath27 ( see the cited papers ) , and we construct an analog of this transform for manifolds with singularities , thus proving the equivalence ( and even an almost isomorphism ) of quantizations in this case . this naturally results in an index formula of the form for elliptic operators on manifolds with conical singularities . this algebraic index formula has several advantages : * first , it expresses the index via the principal symbol alone and is homotopy invariant ; * second , it is valid for arbitrary elliptic symbols , not just for symbols satisfying some symmetry conditions ; * third , most importantly , it is expressed in terms of the cyclic cohomology class @xmath16 $ ] of the algebra @xmath8 . it is the cyclic cohomology of the symbol algebra that replaces the homology of a manifold as one passes from the algebra of functions on a smooth manifold to more general symbol algebras . a serious disadvantage of this formula ( and in general of the index formula for toeplitz quantizations given by projections of general form ) is that it fails to be local . at the same time , if the chosen projection is the positive spectral projection of an unbounded local operator , then a different representation of the index cocycle is possible , which results in more traditional , local index formulas . ( see connes - moscovici @xcite . ) in this connection , the construction of a local index formula involves the natural problem of finding an unbounded local operator such that the toeplitz quantization generated by its positive spectral projection is equivalent to the pseudodifferential quantization on a manifold with conical singularities . in the smooth case , the desired operator is the dirac operator . more precisely , the natural self - adjoint dirac operator @xmath28 acting on sections of the spinor bundle on the odd - dimensional contact manifold @xmath25 is associated with the almost complex structure on the distribution of contact hyperplanes . then the index of a pseudodifferential operator @xmath29 proves to be equal to the index of the toeplitz operator constructed from the symbol @xmath30 and the positive spectral projection of the dirac operator ( see baum - douglas @xcite ) : @xmath31 where @xmath32 is the natural projection . this equality of indices shows that ( at least , modulo torsion elements in the group @xmath33 ) the quantization given by the positive spectral projection @xmath34 of the dirac operator is equivalent to the pseudodifferential quantization and hence to the quantization with the help of the szeg caldern projection . there are @xmath0-theoretic proofs of this fact . ( one of then is based on the atiyah singer theorem , e.g. , see kaminker @xcite , and another , by baum - douglas - taylor @xcite , uses some facts concerning the @xmath35-neumann problem . ) both these approaches to the proof of encounter serious difficulties in the case of manifolds with singularities . the main difficulty is that so far one does not have even a hypothetical candidate for the `` dirac operator '' ( i.e. , an operator for which holds ) for the case of manifolds with conical singularities . a prototype for a possible generalization of formula to the conical case is the proof of ( as a special case of more general index formulas for toeplitz operators and quantized contact transformations on general contact manifolds ) due to epstein , melrose , and mendoza ( see @xcite and also @xcite ) . their proof _ relates the szeg caldern projection and the positive spectral projection of the dirac operator by a finite chain of transformations _ and hence is of interest to us as a model for the possible definition of the dirac operator in the conical case . indeed , we have already constructed a counterpart of the szeg caldern projection for manifolds with conical singularities and proved the equivalence of the toeplitz quantization and the pseudodifferential quantization . the epstein melrose mendoza construction comprises three steps ( we give more detail in the appendix ) : 1 . constructing a resolution of the szeg caldern projection ( in the case of the cosphere bundle of a smooth manifold , the resolution is given by the kohn rossi complex , e.g. , see folland - kohn @xcite ) ; 2 . proving that the toeplitz quantization associated with the positive spectral projection of the operator @xmath36 obtained as the standard roll - up of this resolution produces the operator with the same index as the toeplitz quantization associated with the szeg caldern projection ; 3 . deforming the self - adjoint operator @xmath36 to the dirac operator @xmath28 . in the present paper , we show that step 2 of this construction is a special case of a general assertion that permits one , starting from a toeplitz quantization of an arbitrary @xmath1-algebra @xmath2 , to obtain new equivalent ( i.e. , defining the same class in @xmath33 ) toeplitz quantizations associated with positive spectral projections of self - adjoint operators . the construction goes as follows : for a projection @xmath5 almost commuting with the action of @xmath2 , one takes a finite - length resolution in hilbert @xmath2-modules . rolling up the resolution in the standard way and adding the projection itself and the projection onto the cokernel in the last term of the resolution , one obtains a self - adjoint operator in the direct sum of all hilbert modules occurring in the resolution . the positive spectral projection of this self - adjoint operator is the desired new projection . in conclusion , we note that the so far open problem of defining the dirac operator @xmath28 in the conical case is undoubtedly of interest in that such an operator is a `` fundamental cycle '' of the algebra @xmath2 ( the noncommutative manifold ) and , in particular , defines the poincar duality on it : @xmath37 in the smooth case ( @xmath38 ) , this is reduced to the isomorphism @xmath39 the paper is organized as follows . in the first section , we construct a toeplitz representation of the pseudodifferential quantization on manifolds with conical singularities . section 2 contains the above - mentioned construction of equivalent quantizations from resolutions of projections . first we consider the case in which the resolution is formed by bounded operators and then the case in which the resolution is formed by unbounded operators . in the appendix , we briefly describe the construction of @xcite for the special case in which the contact manifold in question is the cosphere bundle of a smooth compact manifold without boundary . throughout the paper , we assume that the reader is acquainted with the notions and definitions of @xmath0-theory of operator algebras and related topics ( e.g. , see @xcite ) . on the other hand , the necessary definitions and facts of the theory of ( pseudo)differential operators on manifolds with conical singularities are given in subsection 1.1 together with appropriate bibliographic references .
|
a new approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of-theory of algebras and cyclic cohomology . the equivalence of toeplitz and pseudodifferential quantizations , well known in the case of smooth closed manifolds , is extended to the case of manifolds with conical singularities . we describe a general construction that permits one , for a given toeplitz quantization of a-algebra , to obtain a new equivalent toeplitz quantization provided that a resolution of the projection determining the original quantization is given .
|
a new approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of-theory of algebras and cyclic cohomology . the equivalence of toeplitz and pseudodifferential quantizations , well known in the case of smooth closed manifolds , is extended to the case of manifolds with conical singularities . we describe a general construction that permits one , for a given toeplitz quantization of a-algebra , to obtain a new equivalent toeplitz quantization provided that a resolution of the projection determining the original quantization is given .
|
1008.1898
|
i
|
the concept of a soliton as a localized particle - like excitation that preserves its shape can be extended to systems that are far from thermodynamic equilibrium through the concept of a `` dissipative soliton '' @xcite . this allows us to analyze a broad range of physical , chemical , and biological nonlinear systems in which localized excitations are observed . driven magnetic systems , especially those of technological interest , exhibit strongly nonlinear dynamics and are an ideal experimental domain for exploring the dissipative soliton model . in this paper , we provide an analytical theory for a novel , localized oscillation mode in a spin torque oscillator with a free layer having perpendicular magnetic anisotropy . the salient features of this mode include a frequency well below that of uniform ferromagnetic resonance , a weak dependence of frequency on bias current , and a precession angle at the maximal value of @xmath0 . combining numerical micromagnetic simulations with an asymptotic analysis of the equations of motion , we identify this mode as a dynamic , dissipative magnetic soliton that is closely related to the `` magnon droplet '' predicted by ivanov and kosevich in 1977 @xcite . the mode central region exhibits magnetization pointing nearly opposite to its equilibrium direction with a perimeter manifesting @xmath0 precession . from our asymptotic analysis , we derive conditions on perpendicular anisotropy and bias current for the nucleation and existence of the dissipative droplet . using our numerical simulations , we analyze the stability of the dissipative droplet soliton as a function of applied magnetic field , bias current , and spin torque asymmetry . solitons in conservative systems occur when nonlinear terms in the equation of motion balance the effects of dispersion @xcite . a classic example is a light pulse moving in a lossless optical fiber : the change of refractive index with frequency ( dispersion ) tends to make the pulse spread out , but for a certain pulse shape , the change of refractive index with light intensity due to the optical kerr effect ( nonlinearity ) exactly balances the dispersion . pulses having this shape can propagate without spreading and are called solitary waves or solitons . the balance between nonlinearity and dispersion typically allows for the existence of a continuous family of solitons that can be excited in the system , rather than a single solution . in the optical fiber example , the family can be parametrized , for example , by pulse amplitude , and there is a continuous range of amplitudes that satisfy the soliton balancing condition . dissipative solitons @xcite are characterized by an additional balancing condition between gain and loss that typically allows only a single solution for a given set of external parameters . although conservative soliton models can explain weakly nonlinear behavior seen in magnetic systems of exceptionally low damping @xcite , damping is not a small effect for many magnetic systems of both fundamental and technological interest . by combining classical soliton theory with bifurcation theory of nonlinear dynamics and concepts of self - organization @xcite , the dissipative soliton concept provides a framework for describing a broad range of soliton - like behaviors . here , we apply this concept to a nanoscale ferromagnetic system in which both damping and a driving force ( spin torque ) are important . spin torque @xcite occurs when a current is driven through a structure with alternating magnetic and nonmagnetic layers in which spin - dependent conductance at the interfaces results in a spin - polarized electron flow . when the polarized electrons enter a ferromagnetic layer whose magnetization @xmath1 is not collinear with the electron spins , the transmitted spins are rotated toward @xmath1 and the angular momentum absorbed by the ferromagnetic layer is known as the spin torque . typical devices have two ferromagnetic layers through which current is driven : a thick `` fixed '' layer that determines the direction of electron polarization , and a thin `` free '' layer whose orientation can be readily changed by spin torque . for current of the appropriate polarity , spin torque opposes the intrinsic damping torque in the system , and currents above a threshold produce dynamic states in which @xmath1 of the free layer can be manipulated without applying a magnetic field . this effect has been used to control switching of nanoscale magnetic elements @xcite , with potential applications in computer memory and data storage . the effect has also been used to produce coherent , frequency - tunable microwave oscillations @xcite in a nanoscale device known as a spin torque oscillator ( sto ) , with potential applications in integrated microwave circuits for mixing and active phase control . recent reviews cover the physics of spin torque @xcite and its possible applications @xcite . the equations of motion for @xmath1 in the presence of spin torque ( presented below ) are inherently nonlinear , and their full solution for a general case is often studied by use of numerical methods . analytical methods can sometimes be applied by invoking restrictions such as high symmetry , spatially uniform @xmath1 ( the `` macrospin '' model ) , and small precession amplitude ( small angle between @xmath1 and its equilibrium direction ) . these restricted cases have been used to explain experimental results with mixed success . the local nature of spin torque allows it to drive large amplitude excitations in which @xmath1 varies on the scale of the magnetic exchange length ( typically a few nanometers ) , something that applied magnetic fields can not do . as we show here , this regime of strongly nonlinear , strongly nonuniform , sustained magnetodynamics is amenable to theoretical examination using numerical and analytical approaches . this regime is also experimentally accessible in stos . we note that a different type of magnetic soliton generated by spin torque , called a spin wave `` bullet '' , was predicted by slavin and tiberkevich in 2005 to occur in the point - contact geometry with magnetic films exhibiting in - plane oriented anisotropy and in - plane applied magnetic field @xcite . for this case , the precession frequency decreases with increasing current , which can result in localization if the frequency falls below the bottom of the spin wave band at the ferromagnetic resonance ( fmr ) frequency . the weakly nonlinear bullet soliton is a solution to a nonlinear schrdinger type equation with third - order nonlinearity in the excitation amplitude . as such , its predicted experimental signature consists of subtle shifts in microwave output frequency and threshold current relative to that expected for a non - localized mode . in contrast , the droplet soliton studied here exhibits dramatic differences in behavior from that of a non - localized mode . this is due to the fact that it is a strongly nonlinear solution of the full equations of motion , rather than simply a third order expansion . domain walls @xcite , magnetic bubbles @xcite , and vortices @xcite are examples of well studied , strongly nonlinear , localized structures that occur in magnetic materials . the droplet differs from these static structures in that it is inherently dynamic ; the frequency of spin precession within the droplet is always greater than zero . in this work , we focus on the two - dimensional ( 2d ) , non - topological droplet , but we note that droplets in two and three - dimensions come in topological flavors as well @xcite . we begin in the next section by presenting an asymptotic analysis of the model equations for the dissipative droplet in a high - symmetry case . we will also derive the droplet s frequency _ vs. _ current relation in this section . section [ sec : dropl - phys - pert ] is devoted to the study of droplets in physically realistic situations incorporating the current - induced oersted field as well as canting of the applied field and fixed layer . section [ sec : excitation - droplet ] details experimentally accessible nucleation conditions for a droplet that take advantage of a small amplitude instability . in sec . [ sec : discussion ] , we discuss possible extensions of the theory and we relate the droplet to other excitations in thin magnetic films . we conclude in sec . [ sec : conclusion ] with a summary of the droplet s unique properties . appendices [ sec : modul - inst ] and [ sec : numerical - method ] provide details of our stability calculation and numerical method , respectively .
|
a novel type of solitary wave is predicted to form in spin torque oscillators when the free layer has a sufficiently large perpendicular anisotropy . in this structure , which is a dissipative version of the conservative droplet soliton originally studied in 1977 by ivanov and kosevich asymptotic methods are used to derive conditions on perpendicular anisotropy strength and applied current under which a dissipative droplet can be nucleated and sustained . numerical methods are used to confirm the stability of the droplet against various perturbations that are likely in experiments , including tilting of the applied field , non - zero spin torque asymmetry , and non - trivial oersted fields . under certain conditions ,
|
a novel type of solitary wave is predicted to form in spin torque oscillators when the free layer has a sufficiently large perpendicular anisotropy . in this structure , which is a dissipative version of the conservative droplet soliton originally studied in 1977 by ivanov and kosevich , spin torque counteracts the damping that would otherwise destroy the mode . asymptotic methods are used to derive conditions on perpendicular anisotropy strength and applied current under which a dissipative droplet can be nucleated and sustained . numerical methods are used to confirm the stability of the droplet against various perturbations that are likely in experiments , including tilting of the applied field , non - zero spin torque asymmetry , and non - trivial oersted fields . under certain conditions , the droplet experiences a drift instability in which it propagates away from the nanocontact and is then destroyed by damping .
|
1004.3120
|
i
|
muscle is a bundle of cylindrical muscle fibres . one muscle fibre enclose hundreds of thinner cylindrical fibre called fibril . a fibril is a train of sarcommeres connected head to tail . the sarcomere , as seen by light microscopy , is a composite structure of actin filament and myosin filament(fig . [ muscle ] ) . the two filaments are bridged by long myosin protein called myosin molecule motor(see ref . @xcite@xcite or biology textbook for a more detail review ) . the longitudinal section of filament array in muscle fibre . myosin molecules are long polymers as cross - bridges between the actin filament and myosin filaments.,scaledwidth=48.0% ] the length of a sarcomere is observed to decrease during muscle contraction . huxley modeled the muscle contraction as mutual movement between actin filament and myosin filament@xcite . when the myosin molecule in the myosin filament is excited up to attach on the actin filament , myosin molecule would combine with adenosine triphosphate molecules , and convert energy from adenosine triphosphate hydrolysis into mechanical force , this is the attached state . in the detached state , myosin molecule is at rest and doing nothing . the arrangement of myosin molecules along the length of filament is incommensurate . a stochastic model was introduced to describe the cooperative behavior of molecules motors which has disordered arrangement along a backbone@xcite . experiment research is beyond the description of classical mechanics two - state cross - bridge model@xcite which assumed the probability of the two states detached state and attached state satisfy a pair of coupled differential equations . myosin molecules looks like a long arm ended by a head domain . the myosin in muscle cell is usually called myosin - ii , for it has two heavy chains in the head domain . a second kind of myosin molecule with single heavy chain was found in non - muscle cells , they are termed as myosin - i . the average length of myosin - ii is about 160 nm@xcite . visible light corresponds to a wavelength range of 400 - 700 nm . the electromagnetic wavelength comparable to the length of myosin - ii falls in ultraviolet region . a muscle fibre surrounded by membrane is under control of nerve cell . membrane is a filter system which is permeable for certain ions but is impermeable for other ions . the imbalance distribution of ions across the membrane results in an electric potential difference of -60 to -90 mv . the electric signals from nerve cell can modify the permeability of membrane . an active muscle would generate electric signal . scientist use this electric signal to represent the tension inside the muscle . if we insert a needle with two fine - wire electrodes into the muscle , the electric activity of muscle can be detected and recorded by an oscilloscope . this electromyogram is in widespread use for medical examination of muscle . electromagnetic wave is termed as photon in quantum mechanics . the electric signal generated by active muscle is physically equivalent to a wave packet of photons . quantum scattering between photons and molecules within biological system is a common phenomena , so does in muscle . molecule only absorb photons at resonance frequency . we assume there exist a one - to - one correspondence between conformational change of motor molecule and the hopping between quantum states of molecule . we define the excited quantum level as the attached states in which the motor molecule is extended to a longe shape . the detached state is assumed to be equivalent to a quantum states with lower energy . the motor molecule becomes shorter in detached states . besides the two quantum states of molecules , another quantum state of photon is introduced to induce the hopping between the two quantum level . photon can propagate among different molecules . it is photon that couples different motor molecules to work together . quantum physics has many techniques to control quantum states , thus it is possible to control the conformation of molecule motors by electromagnetic wave . muscle fibre can also be stimulated by chemical solutions which involves many complicate biochemistry@xcite . there was an argument on chemical reactions in muscle using quantum mechanics@xcite , it is not related to my mathematical physics model . the time scale in my modeling is much larger than the chemical reaction . we only focus on the electrical stimulation of muscle fibre without adding any external chemical solutions . to make this mathematical physics model work for a real muscle fibre system , the following assumptions are necessary : ( 1 ) the strength of electric field should be so strong that it dominates the conformation of molecules over thermodynamic fluctuations . ( 2 ) for the muscle fibre without any external input of stimulus chemical ions , the conformational change of motor molecules governed by electromagnetic field is much larger than that induced by chemistry . this is because electric field is distributed across the whole space , it exerts a global force on the motor molecules . while chemical ion only interact with molecule locally , it does not come into action unless it finds the correct binding site . ( 3 ) the spatial conformation of motor molecule can be modified by concentration of some chemical ions . we simplified the muscle fibre as a one dimensional chain of many giant quantum particles which represents molecule motors . the velocity of sliding actin filament acts as environmental parameter . these giant quantum particles are excited by absorbing photons , and relaxed by emitting photons . we arrange these particles regularly along a one dimensional chain . the wavelength of the stimulus electromagnetic pulse is assumed to be much larger than a sarcomere . the configuration of spatial arrangement within a sarcomere is supposed to has no influence on the output physics . the article is organized as follows : in section ii , we present the quantum hamiltonian for deriving analogous force - velocity relation with respect to the coresponding macroscopic quantity . the empirical hill s relation is consistent with the steady solution of heisenberg equations . we predict the force velocity relation for the slow release case and the non - steady state solution . in section iii , we study how the force decays after the stimulus of electric pulse . exact solution of quantum two level model reproduced similar tension activation curves of cardiac muscle . the analytical solution of quantum three - level model can generate most similar curves of tension transients of insect flight muscle . in section iv , the tension transients curve of skeletal muscle is mathematically reproduced from the point view of quantum coherent state . the experiment curve coincides with the third order projection of coherent state . in section v , we derived the self - coupled quantum hamiltonian for solving strongly coupled differential equations . for this case , the sliding velocity depends on the internal states of the motor molecule . the last section is a brief summary and outlook .
|
we proposed quantum many - particle hamiltonian to predict the force - velocity relation for the slow release of muscle fibre which has no empirical relation yet , it is much more complicate than hyperbolic relation . using the same hamiltonian , we predicted the mathematical force - velocity relation when the muscle is stimulated by alternative electric current . mathematically modeling electric stimulus as photons exciting a quantum three - level particle reproduced most tension transient curves of water bug .
|
a quantum chain model of many molecule motors is proposed as a mathematical physics theory on the microscopic modeling of classical force - velocity relation and tension transients of muscle fibre . we proposed quantum many - particle hamiltonian to predict the force - velocity relation for the slow release of muscle fibre which has no empirical relation yet , it is much more complicate than hyperbolic relation . using the same hamiltonian , we predicted the mathematical force - velocity relation when the muscle is stimulated by alternative electric current . the discrepancy between input electric frequency and the muscle oscillation frequency has a physical understanding by doppler effect in this quantum chain model . further more , we apply quantum physics phenomena to explore the tension time course of cardiac muscle and insect flight muscle . most of the experimental tension transients curves found their correspondence in the theoretical output of quantum two - level and three - level model . mathematically modeling electric stimulus as photons exciting a quantum three - level particle reproduced most tension transient curves of water bug .
|
q-bio0511004
|
i
|
the choice of the coordinates used to describe a molecule is an important issue if computational considerations are to be taken into account and the efficiency of the simulations is pursued . this choice also affects the coding of applications . if cumbersomely defined coordinates are used , an unnecessary complexity may be added to the design of monte carlo movements , the construction and pruning of a database of structures @xcite or the programming of molecular visualization and manipulation tools . suitable coordinates frequently used to describe arbitrary conformations of molecules are the so - called `` internal '' or `` valence - type '' coordinates @xcite . their adequacy stems from a number of characteristics : first , they are closely related to chemically meaningful structural parameters , such as bond lengths or bond angles ; second , they are local , in the sense that each one of them involves only a small number of ( normally close ) atoms in its definition ; and finally , there are only @xmath2 of them ( where @xmath3 is the number of atoms in the molecule ) , in such a way that the overall rotation and translation have been naturally removed . there also exists a family of coordinates @xcite , extensively used in the inner calculations of many quantum chemistry packages ( such as gaussian @xcite or gamess @xcite ) and based on the `` natural internal coordinates '' originally proposed by pulay and coworkers @xcite , which are defined through linear combinations of the original internals . these coordinates are specially designed to describe normal - mode vibrations in the immediate neighbourhood of energy minima and represent the best choice for accelerating convergence of geometry optimizations in a particular basin of attraction , via diagonal estimation of the hessian matrix @xcite . accordingly , they maximally separate hard and soft movements in these conditions . however , if the conformation of the molecule is far from a minimum , this type of coordinates lose great part of their meaning and they introduce many computational difficulties without increasing the efficiency . also , some of the definitions are _ redundant _ @xcite , i.e. , they use a number of coordinates larger than the number of degrees of freedom . in this work , we will only discuss coordinates , such as internals or cartesian , that may be conveniently used to specify an _ arbitrary _ conformation of the system and that can be directly related to simple geometrical variables . in macromolecules , such as proteins , the number of degrees of freedom is the main limiting factor when one tries to predict their behaviour via computer modeling . therefore , it is also advisable that the set of coordinates chosen allows for a direct implementation of physically meaningful constraints that reduce the dimensionality of the conformational space considered . most of the expressions used in statistical mechanics or in molecular dynamics are best written in cartesian coordinates , however , the implementation of constraints naturally appearing is far from being straightforward in these coordinates . in internal coordinates , on the contrary , the approximate separation of hard and soft movements of the system allows to easily constrain the molecule @xcite by setting the hard coordinates ( those that require a considerable amount of energy to change noticeably ) to constant values or to particular functions of the soft coordinates . moreover , in internal coordinates ( and appealing to some reasonable approximations ) , the statistical mechanics formulae for the constrained system may be written in convenient closed form @xcite . still , although the bond lengths and bond angles are customarily regarded as hard and their definition is unproblematic , the same is not true for dihedral angles . some definitions of dihedrals may lead to difficulties or to worse separation of hard and soft modes . let us exemplify this with a particular case : consider the definition of z - matrix - like @xcite internal coordinates for the hco - l - ala - nh@xmath1 molecule in fig . [ fig : num_ala ] . imagine that we `` position '' ( i.e. , we write the corresponding z - matrix row ) every atom up to the hydrogen denoted by h@xmath4 and that we are now prepared to position the hydrogens in the side chain ( h@xmath5 , h@xmath6 and h@xmath7 ) via one bond length , one bond angle and one dihedral for each one of themwe will denote by @xmath8 the bond length between atoms @xmath9 and @xmath10 ; by @xmath11 , the bond angle between the vectors @xmath12 and @xmath13 ; and by @xmath14 the dihedral angle between the plane defined by the atoms @xmath9 , @xmath10 and @xmath15 and the one defined by @xmath10 , @xmath15 and @xmath16 . ] . a choice frequently seen in the literature @xcite is the one shown in table [ tab : bad_coord ] . cccc atom name & bond length & bond angle & dihedral angle + + h@xmath5 & ( 10,8 ) & ( 10,8,5 ) & @xmath17(10,8,5,3 ) + h@xmath6 & ( 11,8 ) & ( 11,8,5 ) & @xmath18(11,8,5,3 ) + h@xmath7 & ( 12,8 ) & ( 12,8,5 ) & @xmath19(12,8,5,3 ) if we now perform the _ gedanken experiment _ that consists of taking a typical conformation of the molecule and slightly moving each internal coordinate at a time while keeping the rest constant , we find that any one of the three dihedrals in the previous definition is a hard coordinate , since moving one of them while keeping the other two constant distorts the internal structure of the methyl group . hence , in these coordinates , the soft rotameric degree of freedom @xmath0 , which we know , for chemical arguments , that must existaccording to our calculations , at the rhf/6 - 31+g(d ) level of the theory , the barrier for crossing from one of the three equivalent minima to any of the other two ranges from 3.1 to 6.8 kcal / mol , depending on the values of the ramachandran angles @xmath20 and @xmath21 . compare with the barriers in @xmath20 or @xmath21 which may be as large as 20 kcal / mol depending on the region of the ramachandran map explored . ] , is ill - represented . in fact , it must be described as a _ concerted _ movement of the three dihedrals . in references @xcite , this fact is recognized and the concept of `` related dihedrals '' is introduced , however , no action is taken to change the definition of the coordinates . in this work , using the ideas of r. abagyan and coworkers @xcite , we define a set of rules to uniquely and systematically number the groups , the atoms and define the internal coordinates of organic molecules and , particularly , of polypeptidesiupac conventions only define a numeration system for the groups , for the branches and for some selected dihedral angles . they focus on functional considerations and not in computational problems . for related documents and references , see http://www.chem.qmul.ac.uk/ iupac / jcbn/. ] . the main difference with other z - matrix - like coordinates normally used in the literature @xcite is that , instead of positioning each atom with a bond length , a bond angle and a dihedral angle , we use normal dihedral angles ( called , from now on , `` principal dihedrals '' ) only to fix the orientation of whole groups and a somewhat non - standard type of dihedrals , termed `` phase dihedrals '' by r. abagyan and coworkers @xcite ( see fig . [ fig : dihedrals ] ) , to describe the covalent structure inside a groupanother option may be to use , as a third internal coordinate for each atom , another bond angle . this is rather awkward , however , since two bond angles and a bond length do not specify the position of a point in space . any values of these three coordinates ( except for irrelevant degenerate cases ) are compatible with two different symmetrical positions and a fourth number must be provided to break the ambiguity . ] . this allows to approximately separate soft and hard movements of the molecule using only topological information ( i.e. , not knowing the exact form of the potential ) and to easily implement constraints by forcing the coordinates that correspond to hard movements to take constant values or ones that depend on the soft coordinatesin reference @xcite they correctly take this approach into account using out - of - plane angles instead of phase dihedrals , however , they do not describe any rules for a general definition and their numeration of the atoms is non - modular , as it proceeds first through the backbone ( see sec . [ sec : definition_pro ] ) . ] . in addition , the coordinates herein defined , are straightforwardly cast into z - matrix form and may be directly implemented in any quantum chemistry package , such as gaussian @xcite or gamess @xcite . this is due to the fact that , although they involve atoms whose covalent structure is different , the mathematical construction of the two types of angles in fig . [ fig : dihedrals ] is exactly the same , and the phase dihedrals are treated like principal ones without any problem by the applications . a number of perl scripts are provided that number the atoms and generate the coordinates herein defined for polypeptide chains . the applications read a sequence file in which the different ionization states of the titratable side chains , the tautomeric forms of histidine and several terminal groups may be specified . then , an output file is generated with the symbolic definition of the z - matrix of the molecule which may be directly pasted into the input files of gaussian @xcite or gamess @xcite ( and , upon slight modifications , of any quantum chemistry package that is capable of reading z - matrix format ) . these scripts may be found at http://neptuno.unizar.es / files / public / gen_sasmic/. now , if we redo the example in table [ tab : bad_coord ] using phase dihedrals , we must write the rows of the z - matrix for the hydrogens in the side chain as shown in table [ tab : good_coord ] . cccc atom name & bond length & bond angle & dihedral angle + + h@xmath5 & ( 10,8 ) & ( 10,8,5 ) & @xmath22*(10,8,5,3 ) * + h@xmath6 & ( 11,8 ) & ( 11,8,5 ) & @xmath23(11,8,5 , ) + h@xmath7 & ( 12,8 ) & ( 12,8,5 ) & @xmath24(12,8,5 , ) where the angle * ( 10,8,5,3 ) * is now the principal dihedral @xmath0 describing the relative rotation of the methyl group around the bond length ( 8,5 ) and the other two are phase dihedrals that describe the internal structure of the group and that are _ pure _ hard coordinates ( as far as can be told only from topological information ) . however , one must point out that , although all bond lengths , bond angles and phase dihedrals may be regarded as hard coordinates , not all the principal dihedrals will be soft . examples of hard principal dihedrals are the ones that describe the rotation around a double bond ( or a triple one ) or some of the principal dihedrals in cyclic parts of molecules . the _ physical approach _ described in this section , which should be taken into account when designing internal coordinates , is embodied in a set of rules for any organic molecule in sec . [ sec : definition_gen ] , with a slightly different prescription for polypeptide chains in sec . [ sec : definition_pro ] . the systematic numeration introduced facilitates the computational treatment of this type of systems and the rules given for polypeptide chains ensure modularity @xcite , i.e. , allows to add any residue with minimal modification of the already existing notation and to easily construct databases of structures or of potential energy surfaces ( pes ) . the characteristics aforementioned have led us to term the coordinates herein defined _ systematic , approximately separable and modular internal coordinates _ ( sasmic ) . in this work , we will only deal with the numeration of one isolated molecule , however , the procedure described may be easily generalized ( and will be in future works ) to systems of many molecules ( an important example being a macromolecular solute in a bath of solvent molecules ) . this could be done using _ ghost atoms _ in a similar manner to what is done in ref . @xcite , to position the center of mass of the system , and in refs . @xcite , to actually define the coordinates of a system of molecules . finally , in sec . [ sec : application ] , we use the new coordinates and ab initio quantum mechanical calculations in order to evaluate the approximation of the free energy , obtained from `` integrating out '' the rotameric degree of freedom @xmath0 , via the typical pes in the protected dipeptide hco - l - ala - nh@xmath1 . this will be relevant to design effective polypeptide potentials . we also present a small part of the hessian matrix in two different sets of coordinates to illustrate the approximate separation of soft and hard movements when the sasmic defined in this work are used . [ sec : conclusions ] is devoted to the conclusions .
|
a set of rules is defined to systematically number the groups and the atoms of organic molecules and , particularly , of polypeptides in a modular manner . supported by this numeration , a set of internal coordinates is defined . these coordinates ( termed systematic , approximately separable and modular internal coordinates , sasmic ) are straightforwardly written in z - matrix form and may be directly implemented in typical quantum chemistry packages . coordinates normally used in the literature is that normal dihedral angles ( `` principal dihedrals '' in this work ) are only used to fix the orientation of whole groups and a somewhat non - standard type of dihedrals , termed `` phase dihedrals '' , are used to describe the covalent structure inside the groups . this _ physical approach _ allows to approximately separate soft and hard movements of the molecule using only topological information and to directly implement constraints . as an application , we use the coordinates defined and ab initio quantum mechanical calculations to assess the commonly assumed approximation of the free energy , obtained from `` integrating out '' the side chain degree of freedom , by the potential energy surface ( pes ) in the protected dipeptide hco - l - ala - nh . we also present a sub - box of the hessian matrix in two different sets of coordinates to illustrate the approximate separation of soft and hard movements when the coordinates defined in this work are used . +
|
a set of rules is defined to systematically number the groups and the atoms of organic molecules and , particularly , of polypeptides in a modular manner . supported by this numeration , a set of internal coordinates is defined . these coordinates ( termed systematic , approximately separable and modular internal coordinates , sasmic ) are straightforwardly written in z - matrix form and may be directly implemented in typical quantum chemistry packages . a number of perl scripts that automatically generate the z - matrix files for polypeptides are provided as supplementary material . the main difference with other z - matrix - like coordinates normally used in the literature is that normal dihedral angles ( `` principal dihedrals '' in this work ) are only used to fix the orientation of whole groups and a somewhat non - standard type of dihedrals , termed `` phase dihedrals '' , are used to describe the covalent structure inside the groups . this _ physical approach _ allows to approximately separate soft and hard movements of the molecule using only topological information and to directly implement constraints . as an application , we use the coordinates defined and ab initio quantum mechanical calculations to assess the commonly assumed approximation of the free energy , obtained from `` integrating out '' the side chain degree of freedom , by the potential energy surface ( pes ) in the protected dipeptide hco - l - ala - nh . we also present a sub - box of the hessian matrix in two different sets of coordinates to illustrate the approximate separation of soft and hard movements when the coordinates defined in this work are used . + * pacs : * 87.14.ee , 87.15.-v , 87.15.aa , 87.15.cc , 89.75.-k +
|
hep-th0410229
|
i
|
while de sitter space shares the same number of isometries as minkowski space , field theories exhibit some surprising properties in this simplest of curved backgrounds . an immediate example is the enormous stretching of scales in de sitter space which naturally connects short distances in the past to large distances today . this rapid expansion is a familiar and very appealing feature of inflation @xcite . during the slow - roll regime of inflation , for which de sitter space is an idealization , quantum fluctuations grow exponentially large and eventually seed the large scale structure of the universe . depending upon the hubble scale and the duration of inflation , this large scale structure could have been determined by fluctuations which occurred at or well below the planck scale . most models of inflation produce far more than the necessary sixty @xmath2-folds necessary to solve the horizon problem and this connection between large scales and potentially planckian physics has been called the `` transplanckian problem '' of inflation @xcite , although it has recently been viewed as more of an opportunity , since it could allow the observation of physics well beyond experimentally accessible scales , most typically within an order of magnitude or two above the hubble scale during inflation . non - thermal features of the state for the field driving inflation tend to provide a more robust signal of these transplanckian effects @xcite . a further difference from minkowski space is the existence of a much richer family of invariant or covariant states in de sitter space . for a free scalar field in a de sitter background , the states invariant under the @xmath3 isometry group can be distinguished by a complex parameter @xmath1 , although only real values of @xmath1 correspond to @xmath4 invariant theories @xcite . these @xmath1-vacua are not the lowest energy eigenstates of a globally conserved hamiltonian , as is the case of the standard poincar ' e - invariant vacuum of minkowski space , since de sitter space does not admit a globally defined time - like killing vector . nevertheless , a unique element in this infinite family , the bunch - davies vacuum @xcite , can be selected by demanding that at short distances or as the curvature of the de sitter space is taken to zero the state should match with the vacuum of minkowski space . both of these features emphasize the need for understanding quantum field theory particularly the ideas of decoupling and renormalization in an expanding background starting from a non - standard state . for this purpose the @xmath1-vacua provide an ideal test case to study how these ideas are to be modified in such a setting since the high amount of symmetry of these non - thermal states allows them to be readily analyzed analytically . it was recently realized that for a scalar field in an @xmath1-state , a point source propagator does not produce a well behaved perturbation theory @xcite . one method for expressing this pathological behavior is to impose a cutoff @xmath5 on physical three - momentum of the theory . loop processes then diverge as @xmath6 in such a way as can not be cancelled by simple counterterms . for example , the one - loop correction to the self - energy in a @xmath7 theory diverges linearly with @xmath5 and the dependence of this divergent term on the external momentum does not match that of a @xmath8 counterterm @xcite . the resolution of these divergences came with the realization that the propagator should be modified for these states to be the green s function for _ two _ point sources @xcite . in this article , we examine the structure and the properties of a spin @xmath0 fermion in a de sitter background , which possesses its own one - parameter set of covariant states @xcite . one reason for doing so is to learn whether the double source construction for the scalar field can be circumvented by using fermionic loops to cancel the non - renormalizable divergences from bosonic loops . while a fermion loop correction to a scalar self - energy also diverges linearly with the cutoff @xmath5 , here we show that this divergence can not be cancelled by that of the bosonic loop , even allowing an arbitrary fine - tuning of the relative values of @xmath1 for the scalar and the fermion fields . as with a bosonic @xmath1-vacuum , the peculiar divergences in fermion loops arise from an inconsistency between the single - source propagator and the fermionic @xmath1-state . in some sense , the physical setting resembles a field theory where we have imposed boundary conditions along an initial time surface @xcite . there , we must also modify the propagator by adding an image source to obtain a consistent perturbation theory ; any new divergences that result from this modification only appear as counterterms restricted to the initial surface . the bulk theory is unchanged . in de sitter space , the inconsistency is also resolved by adding a new source term in the definition of the propagator when in an @xmath1-state . the remarkable property of de sitter space is that there exists a special point , the antipode , at which a source can be placed without breaking the @xmath3 symmetry properties of the state . for a fermion , the extra antipodal source entails some additional dirac structure . the next section derives the @xmath1-states for a spin @xmath0 dirac field in a de sitter background . the @xmath1-propagator for a point source is developed in sec . [ pointpropagation ] . we then show in sec . [ fermiloop ] that a theory with a yukawa coupling produces divergences in the one - loop corrections to the scalar propagator which can not be cancelled by adding simple counterterms to the lagrangian nor do they cancel divergences from analogous graphs where a scalar loop replaces the fermion . section [ antiprop ] shows that the these divergences can be avoided by modifying the propagator , adding an additional source at the antipode , resulting in a renormalizable theory . the final section concludes with comments on the relation between @xmath1-vacua and the problem of quantizing a theory with a specified initial state .
|
a spin particle propagating in a de sitter background has a one parameter family of states which transform covariantly under the isometry group of the background . we shall show how using a point - source propagator for a fermion in an-state produces divergent perturbative corrections . these corrections can not be used to cancel similar divergences arising from scalar fields in bosonic-vacua since they have an incompatible dependence on the external momenta . the theory can be regularized by modifying the propagator to include an antipodal source .
|
a spin particle propagating in a de sitter background has a one parameter family of states which transform covariantly under the isometry group of the background . these states are the fermionic analogues of the-vacua for a scalar field . we shall show how using a point - source propagator for a fermion in an-state produces divergent perturbative corrections . these corrections can not be used to cancel similar divergences arising from scalar fields in bosonic-vacua since they have an incompatible dependence on the external momenta . the theory can be regularized by modifying the propagator to include an antipodal source .
|
cond-mat9809351
|
i
|
in colloidal suspensions the interaction between the mesoscopic dissolved particles and the solvent is of basic importance @xcite . for example , the solvent generates effective interactions between the colloidal particles which can even lead to flocculation . the richness of the physical properties of these systems is mainly based on the possibility to tune these effective interactions over wide ranges of strength and form of the interaction potential . traditionally this tuning is accomplished by changing the chemical composition of the solvent , e.g. , by adding salt , polymers , or other components @xcite . compared with such modifications , changes of the temperature or pressure typically result only in minor changes of the effective interactions . this , however , is only true as long as the solvent is not thermodynamically close to a phase transition of its own . for example , if the solvent consists of a binary liquid mixture close to a _ first - order _ demixing transition into a a - rich and a b - rich liquid phase , even slight changes of the temperature or of the partial pressures of the two species a and b can lead to massive changes of the effective interactions between dissolved colloid particles induced by the occurrence of wetting transitions . they lead to wetting films of the preferred phase coating the colloidal particles @xcite . these wetting films can snap into bridges if the particles come close to each other leading to flocculation @xcite . for charged colloidal particles such as silica spheres immersed in the binary liquid mixture of water and 2,6-lutidine @xcite flocculation can also be influenced by screening effects generated by the adsorbed layers @xcite . similarly drastic effects can occur if the solvent is brought close to a _ critical point_. the inevitable preference of the surfaces of the colloidal particles for one of the two solvent species of a binary liquid mixture near its critical demixing point or for the liquid phase of a one- or two - component solvent fluid near its liquid - vapor critical point results into the presence of effective surface fields leading to pronounced adsorption profiles of the preferred component . this so - called ` critical adsorption ' becomes particularly long - ranged due to the correlation effects induced by the critical fluctuations of the order parameter of the solvent . in the case of a planar wall critical adsorption has been studied in much detail @xcite . asymptotically close to the critical point @xmath0 it is characterized by a surface field which is infinitely large so that the order parameter profile actually diverges upon approaching the wall up to atomic distances @xmath1 . as compared to a planar surface critical adsorption on a spherical particle is expected to exhibit important differences in behavior because the confining surface has a positive curvature and because a sphere represents only a quasi - zero - dimensional defect floating in the critical fluid . the interference of critical adsorption on neighboring spheres gives rise to the so - called critical casimir forces @xcite which have been argued to contribute to the occurrence of flocculation near @xmath0 @xcite . a quantitative understanding of these phenomena requires the knowledge of the critical adsorption profiles near the colloidal particles and the resulting effective free energy of interaction in the whole vicinity of the critical point , i.e. , as function of both the reduced temperature @xmath2 and the field @xmath3 conjugate to the order parameter . this ambitious goal has not yet been accomplished . instead , the introduction of a surface curvature has limited the knowledge of the corresponding critical adsorption so far to the case of spheres for the particular thermodynamic state @xmath4 of the solvent @xcite . only recently at least the temperature dependence of the critical casimir force between a sphere and a planar container wall has been addressed @xcite . thus the present study of the temperature dependence of critical adsorption on a single sphere contributes one step towards reaching the aforementioned general goal . apart from spherical particles , also rodlike particles play an important role @xcite . rodlike objects are provided , e.g. , by fibers or colloidal rods @xcite , semiflexible polymers with a large persistence length such as actin @xcite , microtubuli @xcite , and carbon nanotubes @xcite . moreover the knowledge of the general curvature dependence of critical adsorption is also relevant , e.g. , for curved membranes @xcite dissolved in a fluid near criticality or for the liquid - vapor interface between a binary liquid mixture near its critical demixing point and its noncritical vapor , which exhibits rippled configurations due to the occurrence of capillary waves @xcite . near criticality the relevant length scales of the solvent structures are dominated by the diverging bulk correlation length @xmath5 , where @xmath6 is the standard universal bulk critical exponent ; @xmath7 and @xmath8 are nonuniversal amplitudes in the one- ( @xmath9 ) and two - phase region ( @xmath10 ) , respectively , with values typically in the order of a few . in practice the correlation length can span the range between @xmath11 - @xmath12 depending on @xmath13 . in the present context this length scale is played off against the length scale @xmath14 of the radius of the dissolved particles . we note that the available systems can realize both the limit @xmath15 as well as the opposite limit @xmath16 . in the case of ludox silica particles @xmath17 @xcite so that the limit @xmath16 can be easily achieved even with the upper limits for @xmath18 set by finite experimental resolutions . the ratio of the length @xmath19 and the radius @xmath14 of rodlike particles can be quite large , in conjunction with a small radius such as @xmath20 in the case of colloidal boehmite rods @xcite . in this work we consider _ long _ rods , i.e. , @xmath21 , and neglect effects which may arise due to their finite length @xmath19 . in the present contribution we investigate systematically the temperature dependence of the critical adsorption on a single spherical or rodlike particle , i.e. , the case @xmath22 . ( the generalization to the case @xmath23 is straightforward but tedious . ) in order to be able to treat spheres and cylinders in a unified way within a field - theoretical approach and for general spatial dimensions @xmath24 it is helpful to consider the particle shape of a _ generalized cylinder _ @xmath25 @xcite with an infinitely extended ` axis ' of dimension @xmath26 . the ` axis ' can be the axis of an ordinary infinitely elongated cylinder ( @xmath27 ) , or the midplane of a slab ( @xmath28 ) , or the center of a sphere ( @xmath29 ) . for general integer @xmath24 and @xmath26 the explicit form of @xmath25 is @xmath30 with @xmath31 and @xmath32 perpendicular and parallel to the axis , respectively ( see fig.[fig_cyl ] ) . note that @xmath31 is a @xmath33-dimensional vector with @xmath34 the radius @xmath14 of the generalized cylinder @xmath25 is the radius in the cases of an ordinary cylinder or a sphere and it is half of the thickness in the case of a slab . for the slab the geometry reduces to the much studied case of ( two decoupled ) half spaces . the generalization of @xmath24 to values different from three is introduced for technical reasons because @xmath35 marks the upper critical dimension for the relevance of fluctuations of the order parameter leading to a behavior different from that obtained from mean - field theory valid for @xmath36 . in sec.[sec_op ] we discuss the general scaling properties of the local order parameter profiles for critical adsorption on spheres and cylinders , in particular the behavior close to the particle surfaces and for small particle radii , respectively . in sec.[sec_excess ] we consider the corresponding properties of the excess adsorption . in sec.[sec_mf ] we present explicit results both for the order parameter profiles and for the excess adsorption in mean - field approximation . section [ summary ] contains our conclusions . in appendix [ app_ex ] we discuss the two - point correlation function near a microscopically thin ` needle ' at criticality . in appendix [ app_neu ] we determine a universal amplitude and a universal scaling function as needed in secs.[sec_op ] and [ sec_excess ] , respectively . in appendix [ appb ] , finally , we consider the general curvature dependence of the excess adsorption . ( 16,10 ) ( -0.7,-1.3 ) ( 8.0,-1.3 )
|
the temperature dependence of the corresponding order parameter profiles and of the excess adsorption are calculated explicitly . critical adsorption on elongated rods is substantially more pronounced than on spherical particles . it turns out that , within the context of critical phenomena in confined geometries , critical adsorption on a microscopically thin ` needle ' represents a distinct universality class of its own . under favorable conditions the results are relevant for the flocculation of colloidal particles .
|
a systematic fieldtheoretic description of critical adsorption on curved objects such as spherical or rodlike colloidal particles immersed in a fluid near criticality is presented . the temperature dependence of the corresponding order parameter profiles and of the excess adsorption are calculated explicitly . critical adsorption on elongated rods is substantially more pronounced than on spherical particles . it turns out that , within the context of critical phenomena in confined geometries , critical adsorption on a microscopically thin ` needle ' represents a distinct universality class of its own . under favorable conditions the results are relevant for the flocculation of colloidal particles . = 23.0 true cm = 16.0 true cm = 0 true cm -0.25 cm 0 cm 1.5 cm 1.5em 0.5 mm
|
1607.05530
|
i
|
in 1894 , with their pioneering work on free - recall experiments , binet and henry introduced a key tool for the controlled investigation of short - term memory ( binet and henry , 1894 ) . in its traditional form , a free - recall experiment is performed by presenting the subject with a list of words and then requesting him or her to recall it in any order ( murdock , 1960 , 1962 ; roberts , 1972 ; standing , 1973 ) . several types of effects have been reported : \1 ) effects depending on the lexical properties of individual words . in particular , lists of short words are recalled better than lists of long ones , a fact known in the literature as the word - length effect ( baddeley et al . , 1975 ; russo and grammatopoulou , 2003 ; tehan and tolan , 2007 ; bhatarah et al . , 2009 ) . \2 ) effects in which the recall probability depends on the absolute position of words in the list . it has been observed that the first and last words in the list are recalled more easily ( `` primacy '' and `` recency '' effects ) . \3 ) effects depending on the relative position of words with respect to each other . most notably , the recall probabilities of contiguous words correlate positively , a fact known as the contiguity effect ( murdock , 1960 , 1962 ) . the need to understand serial - position effects led to the devising of retrieved - context models , such as the temporal context model ( howard and kanaha , 2002 ) . in these models the recall process , rather than retrieving a word directly , retrieves the context associated to the word first . within this scenario , recency effects appear because the context at the time of the `` memory test '' is most similar to the context associated with recent items . when an item is retrieved , it reinstates the context active when that item was presented . because this context overlaps with the encoding context of the items neighbors , a contiguity effect results . through these models , serial - position effects have been substantially understood over the past fifteen years ( howard and kahana , 2002 , 2002b ; sederberg , howard , and kahana , 2008 ; polyn , norman , and kahana , 2009 , 2009b ; lohnas , polyn , and kahana , 2015 ; kahana , 2012 ) . the same can not be said , however , about the word - length effect . the word - length effect ( wle ) has been a traditional testing ground for models of short - term memory ( campoy , 2011 ; jalbert et al . , 2011 ) , and it has played a key role in establishing the working - memory paradigm and the phonological loop hypothesis ( baddeley and hitch . , 1974 ) . the standard account of the effect ( baddeley , 2007 ) relies on a trade - off between memory decay ( in the phonological store ) and subvocal rehearsal via an articulatory control process . because shorter words take less time to rehearse , more decaying traces of them can be refreshed than decaying traces of long items , and , therefore , more short items can be recalled . this picture , however , is not able to account for all experimental observations concerning this effect , and has been repeatedly called into question . in ( neath et al . , 2003 ) , it was shown that with words having the same number of syllables but different pronunciation times , no unambiguous wle arises . this result ( extended in jalbert et al . 2011 ) suggests that the effect depends on the number of syllables , and not on the time it takes to pronounce them . experiments have also been performed in conditions where there was a delay between lists , making subvocal rehearsal possible in the interval . no appreciable difference in recall probabilities was found ( campoy , 2008 ) . in the same study , experiments were performed in which subvocal rehearsal was prevented by a high presentation rate . no delay was allowed between the presentation of word lists and the memory test . yet , the wle occurred unperturbed . in the 2011 paper i just cited , jalbert et al . concluded : `` the wle may be better explained by the differences in linguistic and lexical properties of short and long words rather than by length per se '' . in the meanwhile , within the fields of experimental and computational linguistics , progress has been made in understanding the role of word length in verbal processing . over the years , it has emerged that words with different lengths tend to have different semantic properties . the idea was first put forth in pedagogical studies ( klare , 1988 ) . in ( elts , 1995 ) , a correlation coefficient of 0.96 was found between a noun s length and its average tendency to be used as a technical term ( `` terminologicality '' ) . mikk et al . ( 2000 ) , using data on the human - assessed complexity of a large sample of words , found a correlation coefficient 0.86 between words length and their semantic complexity . pinning down the precise semantic property that correlates to word length has proven difficult . already in ( greenberg , 1966 ) it was argued that a word s length correlates positively to its conceptual `` markedness '' of meaning . various notions of markedness have subsequently been discussed in the literature ( haspelmath , 2006 ) . piantadosi et al . ( piantadosi et al . , 2011 , 2011b ) and later mahowald et al . , ( mahowald et al . , 2012 ) reported that the length of words correlates positively with their contextual information rate . more recently , lewis and frank ( 2016 ) have carried out a comprehensive experimental study across 80 languages . they found that , in all the languages considered , judgments of conceptual complexity for a sample of real words correlate highly with their length , and they even control for frequency , familiarity , imageability , and concreteness . their conclusion is : `` while word lengths are systematically related to usage @xmath0 both frequency and contextual predictability @xmath0 our results reveal a systematic relationship with meaning as well '' . in the light of these findings , it would be a natural step to attempt an explanation of the wle in terms of the semantic differences among words . however , no such approach seems to have been attempted in the literature . recently , new aspects of the wle have emerged through the analysis of a large set of data from experiments by miller and al ( miller et al . , 2012 ) . the data analysis was performed by katkov et al . ( katkov et al . , 2014 ) , who found no negative correlation between total length of presented items and number of recalled words , thus disproving both rehearsal - time theories and hypotheses based on the increasing complexity of longer items . moreover , they reported an inversion of the effect in mixed lists , that is , lists where words are selected irrespectively of their length . they observed that , in this type of lists , the mean values of recall probabilities allow to establish an increasing trend . long words are recalled better than short ones . an `` inverse '' wle had been previously reported by at least two groups , but in somewhat less general circumstances : one of them ( hulme et al . , 2006 ) embedded strictly pure lists with a single word of a different type , while the results of xu et al . ( xu et al . , 2009 ) , may not bear direct comparison with data in languages other than chinese . if the inversion of the wle for mixed lists will be confirmed by further experiments , it will have to be taken into account by every general theory of the standard wle . let us consider , therefore , what requirements a model should fulfill to explain both phenomena simultaneously . call @xmath1 the fraction of long words in the list ; call @xmath2 the probability of recalling successfully a given long word from a list in which a fraction @xmath1 of words are long ; and let @xmath3 be the probability of recalling successfully a short word , from a list with a fraction @xmath1 of long words . obviously , the function @xmath2 is only defined for @xmath4 , and the function @xmath3 only for @xmath5 . for @xmath60 , 1[$ ] , both functions are defined . theorists would have to reconcile two observations on the curves @xmath7 : \1 . @xmath8 \2 . @xmath9 for @xmath60 , 1[$ ] . the only way these two inequalities can be simultaneously satisfied is if _ both _ @xmath10 and @xmath11 are , on the whole , decreasing functions of @xmath1 . the simplest choice of these curves compatible with experiments is one where both are monotonously decreasing , that is : @xmath12 [ fig1 ] ( -225,5)@xmath13 ( -50,5 ) @xmath14 ( -200,146 ) @xmath10 ( -200,97 ) @xmath11 this means that , whenever we replace a short word of the list with a long word , we are lowering the recall probability of all the words in the list , both long and short . a higher number of long words makes every single word in the list harder to recall . it is difficult to imagine how this could ensue from the different @xmath15 of long and short words . the question is then : can equation ( [ derivatives ] ) result directly from the different semantic properties of long and short words ? in this paper , i will show that the answer is positive as long as one models carefully the process of verbal perception . a suitable way of doing so is demonstrated in the next section . in section iii , i employ a retrieved - context description of verbal recall to derive both wles ( standard and inverse ) . in section iv , i test two key predictions of the theory against data from the peers experiment of kahana et al . ( lohnas and kahana , 2013 ; healey and kahana , 2016 ) . in section v , i list five experiments designed to test further predictions of the theory . in the conclusions , i sketch a possible interpretation of the results .
|
the observation that lists of short words are recalled better than lists of long ones has been a long - standing subject of controversy , further complicated by the apparent inversion of the effect for mixed lists . in the framework here proposed , these behaviors emerge as an effect of the different level of localization of short and long words in semantic space . events corresponding to the recognition of a nonlocal word have a clustering property in phase space , which facilitates associative retrieval . finally , an interpretation of the above results is presented .
|
a theoretical framework is proposed for the understanding of verbal perception the conversion of words into meaning , modeled as a compromise between lexical demands and contextual constraints and the theory is tested against experiments on short - term memory . the observation that lists of short words are recalled better than lists of long ones has been a long - standing subject of controversy , further complicated by the apparent inversion of the effect for mixed lists . in the framework here proposed , these behaviors emerge as an effect of the different level of localization of short and long words in semantic space . events corresponding to the recognition of a nonlocal word have a clustering property in phase space , which facilitates associative retrieval . the standard word - length effect arises directly from this property , and the inverse effect from its breakdown . an analysis of data from the peers experiments ( healey and kahana , 2016 ) confirms the main predictions of the theory . further predictions are listed and new experiments are proposed . finally , an interpretation of the above results is presented .
|
1005.2147
|
i
|
it is usually claimed that the quantum state of the universe is described by a wavefunction @xcite which should account for all the physical information which can be extracted from the universe . two main approaches have been followed in order to obtain such a wavefunction of the universe . first , it can be obtained as the solution of the wheeler - de witt equation @xcite . this can actually be seen as a schrdinger equation in which no time derivative appears in order to satisfy the time invariance required by the general relativity theory @xcite . secondly , a path integral approach can also be taken @xcite . then , the wavefunction of the universe is given by a sum over all geometries and field configurations which can be matched with a given value of the field configuration , defined in a spatial section of the whole space - time manifold . we make a wick rotation to euclidean time in order to obtain well - defined integrals , and then rotate back to lorentzian time to obtain the final results . in both cases , some boundary conditions have to be imposed @xcite . the hartle - hawking no - boundary proposal @xcite represents a universe which is created from `` nothing '' , meaning by that the absence of space , time and matter . vilenkin @xcite considered a universe which is also created from nothing , but through a quantum tunneling transition instead . vilenkin argued that the idea of a universe being created from nothing is not crazier than the creation of particle in other quantum theories . thus , it seems appropriate to consider that the universe may be created as a quantum fluctuation of the gravitational vacuum . once a universe nucleates in the space - time foam @xcite , it may bubble and eventually jump into an inflationary period @xcite , which is supposed to be the origin of our current universe . therefore , we must consider a multiverse in which changes in the topology of space - time are allowed . a variety of multiverse hypotheses have been recently considered from different cosmological viewpoints @xcite . the specific meaning which is ascribed to a single universe depends on the formalism of the relevant theory . some well - known examples include the everett s many - world interpretation of the quantum theory @xcite , the chaotic inflationary multiverse @xcite or the landscape in string theory @xcite , among others ( for an exhaustive review , see ref . @xcite and references therein ) . in this paper we only consider the case of a multiverse which is described by a set of quantum oscillators , each one representing a causally disconnected region of the whole space - time . topology changes were first claimed to appear in the quantum physics of black hole evaporation @xcite , and so first arose as a result of trying to take quantum physics seriously as a description of the whole universe . on pure cosmological grounds , the transition from a matter dominated universe into a space - time filled with phantom energy provides us with another example of a bifurcating topology originated from the big rip singularity which splits space - time into two disconnected regions . creation of universes and therefore topology changes are naturally contemplated in the third quantization formalism @xcite . this consists of a further quantization of the wavefunction of the universe similarly to how quantum field theory is constructed from the schrdinger wavefunction of matter fields . the computations are difficult to perform in the general case . in this paper , however , a simplified model will be presented in which such computations can in fact be carried out . moreover , the general scheme of the third quantization formalism can be applied to both , a multiverse made up of parent universes and a space - time foam filled with virtual baby universes @xcite . parent universes are defined to be large space - time regions with a hubble length of the same order as the hubble length of our universe . baby universes are considered to be virtual fluctuations of the metric in the vacuum of gravity , and their contribution to calculations of the gravitational field is extremely important at the planck length . the gravitational vacuum would be also populated with virtual lorentzian and euclidean wormholes . the former can be seen as solutions of the einstein s equations with at least two asymptotically flat regions , connecting two separate parts either of the same universe or of two different universes . the latter can be considered as euclidean sectors of a friedman space - time @xcite , whose quantum states can be seen as the exponentially decaying versions of the oscillatory universes from which they were wick rotated . furthermore , the vacuum state of gravity might also be observable as far as it could induce a loss of quantum coherence in the matter fields . the debate was centered ( see e.g. refs . @xcite and @xcite ) around the assumption of taking doubly or even multiply connected wormholes instead of simply connected ones in the quantum vacuum of gravity . equivalently , it can be placed in terms of whether virtual baby universes are created as single universes or in pairs . in the theory of the quantum multiverse , in which it has been shown that general topological changes may occur , the loss of quantum coherence in the matter field sector seems to be unavoidable . our aim in this paper is then to apply the third quantization formalism given in ref . @xcite to a simplified model of the universe , which nevertheless retains the fundamental features of the quantum theory when it is applied to the universe as a whole . it will be shown that in such a model a well - defined quantum state for the multiverse can be obtained , which satisfies the usual boundary conditions . squeezed states @xcite , which are usually interpreted as quantum states without any classical analog @xcite , are found in the context of the multiverse , and in particular they appear in the context of accelerated universes . this seems to give support to the idea that the acceleration is due to distinctively quantum mechanical effects @xcite . furthermore , the third quantization formalism used in this paper shows some advantages respect to other approaches which are usually taken in quantum cosmology . for instance , it can be demonstrated that the quantum state of the multiverse may be given in terms of the states of a quantum harmonic oscillator . dealing with frequencies instead of potentials simplifies the computations . moreover , neither a perturbative nor a semiclassical approximation needs to be taken in the model presented in this paper , and therefore the state of the multiverse which is obtained can describe the global state of the multiverse as a whole . before going any further , however , some caveats and comments should be made about the ideas and results considered in this paper . first of all , dealing with squeezed states in the context of the quantum state of the multiverse would , at first sight , seem somewhat extraneous . actually , squeezed states can be readily interpreted both in quantum optics and in a space - time foam made up of baby universes , which are both deployed in a common space - time . baby universes can in fact be taken as tiny particles representing small perturbations of the space - time field , whose quantum state affects the vacuum state of the matter field sector , and so this theory may turn out to be testable . it is more difficult to imagine how we could test a theory of a multiverse made up of parent universes . it looks quite different even though wormhole communications channels may exist through which microscopic particles could travel from one universe to other @xcite . furthermore , some correlations among the quantum state of different universes could also be considered . for instance , if the creation of universes would be produced in entangled pairs , whose correlations might induce some observable effects on the state of each individual universe in the pair @xcite . squeezed states have already been studied in other cosmological contexts such as the inflationary universe and gravitational waves @xcite . it was shown @xcite that the gravitational vacuum is populated by gravitons which evolve into a highly squeezed state . in this paper it is also demonstrated in the context of a third quantization formalism that a space - time foam made up of baby universes is in a squeezed state . if such a result could be extrapolated to parent universes , these universes might not be independent but quantum mechanically correlated , too . the paper can then be outlined as follows : a discussion with the main features of the third quantization formalism has been also added as sec . ii . in sec . iii , it is obtained the wavefunction for a quantum multiverse made up of friedmann space - times filled with an homogeneous and isotropic fluid , and the appropriate fock space is defined . we then analyze the two interesting limits of the quantum state of both a large parent universe and the quantum gravitational vacuum , which turn out to be described by a squeezed state with no classical analog . in sec . iv , the quantum state of the multiverse is represented by a density matrix rather than by a wavefunction , accounting thus for mixed states as well as pure ones . three representations are employed : first , the second quantized wavefunction is used as the configuration variable , the usual boundary conditions of refs . @xcite are applied and a probability interpretation can then be used . secondly , a general squeezed number representation is taken in order to study the quantum state of the multiverse . parent universes will be represented by number states . we show that the quantum state for the vacuum of baby universes is given by a squeezed state , and that high order correlations appear among them . for the sake of completeness , we have also used the usual @xmath0 representation in terms of coherent states . in sec . v , we compare the results obtained in this paper with previous works which have considered or criticized the third quantization formalism . the conclusions are collected in sec . vi , together with some comments to extend the model to a two - dimensional wave equation which would explicitly account for other matter fields .
|
a third quantization formalism is applied to a simplified multiverse scenario . a well defined quantum state of the multiverse is obtained which agrees with standard boundary condition proposals . these states are found to be squeezed , and related to accelerating universes : they share similar properties to those obtained previously by grishchuk and siderov . we also comment on related works that have criticized the third quantization approach .
|
a third quantization formalism is applied to a simplified multiverse scenario . a well defined quantum state of the multiverse is obtained which agrees with standard boundary condition proposals . these states are found to be squeezed , and related to accelerating universes : they share similar properties to those obtained previously by grishchuk and siderov . we also comment on related works that have criticized the third quantization approach .
|
0805.3814
|
c
|
in conclusion , we have studied the extension to two spatial dimensions of a one - dimensional model describing coupled optical waveguides going beyond the lowest - order approximation and including nonlinear coupling , which , e.g. , is relevant for arrays of linear waveguides embedded in nonlinear media . the phenomenology of the 1d model presented in a series of papers @xcite has been revisited in the 2d setting . especially , we have shown that a vanishing energy difference between @xmath59-site and @xmath60-site stationary solutions and a subsequent small peierls - nabarro energy barrier , taking into account also the existence of asymmetric intermediate solutions , does _ not _ result in good mobility of highly localized modes . the main reason , and the difference to models that show good mobility @xcite , is that , despite small energy differences , the stationary solutions are still in some sense far apart . rather , the bifurcation points where the fundamental modes change their stability and where they exhibit a depinning mode promoting mobility are , although close in hamiltonian ( @xmath29 ) , far apart in norm ( @xmath31 ) ( figs . [ fig : bifurc3 ] and [ fig : bifurc5 ] ) . from this we conclude that for the stationary solutions that have near equal values of both hamiltonian and norm there is likely no trajectory in phase space passing close to them both . thus , in order to have good mobility of narrow modes in discrete systems , we need _ not only _ a small pn - barrier but also an exchange of stability between @xmath59-site and @xmath60-site solutions taking place _ in the vicinity_. we therefore also conjecture that the size of the oscillating background accompanying travelling modes @xcite is not only related to the size of the pn - barrier but also to the difference in other conserved quantities ( like @xmath31 ) of the involved stationary modes measured at the points where their stability is changed . the lack of mobility near a zero of the energy difference between fundamental modes was observed also in @xcite , but no further analysis was carried out . effects of a more delicate balance between linear and nonlinear coupling terms in eq . ( [ eq : xdnls ] ) have also been discussed . these include the existence and stability of compact solutions , both discrete breathers and discrete vortex solitons . the latter have to our knowledge not been previously reported . mathematically , these solutions are not robust with respect to parameter variation and require a balance of the coupling parameters in the equation for exact compactness , but near perfect localization persists in the neighborhood of the exact parameter values . an interesting question is if they can prevail also beyond the tight - binding approximation of eq . ( [ eq : xdnls ] ) . the answer is likely that they are only exact solutions for the present model , but that they will correspond to modes of a higher degree of localization in more accurate models , i.e. , they give an indication in which parameter regimes high localization can be achieved . as discussed in sec . [ sec : compact ] the compact solutions do not represent a true compactification in the real system . additional interesting effects present also in 1d arise from a non - trivial dependence of the power flow in the lattice on the amplitude and phase difference of an excitation . particularly , the current can become zero for a non - trivial phase twist which also in 2d leads to the existence of complex localized stationary modes with an open geometry , that may even be stable . further , we have shown how the transversal flow of power in the array of waveguides can be controlled with great flexibility by excitation with plane waves , and in particular explicitly demonstrated how the transport _ direction _ may be continuously tuned by _ amplitude _ variation . this may have applications for power - coupling devices . however , the use of a 2d array of the type studied here for multiport switching is discouraged due to the poor mobility of localized modes . for this purpose a 1d array @xcite or an array exhibiting a saturable @xcite or quadratic @xcite ( see discussion below ) nonlinearity instead of the kerr nonlinearity is better suited . in @xcite , abdullaev et.al extended the wannier - function approach of @xcite for the continuous 1d nls equation with periodic linear potential to the case with periodic variation also for the nonlinearity coefficient ( which is the case also considered in our work ) . within the tight - binding approximation , they derived under quite general conditions a lattice equation , which in a special case ( @xmath196 in @xcite ) is equivalent to the 1d version of our eq . ( [ eq : xdnls ] ) as derived in @xcite . an important conclusion of @xcite is the crucial importance to include also the nonlinear coupling terms ( corresponding to our @xmath197 and @xmath198 ) , as they for specific choices of nonlinear interactions were shown to be comparable with , or even stronger than , the on - site nonlinearity . thus , this supports the soundness of our approach to consider variation of the parameters @xmath197 and @xmath198 over a rather large range of values . several works discussing properties of moving solitons and ( vanishing ) pn - barriers in 1d generalized dnls models have appeared . dmitriev , khare et al . @xcite found analytically exact stationary and moving solutions for some of the exceptional , translationally invariant discretizations of @xcite . whether these are of relevance for any physically realizable system is , to our knowledge , still unclear . pelinovsky et al . @xcite developed a mathematical technique for analysis of persistence of traveling single - humped localized solutions from a certain limit , and found specifically that while travelling solutions terminated when continued from the integrable al - limit of the salerno model ( corresponding to the development of a resonant tail @xcite ) , they generally persisted in the translationally invariant models . oxtoby and barashenkov @xcite used the method of asymptotics beyond all orders to evaluate the amplitude of radiation from a moving small - amplitude soliton in the saturable dnls equation , and found it to be completely suppressed at certain ` sliding velocities ' , where , similarly as in @xcite , it was interpreted as an embedded soliton . the properties of the travelling solitary waves in the saturable model were also analyzed numerically in more detail by melvin et al . in @xcite . for the two - dimensional case , susanto et al . @xcite studied the mobility of discrete solitons in a square lattice with _ quadratic _ ( second - harmonic - generating ) nonlinearity . in this case , due to the absence of collapse instability in the continuum limit , good mobility of stable solutions was observed as long as the coupling constants were not too small ( ` quasicontinuum regime ' ) . in this regime , mobility in arbitrary directions ( not necessarily axial or diagonal ) was observed , which is not surprising since discreteness effects are smoothened out for broad , continuum - like solitons . however , no non - trivial zeros of the pn barrier were found for this model , and consequently strongly localized solutions were reported to be immobile . thus , in many aspects , the mobility properties of the 2d model with quadratic nonlinearity is similar to those of the 1d cubic on - site dnls model . finally , a very recent preprint by chong et al . @xcite extended the analysis of the on - site cubic - quintic dnls model of @xcite to higher dimensions . as concerns the two - dimensional mobility , the results were shown to be very similar to that of the saturable model @xcite : enhanced mobility was found in regimes close to stability inversion and associated with appearance of asymmetric intermediate solutions and low pn barrier . thus , as far as we are aware , the present work still provides the only known explicit example of a model where these properties do _ not _ lead to an enhanced mobility . m would like to thank the mathematics department at heriot - watt university , edinburgh , for its hospitality and especially j.c . eilbeck for guidance , support and very useful discussions . this work was partly carried out under the hpc - europa project ( rii3-ct-2003 - 506079 ) , with the support of the european community - research infrastructure action under the fp6 `` structuring the european research area '' programme . partial support from the swedish research council is acknowledged . pelinovsky , nonlinearity * 19 * , 2695 ( 2006 ) . kevrekidis , s.v . dmitriev and a.a . sukhorukov , mathematics and computers in simulation * 74 * , 343 ( 2007 ) . a. maluckov , m. stepi , d. kip and lj . hadievski , eur . j. b * 45 * , 539 ( 2005 ) . m. ster , _ stability and mobility of localized and extended excitations in nonlinear schrdinger models _ , doctoral thesis , linkping studies in science and technology . dissertations , no . 1072 , isbn 978 - 91 - 85715 - 83 - 1 , linkping university ( 2007 ) ; published electronically at http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-8091 . m. ster and m. johansson , in _ proceedings of the conference on localization and energy transfer in nonlinear systems , san lorenzo de escorial , madrid , spain , 2002 _ , edited by l. vzquez , m.p . zorzano and r.s . mackay ( world scientific , singapore , 2003 ) .
|
we find that despite a vanishing energy difference ( peierls - nabarro barrier ) of fundamental stationary modes the mobility of localized excitations is very poor . this is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes . at these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points . the control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power - coupling devices . in certain parameter regimes , the direction of a stable propagating plane - wave current is shown to be continuously tunable by amplitude variation ( with fixed phase gradient ) . nonlinear coupling , peierls - nabarro potential , mobility , inversion of stability , modulational instability , power currents . 42.65.wi , 42.82.et , 63.20.pw , 05.60.-k
|
a two - dimensional nonlinear schrdinger lattice with nonlinear coupling , modelling a square array of weakly coupled linear optical waveguides embedded in a nonlinear kerr material , is studied . we find that despite a vanishing energy difference ( peierls - nabarro barrier ) of fundamental stationary modes the mobility of localized excitations is very poor . this is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes . at these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points . the control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power - coupling devices . in certain parameter regimes , the direction of a stable propagating plane - wave current is shown to be continuously tunable by amplitude variation ( with fixed phase gradient ) . more exotic effects of the nonlinear coupling terms like compact discrete breathers and vortices , and stationary complex modes with non - trivial phase relations are also briefly discussed . regimes of dynamical linear stability are found for all these types of solutions . , nonlinear coupling , peierls - nabarro potential , mobility , inversion of stability , modulational instability , power currents . 42.65.wi , 42.82.et , 63.20.pw , 05.60.-k
|
1601.01886
|
i
|
a _ repetition _ of length @xmath7 ( @xmath8 ) in a sequence of symbols is a subsequence of consecutive terms of the form @xmath9 . a sequence is _ nonrepetitive _ ( or _ square - free _ ) if it does not contain a repetition of any length . in 1906 thue proved that there exist arbitrarily long nonrepetitive sequences over an alphabet of size @xmath10 ( see @xcite ) . the method discovered by thue is constructive and uses substitutions over a given set of symbols . a different approach to creating long nonrepetitive sequences was recently introduced by grytczuk , kozik , and micek @xcite : generate a sequence by iteratively appending a random symbol at the end , and each time a repetition appears erase the repeated block . ( for instance , if the sequence generated so far is @xmath11 and we add @xmath12 , then we erase the last two symbols , bringing us back to @xmath13 . ) by a simple counting argument one can prove that with positive probability the length of the constructed sequence eventually exceeds any finite bound , provided the alphabet has size at least @xmath0 . this is one more than in thue s result but the proof is more flexible and can be adapted to other settings . for instance , it led to a very short proof that for every @xmath14 and every sequence of sets @xmath15 , each of size at least @xmath0 , there exists a nonrepetitive sequence @xmath16 where @xmath17 for all @xmath18 ( see @xcite ) , a theorem first proved by grytczuk , przybyo , and zhu @xcite via an intricate application of the lefthanded local lemma . whether the analogous statement for lists of size @xmath10 is true remains an exciting open problem . in this paper we make use of the above - mentioned approach to color trees nonrepetitively . given an ( undirected , simple ) graph @xmath19 , we denote by @xmath20 and @xmath21 its vertex set and edge set , respectively . a coloring @xmath22 of the vertices of @xmath19 is _ nonrepetitive _ if there is no repetition in the color sequence of any path in @xmath19 ; that is , @xmath23 is nonrepetitive if for every path @xmath24 with an even number of vertices the sequence of colors on the first half of @xmath24 is distinct from the sequence of colors on the second half of @xmath24 . ( we remark that all paths in this paper are simple , that is , contain no repeated vertex . ) the minimum number of colors used in a nonrepetitive coloring of @xmath19 is called the _ thue chromatic number _ of @xmath19 and is denoted by @xmath25 . now , given a graph @xmath19 , suppose that each vertex @xmath26 has a preassigned list of available colors @xmath27 . a coloring of @xmath19 with these lists is a coloring @xmath23 of @xmath19 such that @xmath28 for each vertex @xmath26 . the _ thue choice number _ of @xmath19 , denoted by @xmath29 , is the minimum @xmath2 such that , for every list assignment @xmath30 with @xmath31 for each @xmath26 , there is a nonrepetitive coloring of @xmath19 with these lists . similarly as for many graph coloring parameters , the thue chromatic ( choice ) number can be bounded from above by a function of the maximum degree : alon , grytczuk , hauszczak , and riordan @xcite proved that for every graph @xmath19 with maximum degree @xmath32 we have @xmath33 for some absolute constant @xmath12 . a number of subsequent works @xcite focused on reducing the value of the constant @xmath12 , the current best bound being @xmath34 ( see @xcite ) . et al . _ @xcite also showed that there are graphs with maximum degree @xmath32 with @xmath35 . ( whether this can be improved by a @xmath36 factor remains an open problem . ) it is not difficult to show that every tree has thue chromatic number at most @xmath0 ( see @xcite ) , which is best possible . this result was generalized to graphs of bounded treewidth by kndgen and pelsmajer @xcite . they proved that @xmath37 for every graph @xmath19 of treewidth @xmath4 . it is not known whether this upper bound can be improved to a polynomial in @xmath4 . however , if one considers graphs of pathwidth @xmath4 instead , a polynomial bound is known : it was shown by dujmovi _ _ @xcite that @xmath38 for every graph @xmath19 of pathwidth @xmath4 . ( we note that quadratic might not be the right order of magnitude here . ) probably the most intriguing open problem regarding the thue chromatic number is whether it is bounded for all planar graphs , a question originally asked by grytczuk @xcite . a @xmath39 upper bound is known @xcite , and from below ochem constructed a planar graph requiring @xmath40 colors ( see @xcite ) . the main focus of this paper is the list version of the parameter , the thue choice number . as mentioned at the beginning of the introduction , we have @xmath41 for every path @xmath24 , and it is open whether this bound can be improved to @xmath10 . fiorenzi , ochem , ossona de mendez , and zhu @xcite gave the first example of a class of graphs where the thue chromatic and thue choice numbers behave very differently : while trees have thue chromatic number at most @xmath0 , they showed that the thue choice number of trees is unbounded . clearly , trees with large thue choice number must have large maximum degree , and in fact one can deduce from the proof in @xcite that there are trees with maximum degree @xmath32 and thue choice number @xmath42 . kozik and micek @xcite subsequently showed that a better - than - quadratic upper bound in terms of the maximum degree exists for trees : for every @xmath43 there exists @xmath44 such that @xmath45 for every tree @xmath46 of maximum degree @xmath32 . ( bridging the significant gap between the upper and lower bounds remains an open problem . ) note that graphs of bounded treewidth have unbounded thue choice number since this is already the case for trees . on the other hand , dujmovi _ et al . _ @xcite observed that @xmath29 is bounded when @xmath19 is a graph of pathwidth @xmath47 . this prompted the authors of @xcite to ask whether @xmath29 is bounded more generally when @xmath19 has bounded pathwidth ( which is the case for the thue chromatic number ) . also , since connected graphs @xmath19 of pathwidth @xmath47 are caterpillars , and thus trees in particular , they also asked the same question but with @xmath19 moreover required to be a tree . a second motivation for the latter question was that the trees with arbitrarily large thue choice number constructed by fiorenzi _ et al . _ @xcite also have unbounded pathwidth . in this paper we answer both questions . first , we give a simple construction showing that the thue choice number is unbounded for graphs of bounded pathwidth ; in fact , this is true even for graphs of pathwidth @xmath6 ( which is best possible as noted above ) : [ thm : pw2 ] for every @xmath48 , there is a graph @xmath19 of pathwidth @xmath6 with @xmath49 . next , we address the case of trees and prove that their thue choice number is bounded from above by a function of their pathwidth : [ thm : tree ] there is a function @xmath50 such that @xmath51 for every tree @xmath46 of pathwidth @xmath4 . the proof of theorem [ thm : tree ] combines an induction on the pathwidth with the algorithmic method of grytczuk _ et al . _ @xcite to produce arbitrarily long nonrepetitive sequences described at the beginning of the introduction . this method , which finds its roots in the celebrated algorithmic proof of the local lemma by moser and tardos @xcite , was extended to produce nonrepetitive colorings of graphs ( in @xcite ) and trees ( in @xcite ) . part of our proof consists in adapting the ideas from @xcite to the situation under consideration . we note that the bounding function @xmath52 in theorem [ thm : tree ] stemming from our proof is quite large , it is doubly exponential in @xmath4 . the paper is organized as follows : in section [ sec : definitions ] we introduce definitions and terminology . then we prove theorem [ thm : pw2 ] in section [ sec : pw2 ] , and theorem [ thm : tree ] in section [ sec : trees ] .
|
a vertex coloring of a graph is _ nonrepetitive _ if there is no path in the graph whose first half receives the same sequence of colors as the second half . while every tree can be nonrepetitively colored with a bounded number of colors ( colors is enough ) , fiorenzi , ochem , ossona de mendez , and zhu recently showed that this does not extend to the list version of the problem , that is , for every _ : there exists a function such that every tree of pathwidth is nonrepetitively-choosable . we also show that such a property is specific to trees by constructing a family of pathwidth- graphs that are not nonrepetitively-choosable for any fixed .
|
a vertex coloring of a graph is _ nonrepetitive _ if there is no path in the graph whose first half receives the same sequence of colors as the second half . while every tree can be nonrepetitively colored with a bounded number of colors ( colors is enough ) , fiorenzi , ochem , ossona de mendez , and zhu recently showed that this does not extend to the list version of the problem , that is , for every there is a tree that is not nonrepetitively-choosable . in this paper we prove the following positive result , which complements the result of fiorenzi _ et al . _ : there exists a function such that every tree of pathwidth is nonrepetitively-choosable . we also show that such a property is specific to trees by constructing a family of pathwidth- graphs that are not nonrepetitively-choosable for any fixed .
|
0802.0326
|
c
|
ab dor , speedy mic and rst 137b represent young ( @xmath0 100myr ) rapidly rotating ( p@xmath7612hr ) late - type stars lying at the saturated - supersaturated corona boundary region . based on an analysis of high resolution _ chandra _ x - ray spectra of these stars and subsequent comparison with results for other active stars culled from the literature we draw the following conclusions . 1 . the temperature structures of ab dor , speedy mic and rst 137b all peak at @xmath30[k]@xmath107.0 - 7.1 , though the overall dem shapes are slightly different . if the dem trends observed here in only three stars can be generalised , they hint that as supersaturation is reached the dem slope below the temperature of peak dem becomes shallower , while the dem drop - off above this temperature becomes more pronounced . 2 . in the context of the larger stellar sample , we observe that in dwarf single and binary stars coronal thermal structure shows an increase in the emission of plasma at high temperatures ( @xmath77 ) as the rossby number decreases and approaches the saturated - supersaturated boundary . however , once the supersaturated region is reached this trend inverts ; supersaturated stars maintain a smaller fraction of coronal plasma at and above 10 million k than stars of higher rossby number . this result is consistent with the tentative generalised dem behaviour outlined in ( 1 ) . 3 . all three of the stars studied in detail here show evidence for an inverse of the solar - like fip effect , with smaller coronal abundances of the low fip elements mg , si and fe , relative to the high fip elements s , o and ne . this is consistent with existing coronal abundance studies of other active stars . the stellar sample shows that coronal fe abundance is inversely correlated with @xmath1 , and for dwarfs is also well - correlated with rossby number . the fe abundance is seen to decline slowly with rising @xmath1 , but declines sharply at @xmath4 . coronal o abundances average at values of [ o / h]@xmath78 . comparison of coronal and photospheric values for some of the sample suggests that active stellar coronae are in general slightly depleted in o relative to their photospheres . . the are no obvious trends of o abundance with activity indicators . derived coronal o abundances are perhaps very weakly correlated with the coronal temperature index @xmath49 with hotter coronae possibly exhibiting larger o abundances . rs cvn - type binaries exhibit systematically larger o abundances than dwarfs ; this could be partially due to galactic evolutionary differences in [ o / fe ] between dwarf and rs cvn samples . 7 the coronal o / fe ratio for dwarfs shows a strong trend of increasing o / fe with decreasing rossby number , and appears to saturate at the supersaturation boundary with a value of [ o / fe]@xmath79 . similar correlations are seen with o / fe increasing as a function of coronal temperature index , as revealed in earlier work , and with increasing @xmath1 . the range in o / fe variations attributable to differences in coronal properties among the sample is about a factor of 10 . dga and wb were supported by _ chandra _ grants go1 - 2006x and go1 - 2012x . ll was supported by nasa aisrp contract nag5 - 9322 ; we thank this program for providing financial assistance for the development of the pintofale package . we also thank the chianti project for making publicly available the results of their substantial effort in assembling atomic data useful for coronal plasma analysis . jjd and vk were supported by nasa contract nas8 - 39073 to the _ chandra_. , c. , barklem , p. s. , lambert , d. l. , & cunha , k. 2004 , , 420 , 183 , c. , drake , j. j. , maggio , a. , peres , g. , sciortino , s. , & harnden , f. r. 2004 , , 609 , 925 , m. , grevesse , n. , & sauval , a. j. 2005 , in asp conf . ser . 336 : cosmic abundances as records of stellar evolution and nucleosynthesis , 25 , m. , grevesse , n. , sauval , a. j. , allende prieto , c. , & kiselman , d. 2004 , , 417 , 751 , m. , g " udel , m. , sres , a. , raassen , a. j. j. , & mewe , r. 2003 , , 398 , 1137 , m. , telleschi , a. , gdel , m. , skinner , s. l. , pallavicini , r. , & mitra - kraev , u. 2004 , , 617 , 531 , b. , garca - alvarez , d. , drake , j. j. , lin , l. , & kashyap , v. l. 2007 , , submitted , j. r. 2005 , , 364 , 137 , j. r. , collier cameron , a. , james , d. j. , & donati , j .- f . 2001 , , 324 , 231 , t. , feltzing , s. , & lundstrm , i. 2004 , , 415 , 155 , s. v. , berdyugin , a. v. , ilyin , i. , & tuominen , i. 1998 , , 340 , 437 , a. m. 1989 , , 336 , 798 , a. c. , et al . 2001 , , 365 , l324 , g. e. , kellett , b. j. , jeffries , r. d. , innis , j. l. , matthews , l. , anders , g. j. , & coates , d. w. 1992 , in asp conf . ser . 26 : cool stars , stellar systems , and the sun , 80 , r. , cayrel de strobel , g. , & campbell , b. 1985 , , 146 , 249 , a. , bedford , d. k. , rucinski , s. m. , vilhu , o. , & white , n. e. 1988 , , 231 , 131 , a. & foing , b. h. 1997 , the observatory , 117 , 218 , g. , kurster , m. , pagano , i. , & rodono , m. 1997 , informational bulletin on variable stars , 4419 , 1 , k. p. , landi , e. , young , p. r. , & del zanna , g. 2001 , , 134 , 331 , j. j. 1998 , , 496 , l33 , j. j. 2003 , advances in space research , 32 , 945 . 2004 , chandra news , j. j. , brickhouse , n. s. , kashyap , v. , laming , j. m. , huenemoerder , d. p. , smith , r. , & wargelin , b. j. 2001 , , 548 , l81 , j. j. , chung , s .- m . , kashyap , v. l. , & garca - alvarez , d. 2007a , , submitted , j. j. , laming , j. m. , lin , l. , & kashyap , v. l. 2007b , , submitted , j. j. , laming , j. m. , & widing , k. g. 1997 , , 478 , 403 , j. j. & smith , g. 1993 , , 412 , 797 . 1993 , , 412 , 797 , j. j. & testa , p. 2005 , , 436 , 525 , b. r. & latour , j. 1978 , geophysical and astrophysical fluid dynamics , 9 , 241 , f. , reale , f. , micela , g. , sciortino , s. , maggio , a. , & matsumoto , h. 2000 , , 353 , 987 , m. , audard , m. , briggs , k. , haberl , f. , magee , h. , maggio , a. , mewe , r. , pallavicini , r. , & pye , j. 2001 , , 365 , l336 , d. , drake , j. j. , ball , b. , lin , l. , & kashyap , v. l. 2006 , , 638 , 1028 , d. , drake , j. j. , lin , l. , kashyap , v. l. , & ball , b. 2005 , , 621 , 1009 , t. , ottmann , r. , & reetz , j. 1999 , , 344 , 221 , n. & sauval , a. j. 1998 , space science reviews , 85 , 161 , m. 2004 , , 12 , 71 , m. , audard , m. , magee , h. , franciosini , e. , grosso , n. , cordova , f. a. , pallavicini , r. , & mewe , r. 2001 , , 365 , l344 , a. , schmitt , j. h. m. m. , schultz , m. , ruediger , g. , & stepien , k. 1995 , , 294 , 515 , d. p. , canizares , c. r. , drake , j. j. , & sanz - forcada , j. 2003 , , 595 , 1131 , d. p. , canizares , c. r. , & schulz , n. s. 2001 , , 559 , 1135 , d. p. , testa , p. , & buzasi , d. l. 2006 , , 650 , 1119 , j. l. , coates , d. w. , & thompson , k. 1988 , , 233 , 887 , m. 2004 , , 414 , l5 , m. & unruh , y. c. 1999 , , 346 , 883 , m. , brandner , w. , lenzen , r. , close , l. , nielsen , e. , hartung , m. , henning , t. & bouy , h. 2007 , , 462 , 615 , r. d. 1993 , , 262 , 369 , r. d. & maxted , p. f. l. 2005 , astronomische nachrichten , 326 , 944 , c. & montesinos , b. 1991 , , 252 , 21p , v. & drake , j. j. 1998 , , 503 , 450 . 2000 , bulletin of the astronomical society of india , 28 , 475 , j. r. , soderblom , d. r. , fischer , d. , & jones , b. f. 2000 , , 533 , 944 , m. , schmitt , j. h. m. m. , cutispoto , g. , & dennerl , k. 1997 , , 320 , 831 , j. m. 2004 , , 614 , 1063 , j. 1993 , , 405 , l33 , j. , montes , d. , crespo - chacn , i. & fernndez - figueroa , m. j. 2006 , , 643 , 1160 , k. l. , stauffer , j. r. , & mamajek , e. e. 2005 , , 628 , l69 , a. , drake , j. j. , kashyap , v. , harnden , jr . , f. r. , micela , g. , peres , g. , & sciortino , s. 2004 , , 613 , 548 , a. , pallavicini , r. , reale , f. , & tagliaferri , g. 2000 , , 356 , 627 , v. v. 2003 , , 126 , 1996 , a. , micela , g. , peres , g. , & sciortino , s. 2003 , , 407 , l63 , e. l. & brandner , w. 1995 , , 294 , 744 , p. , mazzitelli , g. , colafrancesco , s. , & vittorio , n. 1998 , , 133 , 403 , r. , kaastra , j. s. , white , s. m. , & pallavicini , r. 1996 , , 315 , 170 , d. , lpez - santiago , j. , glvez , m. c. , fernndez - figueroa , m. j. , de castro , e. , & cornide , m. 2001 , , 328 , 45 , h. w. , et al . 2000 , , 538 , l1 , t. , g. , m. , favata , f. , katz , d. , & pilliteri , i. 2003 , , s. a. , sandmann , w. s. , & pettersen , b. r. 1992 , , 104 , 1045 , b. , mayor , m. , andersen , j. , holmberg , j. , pont , f. , jrgensen , b. r. , olsen , e. h. , udry , s. , & mowlavi , n. 2004 , , 418 , 989 , r. w. , hartmann , l. w. , baliunas , s. l. , duncan , d. k. , & vaughan , a. h. 1984 , , 279 , 763 , v. g. and jilinski , e. and de la reza , r. & bazzanella , b. 2007 , , 377 , 441 , r. a. , ayres , t. r. , brown , a. , linsky , j. l. , & krishnamurthi , a. 2003 , , 582 , 1073 , r. , pfeiffer , m. j. , & gehren , t. 1998 , , 338 , 661 , m. w. 1981 , , 104 , 33 , m. a. c. , et al . 1997 , , 323 , l49 , c. f. , randich , s. , stauffer , j. r. , schmitt , j. h. m. m. , & simon , t. 1996 , , 112 , 1570 , s. 1998 , in asp conf . ser . 154 : cool stars , stellar systems , and the sun , 501 , s. , giampapa , m. s. , & pallavicini , r. 1994 , , 283 , 893 , j. & schmitt , j. h. m. m. 2005 , , 435 , 1073 , h. j. , flynn , c. , scalo , j. , hnninen , j. , maciel , w. j. , & hensler , g. 2004 , , 423 , 517 , s. m. , mewe , r. , kaastra , j. s. , vilhu , o. , & white , s. m. 1995 , , 449 , 900 , r. d. , neukirch , t. , & jardine , m. 2005 , , 433 , 323 , j. , favata , f. , & micela , g. 2004 , , 416 , 281 . 2006 , , 445 , 673 , j. , maggio , a. , & micela , g. 2003 , , 408 , 1087 , j. h. m. m. 1990 , advances in space research , 10 , 115 , s. c. , king , j. r. , hobbs , l. m. , & pinsonneault , m. h. 2004 , , 602 , l117 , k. p. , drake , s. a. , gotthelf , e. v. , & white , n. e. 1999 , , 512 , 874 , a. 1972 , , 171 , 565 , j. r. & hartmann , l. w. 1987 , , 318 , 337 , j. r. , hartmann , l. w. , prosser , c. f. , randich , s. , balachandran , s. , patten , b. m. , simon , t. , & giampapa , m. 1997 , , 479 , 776 , k. , schmitt , j. h. m. m. , & voges , w. 2001 , , 370 , 157 , b. , burwitz , v. , audard , m. , gdel , m. , ness , j .- u . , grosso , n. , neuhuser , r. , schmitt , j. h. m. m. , predehl , p. , & aschenbach , b. 2002 , , 392 , 585 , a. , gdel , m. , briggs , k. , audard , m. , ness , j .- u . , & skinner , s. l. 2005 , , 622 , 653 , o. & monier , r. 1999 , , 351 , 247 , o. , ambruster , c. w. , neff , j. e. , linsky , j. l. , brandenburg , a. , ilyin , i. v. , & shakhovskaya , n. i. 1989 , , 222 , 179 , o. , gustafsson , b. , & edvardsson , b. 1987 , , 320 , 850 , o. , gustafsson , b. , & walter , f. m. 1991 , , 241 , 167 , o. & linsky , j. l. 1987 , , 99 , 1071 , o. , muhli , p. , mewe , r. , & hakala , p. 2001 , , 375 , 492 , b. e. & linsky , j. l. 2006 , , 643 , 444 , b. , song , i. , & bessell , m. s. 2004 , , 613 , l65
|
plasma emission measure distributions as a function of temperature show broad peaks at k. differences between stars suggest that as supersaturation is reached the dem slope below the temperature of peak dem becomes shallower , while the dem drop - off above this temperature becomes more pronounced . a larger sample comprising our three targets and 22 active stars studied in the recent literature reveals a general increase of plasma at toward the saturated - supersaturated boundary but a decline beyond this among supersaturated stars . all three of the stars studied in detail here show lower coronal abundances of the low fip elements mg , si and fe , relative to the high fip elements s , o and ne , as compared to the solar mixture . the coronal fe abundances of the stellar sample are inversely correlated with , declining slowly with rising , but with a much more sharp decline at . for dwarfs the fe abundance is also well - correlated with rossby number . the coronal o / fe ratios for dwarfs show a clear increase with decreasing rossby number , apparently reaching saturation at [ o / fe]=0.5 at the coronal supersaturation boundary . the range in o / fe variations attributable to outer atmosphere chemical fractionation in our sample is about a factor of 10 .
|
ab dor , speedy mic and rst 137b are in their early post - t tauri evolutionary phase ( 100 myr ) , at the age of fastest rotation in the life of late - type stars . they straddle the coronal saturation - supersaturation boundary first defined by young stars in open clusters . high resolution _ chandra _ x - ray spectra have been analysed to study their coronal properties as a function of coronal activity parameters _ rossby number _ , and a coronal temperature index . plasma emission measure distributions as a function of temperature show broad peaks at k. differences between stars suggest that as supersaturation is reached the dem slope below the temperature of peak dem becomes shallower , while the dem drop - off above this temperature becomes more pronounced . a larger sample comprising our three targets and 22 active stars studied in the recent literature reveals a general increase of plasma at toward the saturated - supersaturated boundary but a decline beyond this among supersaturated stars . all three of the stars studied in detail here show lower coronal abundances of the low fip elements mg , si and fe , relative to the high fip elements s , o and ne , as compared to the solar mixture . the coronal fe abundances of the stellar sample are inversely correlated with , declining slowly with rising , but with a much more sharp decline at . for dwarfs the fe abundance is also well - correlated with rossby number . coronal o abundances appear lower than photospheric expectations by up to dex , but with no obvious trends with activity indices . the coronal o / fe ratios for dwarfs show a clear increase with decreasing rossby number , apparently reaching saturation at [ o / fe]=0.5 at the coronal supersaturation boundary . similar increases in o / fe with increasing coronal temperature and are seen . the range in o / fe variations attributable to outer atmosphere chemical fractionation in our sample is about a factor of 10 .
|
0708.2818
|
i
|
ads / cft correspondence @xcite is a useful framework to analyze strongly coupled yang - mills ( ym ) theories , and its application to quark - hadron physics is one of the important subjects from the phenomenological point of view . since macroscopic properties of quark - hadron systems , based on thermodynamics and hydrodynamics , are as important as their microscopic nature , it is quite significant to establish holographic descriptions of thermodynamics and hydrodynamics of the ym theories . finite - temperature ads / cft has been initiated in ref . hydrodynamic quantities of static ym - theory plasma have been computed in ads / cft ( see , for example , reviews @xcite and the references therein ) , holographic description of hydrodynamics of time - dependent ym - theory fluid has been investigated in refs . @xcite . for a complete description of the thermodynamic and hydrodynamic properties , we need to introduce chemical potentials of the conserved charges to the framework . since the lattice gauge theory has a technical difficulty in introduction of finite baryon chemical potential , it is quite significant to establish a holographic description of baryon chemical potential . a holographic description of r - charge chemical potential has been proposed in refs . @xcite and isospin chemical potential in ads / cft has also been studied in refs . however attempts to introduce a baryon chemical potential to ads / cft have been just started recently @xcite , and there still exists a point which is under debate @xcite . the issue under the question is existence of so - called minkowski - embedding phase in the d3-d7 systems @xcite . for recent studies on finite baryon density systems in holographic frameworks , @xcite . in this paper , we present a closed framework of ads / cft with finite @xmath0-charge chemical potential .- charge chemical potential and baryon chemical potential . see for details , section [ discussion ] . ] although our approach has wide overlap with what have been discussed in refs . @xcite , the following points will be clarified in this paper : * a standard dictionary of ads / cft implies that the chemical potential is given by the boundary value of the zeroth component of the bulk @xmath1-gauge field @xcite . however , this identification is not manifestly gauge invariant . a manifestly gauge - invariant identification of the chemical potential and the @xmath1-gauge field has been proposed in refs . we clarify how the gauge - invariant formulation emerges starting from the standard ads / cft dictionary . * the minkowski embeddings at finite baryon density is claimed to be unphysical in ref . however , we will show their necessity and physical significance within the context of our framework . it will be shown that the model with only the black - hole embeddings proposed in ref . @xcite has a serious problem : the model lacks the low - temperature and the low - chemical - potential region of the parameter space in the grand - canonical ensemble . we call this `` incomplete - ness problem '' in this paper . * the incomplete - ness of the model with only the black - hole embeddings can also be seen in terms of thermodynamic instability in the canonical ensemble . there is a parameter region where thermodynamically stable black - hole embeddings do not exist in the canonical ensemble . the minkowski embeddings provide a stable final state in that case . * we will present a possible physical picture which clarify the difference between the model of ref . @xcite and that of this paper . we will also propose an idea which may remedy the incomplete - ness problem in the model of ref . @xcite . although we will work out along the d3-d7 systems , the formalism of the @xmath0-charge chemical potential given in this paper is applicable to general setups of ads / cft with flavor branes . the organization of the paper is as follows . in section [ basic ] , we present a basic setup and notations in our framework . a closed formulation of ads / cft with finite @xmath0-charge chemical potential is given in section [ formulation ] . the numerical results that support consistency of our framework is presented in section [ numerical ] . the incomplete - ness of the model proposed in ref . @xcite is also pointed out there . in section [ evidences ] , we re - examine the importance of the minkowski embeddings from the viewpoint of thermodynamic stability in the canonical ensemble . in the discussion section , we propose a possible physical interpretation of our framework . we also discuss how the discrepancy between ref . @xcite and ref . @xcite can be interpreted . a possible improvement to remedy the incomplete - ness problem of the model of ref . @xcite is also discussed .
|
abstract we present a closed framework of ads / cft with finite-charge chemical potential . we show how the gauge - invariant identification of the chemical potential with the bulk gauge field emerges from the standard ads / cft dictionary . physical importance and necessity of the minkowski embeddings within the present framework is also shown numerically in the d3-d7 systems .
|
abstract we present a closed framework of ads / cft with finite-charge chemical potential . we show how the gauge - invariant identification of the chemical potential with the bulk gauge field emerges from the standard ads / cft dictionary . physical importance and necessity of the minkowski embeddings within the present framework is also shown numerically in the d3-d7 systems . we point out that the d3-d7 model with only the black - hole embeddings does not have the low - temperature and low - chemical - potential region in the grand - canonical ensemble , hence it is incomplete . a physical interpretation that explains these numerical results is also proposed . 0.5 cm
|
0708.2818
|
r
|
now , we would like to bring our attention to the claim presented in ref . @xcite . the authors of ref . @xcite pointed out the necessity of the charged source at @xmath117 , and they provided it by adding @xmath0-charge carrying objects to the system . their idea is to put fundamental strings ( f1 s ) between the d7-branes and the black - hole horizon , that are interpreted as quarks . then , they found that the minkowski embeddings in ref . @xcite are unstable due to the tension of the f1 s and there is no way to keep the d7-branes off the horizon . this is why the minkowski embeddings at finite baryon - number density are concluded to be unphysical in ref . @xcite . however , one should notice that this is not what we are doing in the present paper . the necessary charged source term has been introduced as an _ external _ source to the d7-brane dbi theory at ( [ d7+source ] ) and we have not added any corresponding nambu - goto action of the f1 there . this means the system we are dealing with is something different from that in ref . since we have not introduced the additional f1 s , we expect that in the present framework , the minkowski embeddings are physical as well . in this section , we will show that it is indeed the case . we will present numerical results , @xmath118 so that @xmath119 and @xmath120 in the numerical analysis . the numerical values of the free - energy densities @xmath102 and @xmath90 are in the unit of @xmath121 , those of the chemical potential and @xmath47 are in the unit of @xmath122 and @xmath49 , respectively . the unit of the entropy density and the quark condensate is also @xmath49 in the numerical analysis . ] which indicate consistency of the present formalism and the necessity of the minkowski embeddings . a physical interpretation of our setup will be proposed in section [ discussion ] . in order to show a consistency of our framework , we will employ @xmath123-@xmath47 diagrams where the relationship between @xmath60 and @xmath47 obtained from ( [ f - tilde ] ) is drawn . ( we may use @xmath123 as the meaning of @xmath60 . ) let us present basic explanations on the @xmath123-@xmath47 diagram in this subsection as preparation . an example of the @xmath123-@xmath47 diagram is given at fig . [ fig : ptq01 ] ( or at fig . [ fig : ptm01 ] ) @xcite . the d7-brane solutions that belong to the minkowski embeddings start from @xmath124 ( which is the origin of the plane ) and go through @xmath125 and @xmath126 until @xmath127 where the line meets the vertical axes . the black - hole embeddings exactly start from @xmath127 where the minkowski embeddings terminate , and go through @xmath128 , @xmath102 , @xmath129 , @xmath130 , @xmath131 and extend to the large-@xmath47 and large-@xmath123 region . it is worthwhile checking a consistency of the @xmath123-@xmath47 diagram . [ fig : fq01 ] and fig . [ fig : am01 ] show the relationship between the helmholtz free energy density and @xmath47 , and that between the grand potential density and @xmath123 , respectively . although the thermodynamic potentials obtained from all the possible solutions are indicated , what we should take is the line which has the minimum value . then , we find first - order phase transitions . let us see fig . [ fig : am01 ] , for example . the phase transition is a jump between a minkowski embedding and a black - hole embedding at the critical chemical potential @xmath132 . since the grand potentials in both embeddings have the same values at @xmath132 , the integral @xmath133 below @xmath132 and that above @xmath132 have to be same because the integrals compute @xmath134 by virtue of @xmath135 . this is the maxwell construction . one can see that the maxwell construction works in a non - trivial way in good accuracy in fig . [ fig : ptm01 ] . we have checked numerically that the area of the shaded regions marked `` @xmath136 '' agrees with the area of the regions with `` @xmath137 '' , due to a non - trivial collaboration of the minkowski and the black - hole embeddings . we can also see that the maxwell construction works in the canonical ensemble in fig . [ fig : ptq01 ] , again by virtue of the collaboration of the two types of the embeddings . now , we are ready to make some important comments based on the numerical results . fig . [ fig : muq ] shows that how the @xmath123-@xmath47 diagram is deformed if we make the temperature lower . the temperature goes down as we move from fig . [ fig : muq01 ] to fig . [ fig : muq04 ] . an important feature is that the cusp @xmath130 located at the origin on fig . [ fig : muq01 ] `` goes up '' ( fig . [ fig : muq02 ] ) and disappears ( fig . [ fig : muq03 ] and fig . [ fig : muq04 ] ) along the cooling process . the cusp on fig . [ fig : muq02 ] gives the minimum value of the chemical potential within the black - hole embeddings . this means that the low - chemical - potential region disappears from the parameter space of the theory at the sufficiently low temperature if we abandon the minkowski embeddings . we can also interpret the behaviour of the diagram in a different way . suppose that we are in the grand - canonical ensemble and we examine a process in which we vary the temperature with maintaining the chemical potential . then , we encounter a problem that there is no low - temperature region in our parameter space at sufficiently small chemical potential . the lack of the low - temperature region can be seen more clearly in fig . [ mubranch ] , where the d7-brane solutions on the @xmath41-@xmath138 plane are given . the minkowski embeddings ( @xmath39 ) and the black - hole embeddings ( @xmath139 ) are connected at @xmath140 there . notice that all the temperature region is covered by using the two types of embeddings . however , if we abandon the minkowski embeddings , the theory can not cover the low - temperature region . ( recall that @xmath141 . ) we can conclude from the above that if we abandon the minwkowski embeddings , the low - temperature and the low - chemical - potential region in the grand - canonical ensemble does not exist in the formalism ; we have no way to introduce the flavor degree of freedom in that region . this is an incomplete - ness of the formalism . on the other hand , if we include the minkowski embeddings together with the black - hole embeddings , all the parameter region of the theory is covered in harmony of the two types of embeddings . we can also see that the two embeddings are connected into a single family of the solutions on fig . [ fig : ptc ] , fig . [ fig : muq ] and fig . [ mubranch ] . we present some numerical results in the grand - canonical ensemble in appendix [ grand - more ] to make the above points vivid . incidentally , we can understand why we did not encounter the above `` incomplete - ness '' problem manifestly in ref . let us look at fig . [ fig : muq ] again . we notice that the entire region of @xmath47 can always be covered even if we use only the black - hole branch . this means that if we examine the process in which we vary the temperature with fixing @xmath47 , we have always at least one solution inside the black - hole branch at any temperature ; all the region on the @xmath47-@xmath142 plane can be covered even if we use only the black - hole embeddings . namely , we do not have a missing parameter region in the _ canonical _ ensemble . the reason why we do not see the incomplete - ness in ref . @xcite in a manifest way is that the analysis is given in the canonical ensemble there . however , our claim is that the canonical ensemble with only the black - hole embeddings is transformed to the incomplete formalism of the grand - canonical ensemble through the legendre transformation : the formalism lacks something necessary even in the canonical ensemble . in the next section , we will see on this point and we will find the minkowski embeddings play an important role in the canonical ensemble as well .
|
we point out that the d3-d7 model with only the black - hole embeddings does not have the low - temperature and low - chemical - potential region in the grand - canonical ensemble , hence it is incomplete . a physical interpretation that explains these numerical results is also proposed . 0.5 cm
|
abstract we present a closed framework of ads / cft with finite-charge chemical potential . we show how the gauge - invariant identification of the chemical potential with the bulk gauge field emerges from the standard ads / cft dictionary . physical importance and necessity of the minkowski embeddings within the present framework is also shown numerically in the d3-d7 systems . we point out that the d3-d7 model with only the black - hole embeddings does not have the low - temperature and low - chemical - potential region in the grand - canonical ensemble , hence it is incomplete . a physical interpretation that explains these numerical results is also proposed . 0.5 cm
|
astro-ph0405170
|
i
|
most galactic nuclei are now believed to harbor supermassive black holes . these black holes are all accreting gas at a minimum from the interstellar medium proximate to the event horizon , and in some cases from dense disks of gas in orbit around the hole . the observational signatures of this accretion , however , differ dramatically from galaxy to galaxy . the black hole in the galactic center , for example , has an x - ray luminosity in quiescence of only @xmath0 , reaching @xmath1 during flaring states ( @xcite , 2003 ; @xcite ) . many nearby elliptical galaxies also show nuclear emission that is much weaker than might be expected on the basis of simple estimates of the black hole accretion rate ( @xcite ; @xcite ; @xcite ) . in stark contrast to these feeble displays , accretion onto black holes in quasars powers the most luminous steady sources in the universe . at a minimum , a theory of accretion needs to account for this dichotomy , and to explain at least some of the many secondary phenomena ( jets , outflows , variability , etc . ) associated with active galactic nuclei ( agn ) . more ambitiously , one might hope to understand the role of accretion in actually forming supermassive black holes . black hole formation is a difficult theoretical problem which , although currently untroubled by direct observations , is receiving increasing attention . in this chapter , i discuss selected aspects of the theory of accretion onto supermassive black holes , with an emphasis on the physical processes that drive accretion and determine the qualitative properties of the flow . in [ secpjatransport ] , i outline the mechanisms which can lead to angular momentum transport within an accretion flow , thereby allowing rotationally supported gas to flow inward and liberate gravitational potential energy . close to the black hole , turbulence driven by magnetohydrodynamic instabilities is probably the dominant mechanism for transport . further out , other processes , such as gravitational instabilities , are likely to assume that role . angular momentum transport , however , is only part of the story . although central to our understanding of accretion , knowledge of its origin does not suffice to explain why the galactic center looks nothing like a powerful agn . for that we need to consider the distinction between geometrically thin accretion disks , in which the gas can radiate efficiently and cool to sub - virial temperatures , and hot thick disks , which are radiatively inefficient and vulnerable to rapid mass loss . these questions are addressed in [ secpjaaccrate ] . subsequent sections examine the status of several open questions in the study of black hole accretion , including the dynamics of gas executing its final plunge into the hole , the origin of variability , and the evolution of disks that are warped or eccentric .
|
accretion onto supermassive black holes produces both the dramatic phenomena associated with active galactic nuclei and the underwhelming displays seen in the galactic center and most other nearby galaxies . i review selected aspects of the current theoretical understanding of black hole accretion , emphasizing the role of magnetohydrodynamic turbulence and gravitational instabilities in driving the actual accretion and the importance of the efficacy of cooling in determining the structure and observational appearance of the accretion flow . ongoing investigations into the dynamics of the plunging region , the origin of variability in the accretion process , and the evolution of warped , twisted , or eccentric disks are summarized .
|
accretion onto supermassive black holes produces both the dramatic phenomena associated with active galactic nuclei and the underwhelming displays seen in the galactic center and most other nearby galaxies . i review selected aspects of the current theoretical understanding of black hole accretion , emphasizing the role of magnetohydrodynamic turbulence and gravitational instabilities in driving the actual accretion and the importance of the efficacy of cooling in determining the structure and observational appearance of the accretion flow . ongoing investigations into the dynamics of the plunging region , the origin of variability in the accretion process , and the evolution of warped , twisted , or eccentric disks are summarized .
|
1404.0184
|
i
|
the first modern explanation of the chemical bond between two neutral hydrogen atoms was put forth in 1927 by heitler and london @xcite , and is one of the earliest applications of quantum theory , specifically that of schrdinger s wave mechanics formulation in 1926 @xcite . heitler and london showed that by accounting for pauli s exclusion principle @xcite in combining atomic hydrogen wavefunctions to construct molecular wavefunctions , the existence of a bound molecular state is explained . despite their calculated binding energy being off by some 30% from the contemporary experimental value , their pioneering quantum mechanical calculation for the stability of molecular hydrogen ushered the era of quantum chemistry . it is interesting to note that the born - oppenheimer approximation @xcite was also proposed in 1927 , and this approach of separating electronic and nuclear motions has largely shaped molecular theory since . the next breakthrough in _ ab initio _ potential calculations for h@xmath0 was achieved by james and coolidge in 1933 in their treatment of the ( @xmath8 ) ground state @xcite . using two - electron wave functions with explicitly correlated electrons , an approach introduced by hylleraas for the helium atom @xcite , they transcended the concept of electrons being in individual states as used in the hartree - fock method . the james - coolidge solution relied on the variational method to determine the correct nonlinear parameters in combining the wave functions . with a set of only 13 of these wave functions , taken as a truncated basis to represent the total hilbert space of infinite dimension , they improved the minimum energy in the born - oppenheimer potential of the @xmath9 state to @xmath10 @xmath2 . this was a substantial improvement of about @xmath11 @xmath2 with respect to the best theoretical values available at the time . over the years improvements on the accuracy has been obtained @xcite , and important methodical reviews can be found in refs . the achievement of the initial studies of james and coolidge @xcite can best be appreciated considering that further improvement in the calculated potential has been only 222 @xmath2 since then , obtained by wolniewicz in 1995 with essentially the same method but with a basis of 883 wave functions @xcite . at present , the born - oppenheimer potential energy can be evaluated to accuracies better than 15 digits using more than 22,000 basis functions @xcite , made possible by developments in numerical procedures and improvements in computing power . the precision of the calculated born - oppenheimer energy may be considered exact for the purpose of comparisons with experiment . corrections beyond the born - oppenheimer approximation need to be evaluated to improve upon the accuracy of the _ ab initio _ values . in addition to adiabatic and nonadiabatic effects comprising the non - relativistic born - oppenheimer corrections , it is also necessary to account for accurate relativistic and radiative or quantum electrodynamic ( qed ) corrections . until up to 2010 , the work of wolniewicz @xcite that included estimates of radiative corrections , had constituted the state - of - the - art for calculations of level energies in the @xmath8 ground state of molecular hydrogen . this led to a calculated energy of the actual ground state ( or equivalently the dissociation limit ) to an accuracy 0.01 @xmath2 . the recent work of pachucki , komasa and co - workers has achieved breakthroughs in the evaluation of nonadiabatic effects @xcite as well as relativistic and radiative corrections @xcite , resulting in accurate level energies of @xmath8 rovibrational levels @xcite . in fig . [ energycontribution ] the different contributions to the level energy of the lowest quantum state ( @xmath12 ) with respect to the dissociation energy of molecular hydrogen are represented graphically to give an impression of the scale of the corrections . ) as corrections to the born - oppenheimer approximation level energy , with respect to the dissociation limit , of the @xmath13 , @xmath14 state for h@xmath15 , hd and d@xmath15 . bo : born - oppenheimer energy ; ad : adiabatic ; nad : nonadiabatic ; rel : relativistic ; qed : radiative corrections . ] theoretical and experimental efforts on the determination of ground state level energies in molecular hydrogen mutually stimulated improvements on both fronts as soon as more accurate values were obtained . as an illustration , consider the dissociation energy of the @xmath8 ground state , a benchmark quantity for the comparison of experiment and theory . the experimental determination of the dissociation energy by witmer @xcite in 1926 already gave results within 3@xmath16 of the modern value , an order of magnitude better than the heitler - london calculations as mentioned . the james - coolidge calculations in 1933 @xcite resulted in a dissociation energy that is within @xmath17 of the present value , matched later by the experimental determination by beutler in 1935 @xcite that was also accurate to within @xmath17 . this lively dynamics continued through the 1960s-1970s , between the experimental efforts of herzberg and co - workers @xcite and theoretical efforts by koos and co - workers @xcite . in the middle of the 1990s , the theoretical result of wolniewicz @xcite for the dissociation energy was expressed in 8 significant digits , although the uncertainty was not explicitly mentioned . eyler and co - workers determined the dissociation limit to an accuracy of 0.01 @xmath2 @xcite in 2004 , improving upon their previous result @xcite using the same method . the most accurate experimental dissociation energy for h@xmath15 was obtained in 2009 by liu _ et al . _ @xcite , and later extended to d@xmath15 @xcite and hd @xcite . remarkably , accurate theoretical values for h@xmath15 and d@xmath15 dissociation energies @xcite as well as hd @xcite were presented a short time thereafter . _ ab initio _ theory can also be tested through a comparison with the experimental determinations of level splittings in the rovibrational manifold of the ground state . herzberg first predicted , in 1938 , that it should be possible to record rovibrational transitions in the ground state manifold @xcite , and later discovered the quadrupole spectrum in 1949 by photographing a total of eight lines in the ( 2,0 ) and ( 3,0 ) bands @xcite . subsequently the quadrupole spectrum including the fundamental ( 1,0 ) band was investigated by several other groups , for example by rank and co - workers @xcite . the measurements by bragg _ et al . _ @xcite greatly improved the accuracy of the spectroscopy of the quadrupole bands and was until recent years considered as the most accurate work on the direct measurement of the vibrational splittings . laser - based direct excitation of the weaker ( 4,0 ) and ( 5,0 ) overtone quadrupole bands was performed in the visible domain @xcite . later investigations using cavity - ring down spectroscopy on the h@xmath15 ( 3,0 ) overtone band were carried out by robie _ et al . _ @xcite using a pulsed source and hu _ et al . _ @xcite using a cw source . campargue and co - workers have recently performed high - resolution determinations of the ( 2,0 ) overtone bands of h@xmath15 @xcite and d@xmath15 @xcite using quantum cascade lasers . maddaloni _ et al . _ @xcite performed precision measurements using cavity - ring down techniques for the fundamental band of d@xmath15 . the quadrupole excitations in the ground electronic state described above have very low transition probabilities . the ground state energy splittings can be determined indirectly from appropriate combinations of dipole - allowed transitions between ground state and excited electronic states . for example , the strongest molecular hydrogen transitions in lyman ( @xmath18 @xmath19 ) and werner ( @xmath20@xmath19 ) bands have been used to derive ground state rovibrational constants . using this approach , stanke _ et al . _ @xcite derived accurate ground state molecular constants based largely on the experimental data of dabrowski @xcite but also including quadrupolar transitions . the natural linewidths of transitions in the lyman and werner bands ultimately limit the accuracy that can be achieved @xcite . in contrast , the rovibrational levels of the lowest - lying excited singlet gerade state @xmath21 of molecular hydrogen have longer natural lifetimes , even up to 150 ns @xcite , since one - photon transitions to the ground state are forbidden . the gerade states can be accessed from the ground state through two - photon spectroscopies , which also allow for more accurate level energy determinations . this first excited singlet gerade state in molecular hydrogen , the @xmath22 state , shown to correspond to a double - well potential @xcite , has been investigated thoroughly over the years . eyler and coworkers performed a number of laser spectroscopic studies of increasing accuracy @xcite . a determination of frequencies of q - branch transitions in the lowest @xmath23 ( 0,0 ) band was performed with improved accuracy by hannemann _ et al . _ the lowest rotational levels in the @xmath22 state derived from the latter study were used as anchor lines , to which a large number of levels in the excited state manifold , obtained from high - resolution fourier - transform studies , were connected to the ground state @xcite . accurate values for level energies of the high rotational states up to @xmath24 in the @xmath25 electronic state were obtained in ref . @xcite using uv two - photon spectroscopy . in this paper , we present accurate experimental and theoretical values for the fundamental vibrational splitting of h@xmath15 , d@xmath15 and hd . this extends a recent report @xcite on the _ rotationless _ ground tone frequencies of hydrogen and its isotopomers , now also including values for @xmath26 and @xmath27 levels . the experimental determination of the fundamental vibrational splitting is based on combination differences of the transition frequencies between the @xmath13 and @xmath21 states , measured by two - photon doppler - free spectroscopy . relies on the measurement of q - line transitions the @xmath23 ( 0,1 ) band obtained in the present study and on the measurement of the q - lines in the @xmath23 ( 0,0 ) band , obtained in ref . the excitation channels are indicated by ( 1 ) and ( 0 ) , respectively . an auxiliary laser beam of 355-nm radiation was used in the rempi detection scheme . the squared moduli of the vibrational wavefunctions are indicated in the inset . ]
|
accurate experimental values for the vibrational ground tone or fundamental vibrational energy splitting of h , hd , and d are presented . a comparison is made with full _ ab initio _ calculations encompassing born - oppenheimer energies , adiabatic and non - adiabatic corrections , as well as relativistic corrections and qed - contributions . astron .
|
accurate experimental values for the vibrational ground tone or fundamental vibrational energy splitting of h , hd , and d are presented . absolute accuracies of are obtained from doppler - free laser spectroscopy applied in a collisionless environment . the vibrational splitting frequencies are derived from the combination difference between separate electronic excitations from the and vibrational states to a common state . the present work on rotational quantum states extends the results reported by dickenson _ et al . _ on [ phys . rev . lett . 110 ( 2013 ) 193601 ] . the experimental procedures leading to this high accuracy are discussed in detail . a comparison is made with full _ ab initio _ calculations encompassing born - oppenheimer energies , adiabatic and non - adiabatic corrections , as well as relativistic corrections and qed - contributions . the present agreement between the experimental results and the calculations provides a stringent test on the application of quantum electrodynamics in molecules . furthermore , the combined experimental - theoretical uncertainty can be interpreted to provide bounds to new interactions beyond the standard model of physics or _ fifth forces _ between hadrons . astron . astroph . molecular hydrogen , fundamental vibration , uv spectroscopy , test of qed
|
1404.0184
|
c
|
the fundamental vibrational energy splitting of h@xmath0 , hd , and d@xmath0 were determined to absolute accuracies of @xmath1 @xmath2 , or a relative accuracy of a few parts in @xmath105 . the vibrational splitting frequencies are derived from the combination difference between separate electronic excitations from the @xmath3 and @xmath4 vibrational states to a common @xmath5 state . doppler - free laser spectroscopic investigation applied in a collisionless molecular beam environment leads to high accuracy , where pressure effects are negligible in contrast to studies based on gas cells . the excellent agreement between the experimental results and the calculations provides a stringent test on the application of quantum electrodynamics in molecules , and can be used to provide bounds to new interactions . upper bounds derived from molecular hydrogen indicate that the interaction strength of possible fifth forces must be at least 8 orders of magnitude weaker than the electromagnetic strength , for a fifth - force interaction range in the order of typical internuclear distances of @xmath106 . this brings molecular spectroscopy studies again to the forefront of physics , reminiscent of the early days of quantum mechanics .
|
absolute accuracies of are obtained from doppler - free laser spectroscopy applied in a collisionless environment . the vibrational splitting frequencies are derived from the combination difference between separate electronic excitations from the and vibrational states to a common state . the experimental procedures leading to this high accuracy are discussed in detail . the present agreement between the experimental results and the calculations provides a stringent test on the application of quantum electrodynamics in molecules .
|
accurate experimental values for the vibrational ground tone or fundamental vibrational energy splitting of h , hd , and d are presented . absolute accuracies of are obtained from doppler - free laser spectroscopy applied in a collisionless environment . the vibrational splitting frequencies are derived from the combination difference between separate electronic excitations from the and vibrational states to a common state . the present work on rotational quantum states extends the results reported by dickenson _ et al . _ on [ phys . rev . lett . 110 ( 2013 ) 193601 ] . the experimental procedures leading to this high accuracy are discussed in detail . a comparison is made with full _ ab initio _ calculations encompassing born - oppenheimer energies , adiabatic and non - adiabatic corrections , as well as relativistic corrections and qed - contributions . the present agreement between the experimental results and the calculations provides a stringent test on the application of quantum electrodynamics in molecules . furthermore , the combined experimental - theoretical uncertainty can be interpreted to provide bounds to new interactions beyond the standard model of physics or _ fifth forces _ between hadrons . astron . astroph . molecular hydrogen , fundamental vibration , uv spectroscopy , test of qed
|
1112.6212
|
i
|
adaptive network consists of a collection of agents that are interconnected to each other and solve distributed estimation and inference problems in a collaborative manner . two useful strategies that enable adaptation and learning over such networks in real - time are the incremental strategy @xcite and the diffusion strategy @xcite . incremental strategies rely on the use of a hamiltonian cycle , i.e. , a cyclic path that covers all nodes in the network , which is generally difficult to enforce since determining a hamiltonian cycle is an np - hard problem . in addition , cyclic trajectories are not robust to node or link failure . in comparison , diffusion strategies are scalable , robust , and able to match well the performance of incremental networks . in adaptive diffusion implementations , information is processed locally at the nodes and then diffused in real - time across the network . diffusion strategies were originally proposed in @xcite and further extended and studied in @xcite . they have been applied to model self - organized and complex behavior encountered in biological networks , such as fish schooling @xcite , bird flight formations @xcite , and bee swarming @xcite . diffusion strategies have also been applied to online learning of gaussian mixture models @xcite and to general distributed optimization problems @xcite . there have also been several useful works in the literature on distributed consensus - type strategies , with application to multi - agent formations and distributed processing @xcite . the main difference between these works and the diffusion approach of @xcite is the latter s emphasis on the role of adaptation and learning over networks . in the original diffusion least - mean - squares ( lms ) strategy @xcite , the weight estimates that are exchanged among the nodes can be subject to quantization errors and additive noise over the communication links . studying the degradation in mean - square performance that results from these particular perturbations can be pursued , for both incremental and diffusion strategies , by extending the mean - square analysis already presented in @xcite , in the same manner that the tracking analysis of conventional stand - alone adaptive filters was obtained from the counterpart results in the stationary case ( as explained in ( * ? ? ? * ch . 21 ) ) . useful results along these lines , which study the effect of link noise during the exchange of the weight estimates , already appear for the traditional diffusion algorithm in the works @xcite and for consensus - based algorithms in @xcite . in this paper , our objective is to go beyond these earlier studies by taking into account additional effects , and by considering a more general algorithmic structure . the reason for this level of generality is because the analytical results will help reveal which noise sources influence the network performance more seriously , in what manner , and at what stage of the adaptation process . the results will suggest important remedies and mechanisms to adapt the combination weights in real - time . some of these insights are hard to get if one focuses solely on noise during the exchange of the weight estimates . the analysis will further show that noise during the exchange of the regression data plays a more critical role than other sources of imperfection : this particular noise alters the learning dynamics and modes of the network , and biases the weight estimates . noises related to the exchange of other pieces of information do not alter the dynamics of the network but contribute to the deterioration of the network performance . to arrive at these results , in this paper , we first consider a generalized analysis that applies to a broad class of diffusion adaptation strategies ( see further ahead ; this class includes the original diffusion strategies and as two special cases ) . the analysis allows us to account for various sources of information noise over the communication links . we allow for noisy exchanges during _ each _ of the three processing steps of the adaptive diffusion algorithm ( the two combination steps and and the adaptation step ) . in this way , we are able to examine how the three sets of combination coefficients @xmath0 in influence the propagation of the noise signals through the network dynamics . our results further reveal how the network mean - square - error performance is dependent on these combination weights . following this line of reasoning , the analysis leads to algorithms and further ahead for choosing the combination coefficients to improve the steady - state network performance . it should be noted that several combination rules , such as the metropolis rule @xcite and the maximum degree rule @xcite , were proposed previously in the literature especially in the context of consensus - based iterations @xcite . these schemes , however , usually suffer performance degradation in the presence of noisy information exchange since they ignore the network noise profile @xcite . when the noise variance differs across the nodes , it becomes necessary to design combination rules that are aware of this variation as outlined further ahead in section vi - b . moreover , in a mobile network @xcite where nodes are on the move and where neighborhoods evolve over time , it is even more critical to employ adaptive combination strategies that are able to track the variations in the noise profile in order to cope with such dynamic environments . this issue is taken up in section vi - c . we use lowercase letters to denote vectors , uppercase letters for matrices , plain letters for deterministic variables , and boldface letters for random variables . we also use @xmath1 to denote conjugate transposition , @xmath2 for the trace of its matrix argument , @xmath3 for the spectral radius of its matrix argument , @xmath4 for the kronecker product , and @xmath5 for a vector formed by stacking the columns of its matrix argument . we further use @xmath6 to denote a ( block ) diagonal matrix formed from its arguments , and @xmath7 to denote a column vector formed by stacking its arguments on top of each other . all vectors in our treatment are column vectors , with the exception of the regression vectors , @xmath8 , and the associated noise signals , @xmath9 , which are taken to be row vectors for convenience of presentation .
|
adaptive networks rely on in - network and collaborative processing among distributed agents to deliver enhanced performance in estimation and inference tasks . information is exchanged among the nodes , usually over noisy links . the combination weights that are used by the nodes to fuse information from their neighbors play a critical role in influencing the adaptation and tracking abilities of the network . , the analysis reveals that link noise over the regression data modifies the dynamics of the network evolution in a distinct way , and leads to biased estimates in steady - state . the analysis also reveals how the network mean - square performance is dependent on the combination weights . diffusion adaptation , adaptive networks , imperfect information exchange , tracking behavior , diffusion lms , combination weights , energy conservation .
|
adaptive networks rely on in - network and collaborative processing among distributed agents to deliver enhanced performance in estimation and inference tasks . information is exchanged among the nodes , usually over noisy links . the combination weights that are used by the nodes to fuse information from their neighbors play a critical role in influencing the adaptation and tracking abilities of the network . this paper first investigates the mean - square performance of general adaptive diffusion algorithms in the presence of various sources of imperfect information exchanges , quantization errors , and model non - stationarities . among other results , the analysis reveals that link noise over the regression data modifies the dynamics of the network evolution in a distinct way , and leads to biased estimates in steady - state . the analysis also reveals how the network mean - square performance is dependent on the combination weights . we use these observations to show how the combination weights can be optimized and adapted . simulation results illustrate the theoretical findings and match well with theory . diffusion adaptation , adaptive networks , imperfect information exchange , tracking behavior , diffusion lms , combination weights , energy conservation .
|
1112.6212
|
c
|
in this work we investigated the performance of diffusion algorithms under several sources of noise during information exchange and under non - stationary environments . we first showed that , on one hand , the link noise over the regression data biases the estimators and deteriorates the conditions for mean and mean - square convergence . on the other hand , diffusion strategies can still stabilize the mean and mean - square convergence of the network with noisy information exchange . we derived analytical expressions for the network msd and emse and used these expressions to motivate the choice of combination weights that help ameliorate the effect of information - exchange noise and improve network performance . we also extended the results to the non - stationary scenario where the unknown parameter @xmath16 is changing over time . simulation results illustrate the theoretical findings and how well they match with theory .
|
this paper first investigates the mean - square performance of general adaptive diffusion algorithms in the presence of various sources of imperfect information exchanges , quantization errors , and model non - stationarities . among other results we use these observations to show how the combination weights can be optimized and adapted . simulation results illustrate the theoretical findings and match well with theory .
|
adaptive networks rely on in - network and collaborative processing among distributed agents to deliver enhanced performance in estimation and inference tasks . information is exchanged among the nodes , usually over noisy links . the combination weights that are used by the nodes to fuse information from their neighbors play a critical role in influencing the adaptation and tracking abilities of the network . this paper first investigates the mean - square performance of general adaptive diffusion algorithms in the presence of various sources of imperfect information exchanges , quantization errors , and model non - stationarities . among other results , the analysis reveals that link noise over the regression data modifies the dynamics of the network evolution in a distinct way , and leads to biased estimates in steady - state . the analysis also reveals how the network mean - square performance is dependent on the combination weights . we use these observations to show how the combination weights can be optimized and adapted . simulation results illustrate the theoretical findings and match well with theory . diffusion adaptation , adaptive networks , imperfect information exchange , tracking behavior , diffusion lms , combination weights , energy conservation .
|
1610.02457
|
i
|
aiming at deriving a statistically well - justified galactic center distance based on a large variety of tracers and reducing any occurrence of publication bias , we embarked on an extensive data - mining effort of the scientific literature , eventually yielding 273 new or revised @xmath0 estimates published since records began in october 1918 until june 2016 . our large database of galactic center distance estimates , a fully linked version of which is made available to the scientific community , allowed us to explore the pros and cons of a variety of different approaches used to determine the distance to the galactic center . we separated our compilation into direct and indirect distance measurements . the former include distances such as those based on orbital modeling of the so - called s stars orbiting sgr a * , the closest visual counterpart of the milky way s central supermassive black hole , as well as those relying on statistical parallaxes of either the nuclear star cluster or the stellar population in the inner galactic core . careful assessment of the body of published galactic center distances based on these methods resulted in our galactic center distance recommendation of @xmath287 kpc . a much larger body of galactic center distance determinations is based on indirect methods , either those relying on centroid determinations for a range of different tracer populations ( e.g. , globular clusters , cepheid , rr lyrae , or mira variables , or red clump stars ) or measurements based on kinematic observations of objects at the solar circle , combined with a mass and/or rotational model of the milky way . the latter approaches are affected by significantly larger uncertainties than the former , while the central , mean galactic center distances based on the kinematic methods are systematically smaller than those based on centroid determinations . most centroid - based distances are in good agreement with those resulting from the direct methods . we did not find any conclusive evidence of the presence of a bandwagon effect in the post-1990 galactic center distance measurements , neither among the direct distance estimates nor among those based on centroid determinations . our set of kinematics - based distance measurements can not be used to explore this issue given the significant uncertainties associated with the latter methods . however , these latter measurement can indeed be used to constrain the galaxy s rotation velocity at the solar galactocentric distance using the @xmath264 ratios employed by their respective authors . we found a gradual increase in the mean value of @xmath2 from @xmath288 km s@xmath5 in the 1990s to @xmath289 km s@xmath5 more recently . ( both values represent the means of the distributions of rotation speeds ; their standard deviations are of order 2030 km s@xmath5 . ) our results thus imply that the iau - recommended galactic center distance ( @xmath290 kpc ) needs a downward adjustment , while the recommendation for the galactic rotation velocity at the solar circle ( @xmath4 km s@xmath5 ) requires an upward revision . finally , in view of the recent _ gaia _ data release 1 ( lindegren et al . 2016 ) , this is an opportune time to consider the impact on galactic center distance determinations of the improved parallax measurements of upwards of a billion stars that the mission will provide by the end of its nominal five - year duration . the final _ gaia _ catalog will map a significant fraction of the galactic volume , with particularly high ( distance - dependent ) parallax precision for stars in the solar neighborhood , gradually decreasing toward the galactic center . because of the high extinction and significant crowding in the galaxy s inner regions , _ gaia _ s optical ( @xmath291-band ) measurements will reach the galaxy s central regions , but not the center itself . it is anticipated that _ gaia _ s direct distance determinations of stars near the galactic center will be accurate to approximately 20% , i.e. , significantly worse than the distance estimates provided by any of the direct methods discussed in section [ direct.sec ] or the cepheid - based result of matsunaga et al . however , _ gaia s _ homogeneous and extensive final catalog will undoubtedly facilitate a significantly improved _ statistical _ determination of @xmath0 based on kinematic measurements at as well as inside the solar circle . in addition , _ gaia _ will provide improved distance measurements to large numbers of standard candles , thus almost certainly improving their zero - point accuracies , which in turn will allow us to test for the effects of population differences associated with the use of secondary distance tracers . _ gaia _ is currently among the best - placed facilities to resolve the remaining uncertainties in the galactic distance scale , which is no longer seriously affected by statistical uncertainties . the prevailing uncertainties preventing us from determining a more accurate distance to the galactic center are systematic in nature . among the latter , the effects of not just variations in the extinction , but of possible variations in the prevailing extinction _ law _ are among the most important stumbling blocks . the data set analyzed in this paper is , unfortunately , unsuitable for systematic studies of the uncertainties introduced by variations in the extinction law ( e.g. , nishiyama et al . 2009 ; nataf et al . 2016 ; and references therein ) . this is particularly so , because the vast majority of the individual data points retrieved from the literature have been extinction - corrected by their respective authors using the most appropriate approaches available at the time of their analyses . this implies the inherent presence of intrinsic inhomogeneities in the extinction corrections , which are impossible to fully correct for at the present time . in addition , we point out that current empirical estimates of reddening laws in the local universe are mainly based on optical and near - infrared photometry . to further constrain possible systematic effects , independent approaches based on either analyses of diffuse interstellar bands ( e.g. , munari & zwitter 1997 ; wallerstein et al . 2007 ; munari et al . 2008 ; kashuba et al . 2016 ) or spectroscopic studies ( e.g. , kudritzki et al . 2012 ) may shed new light on this long - standing problem . indeed , detailed , large - scale studies such as that of nataf et al . ( 2016 ) , perhaps combined with high spatial resolution observations of carefully selected , homogeneous samples of galactic center objects at near- to mid - infrared wavelengths ( such as those anticipated to result from _ wfirst _ operation or from campaigns with the next generation of 30 m - class ground - based telescopes ) , seem most promising to overcome the remaining systematics in the reddening laws .
|
aiming at deriving a statistically well - justified galactic center distance , , and reducing any occurrence of publication bias , we compiled the most comprehensive and most complete database of galactic center distances available to date , containing 273 new or revised estimates published since records began in october 1918 until june 2016 . we separate our compilation into direct and indirect distance measurements . the latter include a large body of estimates that rely on centroid determinations for a range of tracer populations as well as measurements based on kinematic observations of objects at the solar circle , combined with a mass and/or rotational model of the milky way . careful assessment of the galactic center distances resulting from orbital modeling and statistical parallax measurements in the galactic nucleus yields our final galactic center distance recommendation of kpc . the centroid - based distances are in good agreement with this recommendation . the kinematics - based distance estimates are affected by significantly larger uncertainties , but they can be used to constrain the galaxy s rotation velocity at the solar galactocentric distance , . our results imply that the international astronomical union - recommended galactic center distance ( kpc ) needs a downward adjustment , while its recommendation ( km s ) requires a substantial upward revision .
|
aiming at deriving a statistically well - justified galactic center distance , , and reducing any occurrence of publication bias , we compiled the most comprehensive and most complete database of galactic center distances available to date , containing 273 new or revised estimates published since records began in october 1918 until june 2016 . we separate our compilation into direct and indirect distance measurements . the latter include a large body of estimates that rely on centroid determinations for a range of tracer populations as well as measurements based on kinematic observations of objects at the solar circle , combined with a mass and/or rotational model of the milky way . careful assessment of the galactic center distances resulting from orbital modeling and statistical parallax measurements in the galactic nucleus yields our final galactic center distance recommendation of kpc . the centroid - based distances are in good agreement with this recommendation . neither the direct measurements nor the post-1990 centroid - based distance determinations suggest that publication bias may be important . the kinematics - based distance estimates are affected by significantly larger uncertainties , but they can be used to constrain the galaxy s rotation velocity at the solar galactocentric distance , . our results imply that the international astronomical union - recommended galactic center distance ( kpc ) needs a downward adjustment , while its recommendation ( km s ) requires a substantial upward revision .
|
1506.07444
|
c
|
the main results are summarized here . \1 ) approximate solutions in closed form are obtained for the 5-dimensional bohr hamiltonian with the woods saxon potential , using the pekeris approximation and the exact solutions of an extended woods saxon potential in one dimension , featuring a dip near its surface . \2 ) applying the results to several @xmath0-unstable and prolate deformed nuclei , we find that the ws potential can describe the ground state bands and the @xmath1 bands equally well as other potentials ( davidson , kratzer , morse ) , if the `` well size '' and the diffuseness are large enough ( at least 1.9 and 0.44 respectively ) , but it fails to describe the @xmath2 bands , apparently because of its lack of a hard core . several ( forty - four ) examples of deformed nuclei satisfying this condition have been found , but on the other hand all @xmath0-unstable nuclei considered violate this condition . \3 ) the form of the potentials coming out from the fits exhibits a very large dip near the surface . in other words , the bohr equation forces the parameters of the ws potential to obtain values producing a very large dip near its surface , so that its overall shape around its minimum largely resembles the shape around the minimum of the davidson , or the kratzer , or the morse potential . \4 ) the present results suggest that potentials used in the bohr hamiltonian can provide satisfactory results for nuclear spectra if they possess two features , a hard core and a deep oscillator - like minimum . they also suggest that the lack of a hard core does not decisively affect the description of the ground state and @xmath1 bands , but destroys the ability of the potential to describe @xmath2 bands . concerning the position of the quasi-@xmath2 bands , which are not reproduced by the woods saxon potential , the following comments apply . \1 ) the quasi-@xmath2 bandheads move to higher energies ( normalized to the energy of the first excited state ) both in @xmath0-unstable and in deformed nuclei , if the left wall of the infinite well potential used in the e(5 ) and x(5 ) critical point symmetries is gradually moved to the right of the origin of the @xmath55-axis , approaching the right wall @xcite . \2 ) the interlevel spacings within the @xmath2- band , which are known to be overestimated in the framework of the x(5 ) critical point symmetry , get fixed by allowing the right wall of the infinite well potential to be sloped to the right @xcite . it should be mentioned here that the identification of a symmetry underlying the x(5 ) special solution of the bohr hamiltonian remains an open problem . analytical wave functions for the bohr hamiltonian with the ws potential can be readily obtained by exploiting the similarity of this hamiltonian , after using the pekeris approximation , to the exactly soluble extended ws potential with a dip near its surface . the calculation of b(e2 ) transition rates becomes then a straightforward task , to be addressed in further work . finally , the solution of the bohr hamiltonian with a woods saxon potential within the framework of the algebraic collective model @xcite would offer the opportunity of comparison of the present approximate results to exact numerical solutions .
|
approximate analytical solutions in closed form are obtained for the 5-dimensional bohr hamiltonian with the woods saxon potential , taking advantage of the pekeris approximation and the exactly soluble one - dimensional extended woods comparison to the data for several-unstable and prolate deformed nuclei indicates that the potential can describe well the ground state and bands of many prolate deformed nuclei corresponding to large enough `` well size '' and diffuseness , while it fails in describing the bands , due to its lack of a hard core , as well as in describing-unstable nuclei , because of the small `` well size '' and diffuseness they exhibit . _ keywords _ : bohr hamiltonian , woods - saxon potential , pekeris approximation +
|
approximate analytical solutions in closed form are obtained for the 5-dimensional bohr hamiltonian with the woods saxon potential , taking advantage of the pekeris approximation and the exactly soluble one - dimensional extended woods saxon potential with a dip near its surface . comparison to the data for several-unstable and prolate deformed nuclei indicates that the potential can describe well the ground state and bands of many prolate deformed nuclei corresponding to large enough `` well size '' and diffuseness , while it fails in describing the bands , due to its lack of a hard core , as well as in describing-unstable nuclei , because of the small `` well size '' and diffuseness they exhibit . _ keywords _ : bohr hamiltonian , woods - saxon potential , pekeris approximation +
|
1401.8272
|
i
|
it is a great honor for me to be invited at the seventh international conference on geometry and topology of manifolds , dedicated to the mathematical legacy of charles ehresmann . i enjoyed with great pleasure the hospitality of the mathematical research and conference center of the polish academy of sciences , and i address my warmest thanks to the organizers and to the supporting institutions . around 1923 , lie cartan @xcite introduced the notion of an _ affine connection _ on a manifold . that notion was previously used , in a less general setting , by h. weyl @xcite and rests on the idea of parallel transport due to t. levi - civita @xcite . at the very beginning of @xcite , even before defining explicitly affine connections , lie cartan explains how that concept can be used in newtonian and einsteinian mechanics . he shows that the _ principle of inertia _ ( which is at the foundations of mechanics ) , according to which a material point particle , when no forces act on it , moves along a straight line with a constant velocity , can be expressed locally by the use of an affine connection . under that form , that principle remains valid in ( curved ) einsteinian space - times . cartan even shows that by a suitable adjustment of the connection , a gravity force ( that means , an acceleration field ) can be accounted for , and becomes a part of the geometry of space - time . that result expresses the famous _ equivalence principle _ used by einstein for the foundations of general relativity . as shown by cartan , it is valid for newtonian mechanics as well . then lie cartan presents a thorough geometric study of affine connections ; he defines their curvature and torsion , and discusses the parallel displacement of a frame along a closed loop . he introduces euclidean , galilean and minkowskian connections , for which the group of affine transformations is replaced by a suitable subgroup . in @xcite he introduces more general types of connections associated to transformation groups which are no more subgroups of the group of affine transformations . cartan s ideas were fully formalized by charles ehresmann in the framework of connections on fibre bundles , which he introduced in @xcite . in section 2 we briefly present cartan s intuitive ideas about connections . then in section 3 we describe ehresmann connections on fibre bundles , and in section 4 cartan connections as seen by ehresmann . in section 5 we present with more details examples of cartan connections , including affine , projective and conformal connections . in section 6 , following cartan , we show how a gravitational force can be included in the geometry of space - time by the use of a suitable connection , and we briefly present other applications of connections : geometric quantization , phases in mechanics , nonholonomic or active constraints , maxwell s equations , yang - mills fields .
|
around 1923 , lie cartan introduced affine connections on manifolds and defined the main related concepts : torsion , curvature , holonomy groups . lie cartan extended these concepts for other types of connections on a manifold : euclidean , galilean and minkowskian connections which can be considered as special types of affine connections , the group of affine transformations of the affine tangent space being replaced by a suitable subgroup ; and more generally , conformal and projective connections , associated to a group which is no more a subgroup of the affine group . [ multiblock footnote omitted ]
|
around 1923 , lie cartan introduced affine connections on manifolds and defined the main related concepts : torsion , curvature , holonomy groups . he discussed applications of these concepts in classical and relativistic mechanics ; in particular he explained how parallel transport with respect to a connection can be related to the principle of inertia in galilean mechanics and , more generally , can be used to model the motion of a particle in a gravitational field . in subsequent papers , lie cartan extended these concepts for other types of connections on a manifold : euclidean , galilean and minkowskian connections which can be considered as special types of affine connections , the group of affine transformations of the affine tangent space being replaced by a suitable subgroup ; and more generally , conformal and projective connections , associated to a group which is no more a subgroup of the affine group . around 1950 , charles ehresmann introduced connections on a fibre bundle and , when the bundle has a lie group as structure group , connection forms on the associated principal bundle , with values in the lie algebra of the structure group . he called _ cartan connections _ the various types of connections on a manifold previously introduced by . cartan , and explained how they can be considered as special cases of connections on a fibre bundle with a lie group as structure group : the standard fibre of the bundle is then an homogeneous space ; its dimension is equal to that of the base manifold ; a cartan connection determines an isomorphism of the vector bundle tangent to the the base manifold onto the vector bundle of vertical vectors tangent to the fibres of the bundle along a global section . these works are reviewed and some applications of the theory of connections are sketched . [ multiblock footnote omitted ]
|
1609.04129
|
i
|
a particularly interesting direction within the landscape of topological systems is to realize topological superconductivity . within condensed matter physics there has been a significant effort to find systems that exhibit topological superconductivity as such systems are predicted to harbor the , heretofore , elusive majorana fermions@xcite . there have been a number of proposals that have been predicted to realize topological superconductivity , and these proposals may be grouped into two : ( i ) unconventional superconducting materials such as sr@xmath1ruo@xmath2@xcite or doped superconducting materials such as cu@xmath3bi@xmath4se@xmath5@xcite and ( ii ) proximity - coupled system comprised of a conventional superconductor and a system such as strongly spin - orbit coupled semiconductors@xcite , magnetic adatoms@xcite , or 3d time - reversal invariant ( tri ) topological insulators ( ti)@xcite . while much work has taken place on both groups of proposals , there have been few unambiguous signs of topological superconductivity observed experimentally . in this endeavor , the most promising experimental signatures have come from the second class of proposals , in particular the spin - orbit coupled semiconductors@xcite and magnetic adatoms proximity - coupled with @xmath0-wave superconductors@xcite . nonetheless , it is clear that within each of the proposals to observe topological superconductivity , there is a trend in the components required to produce the unconventional superconductivity : non - zero berry curvature induced by spin - orbit coupling and broken time reversal symmetry by magnetism . of the available platforms within which one may combine these ingredients , one of the well known routes to generate topological superconductivity is via the superconducting proximity effect in a heterostructure sample of a conventional @xmath0-wave superconductor and 3d tri ti@xcite . in the pioneering work of fu and kane , cooper pairs from @xmath0-wave superconductors that are proximity - coupled to 3d tri ti tunnel from the superconductor into the ti resulting in the acquisition of an topological superconductivity that behaves an effective spinless , chiral @xmath6 superconductor without breaking time - reversal symmetry . as compared to proposals using non - ti heterostructures@xcite or intrinsic superconductors@xcite , the fu - kane proposal is attractive as it does not require further assumptions on any of the physical parameters such as cooper pairing amplitudes between different orbitals@xcite or the position of the chemical potential@xcite . to facilitate the generation of chiral edge states , a zeeman field may be introduced to open a gap in the energy spectrum and thereby form a boundary at the surface of the ti@xcite . to this end , topological insulators with magnetic dopants that break time - reversal symmetry are of great interest@xcite as a platform to observe topological superconductivity and chiral edge states . in this work , we study how introducing magnetic dopants affects the proximity induced superconductivity of the 3d ti system . unlike a similar study that has been performed on magnetically - doped ti whose zeeman field is randomly oriented within the ti@xcite , we consider magnetically ordered dopants through the addition of a uniform zeeman splitting term in a thin 3d tri ti sample such as bi@xmath1se@xmath7 to form a magnetic domain via the net exchange field@xcite . as experimental ti samples must be thin for superconductivity to be observed on the surface , we focus on the `` ultrathin '' limit of the ti where the surface states are not well - isolated but hybridized resulting in a gapped surface state spectrum . . the induced order parameter at top ( bottom ) surface is indicated as @xmath8 ( @xmath9 ) . ( b ) after we apply the proper rotation to the system , we obtain two decoupled systems in hybridization basis of the top and bottom surfaces . two individual sectors are referred as to a sector 1 and sector 2 . ( c ) total induced superconducting order parameter is equivalent to a combination of ( spatially ) symmetric and anti - symmetric superconducting order parameter.,scaledwidth=50.0% ] we seek to understand the physics of magnetically - doped , ultrathin ti and its topological phase by analyzing the superconducting order parameter using both analytical and numerical techniques . in section [ sec : phenomenological ] , we introduce a 2d continuum model for the surface states of an ultrathin ti that accounts for the hybridization gap . by applying a series of unitary transformations , we note that the model can be separated into independent sectors , whose relevant pairing potential have a symmetric and anti - symmetric spatial form when superconductivity is added . we then analyze the symmetric and anti - symmetric @xmath0-wave pairing potential at the phenomenological level by assuming a constant induced order parameter , and find that anti - symmetric pairing is dominant for experimentally relevant strengths of the zeeman field . simplifying the hamiltonian with assumed anti - symmetric pairing potential , we find that gap closing points exist and are controlled by three parameters that can be tuned in experiment : the chemical potential , hybridization gap , and zeeman energy . as our system is in @xmath10 class within the altland - zirnbauer classification@xcite , it is characterized by a @xmath11 topological invariant@xcite . thus we analyze the gap closing points and corresponding topological phase by evaluating the chern number to obtain the resulting phase diagram . in section [ sec : results3d ] , we model a more realistic lattice system by self - consistently solving for the superconducting order parameter in a heterostructure of a @xmath0-wave superconductor coupled to a ti using the bogoliubov - de gennes ( bdg ) formalism . our numerical simulation accurately captures the effects of bulk and surface bands that are present in ti and includes the spin dynamics of these bands when magnetism is introduced . an induced superconducting order parameter obtained from bulk states of the ti shows a rapid decay in magnitude with increasing magnetic impurity concentration as the zeeman energy splits the band and suppresses the @xmath0-wave pairing . in contrast , the surface states show an induced order parameter that persists over an experimentally relevant range of the zeeman energy due to their spin - momentum locked nature and non - zero projection of electron pairs into the @xmath0-wave pairing component . moreover , self - consistent calculation shows that the anti - symmetric pairing potential is dominant at non - zero zeeman energy , thereby , we confirm our phenomenological analysis . lastly , in section [ sec : conclusion ] , we summarize our results and provide our concluding remarks .
|
as a promising candidate system to realize topological superconductivity , the system of a 3d topological insulator ( ti ) grown on top of the-wave superconductor has been extensively studied . to access the topological superconductivity experimentally , the 3d ti sample must be thin enough to allow for cooper pair tunneling to the exposed surface of ti . our findings provide a useful guide in choosing relevant parameters to facilitate the observation of topological superconductivity in thin film ti - superconductor hybrid systems .
|
as a promising candidate system to realize topological superconductivity , the system of a 3d topological insulator ( ti ) grown on top of the-wave superconductor has been extensively studied . to access the topological superconductivity experimentally , the 3d ti sample must be thin enough to allow for cooper pair tunneling to the exposed surface of ti . the use of magnetically ordered dopants to break time - reversal symmetry may allow the surface of a ti to host majorana fermion , which are believed to be a signature of topological superconductivity . in this work , we study a magnetically - doped thin film ti - superconductor hybrid systems . considering the proximity induced order parameter in thin film of ti , we analyze the gap closing points of the hamiltonian and draw the phase diagram as a function of relevant parameters : the hybridization gap , zeeman energy , and chemical potential of the ti system . our findings provide a useful guide in choosing relevant parameters to facilitate the observation of topological superconductivity in thin film ti - superconductor hybrid systems . in addition , we further perform numerical analysis on a ti proximity coupled to a-wave superconductor and find that , due to the spin - momentum locked nature of the surface states in ti , the induced-wave order parameter of the surface states persists even at large magnitude of the zeeman energy .
|
1206.4313
|
m
|
to model cloudy t dwarf atmospheres , we modify the am01 cloud model . this model has successfully been used to model the effects of the iron , silicate , and corundum clouds on the spectra of l dwarfs saumon08,stephens09 . here , we do not include the opacity of iron , silicate , and corundum clouds ; based on observed trends , we assume that the opacity of these clouds becomes negligible for the early t dwarfs . we instead include cr , mns , na@xmath0s , zns , and kcl . the am01 approach avoids treating the highly uncertain microphysical processes that create clouds in brown dwarf and planetary atmospheres . instead , it aims to balance the advection and diffusion of each species vapor and condensate at each layer of the atmosphere . it balances the upward transport of vapor and condensate by turbulent mixing in the atmosphere with the downward transport of condensate by sedimentation . this balance is achieved using the equation @xmath15 where @xmath16 is the vertical eddy diffusion coefficient , @xmath17 is the mixing ratio of condensate and vapor , @xmath18 is the mixing ratio of condensate , @xmath19 is the convective velocity scale , and @xmath3 is a parameter that describes the efficiency of sedimentation in the atmosphere . this calculation provides the total amount of condensate at each layer of the atmosphere . the distribution of particle sizes at each level of the atmosphere is represented by a log - normal distribution in which the modal particle size is calculated using the sedimentation flux . a high sedimentation efficiency parameter @xmath3 results in vertically thinner clouds with larger particle sizes , whereas a lower @xmath3 results in more vertically extended clouds with smaller particles sizes . as a result , a higher @xmath3 corresponds to optically thinner clouds and a lower @xmath3 corresponds to optically thicker clouds . the am01 cloud model code computes the available quantity of condensible gas above the cloud base by comparing the local gas abundance ( accounting for upwards transport by mixing via @xmath20 ) to the local condensate vapor pressure @xmath21 . in cases where the formation of condensates does not proceed by homogeneous condensation we nevertheless compute an equivalent vapor pressure curve as described in section [ cloudcondchem ] . the cloud code is coupled to a 1d atmosphere model that calculates the pressure - temperature profile of an atmosphere in radiative - convective equilibrium . the atmosphere models are described in mckay89 , marley96 , burrows97 , mm99 , marley02 , saumon08 , fortney08b . this methodology has been successfully applied to modeling brown dwarfs with both cloudy and clear atmospheres marley96 , marley02 , burrows97 , saumon06 , saumon07 , leggett07a , leggett07b , mainzer07 , blake07 , cushing08 , geballe09 , stephens09 . in the atmosphere model , the thermal radiative transfer is determined using the `` source function technique '' presented in toon89 . within this method , it is possible to include mie scattering of particles as an opacity source in each layer . our opacity database for gases , described extensively in freedman08 , includes all the important absorbers in the atmosphere . this opacity database includes two significant updates since freedman08 , which are described in saumon12 : a new molecular line list for ammonia yurchenko11 and an improved treatment of the pressure - induced opacity of h@xmath0 collisions richard12 . both the cloud model and the chemical equilibrium calculations ( see section [ cc ] ) are coupled with the radiative transfer calculations and the pressure - temperature profile of the atmosphere ; this means that a converged model will have a temperature structure that is self - consistent with the clouds and chemistry . s index of refraction . the real and imaginary parts of the sodium sulfide index of refraction from the two sources used are plotted as a function of wavelength . montaner79 observational data are shown as a blue dashed line . khachai09 calculations are shown as a pink dashed line . the interpolated values used for the mie scattering calculation are shown as pink circles . , width=336 ] s index of refraction . the real and imaginary parts of the sodium sulfide index of refraction from the two sources used are plotted as a function of wavelength . montaner79 observational data are shown as a blue dashed line . khachai09 calculations are shown as a pink dashed line . the interpolated values used for the mie scattering calculation are shown as pink circles . , width=336 ] we calculate the effect of the model cloud distribution on the flux using mie scattering theory to describe the cloud opacity . assuming that particles are spherical and homogeneous , we calculate the scattering and absorption coefficients of each species for each of the particle sizes within the model . in order to make these scattering calculations , we need to understand the optical properties ( the real and imaginary parts of the index of refraction ) of each material . the optical properties were found from a variety of diverse sources , summarized in table [ indexofref ] . to calculate mie scattering within the model atmosphere , we use a grid of optical properties at wavelengths from 0.268 to 227 . where data were not available , we extrapolated the available data , following trends for similar known molecules . the molecules with the most complete published optical properties are zns and kcl , both of which are obtained from querry87 , who tabulates the optical constants for 24 different minerals . optical properties for cr are published in stashchuk84 from 0.26 to 15 . the optical properties from 15 to 227 were linearly extrapolated from these experimental data following the trend of the optical properties of fe . various extrapolations were tested ; the choice of optical properties beyond 15 does not change the results of the calculations in any meaningful way . optical properties for mns are published in huffman67 , from 0.09 to 13 . optical properties from 15 to 227 are extrapolated , following the trends of the other two studied sulfide condensates zns and na@xmath0s . the optical properties for na@xmath0s , the clouds with the largest optical depth , are combined from two different sources . montaner79 provides experimental data in the infrared , from 25 to 198 . khachai09 provides first principles calculations of the optical properties from 0.03 to 91 . in the region of overlap , the montaner79 laboratory values are used . the real and imaginary parts of the index of refraction are plotted in figure [ na2s_n ] . lll kcl & querry87 & 0.22 - 167 + zns & querry87 & 0.22 - 167 + mns & huffman67 & 0.09 - 13 + cr & stashchuk84 & 0.26 - 15 + na@xmath0s & montaner79 & 25 - 198 + & khachai09 & 0.03 - 91 + [ indexofref ] the abundances of molecular , atomic , and ionic species are calculated using thermochemical equilibrium following the models of fegley94 , fegley96 , lodders99 , lodders02,lodders02b , lodders03 , lodders06 , lodders09 . we adopt solar - composition elemental abundances from lodders03 . the differences between lodders03 and newer abundance measurements [ e.g.][]asplund09 are not large enough to significantly alter the condensation temperatures considered in the paper . lodders03 abundances were therefore selected for consistency with previous modeling efforts by our groups . the abundances of condensate - forming elements are listed in table 2 . we assume uniform heavy element abundance ratios over a range of metallicities from [ fe / h ] = -0.5 to [ fe / h ] = + 0.5 in order to explore the metallicity dependence of the condensation temperature expressions . a simplified equilibrium condensation approach is used to calculate saturation vapor pressures and condensation curves ( see figure 3 ) for cr , mns , na@xmath22s , zns , and kcl as a function of pressure , temperature , and metallicity , based upon the more comprehensive thermochemical models of lodders & fegley ( see [ gaschem ] ) and visscher06,visscher10 . in each case , we consider condensation from the most abundant cr- , mn- , na- , zn- , and k - bearing gas phases at the cloud base as predicted by the chemical models . the relative mass of each cloud ( relative to na@xmath22s ) is listed in table 3 , assuming complete removal of available condensate material from the gas phase . lll fe & @xmath23 & fe + si & @xmath24 & mg@xmath0sio@xmath7 , mgsio@xmath6 + mg & @xmath25 & mg@xmath0sio@xmath7 , mgsio@xmath6 + o & @xmath26 & mg@xmath0sio@xmath7 , mgsio@xmath6 , al@xmath0o@xmath6 , h@xmath0o + al & @xmath27 & al@xmath0o@xmath6 , caal@xmath28o@xmath29 , + & & caal@xmath0o@xmath7 , ca@xmath0al@xmath0sio@xmath30 + na & @xmath31 & na@xmath0s + zn & @xmath32 & zns + mn & @xmath33 & mns + s & @xmath34 & na@xmath0s , zns , mns + cr & @xmath35 & cr + k & @xmath36 & kcl + cl & @xmath37 & kcl + [ abundances ] chromium metal is the most refractory of the clouds considered here and condenses from monatomic cr gas via the reaction @xmath38 where ` ( s ) ' indicates a solid phase . the condensation condition for cr - metal is defined by @xmath39 where @xmath40 is the saturation vapor pressure of cr gas in equilibrium with cr - metal and @xmath41 is the partial pressure of cr below the cloud for a solar - composition gas ( @xmath42 , where @xmath43 is the mole fraction abundance of cr and @xmath44 is the total atmospheric pressure ) . upon condensation , the thermodynamic activity of cr - metal is unity and the equilibrium constant ( @xmath45 ) expression for reaction ( [ rxn : cr ] ) can be written as @xmath46 substituting for the temperature - dependent value of @xmath47 , the saturation vapor pressure of cr above the cloud base can be estimated using the expression @xmath48 for @xmath49 in kelvin and @xmath50 in bars . below the cloud , we assume that cr gas is approximately representative of the elemental cr abundance in solar composition gas ( see table [ cloudstable ] ) : @xmath51.\ ] ] the condensation temperature as a function of the total atmospheric pressure ( @xmath44 ) and metallicity can therefore be approximated by setting @xmath52 and rearranging to give @xmath53.\ ] ] this expression yields a condensation temperature near @xmath54 k at 1 bar and solar metallicity [ cf.][]lodders06 , and shows that greater total pressures and/or metallicities will lead to higher condensation temperatures . condensation of cr - metal effectively removes gas - phase chromium from the atmosphere , and the abundances of cr - bearing gases rapidly decrease with altitude above the cloud . our modeling of sulfide condensation chemistry follows that of visscher06 , and the condensation reactions and temperature - dependent expressions presented here are taken from that study . the deepest sulfide cloud expected in brown dwarf atmospheres is mns , which forms via the reaction @xmath55 the formation of the mns cloud is limited by the total manganese abundance , which is 2% of the sulfur abundance in a solar - composition gas . the condensation curve for mns is thus derived by exploring the chemistry of monatomic mn , which is the dominant mn - bearing gas near the cloud base . using results from visscher06 , the saturation vapor pressure of mn above the cloud is given by @xmath56,\ ] ] where the metallicity dependence comes from h@xmath22s ( the dominant s - bearing gas ) remaining in the gas phase above the mns cloud base . by setting @xmath57 , the mns condensation curve is approximated by visscher06 : @xmath58,\ ] ] giving a condensation temperature near @xmath59 k at 1 bar in a solar - metallicity gas . the na@xmath22s cloud is the most massive of the metal sulfide clouds expected to form in brown dwarf atmospheres because na is more abundant than either mn or zn in a solar - composition gas ( see table [ cloudstable ] ) . sodium sulfide condenses via the net thermochemical reaction @xmath60 the mass of the na@xmath22s cloud is limited by the elemental abundance of sodium , which is 13% of the abundance of sulfur in a solar composition gas . using results from visscher06 , the saturation vapor pressure of na above the cloud base is given by @xmath61,\ ] ] where the metallicity dependence results from h@xmath22s remaining in the gas phase above the na@xmath22s cloud and from the stoichiometry of na and h@xmath22s in the condensation reaction . the condensation temperature ( where @xmath62 ) is given by visscher06 : @xmath63,\ ] ] indicating condensation near @xmath64 k at 1 bar in a solar - metallicity gas . the zns cloud layer forms via the reaction @xmath65 the formation of the zns cloud is limited by the total zn abundance , which is 0.3% of the s abundance in a solar - composition gas . using results from visscher06 , the saturation vapor pressure of zn over condensed zns is given by @xmath66\ ] ] the condensation curve ( where @xmath67 ) is approximated by visscher06 : @xmath68,\ ] ] giving a condensation temperature of @xmath69 k at 1 bar in a solar - metallicity gas . our treatment of kcl condensation chemistry is similar to that for cr - metal and the metal sulfides . with decreasing temperatures , kcl replaces neutral k as the dominant k - bearing gas in brown dwarf atmospheres lodders99,lodders06 . the kcl cloud layer is thus expected to form via the net thermochemical reaction @xmath70 and condenses as a solid over the range of conditions considered here . the vapor pressure of kcl above condensed kcl(s ) is given by @xmath71 derived from the equilibrium constant expression for the condensation reaction . the mass of the kcl cloud is limited by the total potassium abundance , which is 70% of the chlorine abundance in a solar - composition gas lodders03 . note that other k - bearing species may remain relatively abundant near cloud condensation temperatures , particularly at higher pressures ( e.g. , see @xcite and @xcite for a more detailed discussion of chemical speciation ) . however , kcl is the dominant k - bearing gas near the cloud base for the relevant @xmath72 conditions expected in cool brown dwarf atmospheres ( see figure 3 ) over the range of metallicities ( -0.5 to + 0.5 dex ) considered here . for simplicity we therefore assume that kcl is approximately representative of the elemental k abundance below the cloud , given by @xmath73.\ ] ] the condensation temperature as a function of pressure and metallicity is estimated by setting @xmath74 and rearranging to give @xmath75,\ ] ] yielding a condensation temperature near @xmath76 k at 1 bar in a solar - metallicity gas [ cf.][]lodders99,lodders06 . in general , the condensation curve expressions demonstrate that condensation temperatures increase with total pressure , as illustrated in figure 4 . furthermore , higher metallicities are expected to result in higher condensation temperatures and more massive cloud layers in brown dwarf atmospheres . in each case , the saturation vapor pressures of cloud - forming species rapidly decrease with altitude above the cloud layers . llc cr & @xmath77 & 0.30 + mns & @xmath78 & 0.36 + na@xmath22s & @xmath79 & @xmath80 + zns & @xmath81 & 0.05 + kcl & @xmath82 & 0.12 + fe & @xmath83 & 20.85 + [ cloudstable ] the am01 model is one method of several that have been applied to cloudy brown dwarf atmospheres . helling08 review various cloud modeling techniques and compare model predictions for various cases . the most important conceptual differences between these approaches lies in the assumptions of how condensed phases interact with the gas . in the chemical equilibrium approach [ e.g.][]allard01 , condensed phases remain in contact with the gas phase and can continue to react with the gas even at temperatures well below the condensation temperature . as an example , when following this approach , fe grains which first condense at temperatures of over 2000 k react with atmospheric @xmath84 to form fes below 1000 k. in the condensation chemistry approach we employ here , the condensed phases are assumed to sediment out of the atmosphere and are not available to interact with gas phases at temperatures below the condensation temperature . thus fe grains form a discrete cloud layer and do not react to form fes . @xmath84 consequently remains in the gas phase and reacts to form condensates as outlined in section 2.4.2 . jupiter is an excellent example of the applicability of this framework , as the presence of h@xmath0s in the observable atmosphere is only possible because fe is sequestered in a deep cloud layer , which prevents the formation of fes which otherwise deplete other gas phase s species fegley94 . the presence of alkali absorption in t dwarfs likewise demonstrates the applicability of condensation chemistry @xcite . a detailed comparison of true equilibrium condensation and cloud condensate removal from equilibrium can be found in fegley94 , lodders06 and references therein . a different approach is taken by helling & woitke helling06 who follow the trajectory of tiny seed particles of tio@xmath0 that are assumed to be emplaced high in the atmosphere and sink downwards . as the seeds fall through the atmosphere they collect condensate material . in helling06 and numerous follow on papers helling08 , witte09 , witte11 , dekok11 this group models the microphysics of grain growth given these conditions . because the background atmosphere is not depleted of gaseous species until the grains fall through the atmosphere , a compositionally very different set of grains are formed . in particular they predict ` dirty ' grains composed of layers of varying condensates . a direct comparison between the predictions of the various cloud modeling schools is often difficult because of differing assumptions of elemental abundances and the background thermal profile . modeling tests in which predictions of the various groups are compared to data would be illuminating , but this is far beyond the scope of the work reported here . in order to calculate absolute magnitudes of the modeled brown dwarfs , we use the results of evolution models which determine the radius of a brown dwarf as it cools and contracts over its lifetime . we use the evolution models of saumon08 with the surface boundary condition from cloudless atmospheres . using a cloudless boundary condition instead of one consistent with these clouds changes the calculated magnitudes of the models very slightly , but does not change the overall trends or results . to analyze the effect of these clouds , we generate a grid of 182 model atmospheres at effective temperatures and surface gravities spanning the full range of t dwarfs . we calculate pressure - temperature profiles and synthetic spectra for atmospheres from 400 to 1300 k ( 50 to 100 k increments ) , with log(@xmath2 ) ( cgs ) of 4.0 , 4.5 , 5.0 , and 5.5 and cloud sedimentation efficiency parameter @xmath3=2 , 3 , 4 , and 5 . for this study , we use only solar metallicity composition . we then compare these , both photometrically and spectroscopically , to observed t dwarfs .
|
are calculated using the am01 cloud model , which is coupled to an atmosphere model to produce atmospheric pressure - temperature profiles in radiative - convective equilibrium . the emergence of sulfide clouds in cool atmospheres , particularly nas , may be a more natural explanation for the `` cloudy '' spectra of these objects , rather than the re - emergence of silicate clouds that wane at the l - to - t transition .
|
as brown dwarfs cool , a variety of species condense in their atmospheres , forming clouds . iron and silicate clouds shape the emergent spectra of l dwarfs , but these clouds dissipate at the l / t transition . a variety of other condensates are expected to form in cooler t dwarf atmospheres . these include cr , mns , nas , zns , and kcl , but the opacity of these optically thinner clouds has not been included in previous atmosphere models . here , we examine their effect on model t and y dwarf atmospheres . the cloud structures and opacities are calculated using the am01 cloud model , which is coupled to an atmosphere model to produce atmospheric pressure - temperature profiles in radiative - convective equilibrium . we generate a suite of models between = 400 and 1300 k , log=4.0 and 5.5 , and condensate sedimentation efficiencies from=2 to 5 . model spectra are compared to two red t dwarfs , ross 458c and ugps 072205 ; models that include clouds are found to match observed spectra significantly better than cloudless models . the emergence of sulfide clouds in cool atmospheres , particularly nas , may be a more natural explanation for the `` cloudy '' spectra of these objects , rather than the re - emergence of silicate clouds that wane at the l - to - t transition . we find that sulfide clouds provide a mechanism to match the near- and mid - infrared colors of observed t dwarfs . our results indicate that including the opacity of condensates in t dwarf atmospheres is necessary to accurately determine the physical characteristics of many of the observed objects .
|
1206.4313
|
i
|
cloud formation is a natural and unavoidable phenomenon in cool substellar atmospheres . at temperatures cooler than those of l dwarfs , chemistry dictates that additional condensates , beyond the silicates and iron , must form . we have examined the effect of including the most abundant of these lower - temperature condensates , cr , mns , na@xmath0s , zns , and kcl , in brown dwarf model atmospheres . within the framework of the am01 cloud model , we have investigated the opacity of these clouds over a wide range of parameter space , across the relevant range of t dwarfs , to the warmest y dwarfs . from our suite of models from 400 to 1300 k , log @xmath2=4.0 to 5.5 , @xmath3=2 to 5 , we have shown the likely role that these low-@xmath1 clouds , dominated by sulfides , play in these cool atmospheres . model spectra were compared to two t dwarfs , ross 458c and ugps 072205 . these two objects have red near - infrared colors and are not well - matched by cloudless models . model spectra that include the sulfide clouds match the observed spectra of both objects more accurately than cloudless models . the photometric colors of the cloudy models were calculated and compared to the full population of brown dwarfs with parallax measurements . this analysis shows that the sulfide clouds provide a mechanism to match the near - infrared colors of observed brown dwarfs . the agreement is particularly good in @xmath10 , while in @xmath8 the models are somewhat too red . wise observations of the coolest t dwarfs and warmest y dwarfs indicate the these new models fit observations better than cloud - free models . we thank rabah khenata and the rest of his team in algeria for providing calculations of optical properties for sodium sulfide . we thank adam burgasser for providing the spectrum for ross 458c . we thank katharina lodders for providing the condensation chemistry tables used in the model calculations as well as other very helpful advice and suggestions . we also thank the anonymous referee for his or her suggestions . based on observations obtained at the gemini observatory , which is operated by the association of universities for research in astronomy , inc . , under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the science and technology facilities council ( united kingdom ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council australia),ministrio da cincia , tecnologia e inovao ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ) .
|
these include cr , mns , nas , zns , and kcl , but the opacity of these optically thinner clouds has not been included in previous atmosphere models . here , we examine their effect on model t and y dwarf atmospheres . the cloud structures and opacities we generate a suite of models between = 400 and 1300 k , log=4.0 and 5.5 , and condensate sedimentation efficiencies from=2 to 5 . we find that sulfide clouds provide a mechanism to match the near- and mid - infrared colors of observed t dwarfs . our results indicate that including the opacity of condensates in t dwarf atmospheres is necessary to accurately determine the physical characteristics of many of the observed objects .
|
as brown dwarfs cool , a variety of species condense in their atmospheres , forming clouds . iron and silicate clouds shape the emergent spectra of l dwarfs , but these clouds dissipate at the l / t transition . a variety of other condensates are expected to form in cooler t dwarf atmospheres . these include cr , mns , nas , zns , and kcl , but the opacity of these optically thinner clouds has not been included in previous atmosphere models . here , we examine their effect on model t and y dwarf atmospheres . the cloud structures and opacities are calculated using the am01 cloud model , which is coupled to an atmosphere model to produce atmospheric pressure - temperature profiles in radiative - convective equilibrium . we generate a suite of models between = 400 and 1300 k , log=4.0 and 5.5 , and condensate sedimentation efficiencies from=2 to 5 . model spectra are compared to two red t dwarfs , ross 458c and ugps 072205 ; models that include clouds are found to match observed spectra significantly better than cloudless models . the emergence of sulfide clouds in cool atmospheres , particularly nas , may be a more natural explanation for the `` cloudy '' spectra of these objects , rather than the re - emergence of silicate clouds that wane at the l - to - t transition . we find that sulfide clouds provide a mechanism to match the near- and mid - infrared colors of observed t dwarfs . our results indicate that including the opacity of condensates in t dwarf atmospheres is necessary to accurately determine the physical characteristics of many of the observed objects .
|
1407.4820
|
i
|
during the brief history of extrasolar planet investigations , our understanding of the relative populations of different types of planets has been limited by the observational biases of the techniques employed . with the advent of sophisticated transit searches and hypersensitive radial velocity measurements , significant progress has been made discovering various types of planets that orbit stars with periods up to a few years . less progress has been made in discovering planets in longer orbits , and particularly around nearby m dwarfs , which account for at least 74% of the stellar population within 10 pc @xcite . m dwarfs offer fertile ground for companion searches , as @xcite have inferred that a high fraction of m dwarfs host terrestrial planets at short orbital periods . less is known about the populations of jupiter - mass planets and brown dwarfs around m dwarfs , particularly at orbital periods longer than a few years . to understand how m dwarf planetary systems form and evolve , we must probe the full regime of companion masses and orbital periods . transit techniques are geometrically biased towards companions with small orbits , while radial velocity techniques are biased towards massive companions with short periods that exert large gravitational accelerations on their host stars . direct imaging techniques are limited to young , giant planets at large separations . astrometric techniques , which measure the positions of stars on the plane of the sky , are most sensitive to jovian - type planets in jovian - type orbits . while radial velocity observing programs are now becoming sensitive to such companions @xcite , the astrometric results presented here have longer observational baselines , of up to 13 years . furthermore , astrometry can detect companions with a large range of inclinations and orientations , and allow for the determination of orbit inclinations and accurate companion masses . to date the majority of nearby extrasolar planets around m dwarfs have been discovered by radial velocity searches , which tend to select the brightest m dwarfs . as discussed in more detail in @xmath2[sec : analysis ] , in ground - based imaging programs the brightest targets generally have the noisiest astrometric residuals due to the short exposures required and the lack of comparably bright reference stars . with the exception of gj 1214 , five m dwarfs in our sample were found to have planets using radial velocity techniques , and are among the brightest targets in our astrometric program . an extreme case is the k dwarf bd @xmath010 3166 , for which we are not sensitive to sub - stellar companions , but for which we provide the first accurate parallax . for comparison , we have included six additional m dwarfs that are less bright , less massive , and closer , and therefore more favorable to companion detection via astrometry . to calibrate our analysis , we have also included three confirmed stellar binaries with clear photocentric perturbations for which we have characterized the orbits . these binaries were chosen from the roughly two dozen binaries in our observing program with clear astrometric perturbations because we have observed multiple orbital periods , and can most accurately characterize the orbits . astrometric solutions for proper motion and parallax are given for each of the 16 systems targeted , plus orbital solutions for three binaries . a detailed analysis of the astrometric residuals is given to search for companions to the 12 m dwarf systems without close stellar companions . periodograms of the astrometric residuals have been generated , along with detection limits based on simulations of 10 million hypothetical companions to each star . these are the first results of a larger recons survey for companions orbiting more than 250 red dwarfs within 25 pc for which we have at least five years of coverage . as observations continue , this sample will grow , further constraining the population of brown dwarf and super - jupiter companions in long period orbits around m dwarfs . finally , to provide context for these results we provide a comprehensive list of the 17 m dwarfs within 25 pc having exoplanets as of 1 july 2014 , including the six targeted in this work .
|
these results complement previously published m dwarf planet occurrence rates by providing astrometrically determined upper mass limits on potential super - jupiter companions at orbits of two years and longer . as part of a continuing survey , these results are consistent with the paucity of super - jupiter and brown dwarf companions we find among the over 250 red dwarfs within 25 pc observed longer than five years in our astrometric program .
|
astrometric measurements are presented for seven nearby stars with previously detected planets : six m dwarfs ( gj 317 , gj 667c , gj 581 , gj 849 , gj 876 , and gj 1214 ) and one k dwarf ( bd 3166 ) . measurements are also presented for six additional nearby m dwarfs without known planets , but which are more favorable to astrometric detections of low mass companions , as well as three binary systems for which we provide astrometric orbit solutions . observations have baselines of three to thirteen years , and were made as part of the recons long - term astrometry and photometry program at the ctio / smarts 0.9 m telescope . we provide trigonometric parallaxes and proper motions for all 16 systems , and perform an extensive analysis of the astrometric residuals to determine the minimum detectable companion mass for the 12 m dwarfs not having close stellar secondaries . for the six m dwarfs with known planets , we are not sensitive to planets , but can rule out the presence of all but the least massive brown dwarfs at periods of 2 12 years . for the six more astrometrically favorable m dwarfs , we conclude that none have brown dwarf companions , and are sensitive to companions with masses as low as 1 for periods longer than two years . in particular , we conclude that proxima centauri has no jovian companions at orbital periods of 2 12 years . these results complement previously published m dwarf planet occurrence rates by providing astrometrically determined upper mass limits on potential super - jupiter companions at orbits of two years and longer . as part of a continuing survey , these results are consistent with the paucity of super - jupiter and brown dwarf companions we find among the over 250 red dwarfs within 25 pc observed longer than five years in our astrometric program .
|
1407.4820
|
r
|
table [ tab1 ] gives the parallax and proper motion results for the 16 systems , with details about the astrometric observations ( filters used , number of seasons observed , number of frames used in reductions , time coverage , span of time , and the number of reference stars ) and results ( relative parallaxes , parallax corrections , absolute parallaxes , proper motions , position angles of the proper motions , and the derived tangential velocities based on relative proper motions and parallaxes ) . all but two of the sixteen systems have parallax errors of @xmath102 mas or less . bd @xmath010 3166 and gj 876 have larger errors due to combinations of faint reference stars and short exposures . corrections to absolute parallax are generally less than @xmath102 mas , so systematics in the corrections should not significantly affect the results . three targets ( gj 317 , gj 667c , and bd @xmath010 3166 , ) have corrections of @xmath103@xmath38-@xmath384 mas due to reddening of the reference stars , which skews their photometric distance estimates . in these cases , we adopt a generic correction of 1.50 @xmath11 0.50 mas . the per observation precision for each target is listed in column 16 , representing the mean of the observation errors in r.a . and decl . the percentage of companions eliminated listed in column 17 is discussed in @xmath2[sec : discuss ] . lccccccccrrrrrrccc + gj 317 & 08 40 59.21 & @xmath023 27 22.6 & r & 5c & 75 & 2009.04 - 2013.38 & 4.35 & 7 & 64.04 @xmath11 1.45 & 1.50 @xmath11 0.50 & 65.54 @xmath11 1.53 & 930.7 @xmath11 1.1 & 330.5 @xmath11 0.13 & 65.5 & 4.99 & & + bd @xmath010 3166 & 10 58 28.79 & @xmath010 46 13.4 & i & 7s & 71 & 2004.43 - 2011.50 & 7.07 & 6 & 13.84 @xmath11 3.04 & 1.50 @xmath11 0.50 & 15.34 @xmath11 3.08 & 185.9 @xmath11 1.5 & 269.1 @xmath11 0.67 & 52.4 & 10.85 & & + gj 581 & 15 19 26.83 & @xmath007 43 20.1 & v & 14s & 267 & 2000.58 - 2013.38 & 12.80 & 11 & 157.67 @xmath11 1.57 & 1.12 @xmath11 0.17 & 158.79 @xmath11 1.58 & 1224.3 @xmath11 0.4 & 266.0 @xmath11 0.03 & 36.5 & 7.93 & 96% & a + gj 1214 & 17 15 18.92 & @xmath3904 57 50.1 & i & 4c & 80 & 2010.39 - 2013.38 & 3.00 & 9 & 68.20 @xmath11 1.26 & 1.88 @xmath11 0.18 & 70.08 @xmath11 1.27 & 945.5 @xmath11 1.4 & 142.0 @xmath11 0.17 & 63.9 & 5.02 & & + gj 667c & 17 18 58.82 & @xmath034 59 48.6 & v & 11s & 140 & 2003.52 - 2013.38 & 9.86 & 5 & 139.38 @xmath11 1.98 & 1.50 @xmath11 0.50 & 140.88 @xmath11 2.04 & 1154.1 @xmath11 0.6 & 101.0 @xmath11 0.05 & 38.8 & 7.66 & 93% & + gj 849 & 22 09 40.34 & @xmath004 38 26.8 & v & 11s & 135 & 2003.52 - 2013.39 & 9.86 & 5 & 113.78 @xmath11 1.97 & 2.27 @xmath11 0.30 & 116.05 @xmath11 1.99 & 1118.0 @xmath11 0.5 & 90.8 @xmath11 0.04 & 45.7 & 9.53 & 81% & + gj 876 & 22 53 16.75 & @xmath014 15 49.2 & v & 11s & 85 & 2003.52 - 2013.39 & 9.87 & 6 & 210.97 @xmath11 3.99 & 2.14 @xmath11 0.57 & 213.11 @xmath11 4.03 & 1149.4 @xmath11 1.1 & 125.7 @xmath11 0.11 & 25.6 & 8.43 & 99% & + + gj 1061 & 03 35 59.72 & @xmath044 30 45.5 & r & 13s & 194 & 1999.62 - 2012.95 & 13.32 & 7 & 269.92 @xmath11 1.29 & 0.94 @xmath11 0.08 & 270.86 @xmath11 1.29 & 827.7 @xmath11 0.3 & 117.7 @xmath11 0.04 & 14.5 & 7.59 & 79% & a + lp 944 - 020 & 03 39 35.25 & @xmath035 25 43.8 & i & 8s & 59 & 2003.95 - 2012.94 & 8.99 & 10 & 154.53 @xmath11 1.03 & 1.36 @xmath11 0.10 & 155.89 @xmath11 1.03 & 408.3 @xmath11 0.3 & 48.5 @xmath11 0.07 & 12.4 & 2.13 & 94% & a + gj 1128 & 09 42 46.36 & @xmath068 53 06.1 & v & 13s & 167 & 2000.23 - 2013.12 & 12.89 & 8 & 153.54 @xmath11 0.75 & 0.73 @xmath11 0.11 & 154.27 @xmath11 0.76 & 1123.0 @xmath11 0.2 & 356.1 @xmath11 0.02 & 34.5 & 3.17 & 80% & a + denis j1048 - 3956 & 10 48 14.56 & @xmath039 56 07.0 & i & 13s & 200 & 2001.15 - 2013.27 & 12.13 & 11 & 247.23 @xmath11 0.60 & 0.85 @xmath11 0.10 & 248.08 @xmath11 0.61 & 1531.6 @xmath11 0.2 & 229.5 @xmath11 0.01 & 29.3 & 2.92 & 97% & a + scr 1138 - 7721 & 11 38 16.76 & @xmath077 21 48.5 & i & 11s & 134 & 2003.25 - 2013.27 & 10.03 & 12 & 119.60 @xmath11 1.01 & 0.81 @xmath11 0.07 & 120.41 @xmath11 1.01 & 2143.3 @xmath11 0.4 & 287.8 @xmath11 0.02 & 84.4 & 4.20 & 69% & a + proxima cen & 14 29 43.02 & @xmath062 40 46.7 & v & 14s & 205 & 2000.57 - 2013.25 & 12.68 & 5 & 766.41 @xmath11 0.91 & 1.72 @xmath11 0.50 & 768.13 @xmath11 1.04 & 3850.8 @xmath11 0.6 & 282.4 @xmath11 0.02 & 23.8 & 4.83 & 99% & a + + lhs 1582ab & 03 43 22.08 & @xmath009 33 50.9 & r & 11s & 102 & 2000.87 - 2012.94 & 12.06 & 7 & 48.84 @xmath11 1.18 & 2.00 @xmath11 0.26 & 50.84 @xmath11 1.21 & 509.4 @xmath11 0.3 & 52.7 @xmath11 0.06 & 47.5 & 3.88 & & a + gj 748ab & 19 12 14.60 & @xmath3902 53 11.0 & v & 10s & 154 & 2004.45 - 2013.39 & 8.95 & 11 & 97.77 @xmath11 1.15 & 2.22 @xmath11 0.41 & 99.99 @xmath11 1.22 & 1857.8 @xmath11 0.5 & 107.4 @xmath11 0.02 & 88.1 & 5.80 & & + lhs 3738ab & 21 58 49.13 & @xmath032 26 25.5 & r & 12s & 151 & 1999.64 - 2012.81 & 13.17 & 12 & 50.82 @xmath11 1.01 & 1.40 @xmath11 0.21 & 52.22 @xmath11 1.03 & 535.2 @xmath11 0.3 & 229.1 @xmath11 0.06 & 48.6 & 2.50 & & a + [ tab1 ] lrrrcrrrcccrrcc + gj 317 & 12.01 & 10.84 & 9.37 & 3 & 7.934 & 7.321 & 7.028 & m3.5 v & 1 & 0.35 & 15.26 @xmath11 0.36 & 9.70 @xmath11 1.53 & 12 & + bd @xmath010 3166 & 10.03 & 9.58 & 9.19 & 3 & 8.611 & 8.300 & 8.124 & k3.0 v & 2 & 0.85 & 65.19 @xmath11 13.64 & & & a + gj 581 & 10.56 & 9.44 & 8.03 & 3 & 6.706 & 6.095 & 5.837 & m3.0 v & 2 & 0.30 & 6.30 @xmath11 0.06 & 6.60 @xmath11 1.03 & 12 & + gj 1214 & 14.71 & 13.27 & 11.50 & 3 & 9.750 & 9.094 & 8.782 & m4.5 v & 1 & 0.14 & 14.27 @xmath11 0.24 & 12.42 @xmath11 2.00 & 12 & + gj 667c & 10.34 & 9.29 & 8.09 & 3 & 6.848 & 6.322 & 6.036 & m1.5 v & 2 & 0.36 & 7.10 @xmath11 0.10 & 9.41 @xmath11 1.49 & 12 & + gj 849 & 10.38 & 9.27 & 7.87 & 3 & 6.510 & 5.899 & 5.594 & m3.0 v & 2 & 0.42 & 8.62 @xmath11 0.15 & 5.73 @xmath11 0.92 & 12 & + gj 876 & 10.18 & 8.97 & 7.40 & 3 & 5.934 & 5.349 & 5.010 & m3.5 v & 2 & 0.27 & 4.69 @xmath11 0.09 & 3.46 @xmath11 0.54 & 12 & + + gj 1061 & 13.09 & 11.45 & 9.47 & 6 & 7.523 & 7.015 & 6.610 & m5.0 v & 3 & 0.11 & 3.69 @xmath11 0.02 & 3.55 @xmath11 0.60 & 12 & + lp 944 - 020 & 18.69 & 16.39 & 13.98 & 3 & 10.725 & 10.017 & 9.548 & m9.0 v & 4 & 0.08 & 6.42 @xmath11 0.04 & 7.04 @xmath11 1.32 & 11 & + gj 1128 & 12.74 & 11.36 & 9.62 & 3 & 7.953 & 7.385 & 7.037 & m4.0 v & 2 & 0.15 & 6.48 @xmath11 0.03 & 6.33 @xmath11 1.00 & 12 & + denis j1048 - 3956 & 17.37 & 14.98 & 12.47 & 4 & 9.538 & 8.905 & 8.447 & m8.0 v & 2 & 0.08 & 4.03 @xmath11 0.01 & 4.48 @xmath11 0.73 & 10 & + scr 1138 - 7721 & 14.78 & 13.20 & 11.24 & 4 & 9.399 & 8.890 & 8.521 & m5.0 v & 3 & 0.11 & 8.31 @xmath11 0.07 & 9.45 @xmath11 1.71 & 12 & + proxima cen & 11.13 & 9.45 & 7.41 & 3 & 5.357 & 4.835 & 4.384 & m5.0 v & 2 & 0.11 & 1.30 @xmath11 0.01 & 1.15 @xmath11 0.18 & 12 & b + + lhs 1582ab & 14.69j & 13.33j & 11.60j & 4 & 9.799j & 9.177j & 8.854j & m4.5 vj & 1 & & 19.67 @xmath11 0.47 & 13.27 @xmath11 2.25 & 12 & c + gj 748ab & 11.10j & 9.95j & 8.47j & 3 & 7.087j & 6.572j & 6.294j & m3.5 vj & 2 & & 10.00 @xmath11 0.12 & 7.69 @xmath11 1.26 & 12 & c + lhs 3738ab & 15.78j & 14.29j & 12.46j & 3 & 10.654j & 10.091j & 9.761j & m4.5 vj & 5 & & 19.15 @xmath11 0.38 & 18.50 @xmath11 2.96 & 12 & c + [ tab2 ] nine of the 16 targets in this paper have parallaxes previously published by recons , and are noted in column 18 . the results presented here supersede those published previously by recons because additional data and improved reduction techniques have been used , as discussed in detail in @xcite . the identical parallax of lp 944 - 020 is also presented in @xcite as part of a study of the stellar hydrogen burning limit . for bd @xmath010 3166 , we did not run simulations because it is too massive and far away for us to detect any type of substellar companion . however , we do provide the first accurate parallax , and conclude that bd @xmath010 3166 is not physically related to the star with a similar proper motion , lp 731 - 076 , that is @xmath40 away @xcite . photometric and spectroscopic results are provided in table [ tab2 ] . @xmath15 photometry was taken using the ctio 0.9 m ( number of nights of photometry in column 5 ) , with errors in @xmath41 mag @xcite . @xmath42 photometry was retrieved from the two micron all sky survey ( 2mass , @xcite ) catalog . spectral types are given in column 9 with references in column 10 . mass estimates were calculated as discussed in @xmath2[sec : analysis ] . the photometric distances are calculated using the @xmath43 distance relations ( number of relations in column 14 ) detailed in @xcite . for systems with photometric and trigonometric distances that agree within the errors , we conclude that they lack nearly equal luminosity companions . the trigonometric distances of gj 748ab and lhs 1582ab are greater than their photometric distances due to companion contributions to the systems total flux . the two distances of lhs 3738ab agree well , indicating that the companion is significantly fainter than the primary . gj 317 and gj 849 have discrepant ( at 3.5@xmath44 and 3.1@xmath44 , respectively ) photometric and trigonometric distances , which does not necessarily mean that these stars have stellar companions , as main sequence stars within the same spectral type can vary somewhat in luminosity . @xcite note a radial velocity drift in their observations of gj 849 . @xcite also note this drift , and constrain the minimum companion mass to @xmath45 . our astrometry would show a photocenter shift for unequal mass components , as discussed in @xmath2[sec : observe ] . only components of roughly equal luminosity and mass would provide the additional flux with no perturbation . such a companion would have been observed to separations as close as 1 in our images , which corresponds to @xmath109 au . at a semimajor axis of 9 au , the orbital period is 29.5 years for an equal mass companion . this results in velocities for each component 9.1 km / s for edge - orbits . thus , for most orbital inclinations , such a stellar companion is ruled out by the radial velocity data . therefore , it is unlikely that a stellar companion similar to the primary is contributing to the overluminosity we observe . figure [ fig4 ] shows the range of periods and masses for which 90% of simulated companions would have been detected , based on the simulations for objects with masses from 0.5 to 80 @xmath1 . as discussed in @xmath2[sec : analysis ] , the masses , periods , and orbital parameters of the simulated companions were assigned randomly from a uniform distribution . for this discussion we set the dividing line between planets and brown dwarfs at 13 @xmath1 , and the dividing line between brown dwarfs and stars at 80 @xmath1 . the bottom panel of figure [ fig4 ] is an inset of the top , showing the best case targets in more detail . the noisy nature of the lines is due to the sizes of the bins used in the simulations . the bin sizes were chosen to achieve a reasonably high resolution , while still having enough simulated companions in each bin . the minimum detectable companion mass is smallest for companions with long periods , which produce the largest amplitude perturbations in the astrometric data . the lower panel indicates that for the best case targets ( stars at close distances and of low mass ) . we are most sensitive to jovian - type planets in jovian - type orbits . for a companion at a given mass and orbital period , the amplitude of the resulting astrometric perturbation depends on the orbital parameters of the system , and the mass and distance of the primary . the detection limits we report are based on simulations of companions with a wide range of masses , periods , and orbital parameters . therefore we give a few representative examples of how the results in figure [ fig4 ] translate into astrometric perturbations in mas . in the case of a face - on , circular orbit , a 20 @xmath1 companion in a 4 year orbit around gj 581 would cause a 16 mas perturbation , while a 15 @xmath1 companion in an 8 year orbit would cause a 20 mas perturbation , and a 10 @xmath1 companion in a 12 yr orbit would cause a 17 mas perturbation . for circular , face - on orbits around proxima centauri , companions of 1.5 , 1 , and 0.5 @xmath1 in orbits of 4 , 8 , and 12 years would cause perturbations of 12 , 13 , and 8 mas , respectively . these values are significantly greater than the per observation precisions listed table [ tab1 ] 7.93 mas for gj 581 and 4.83 mas for proxima . thus , the 90% detection thresholds given in figure [ fig4 ] are reasonable for the four planet hosts observed longer than eight years , column 17 of table [ tab1 ] gives the percentages of simulated _ brown dwarf _ companions , ranging from 81 99% , eliminated with orbits between two and eight years , and masses between 13 and 80 @xmath1 . approximately 92% of all simulated brown dwarfs have been eliminated as companions to those stars known to host exoplanets . for the six more astrometrically favorable targets , we calculate the percentages of simulated _ planetary _ companions eliminated with orbits between two and eight years and masses between 1 and 13 @xmath1 , with results ranging from 69 99% . we have eliminated @xmath1086% of all simulated planets with masses of 1 13 @xmath1 around these six astrometrically favorable stars , and effectively all brown dwarf companions in orbital periods of 2 8 years . photocentric orbital solutions for the three binaries are shown in figure [ fig5 ] with the corresponding orbital parameters given in table [ tab3 ] . from our astrometric data for gj 748 ab , we find an orbital period of 2.504 @xmath11 0.025 years , which is consistent with the two detailed studies of the system by @xcite , who found p = 2.466 @xmath11 0.008 years , and @xcite , who found p = 2.469 @xmath11 0.001 years using hst fine guidance sensor data . however , we determine an eccentricity of 0.06 , which is inconsistent with the value of 0.45 found in both of the hst studies . we utilized the orbit - fitting code described in @xcite and set starting eccentricities of 0.05 to 0.95 in increments of 0.05 ; regardless , our data converged to the e = 0.06 value each time . lrrrrrrr gj 748ab & 2.504@xmath110.025 & 2005.86@xmath110.25 & 28.1@xmath111.6 & 0.06@xmath110.04 & 137.8@xmath119.0 & 218.2@xmath1140.3 & 173.6@xmath1112.1 + lhs 1582ab & 5.309@xmath110.049 & 2001.84@xmath110.14 & 21.9@xmath111.3 & 0.17@xmath110.03 & 143.6@xmath117.5 & 62.0@xmath1114.3 & 97.9@xmath1112.4 + lhs 3738ab & 6.141@xmath110.059 & 2005.73@xmath110.16 & 28.1@xmath111.2 & 0.12@xmath110.02 & 131.8@xmath114.1 & 130.5@xmath1111.2 & 130.5@xmath11 5.1 + [ tab3 ] the discrepancy between our eccentricity and that of the hst studies is likely due to our observations of gj 748 ab having been taken at the two different @xmath8 filters discussed in @xmath2 2.1 . while the two filters are photometrically identical within measurable errors @xcite , they are not astrometrically identical . we have analyzed the astrometric residuals for over 500 targets without detectable perturbations in the three different filters ( @xmath46 and @xmath14 ) over the length of our observing program . the r and i filters are stable , but astrometric offsets in the v filters are evident over the time period when the problematic `` new '' @xmath8 filter was used . these offsets have been mitigated as discussed in @xmath2 2.1 , allowing data from both @xmath8 filters to be used to produce reliable parallax results . in the case of gj 748 ab , we are able to recover the correct period , but the offsets in the residuals are likely contributing to the errant eccentricity . we presently do not possess enough observations to perform a reduction of gj 748 ab without the problematic @xmath8 filter data . we include the current solution because it is our only system for which an accurate period has been published , against which to compare our results . in contrast to our photocenter data , the fgs observations resolve the system into two components at 15 epochs over 1.8 years . they are exquisitely sensitive to the separation and position angle of the secondary from the primary , and are to be preferred to our ground - based results for the eccentricity . a clever suggestion by hugh harris of usno has been suggested to solve this dilemma . because in an unresolved system the center of mass location is unknown , the zero points for the residuals in r.a . and are unknown . by shifting the zero points and fitting the residuals , a different eccentricity may be derived . we await the acquisition of resolved data for several more systems before exploring this technique so that a robust analysis can be accomplished . at present errors on the eccentricities in table [ tab3 ] should be treated with caution . the orbital solutions for lhs 1582ab and lhs 3738ab are updated and improved over those presented in @xcite , which were the first orbits presented for each system . in addition to demonstrating the astrometric detection and characterization of unresolved companions , these results can provide additional dynamically determined masses for m dwarfs , once the systems have been resolved .
|
astrometric measurements are presented for seven nearby stars with previously detected planets : six m dwarfs ( gj 317 , gj 667c , gj 581 , gj 849 , gj 876 , and gj 1214 ) and one k dwarf ( bd 3166 ) . measurements are also presented for six additional nearby m dwarfs without known planets , but which are more favorable to astrometric detections of low mass companions , as well as three binary systems for which we provide astrometric orbit solutions . we provide trigonometric parallaxes and proper motions for all 16 systems , and perform an extensive analysis of the astrometric residuals to determine the minimum detectable companion mass for the 12 m dwarfs not having close stellar secondaries . for the six m dwarfs with known planets , we are not sensitive to planets , but can rule out the presence of all but the least massive brown dwarfs at periods of 2 12 years . for the six more astrometrically favorable m dwarfs , we conclude that none have brown dwarf companions , and are sensitive to companions with masses as low as 1 for periods longer than two years .
|
astrometric measurements are presented for seven nearby stars with previously detected planets : six m dwarfs ( gj 317 , gj 667c , gj 581 , gj 849 , gj 876 , and gj 1214 ) and one k dwarf ( bd 3166 ) . measurements are also presented for six additional nearby m dwarfs without known planets , but which are more favorable to astrometric detections of low mass companions , as well as three binary systems for which we provide astrometric orbit solutions . observations have baselines of three to thirteen years , and were made as part of the recons long - term astrometry and photometry program at the ctio / smarts 0.9 m telescope . we provide trigonometric parallaxes and proper motions for all 16 systems , and perform an extensive analysis of the astrometric residuals to determine the minimum detectable companion mass for the 12 m dwarfs not having close stellar secondaries . for the six m dwarfs with known planets , we are not sensitive to planets , but can rule out the presence of all but the least massive brown dwarfs at periods of 2 12 years . for the six more astrometrically favorable m dwarfs , we conclude that none have brown dwarf companions , and are sensitive to companions with masses as low as 1 for periods longer than two years . in particular , we conclude that proxima centauri has no jovian companions at orbital periods of 2 12 years . these results complement previously published m dwarf planet occurrence rates by providing astrometrically determined upper mass limits on potential super - jupiter companions at orbits of two years and longer . as part of a continuing survey , these results are consistent with the paucity of super - jupiter and brown dwarf companions we find among the over 250 red dwarfs within 25 pc observed longer than five years in our astrometric program .
|
1606.06770
|
i
|
the detailed study of unstable nuclei was a major subject in nuclear physics during recent decades . @xmath0 decay measurements provide not only important information on the structure of the daughter and parent nuclei , but can also be used to inform nuclear astrophysics studies and probe fundamental subatomic symmetries @xcite . the link between experimental results and theory is given by the reduced transition probabilities , @xmath3 . experimental @xmath3 values involve three measured quantities : the half - life , @xmath5 , the @xmath6 value of the transition , which determines the statistical phase space factor @xmath7 , and the branching ratio associated with that transition , @xmath8 . in the standard @xmath9 description of @xmath0 decay , @xmath3 values are related to the fundamental constants of the weak interaction and the matrix elements through this equation : @xmath10 where @xmath11 is a constant and @xmath12 are the vector ( axial ) coupling constants of the weak interaction ; @xmath13 and @xmath14 are the spin and isospin operators , respectively . thus , a comparison of the experimental @xmath3 values with the theoretical ones obtained from the calculated matrix elements is a good test of the nuclear wave functions obtained with model calculations . however , to reproduce the @xmath3 values measured experimentally , the axial - vector coupling constant @xmath15 involved in gamow - teller transitions has to be renormalized @xcite . the effective coupling constant @xmath16 is deduced empirically from experimental results and depends on the mass of the nucleus : the quenching factor is @xmath17 in the @xmath18 shell @xcite , @xmath19 in the @xmath20 shell @xcite , and @xmath21 in the @xmath22 shell @xcite . despite several theoretical approaches attempting to reveal the origin of the quenching factor it is still not fully understood @xcite . another phenomenon which shows the limitations of our theoretical models is the so - called _ @xmath0-decay mirror asymmetry_. if we assume that the nuclear interaction is independent of isospin , the theoretical description of @xmath0 decay is identical for the decay of a proton ( @xmath23 ) or a neutron ( @xmath24 ) inside a nucleus . therefore , the @xmath3 values corresponding to analog transitions should be identical . any potential asymmetries are quantified by the asymmetry parameter @xmath25 , where the @xmath26 refers to the @xmath27 decays in the mirror nuclei . the average value of this parameter is @xmath28 for @xmath18 and @xmath20 shell nuclei @xcite . from a theoretical point of view the mirror asymmetry can have two origins : ( a ) the possible existence of exotic _ second - class currents _ @xcite , which are not allowed within the framework of the standard @xmath9 model of the weak interaction and ( b ) the breaking of the isospin symmetry between the initial or final nuclear states . shell - model calculations were performed to test the isospin non - conserving part of the interaction in @xmath0 decay @xcite . the main contribution to the mirror asymmetry from the nuclear structure was found to be from the difference in the matrix elements of the gamow - teller operator ( @xmath29 ) , because of isospin mixing and/or differences in the radial wave functions . large mirror asymmetries have been reported for transitions involving halo states @xcite . for example , the asymmetry parameter for the @xmath30 mirror decays @xmath31ne@xmath32f and @xmath31n@xmath32o to the first excited states of the respective daughters was measured to be @xmath33 and @xmath34 in two independent experiments @xcite . this result was interpreted as evidence for a proton halo in the first excited state of @xmath31f assuming that the fraction of the @xmath35 component of the valence nucleons remains the same in @xmath31ne and @xmath31n . however , a different interpretation was also given in terms of charge dependent effects which increase the @xmath35 fraction in @xmath31ne by about 50% @xcite . the latter result is also consistent with the high cross section obtained in the fragmentation of @xmath31ne @xcite , suggesting the existence of a halo in @xmath31ne . more recently kanungo _ et al . _ reported the possiblity of a two - proton halo in @xmath31ne @xcite . an extremely large mirror asymmetry was also observed in the mirror decay of @xmath36 isobars @xmath37li@xmath38be and @xmath37c@xmath38b . a value of @xmath39 was reported for the @xmath37li and @xmath37c @xmath0-decay transitions to the 11.8 and 12.2 mev levels of their respective daughters , which is the largest ever measured @xcite . despite the low experimental interaction cross sections measured with various targets in attempts to establish the halo nature of @xmath37c @xcite , recent results at intermediate energies @xcite , together with the anomalous magnetic moment @xcite and theoretical predictions @xcite , make @xmath37c a proton halo candidate . the potential relationship between large mirror asymmetries and halos is therefore clear . precision measurements of mirror asymmetries in states involved in strong , isolated , @xmath0-decay transitions might provide a technique to probe halo nuclei that is complementary to total interaction cross section and momentum distribution measurements in knockout reactions @xcite . moreover , @xmath0 decay of proton - rich nuclei can be used for nuclear astrophysics studies . large @xmath40-values of these nuclei not only allow the population of the bound excited states of the daughter , but also open particle emission channels . some of these levels correspond to astrophysically significant resonances which can not be measured directly because of limited radioactive beam intensities . for example , the @xmath41 reaction @xcite plays an important role in the abundance of the cosmic @xmath1-ray emitter @xmath42 . the effect of this reaction is to reduce the amount of ground state @xmath42 , which is bypassed by the sequence @xmath43 , reducing therefore the intensity of the 1809-kev @xmath1-ray line characteristic of the @xmath42 @xmath0 decay @xcite . thus it is important to constrain the @xmath41 reaction rate . @xmath2p is the most proton - rich bound phosphorus isotope . with a half - life of @xmath44 ms and a @xmath45 value of @xmath46 kev @xcite the @xmath0 decay can be studied over a wide energy interval . @xmath0-delayed @xmath1-rays and protons from excited levels of @xmath2si below and above the proton separation energy of @xmath47 kev @xcite were observed directly in previous experiments @xcite and , more recently , indirectly from the doppler broadening of peaks in the @xmath0-delayed proton-@xmath1 spectrum @xcite . the contribution of novae to the abundance of @xmath42 in the galaxy was recently constrained by using experimental data on the @xmath0 decay of @xmath2p @xcite . in addition , @xmath48 is a candidate to have a proton halo @xcite . phosphorus isotopes are the lightest nuclei expected to have a ground state with a dominant contribution of a @xmath49 orbital . low orbital angular momentum orbitals enhance the halo effect , because higher @xmath50-values give rise to a confining centrifugal barrier . the low separation energy of @xmath2p ( 143(200 ) kev @xcite , 0(90 ) kev@xcite ) , together with the narrow momentum distribution and enhanced cross section observed in proton - knockout reactions @xcite give some experimental evidence for the existence of a proton halo in @xmath2p . in this paper , we present a comprehensive summary of the @xmath0-delayed @xmath1 decay of @xmath2p measured at the national superconducting cyclotron laboratory ( nscl ) at michigan state university during a fruitful experiment for which selected results have already been reported in two separate shorter papers @xcite . in the present work , the gamow - teller strength , @xmath51 , and the experimental @xmath3 values are compared to theoretical calculations and to the decay of the mirror nucleus @xmath2na to investigate the gamow - teller strength and mirror asymmetry , respectively . a potential relationship between the mirror asymmetry and the existence of a proton halo in @xmath2p is also discussed . finally , in the last section , the calculated thermonuclear @xmath52al@xmath53si reaction rate , which was used in ref . @xcite to estimate the contribution of novae to the abundance of galactic @xmath2al , is tabulated for completeness .
|
background : : measurements of decay provide important nuclear structure information that can be used to probe isospin asymmetries and inform nuclear astrophysics studies . a complete -decay scheme was built for the allowed transitions to bound excited states ofsi .
|
background : : measurements of decay provide important nuclear structure information that can be used to probe isospin asymmetries and inform nuclear astrophysics studies . purpose : : to measure the-delayed decay ofp and compare the results with previous experimental results and shell - model calculations . method : : ap fast beam produced using nuclear fragmentation was implanted into a planar germanium detector . its -delayed-ray emission was measured with an array of 16 high - purity germanium detectors . positrons emitted in the decay were detected in coincidence to reduce the background . results : : the absolute intensities ofp -delayed-rays were determined . a total of six new-decay branches and 15 new -ray lines have been observed for the first time inp-decay . a complete -decay scheme was built for the allowed transitions to bound excited states ofsi . values and gamow - teller strengths were also determined for these transitions and compared with shell model calculations and the mirror -decay ofna , revealing significant mirror asymmetries . conclusions : : a very good agreement with theoretical predictions based on the usdb shell model is observed . the significant mirror asymmetry observed for the transition to the first excited state ( ) may be evidence for a proton halo in p .
|
1606.06770
|
r
|
as mentioned in sec . [ sec : intro ] , the data presented in this paper are from the same experiment described in refs . @xcite , but independent sorting and analysis routines were developed and employed . the values extracted are therefore slightly different , but consistent within uncertainties . new values derived in the present work are not intended to supersede those from refs . @xcite , but rather to complement them . in this section , the analysis procedure is described in detail and the experimental results are presented . figure [ fig : spec ] shows the cumulative @xmath1-ray spectrum observed in all the detectors of the sega array in coincidence with a @xmath0-decay signal in the gedssd . we have identified 48 photopeaks , of which 30 are directly related to the decay of @xmath2p . most of the other peaks were assigned to the @xmath0 decay of the main contaminant of the beam , @xmath62al . peaks in the spectrum have been labeled by the @xmath1-ray emitting nuclide . twenty - two of the peaks correspond to @xmath2si , while eight of them correspond to @xmath0-delayed proton decays to excited states of @xmath52al followed by @xmath1-ray emission . in this work we will focus on the decay to levels of @xmath2si as the @xmath52al levels have already been discussed in ref . @xcite . ( upper panel ) energy calibration of sega @xmath1-ray spectra using the @xmath0-delayed @xmath1 rays emitted by @xmath56 . the solid line is the result of a second degree polynomial fit . energies and uncertainties are taken from @xcite . ( lower panel ) residuals of the calibration points with respect to the calibration line . ] the energies of the @xmath1 rays emitted during the experiment were determined from a calibration of the sega array . as mentioned in sect . [ sec : experiment ] and in refs . @xcite a gain - matching procedure was performed to align all the signals coming from the 16 detectors comprising the array . this alignment was done with the strongest background peaks , namely the 1460.8-kev line ( from the @xmath64 decay ) and the 2614.5-kev one ( from the @xmath65 decay ) . the gain - matched cumulative spectrum was then absolutely calibrated _ in situ _ using the well - known energies of the @xmath62al @xmath0-delayed @xmath1 rays emitted by @xmath66 , which cover a wide range in energy from 511 kev to almost 10 mev @xcite . to account for possible non - linearities in the response of the germanium detectors , a second degree polynomial fit was used as a calibration function . results of the calibration are shown in fig . [ fig : calibr ] . the standard deviation for this fit is 0.3 kev , which includes the literature uncertainties associated with the energies of @xmath66 . the systematic uncertainty was estimated from the residuals of room background peaks not included in the fit . the lower panel of fig . [ fig : calibr ] shows that these deviations are below 0.6 kev , with an average of 0.2 kev . based on this , the systematic uncertainty was estimated to be 0.3 kev . sega photopeak efficiency . ( top panel ) results of a geant4 simulation [ solid line ( red ) ] compared to the efficiency measured with absolutely calibrated sources ( black circles ) and the known @xmath66 lines ( empty squares ) . the simulation and the @xmath66 data have been scaled to match the source measurements . ( bottom panel ) ratio between the simulation and the experimental data . the shaded area ( yellow ) shows the adopted uncertainties . ] the @xmath0-particle detection efficiency of the gedssd can be determined by taking the ratio between the number of counts under a certain photopeak in the @xmath0-gated @xmath1-ray singles spectrum and the ungated one . in principle , the @xmath0 efficiency depends on @xmath40 . to investigate this effect , we calculated the ratios between the gated and the ungated spectra for all the @xmath66 peaks , which have different combinations of @xmath40 , and found it to be independent of the end - point energy of the @xmath0 particles , with an average ratio of @xmath67 . because of the different implantation depths for @xmath56 and @xmath48 ( @xmath56 barely penetrates into the gedssd ) , we also calculated the gated to ungated ratios of the strongest peaks of @xmath68 ( 1797 kev ) and its daughter @xmath42 ( 829 kev ) obtaining a constant , average , value for the efficiency of @xmath69 . the singular value for @xmath68 and @xmath42 is explained by their common decay point in the gedssd . to obtain precise measurements of the @xmath1-ray intensities , we determined the photopeak efficiency of sega . the photopeak efficiency was studied over a wide energy range between 400 kev and 8 mev . the results of a geant4 @xcite monte - carlo simulation were compared with the relative intensities of the well - known @xmath66 lines used also in the energy calibration . the high energy lines of this beam contaminant made it possible to benchmark the simulation for energies higher than with standard sources . in addition , the comparison of the simulation to data taken offline with absolutely - calibrated @xmath70 and @xmath71 sources allowed us to scale the simulation to determine the efficiency at any energy . the scaling factor was 0.91 . the statistical uncertainty of this scaling factor was inflated by a scaling factor of @xmath72 yielding an uncertainty of 1.5% , which was propagated into the efficiency . the magnitude of this factor is consistent with geant4 simulations of the scatter associated with coincidence summing effects @xcite . figure [ fig : eff ] shows the adopted efficiency curve compared to the source data , and the @xmath66 peak intensities . the accuracy of this photopeak efficiency was estimated to be @xmath73 for energies below 2800 kev and 5% above that energy . ( top panel ) example of a typical fit to the 1960-kev peak , using the function of eq . ( [ eq : emg ] ) . the dashed line corresponds to the background component of the fit . ( bottom panel ) residuals of the fit in terms of the standard deviation @xmath13 . ] the intensities of the @xmath1 rays emitted in the @xmath0 decay of @xmath48 were obtained from the areas of the photopeaks shown in the spectrum of fig . [ fig : spec ] . we used an exponentially modified gaussian ( emg ) function to describe the peak shape together with a linear function to model the local background : @xmath74\mathrm{erfc}\left[\frac { \sigma^2+\tau(\mu - x)}{\sqrt{2}\sigma\tau}\right ] , f = b+\frac{n}{2\tau}e^{\frac{1}{2\tau}\left ( 2\mu+\frac{\sigma^2}{\tau } -2x \right ) } \mathrm{erfc}\!\left[\frac { \sigma^2+\tau(\mu - x)}{\sqrt{2}\sigma\tau}\right ] , \label{eq : emg}\ ] ] where @xmath75 is a linear background , @xmath76 is the area below the curve , @xmath61 and @xmath13 are the centroid and the width of the gaussian , respectively , and @xmath14 is the decay constant of the exponential ; erfc is the complementary error function . the parameters describing the width of the gaussian ( @xmath13 ) and the exponential constant ( @xmath14 ) were determined by fitting narrow isolated peaks at various energies . the centroids and the areas below the peaks were obtained from the fits . when multiple peaks were very close , a multi - peak fitting function was applied using the same values for the @xmath14 and @xmath13 parameters for all the peaks in the region . in general the fits were very good , with reduced chi - squared ( @xmath77 ) close to unity . in those cases where @xmath77 was bigger than one , the statistical uncertainties were inflated by multiplying them by @xmath72 . [ fig : fit ] shows an example of the fit to the 1960-kev peak . the total number of @xmath48 ions implanted and subsequently decaying in the gedssd is , in principle , needed to obtain an absolute normalization of the @xmath1-ray intensities , and hence the @xmath0 branchings of @xmath68 levels . the number of @xmath1 rays observed at energy @xmath78 is : @xmath79 d d c c d d & & @xmath80 & @xmath81 & & + 1797.1(3 ) & & @xmath82 & @xmath83 & 1797.1(3 ) & + 2786.4(3 ) & < 0.39 & @xmath84 & @xmath82 & 989.0(3 ) & 5.7(3 ) + & & & @xmath83 & 2786.5(4 ) & 3.4(2 ) + 3756.8(3 ) & 1.9(2 ) & @xmath85 & @xmath84 & 970.3(3 ) & 1.15(9 ) + & & & @xmath82 & 1959.8(4 ) & 1.7(1 ) + 4138.6(4 ) & 6.2(4 ) & @xmath86 & @xmath84 & 1352.2(4 ) & 0.48(7 ) + & & & @xmath82 & 2341.2(4 ) & 4.7(3 ) + & & & @xmath83 & 4138.0(5 ) & 1.0(1 ) + 4187.6(4 ) & 4.4(3 ) & @xmath87 & @xmath84 & 1401.3(3 ) & 3.8(2 ) + & & & @xmath82 & 2390.1(4 ) & 2.2(1 ) + 4445.1(4 ) & 0.8(2 ) & @xmath88 & @xmath84 & & 0.08(6 ) + & & & @xmath82 & 2647.7(5 ) & 1.7(1 ) + 4796.4(5 ) & 0.56(9 ) & @xmath89 & @xmath84 & 2999.1(5 ) & 0.56(9 ) + 4810.4(4 ) & 3.1(2 ) & @xmath90 & @xmath84 & 2023.9(3 ) & 3.1(2 ) + 5146.5(6 ) & 0.18(5 ) & @xmath91 & @xmath84 & 2360.0(6 ) & 0.18(5 ) + 5288.9(4 ) & 0.76(7 ) & @xmath92 & @xmath88 & 842.9(3 ) & 0.33(7 ) + & & & @xmath85 & 1532.1(5 ) & 0.43(7 ) + & & & @xmath82 & & < 0.12 + 5517.3(3 ) & 2.7(2 ) & @xmath93 & @xmath88 & 1072.1(5 ) & 0.69(9 ) + & & & @xmath87 & 1329.9(3 ) & 1.4(1 ) + & & & @xmath85 & 1759.7(5 ) & 0.47(6 ) + & & & @xmath84 & 2729.9(5 ) & 0.29(5 ) + 5929.3(6 ) & 0.15(5 ) & @xmath94 & @xmath87 & 1741.7(9 ) & 0.15(5 ) + where @xmath95 is the total number of ions decaying , @xmath96 are the efficiencies to detect @xmath1 rays ( @xmath0 particles ) , and @xmath97 is the absolute @xmath1-ray intensity . to circumvent the uncertainty associated with the total number of ions decaying , we used the ratio of the number of @xmath0 decays of @xmath48 to its daughter @xmath68 [ @xmath98 @xcite , and the absolute intensity of the 829-kev @xmath1-rays emitted in the @xmath0 decay of @xmath68 , @xmath99 $ ] @xcite , to calculate the intensity of the 1797-kev line , which is the most intense @xmath1 ray emitted in the decay of @xmath48 ( see table [ tab : levels ] ) . to do so , we applied eq . ( [ eq : abs_intensity ] ) to these two @xmath1 rays : @xmath100 @xmath101 by taking the ratio between eqs . ( [ eq : intensity_al ] ) and ( [ eq : intensity_si ] ) , the only unknown is the intensity of the 1797-kev @xmath1 ray , because the @xmath0 efficiencies can be obtained from the @xmath0-gated to ungated ratios discussed in sec . [ sec : experiment ] . the value obtained for the intensity of the 1797-kev @xmath1 ray is thus 58(3)% , which is in agreement with the value 52(11)% reported in ref . @xcite and more precise . the rest of the @xmath1-ray intensities were determined with respect to this value by employing the efficiency curve and they are presented in table [ tab : levels ] . we also report an upper limit on the intensity of one @xmath1 ray which was expected to be near the theshold of our sensitivity given the intensity predicted by theory . ( color online ) @xmath0-@xmath1-@xmath1 coincidence spectrum gating on the 1797 kev @xmath1-rays ( blue ) . the hatched histogram ( green ) shows coincidences with continuum background in a relatively broad region above the peak gate . the background bins are 16 kev wide and are normalized to the expected background per 2 kev from random coincidences . the strongest peaks corresponding to @xmath1 rays emitted in coincidence are indicated . ] the 16-fold granularity of sega allowed us to obtain @xmath0-@xmath1-@xmath1 coincidence spectra , which helped to interpret the @xmath48 decay scheme . [ fig : coincidences2 ] shows the gamma coincident spectrum gated on the 1797-kev peak , where we can see several peaks corresponding to @xmath1 rays detected in coincidence . to estimate the background from random coincidences , we have created another histogram gated on the background close to the peak and normalized to the number of counts within the gated regions . at some energies the background estimate is too high . this is because of a contribution from real @xmath1-@xmath1 coincidences involving compton background , which should not be normalized according to the random assumption . [ fig : coincidences ] presents a sample of peaks observed in coincidence when gating on some other intense @xmath1 rays observed . from this sample we can see that the coincidence technique helps to cross - check the decay scheme . for example fig . [ fig : coincidences](a ) shows clearly that the 1401-kev @xmath1 ray is emitted in coincidence with the 989-kev @xmath1 ray , indicating that the former @xmath1 ray comes from a higher - lying level . in the same way , we can see in fig . [ fig : coincidences](b ) that the 1330-kev @xmath1-ray is emitted from a level higher than the 4187-kev level . from the gated spectra , some information can also be extracted from the missing peaks . as fig . [ fig : coincidences](c ) shows , by gating on the 2024-kev @xmath1 ray the 970-kev peak disappears , displaying only the 989-kev peak , which means that the 970-kev @xmath1 ray comes from a level which is not connected with these two levels by any @xmath1-ray cascade . [ fig : coincidences](d ) shows clearly the coincidence between the @xmath1 ray emitted from the first @xmath103 state at 1797 kev to the ground state of @xmath2si and the 2341-kev @xmath1 ray from the third @xmath103 state to the first excited state . these coincidence procedures were systematically analyzed for all possible combinations of @xmath1 rays and the results are summarized in table [ tab : coincidence ] in the form of a 2d matrix , where a checkmark ( ) means the @xmath1 rays were detected in coincidence . the condition for a @xmath1 ray to be listed in coincidence with another is for it to be at least 3@xmath13 above the estimated random - coincidence background . it is worth noting that this background estimate is somewhat conservative , therefore the significance of some of the peaks is underestimated . [ cols="^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^,^",options="header " , ] fig . [ fig : decay ] displays the @xmath48 @xmath0-decay scheme deduced from the results obtained in this experiment . only those levels populated in the @xmath0 decay are represented . this level scheme was built in a self - consistent way by taking into account the @xmath1-ray energies and intensities observed in the singles spectrum of fig . [ fig : spec ] and the @xmath0-@xmath1-@xmath1 coincidence spectra described in sec . [ subsec : coincidences ] . the excitation energies of @xmath68 bound levels , their @xmath0-feedings , the energies of the @xmath1 rays , and the absolute intensities measured in this work are shown in table [ tab : levels ] . level energies of @xmath68 populated in the @xmath0 delayed @xmath1 decay of @xmath48 were obtained from the measured @xmath1-ray energies including a correction for the nuclear recoil . the excitation energy values of the levels listed in table [ tab : levels ] were obtained from the weighted average of all the possible @xmath1-ray cascades coming from that level . to assign spins and parities we compared the deduced level scheme with usdb shell - model calculations and took into account @xmath0-decay angular momentum selection rules , showing a 1 to 1 correspondence for all the levels populated by allowed transitions , with a fair agreement in the level energies within theoretical uncertainties of a few hundred kev ( see fig . [ fig : decay ] ) . the @xmath0 branching ratio to the @xmath105-th excited energy level can be determined from the @xmath1-ray intensities : @xmath106 where @xmath107 represents the total @xmath1-ray intensity observed decaying out of ( into ) the @xmath105-th level . the @xmath0-decay branches deduced from this experiment are given in table [ tab : br ] , where they are also compared to previous measurements of @xmath48 @xmath0 decay @xcite . to investigate the possible missing intensity from the pandemonium effect @xcite , we have used a shell - model calculation to estimate the @xmath1-ray intensities of all possible transitions from bound states feeding each particular level , and found them to be on the order of the uncertainty or ( usually ) much lower . l d d d d d d & & + & & & & & & + 1797 & 41(3 ) & 44(12 ) & 47.22 & 4.89(3 ) & 4.89(17 ) & 4.81 + 2786 & < 0.39 & 3.3(20 ) & 0.37 & & 5.87(72 ) & 6.77 + 3757 & 1.9(2 ) & 2.68(68 ) & 1.17 & 5.94(4 ) & 5.81(15 ) & 6.135 + 3842 & & 1.68(47 ) & & & 6.00(17 ) & + 4139 & 6.2(4 ) & 1.78(75 ) & 2.97 & 5.37(3 ) & 5.93(32 ) & 5.634 + 4188 & 4.4(3 ) & 2.91(71 ) & 8.88 & 5.51(3 ) & 5.71(14 ) & 5.182 + 4445 & 0.8(2 ) & & 1.11 & 6.23(8 ) & & 6.071 + 4796 & 0.56(9 ) & & 0.06 & 6.31(7 ) & & 7.274 + 4810 & 3.1(2 ) & & 4.45 & 5.57(3 ) & & 5.934 + 5147 & 0.18(5 ) & & 0.03 & 6.7(1 ) & & 7.474 + 5289 & 0.76(7 ) & & 0.60 & 6.09(6 ) & & 6.158 + 5517 & 2.7(2 ) & & 3.96 & 5.51(4 ) & & 5.262 + 5929 & 0.15(5 ) & 17.96(90)&10.08 & 6.7(1 ) & 4.60(3)&4.810 +
|
positrons emitted in the decay were detected in coincidence to reduce the background . results : : the absolute intensities ofp -delayed-rays were determined . values and gamow - teller strengths were also determined for these transitions and compared with shell model calculations and the mirror -decay ofna , revealing significant mirror asymmetries .
|
background : : measurements of decay provide important nuclear structure information that can be used to probe isospin asymmetries and inform nuclear astrophysics studies . purpose : : to measure the-delayed decay ofp and compare the results with previous experimental results and shell - model calculations . method : : ap fast beam produced using nuclear fragmentation was implanted into a planar germanium detector . its -delayed-ray emission was measured with an array of 16 high - purity germanium detectors . positrons emitted in the decay were detected in coincidence to reduce the background . results : : the absolute intensities ofp -delayed-rays were determined . a total of six new-decay branches and 15 new -ray lines have been observed for the first time inp-decay . a complete -decay scheme was built for the allowed transitions to bound excited states ofsi . values and gamow - teller strengths were also determined for these transitions and compared with shell model calculations and the mirror -decay ofna , revealing significant mirror asymmetries . conclusions : : a very good agreement with theoretical predictions based on the usdb shell model is observed . the significant mirror asymmetry observed for the transition to the first excited state ( ) may be evidence for a proton halo in p .
|
1606.06770
|
c
|
we have measured the absolute @xmath1-ray intensities and deduced the @xmath0-decay branches for the decay of @xmath2p to bound states and low - lying resonances of @xmath2si . we have observed six new @xmath0-decay branches and 15 @xmath1-ray lines never observed before in @xmath2p @xmath0 decay , likely corresponding to most of all the allowed gamow - teller transitions between the ground state and 5.9 mev . the energies measured for the excited states show good agreement with previous results obtained using various nuclear reactions to populate these states . we have calculated the @xmath132 values of all these new transitions and compared them to usdb shell - model calculations . the reported values show good agreement with the theoretical calculations . in addition , the gamow - teller strength function was calculated and compared to theoretical values , showing that the summed gamow teller strength is locally overestimated with the standard @xmath20 shell quenching of 0.6 . the mirror asymmetry was also investigated by calculating the @xmath0-decay asymmetry parameter @xmath171 for 10 transitions . the significant asymmetries observed , particularly for the transition to the first excited states of @xmath2si and its mirror @xmath2 mg ( @xmath172 ) might be further evidence for the existence of a proton halo in the @xmath2p . finally , we have tabulated the total @xmath52al@xmath53si reaction rate at nova temperatures used to estimate the galactic production of @xmath2al in novae in ref . @xcite . the authors gratefully acknowledge the contributions of the nscl staff . this work is supported by the u.s . national science foundation under grants phy-1102511 , phy-0822648 , phy-1350234 , phy-1404442 , the u.s . department of energy under contract no . de - fg02 - 97er41020 , the u.s . national nuclear security agency under contract no . de - na0000979 and the natural sciences and engineering research council of canada . 58ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrevc.91.025501 [ * * , ( ) ] link:\doibase 10.1103/physrevc.7.930 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0375-9474(73)90840-3 [ * * , ( ) ] link:\doibase 10.1103/physrevc.47.163 [ * * , ( ) ] link:\doibase 10.1103/physrevc.28.1343 [ * * , ( ) ] link:\doibase 10.1103/physrevc.53.r2602 [ * * , ( ) ] http://stacks.iop.org/1742-6596/20/i=1/a=025 [ * * , ( ) ] link:\doibase 10.1140/epja / i2003 - 10218 - 8 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0370-2693(70)90150-4 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.38.321 [ * * , ( ) ] link:\doibase 10.1007/s100500050397 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0375-9474(02)01392-1 [ * * , ( ) ] @noop * * , ( ) \doibase http://dx.doi.org/10.1016/0370-2693(93)91564-4 [ * * , ( ) ] http://stacks.iop.org/0954-3899/24/i=1/a=018 [ * * , ( ) ] link:\doibase 10.1103/physrevc.55.r1633 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0370-2693(94)90585-1 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0375-9474(96)00241-2 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.physletb.2003.07.050 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0375-9474(01)00650-9 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.physletb.2003.09.073 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0375-9474(97)81837-4 [ * * , ( ) ] @noop ) \doibase http://dx.doi.org/10.1016/0375-9474(95)00115-h [ * * , ( ) ] http://stacks.iop.org/0256-307x/27/i=9/a=092101 [ * * , ( ) ] link:\doibase 10.1103/physrevc.52.3013 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevc.79.035803 [ * * , ( ) ] link:\doibase 10.1103/physrevc.53.475 [ * * , ( ) ] link:\doibase 10.1088/1674 - 1137/36/12/003 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0370-2693(83)90950-4 [ * * , ( ) ] link:\doibase 10.1103/physrevc.30.1276 [ * * , ( ) ] link:\doibase 10.1103/physrevc.92.031302 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.111.232503 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0370-2693(96)00634-x [ * * , ( ) ] link:\doibase 10.1103/physrevc.53.r572 [ * * , ( ) ] http://stacks.iop.org/0256-307x/26/i=3/a=032102 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.81.5089 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0168-583x(02)01895-5 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.nima.2009.05.100 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.nima.2013.06.027 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0168-9002(01)00257-1 [ * * , ( ) ] link:\doibase 10.1051/epjconf/20146602072 [ * * , ( ) ] @noop \doibase http://dx.doi.org/10.1016/j.nima.2013.12.044 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.nds.2007.10.001 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0168-9002(03)01368-8 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0168-9002(90)90561-j [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/s0375-9474(97)00613-1 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0370-2693(77)90223-4 [ * * , ( ) ] link:\doibase 10.1103/physrevc.75.062801 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevc.92.035808 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/0375-9474(74)90645-9 [ * * , ( ) ] link:\doibase 10.1103/physrevc.18.401 [ * * , ( ) ] link:\doibase 10.1103/physrevc.71.044309 [ * * , ( ) ] http://link.aps.org/doi/10.1103/physrevc.79.032801 [ * * , ( ) ]
|
a total of six new-decay branches and 15 new -ray lines have been observed for the first time inp-decay . conclusions : : a very good agreement with theoretical predictions based on the usdb shell model is observed . the significant mirror asymmetry observed for the transition to the first excited state ( ) may be evidence for a proton halo in p .
|
background : : measurements of decay provide important nuclear structure information that can be used to probe isospin asymmetries and inform nuclear astrophysics studies . purpose : : to measure the-delayed decay ofp and compare the results with previous experimental results and shell - model calculations . method : : ap fast beam produced using nuclear fragmentation was implanted into a planar germanium detector . its -delayed-ray emission was measured with an array of 16 high - purity germanium detectors . positrons emitted in the decay were detected in coincidence to reduce the background . results : : the absolute intensities ofp -delayed-rays were determined . a total of six new-decay branches and 15 new -ray lines have been observed for the first time inp-decay . a complete -decay scheme was built for the allowed transitions to bound excited states ofsi . values and gamow - teller strengths were also determined for these transitions and compared with shell model calculations and the mirror -decay ofna , revealing significant mirror asymmetries . conclusions : : a very good agreement with theoretical predictions based on the usdb shell model is observed . the significant mirror asymmetry observed for the transition to the first excited state ( ) may be evidence for a proton halo in p .
|
1502.02862
|
c
|
both an empirical wave in free flow ( figs . [ breakdown ] ( b ) and [ nuclei1996_2 ] ) and a localized congested pattern ( wide moving jam in fig . [ breakdown ] ( c ) ) become to be nuclei for traffic breakdown , when they reach the effective location of a highway bottleneck . however , the propagation of a single congested pattern to the effective bottleneck location is sufficient for the inducing of the breakdown at the bottleneck . in contrast , many waves in free flow can propagate through the bottleneck while initiating no breakdown at the bottleneck ( figs . [ nuclei1996 ] and [ nuclei1998 ] ) . the latter empirical result allows us to assume that at a given flow rate in free flow at a highway bottleneck there is a _ critical wave _ related to a _ critical nucleus _ for traffic breakdown . therefore , if a wave is a smaller one than the critical wave for a given flow rate at a highway bottleneck , then no breakdown occurs while the wave propagates through the bottleneck . for example , all waves shown in fig . [ nuclei1996 ] for @xmath93 and in fig . [ nuclei1998 ] for @xmath94 should be smaller than critical waves . however , waves that become to be nuclei for the breakdown at the effective locations of the bottlenecks in figs . [ nuclei1996_2 ] and [ nuclei1998_2 ] should be equal to or larger ones than critical waves for the breakdown at the related bottlenecks , respectively . in contrast with waves in free flow , within a congested pattern the speed is usually smaller than a critical speed required for the breakdown in free flow at a bottleneck . for this reason , at the flow rate satisfying condition @xmath95 , any localized congested pattern becomes to be a nucleus for traffic breakdown , when the pattern reaches the effective bottleneck location . thus a basic difference between _ empirical spontaneous _ [ breakdown ] ( b ) ) and _ empirical induced _ breakdowns ( fig . [ breakdown ] ( c ) ) is as follows : to initiate the spontaneous breakdown at the bottleneck , i.e. , to be a nucleus for the breakdown , a wave in free flow should be equal to or a larger one than a critical wave . in contrast , a localized congested pattern is always a nucleus for the breakdown at the bottleneck , when condition @xmath95 is satisfied , i.e. , when traffic breakdown can occur at the bottleneck . km ) , downstream ( @xmath1 7.9 km ) and upstream of the bottleneck ( @xmath1 5.1 km ) : ( a ) real field traffic data measured on april 15 , 1996 ( fig . [ breakdown ] ( b ) ) . ( b ) real field traffic data measured on march 22 , 2001 ( fig . [ breakdown ] ( c ) ) . [ 15041996_22032001 ] , width=9 ] however , after the breakdown has occurred , characteristics of a congested pattern that has been formed at the bottleneck do not depend on whether the congested pattern has occurred due to empirical spontaneous breakdown or due to empirical induced breakdown . this statement is illustrated by empirical data presented in fig . [ 15041996_22032001 ] . in fig . [ 15041996_22032001 ] ( a ) , a congested pattern at the on - ramp bottleneck ( fig . [ breakdown ] ( b ) ) has occurred due to empirical spontaneous breakdown caused by a wave that becomes to be a nucleus for the breakdown , when the wave is at the effective bottleneck location ( fig . [ nuclei1996_2 ] ) . in contrast , in fig . [ 15041996_22032001 ] ( b ) a congested pattern at the on - ramp bottleneck ( fig . [ breakdown ] ( c ) ) has been induced due to the propagation of a wide moving jam through the bottleneck . empirical studies show that features of congested traffic resulting from the induced breakdown ( at @xmath96 7:07 in fig . [ breakdown ] ( c ) ) are qualitatively identical to those found in congested traffic resulting from empirical spontaneous traffic breakdown ( fig . [ breakdown ] ( b ) ) . in particular , in both cases congested traffic resulting from the breakdown at the bottleneck is self - maintained under free flow conditions downstream of the bottleneck . km ) , downstream ( @xmath1 7.9 km ) and upstream of the bottleneck ( @xmath1 5.1 km ) are shown ; data was measured on june 23 , 1998 ( fig . [ breakdown ] ( d ) ) . ( b ) long - time spillover leading to expanded congested pattern ( right ) measured on march 23 , 2001 and scheme of freeway section of freeway a5-north with three bottlenecks ( left ) . bottlenecks in ( b ) have been explained in sec . 9.2.2 of @xcite . [ 23061998s_23032001n ] , width=9 ] this shows that rather than the nature of traffic breakdown , the terms _ empirical spontaneous _ and _ empirical induced _ traffic breakdowns at a bottleneck distinguish different _ sources _ of a nucleus that occurrence leads to traffic breakdown : in fig . [ breakdown ] ( b ) , the source of empirical spontaneous breakdown is one of the waves in free flow shown in fig . [ nuclei1996_2 ] . in fig . [ breakdown ] ( c ) , the source of empirical induced breakdown is the wide moving jam . in contrast with the wide moving jam shown in fig . [ breakdown ] ( c ) , a wide moving jam shown in fig . [ breakdown ] ( d ) does not induce traffic breakdown at the bottleneck . indeed , in the latter case , after the jam is far away upstream of the bottleneck , free flow returns both at the effective bottleneck location as well as downstream and upstream of the bottleneck ( fig . [ 23061998s_23032001n ] ( a ) ) . because under condition @xmath95 the jam is always a nucleus for traffic breakdown at the bottleneck , the case shown in fig . [ breakdown ] ( d ) should be related to the opposite condition @xmath97 at which no traffic breakdown can occur at the bottleneck . we see that empirical induced traffic breakdown is probably the only one @xmath15method " to find whether traffic breakdown can occur at the bottleneck or not . this emphasizes another difference between empirical spontaneous and empirical induced traffic breakdowns at a highway bottleneck : when waves in free flow propagate though the bottleneck without initiating of the breakdown , we can not state whether all waves are smaller than a critical wave , or condition @xmath97 is satisfied at which no traffic breakdown can occur at the bottleneck . in contrast , when a local congested pattern propagates through the bottleneck without inducing of the breakdown , we can state that the flow rate at the bottleneck is smaller than @xmath5 . the effect of continuous upstream propagation of traffic congestion is often called _ spillback_. when due to this upstream propagation a congested pattern affects an upstream road bottleneck , it is often called _ spillover_. in the cases of the wide moving jams shown in figs . [ breakdown ] ( c ) and [ breakdown ] ( d ) , any of the jams can also be considered the effect of spillover because the jam forces congested traffic at the bottleneck . however , when the jams are far away upstream of the bottleneck , they do not force congested traffic at the bottleneck any more . we can see that there can be at least the following qualitatively different effects due to spillover at a highway bottleneck : \(i ) an empirical induced traffic breakdown occurs due to jam propagation through a bottleneck ( fig . [ breakdown ] ( c ) ) . \(ii ) an expanded congested pattern ( ep ) occurs due to spillover ( fig . [ 23061998s_23032001n ] ( b ) ) @xcite : the ep shown in fig . [ 23061998s_23032001n ] ( b ) appears when an empirical congested pattern that occurs initially at an off - ramp bottleneck propagates upstream ( spillback ) . due to this upstream pattern propagation it forces congested conditions at an upstream on - ramp bottleneck ( labeled by @xmath15on - ramp bottleneck 1 " ) ; this spillover lasts several hours . this case of spillover can not be considered as induced traffic breakdown , because congested traffic at the on - ramp bottleneck is forced by spillover . \(iii ) the jam propagation through a bottleneck leads neither to induced traffic breakdown nor to an ep ( fig . [ breakdown ] ( d ) ) . this effect of spillover shows that the flow rate is smaller than the minimum capacity of free flow at the bottleneck : @xmath98 . an empirical study of real field traffic data allows us to make the following conclusions about physical features of empirical nuclei for spontaneous traffic breakdown in free flow at highway bottlenecks : \1 . in the most real field traffic data measured in 19962014 by road detectors on german freeways , a nucleus for traffic breakdown at a highway bottleneck occurs through an interaction of one of the waves in free flow with a permanent speed disturbance localized at a highway bottleneck . when the wave reaches the location of the disturbance at the bottleneck ( effective bottleneck location ) , spontaneous traffic breakdown , i.e. , phase transition from free flow to synchronized flow occurs . waves in free flow , which can be nuclei for spontaneous traffic breakdown at highway bottlenecks , appear due to oscillations in the percentage of slow vehicles over time . these waves propagate with the average speed of slow vehicles in free flow ( about 8588 km / h for german highways ) . within a wave , the total flow rate is larger and the speed averaged across the highway is smaller than outside the wave . any of the waves in free flow , which can be a nucleus for spontaneous traffic breakdown at a highway bottleneck , exhibits a two - dimensional ( 2d ) asymmetric spatiotemporal structure whose characteristics are different in different highway lanes . microscopic traffic simulations with a stochastic traffic flow model in the framework of three - phase theory explain the empirical findings .
|
based on an empirical study of real field traffic data measured in 19962014 through road detectors installed on german freeways , we reveal physical features of empirical nuclei for spontaneous traffic breakdown in free flow at highway bottlenecks . it is shown that the source of a nucleus for traffic breakdown is the solely difference between empirical spontaneous and induced traffic breakdowns at a highway bottleneck . microscopic traffic simulations with a stochastic traffic flow model in the framework of three - phase theory explain the empirical findings . it turns out that in the most cases , a nucleus for empirical spontaneous traffic breakdown occurs through an interaction of one of waves in free flow with an empirical permanent speed disturbance localized at a highway bottleneck . the wave is a localized structure in free flow , in which the total flow rate is larger and the speed averaged across the highway is smaller than outside the wave . the waves in free flow appear due to oscilations in the percentage of slow vehicles ; these waves propagate with the average speed of slow vehicles in free flow ( about 8588 km / h for german highways ) . any of the waves exhibits a two - dimensional asymmetric spatiotemporal structure : wave s characteristics are different in different highway lanes .
|
based on an empirical study of real field traffic data measured in 19962014 through road detectors installed on german freeways , we reveal physical features of empirical nuclei for spontaneous traffic breakdown in free flow at highway bottlenecks . it is shown that the source of a nucleus for traffic breakdown is the solely difference between empirical spontaneous and induced traffic breakdowns at a highway bottleneck . microscopic traffic simulations with a stochastic traffic flow model in the framework of three - phase theory explain the empirical findings . it turns out that in the most cases , a nucleus for empirical spontaneous traffic breakdown occurs through an interaction of one of waves in free flow with an empirical permanent speed disturbance localized at a highway bottleneck . the wave is a localized structure in free flow , in which the total flow rate is larger and the speed averaged across the highway is smaller than outside the wave . the waves in free flow appear due to oscilations in the percentage of slow vehicles ; these waves propagate with the average speed of slow vehicles in free flow ( about 8588 km / h for german highways ) . any of the waves exhibits a two - dimensional asymmetric spatiotemporal structure : wave s characteristics are different in different highway lanes .
|
1602.08845
|
i
|
big data analytics is a major topic in contemporary data management and machine learning research and practice . many platforms , e.g. , optiml @xcite , graphlab @xcite , systemml @xcite , vowpal wabbit @xcite , simsql @xcite , glade @xcite and libraries , e.g. , madlib @xcite , bismarck @xcite , mllib @xcite , mahout , have been proposed to provide support for distributed / parallel statistical analytics . we can categorize these solutions into general frameworks with machine learning support mahout , spark s mllib , graphlab dedicated machine learning systems systemml , simsql , optiml , vowpal wabbit and frameworks within databases madlib , bismarck , glade . in this paper , we focus on the last category frameworks for in - databse analytics . a common assumption across all these systems is that the number of model parameters or features is small enough to fit in memory . this is made explicit by the representation of the model as an in - memory array data structure . however , due to the explosive growth in data acquisition and the wide adoption of analytics methods , the current trend is to devise models with an ever - increasing number of features_big models_. a report on industrial machine learning cites models with 100 billion features ( 800 gb in size ) as early as 2012 . scientific applications such as ab initio nuclear structure calculations also generate extremely large models with billions of features @xcite . while these are extreme cases , there are many realistic applications that require big models and are forced to limit the number of features they consider because of insufficient memory resources . we provide two such examples in the following . * example 1 : recommender systems . * a class of analytics models with highly - dimensional feature space are the ones in which the dimensionality grows with the number of observations . low - rank matrix factorization ( lmf ) is a typical example . in lmf , the observations are a set of cells in a sparse @xmath0 matrix @xmath1 . the non - empty cells represent the users ratings of items in a certain category such as songs or movies with each row corresponding to a user and each column to an item . in general , every row is sparse since a typical user rates only a very limited set of items . each column is also sparse since only a restricted number of users rate an item . lmf seeks to decompose matrix @xmath1 into the product of two dense low - rank matrices @xmath2 and @xmath3 with dimensions @xmath4 and @xmath5 , respectively , where @xmath6 is the rank . the prediction accuracy increases with @xmath6 . the lmf features are matrices @xmath2 and @xmath3 which grow with the number of users @xmath7 and the number of items @xmath8 , respectively , and the rank @xmath6 . lmf is heavily used in recommender systems , e.g. , netflix , pandora , spotify . for example , spotify applies lmf for 24 million users and 20 million songs , which leads to 4.4 billion features at a relatively small rank of 100 . * example 2 : topic modeling . * n - grams are a common feature vector in topic modeling . they are extracted by considering sequences of 1-word tokens ( unigrams ) , 2-word tokens ( bigrams ) , up to n - word tokens ( n - grams ) from a fixed dictionary . the feature vector consists of the union of unigrams , bigrams , @xmath9 , n - grams . several analytics models can be applied over this feature space , including latent dirichlet allocation , logistic regression , and support vector machines . for the english wikipedia corpus , a feature vector with 25 billion unigrams and 218 billion bigrams can be constructed @xcite . a similar scale can be observed in genomics where topic modeling is applied to genome sequence analysis . * existing solutions . * the standard model representation across all the big data analytics systems we are aware of in - database or not is a memory - resident container data structure , e.g. , ` vector ` or ` map ` . depending on the parallelization strategy , there can be one or more model instances in the system at the same time . hogwild ! @xcite uses a single non - synchronized instance , while averaging techniques @xcite replicate the model for each execution thread . at the scale of models introduced above , a memory - resident solution incurs prohibitive cost if it is feasible at all . in reality , in - database analytics frameworks can not handle much smaller models . for example , madlib and bismarck are built using the udf - uda functionality available in postgresql . the model is stored as an array attribute in a single - column table . postgresql imposes a hard constraint of 1 gb for the size of an attribute , effectively limiting the model size . high performance computing ( hpc ) libraries for efficient sparse linear algebra such as intel mkl , trilinos , cusparse , and cusp are optimized exclusively for in - memory processing , effectively requiring that both the training dataset and the model fit in memory simultaneously . two approaches are possible to cope with insufficient memory secondary storage processing and distributed memory processing . in secondary storage processing , the model is split into partitions large enough to fit in memory and the goal is to optimize the access pattern in order to minimize the number of references to secondary storage . this principle applies between any two layers of the storage hierarchy cache and memory , memory and disk ( or ssd ) , and texture memory and global memory of a gpu . while we are not aware of any secondary storage solution for data analytics , there have been several attempts to optimize the memory access pattern of the spmv kernel . however , they are targeted at minimizing the number of cache misses @xcite or the number of accesses to the gpu global memory @xcite not the number of disk accesses . in distributed memory processing , the big model is partitioned across several machines , with each machine storing a sufficiently small model partition that fits in its local memory . since remote model access requires data transfer , the objective in distributed processing is to minimize the communication between machines . this can not be easily achieved for the spmv kernel due to the non - clustered access pattern . distributed big data analytics systems built around the map - reduce computing paradigm and its generalizations , e.g. , hadoop and spark , require several rounds of repartitioning and aggregation due to their restrictive communication pattern . to the best of our knowledge , parameter server @xcite is the only distributed memory analytics system capable of handling big models directly . in parameter server , the big model can be transfered and replicated across servers . whenever a model entry is accessed , a copy is transferred over the network and replicated locally . modifications to the model are pushed back to the servers asynchronously . the communication has to be implemented explicitly and optimized accordingly . while parameter server supports big models , it does so at the cost of a significant investment in hardware and with considerable network traffic . our focus is on cost - effective single node solutions . * approach & contributions . * in this paper , we propose an in - database solution for big model analytics . the main idea is to offload the model to secondary storage and leverage database techniques for efficient training . the model is represented as a table rather than as an array attribute . this distinction in model representation changes fundamentally how in - database analytics tasks are carried out . we identify _ dot - product _ as the most critical operation affected by the change in model representation . our central contribution is the first _ dot - product join physical database operator _ optimized to execute secondary storage array - relation dot - products effectively . dot - product join is a constrained instance of the spmv kernel @xcite which is widely - studied across many computing areas , including hpc , architecture , and compilers . the paramount challenge we have to address is how to optimally schedule access to the dense relation buffer management based on the non - contiguous feature entries in the sparse arrays . the goal is to minimize the overall number of secondary storage accesses across all the sparse arrays . we prove that this problem is np - hard and propose a practical solution characterized by two technical contributions . the first contribution is to handle the sparse arrays in _ batches with variable size_determined dynamically at runtime . the second contribution is a _ reordering strategy _ for the arrays such that accesses to co - located entries in the dense relation can be shared . our specific contributions can be summarized as follows : we investigate the big model analytics problem and identify dot - product as the critical operation in training generalized linear models ( section [ sec : problem ] ) . we also establish a direct correspondence with the well - studied sparse matrix vector ( spmv ) multiplication problem . we present several alternatives for implementing dot - product in a relational database and discuss their relative advantages and drawbacks ( section [ sec : baseline ] ) . we design the first array - relation dot - product join database operator targeted at secondary storage ( section [ sec : dot - product ] ) . we prove that identifying the optimal access schedule for the dot - product join operator is np - hard ( section [ ssec : reordering : np - hard ] ) and introduce two optimizations dynamic batch processing and reordering to make the operator practical . we devise three batch reordering heuristics lsh , radix , and k - center ( section [ sec : reordering ] ) inspired from optimizations to the spmv kernel and evaluate them thoroughly . we show how the dot - product join operator is integrated in the gradient descent optimization pipeline for training generalized linear models ( section [ sec : dp - gd ] ) . we execute an extensive set of experiments that evaluate each sub - component of the operator and compare our overall solution with alternative dot - product implementations over synthetic and real data ( section [ sec : experiments ] ) . the results show that dot - product join achieves an order of magnitude reduction in execution time over alternative in - database solutions .
|
it generalizes big data analytics which is targeted at how to train memory - resident models over out - of - memory training data . in this paper , we propose an in - database solution for big model analytics . the paramount challenge in designing the dot - product join operator is how to optimally schedule access to the dense relation based on the non - contiguous entries in the sparse arrays . we prove that this problem is np - hard and propose a practical solution characterized by two technical contributions dynamic batch processing and array reordering . we devise three heuristics lsh , radix , and k - center for array reordering and analyze them thoroughly . we execute extensive experiments over synthetic and real data that confirm the minimal overhead the operator incurs when sufficient memory is available and the graceful degradation it suffers as memory becomes scarce . moreover , dot - product join achieves an order of magnitude reduction in execution time over alternative in - database solutions .
|
big model analytics tackles the training of massive models that go beyond the available memory of a single computing device , e.g. , cpu or gpu . it generalizes big data analytics which is targeted at how to train memory - resident models over out - of - memory training data . in this paper , we propose an in - database solution for big model analytics . we identify dot - product as the primary operation for training generalized linear models and introduce the first array - relation dot - product join database operator between a set of sparse arrays and a dense relation . this is a constrained formulation of the extensively studied sparse matrix vector multiplication ( spmv ) kernel . the paramount challenge in designing the dot - product join operator is how to optimally schedule access to the dense relation based on the non - contiguous entries in the sparse arrays . we prove that this problem is np - hard and propose a practical solution characterized by two technical contributions dynamic batch processing and array reordering . we devise three heuristics lsh , radix , and k - center for array reordering and analyze them thoroughly . we execute extensive experiments over synthetic and real data that confirm the minimal overhead the operator incurs when sufficient memory is available and the graceful degradation it suffers as memory becomes scarce . moreover , dot - product join achieves an order of magnitude reduction in execution time over alternative in - database solutions .
|
1604.04789
|
i
|
the world wide power grid can be considered one of the greatest masterpiece of engineering that human being has ever made . moreover , starting from the first industrial revolution , smart grids ( sgs ) are one of the most important breakthrough that science and engineering fields are carrying out . currently , the smart grid concept is founded on a paradigmatic revolution that will permeate many aspects of human life . from the power system point of view , sgs can be considered as a way to transform the electric energy infrastructures from a centralized , producer - controlled network , into a distributed and consumer - interactive system , leveraging the same concepts and technologies that enabled the emergence and spread of the internet . sgs vision promises a power grid infrastructure with increased automation of the grid operations and self - healing capabilities . sgs integrate the renewable energy production , seamlessly balancing energy supply and demand , and can be considered as a key technology for facilitating the spread of electric mobility . to achieve that vision , the current power grid has to be thought as a _ technological ecosystem _ that needs a strong _ injection _ of distributed intelligence @xcite . g. k. venayagamoorthy argues that the current power grid can be considered a spatially and temporally complex , nonlinear and non - stationary system with a lot of uncertainties @xcite . accordingly , the sg can be examined for all intents and purposes as a complex system , and computational intelligence ( ci ) and soft computing ( sc ) techniques @xcite , are widely adopted to face a plethora of applications and problems arising in the sg context . the main ci paradigms for sg related problem solutions are : neuro - fuzzy , neuro - swarm , fuzzy - pso , fuzzy - ga , neuro - genetic @xcite . in fact , in mg related tasks , such as control and flow management , the presence of uncertainty and non - linearity , for example in the power demand profile of a large amount of users or in the power produced by solar or wind sources , makes related problems extremely challenging . hence , sc techniques can help managing the complexity of problems offering reliable solutions , especially in presence of non - linearity @xcite and in presence of storage devices that increase the solution space of the unit commitment problem @xcite . consequently , since linear techniques can not be considered adequate in solving problems whose nature is nonlinear and even stochastic , sc techniques offer a suitable framework introducing learning capabilities in the design of the mg controllers , especially in presence of renewable energy sources and storage . the current research follows our previous work @xcite about an application of what we call classic fuzzy - ga paradigm to the problem of flow control optimization in a microgrid ( mg ) . the mg can be thought as a sub - network of the sg characterized by the presence of autonomous ( often renewable ) energy sources buffered by some type of battery energy storage system ( bess ) and locally controlled in order to achieve smart energy flows management . the flow control task is carried out by a flc with two fuzzy inference systems ( fiss ) of mamdani type . fiss , relaying on approximated reasoning based on fuzzy logic ( fl ) , are in charge to optimize energy flows and the overall accounting profit in energy trading operations with the main - grid . the choice of mamdani type fis is related to its simplicity in incorporating human knowledge in the rb . specifically , in the application at hand the computational overhead introduced by the defuzzification process ( absent in a sugeno type fis ) is not a real problem . furthermore , the hga paradigm with its suitable coding scheme is well suited to optimize the structure of a mandani fis . the current work is focused on two main objectives : @xmath0 improve the mg model , in particular the bess model , @xmath1 optimize the rule base ( rb ) of the flc adopted to control power flows in the mg . as concerns the first goal we move on from an ideal battery adopted in @xcite designing an energy storage device based on a real - world model with suitable efficiency parameters . for the second objective , we designed an optimization method based on a suited genetic algorithm ( ga ) that is in charge of learning the fis parameters , optimizing at the same time both the economic return in energy trading and the cardinality of the fuzzy rb . the adopted optimization algorithm is known as hierarchical genetic algorithm ( hga ) aiming to perform at the same time a fine tuning of the fuzzy membership functions ( mfs ) and the structural optimization of the fiss in the following we will refer on this paradigm as `` fuzzy - hga paradigm '' . in fact , while in our previous work the fis structure is constrained to be fixed , with a predefined number of antecedent and consequent terms and , thereby , with an immutable number of fuzzy rules , the adopted hga scheme allows to relax these constraints . moreover , the standard ga approach deals with a chromosome of fixed length whose encoding scheme leads to a lower flexibility in the rb tuning . a ga algorithm able to emulate a variable length chromosome with a suitable encoding scheme of the fis is ideal for optimizing the number of rules in the given rb . finally , the fuzzy rule optimization can lead to an improved performance of the fis , discarding pre - defined low performing rules . the idea behind hgas is based on the biological inspired gene structure of a chromosome formed in a hierarchical fashion , emulating the encoding approach of the deoxyribonucleic acid ( dna ) . in nature the genes can be classified into two different types : regulatory sequences and structural genes . one of the regulatory sequences found in dna is called the promoter " with the task of activating or deactivating structural genes . therefore the presence of active and inactive genes in the structural genes leads to the idea of a hierarchical structure formulation of the chromosome that consists of _ control genes _ and _ parametric genes_. the activation of the parametric genes is governed by the value of the control genes . accordingly , the strategy suggested by nature can be modeled for solving a number of engineering problems , such as those involving mix integer programming methods @xcite or fuzzy control applications demanding the joint optimization of both fiss parameters and rbs @xcite . in the last case the novelty of a hierarchical coding scheme is based on the definition of suitable genetic operators moving from standard ga algorithms to more advanced ones . the hierarchical encoding scheme allows to code the fiss parameters , more precisely mf parameters , as parametric genes and , at the same time , control genes can be used to activate and deactivate mfs composing a given fuzzy rb , thus tuning the overall number of fuzzy rules . the work is organized as follows . [ sec : related_works ] is a literature review about the use of flc and the fuzzy - ga paradigm in the smart grid context . in sec . [ sec : background ] we introduce the optimization problem and the mg model . [ sec : pf ] clarifies the level of abstraction of the problem , introducing the adopted notation and explaining how the fuzzy controller works . the fuzzy control scheme for a mg , together with the classic fuzzy - ga and fuzzy - hga paradigms are treated in sec . [ sec : fuzzy_c_mg ] and related subsections . in sec . [ sec : experim_ev ] , soon after the introduction of the examined mg scenarios and the algorithm settings , the main results are reported and discussed . finally , conclusions are drawn in sec . [ sec : conclusions ] , while feature works are discussed in sec . [ sec : fws ] .
|
bio - inspired algorithms like genetic algorithms and fuzzy inference systems ( fis ) are nowadays widely adopted as hybrid techniques in commercial and industrial environment . in this paper we present an interesting application of the fuzzy - ga paradigm to smart grids . the main aim consists in performing decision making for power flow management tasks in the proposed microgrid model equipped by renewable sources and an energy storage system , taking into account the economical profit in energy trading with the main - grid . this approach will be referred in the following as fuzzy - hga . _ keywords : _ * microgrid , energy management system , battery energy storage , power flow optimization , storage system management , fuzzy systems , evolutionary computation , hierarchical genetic algorithms .
|
bio - inspired algorithms like genetic algorithms and fuzzy inference systems ( fis ) are nowadays widely adopted as hybrid techniques in commercial and industrial environment . in this paper we present an interesting application of the fuzzy - ga paradigm to smart grids . the main aim consists in performing decision making for power flow management tasks in the proposed microgrid model equipped by renewable sources and an energy storage system , taking into account the economical profit in energy trading with the main - grid . in particular this study focuses on the application of a hierarchical genetic algorithm ( hga ) for tuning the rule base ( rb ) of a fuzzy inference system ( fis ) , trying to discover a minimal fuzzy rules set in a fuzzy logic controller ( flc ) adopted to perform decision making in the microgrid . the hga rationale focuses on a particular encoding scheme , based on control genes and parametric genes applied to the optimization of the fis parameters , allowing to perform a reduction in the structural complexity of the rb . this approach will be referred in the following as fuzzy - hga . results are compared with a simpler approach based on a classic fuzzy - ga scheme , where both fis parameters and rule weights are tuned , while the number of fuzzy rules is fixed in advance . experiments shows how the fuzzy - hga approach adopted for the synthesis of the proposed controller outperforms the classic fuzzy - ga scheme , increasing the accounting profit by 67% in the considered energy trading problem yielding at the same time a simpler rb . * _ keywords : _ * microgrid , energy management system , battery energy storage , power flow optimization , storage system management , fuzzy systems , evolutionary computation , hierarchical genetic algorithms .
|
1604.04789
|
c
|
computational intelligence techniques are today a consolidated framework for solving engineering problems such as challenges arising in smart grid context . in this paper we study a portion of a smart grid , acting as a microgrid , characterized by the presence of renewable sources and equipped with a bess unit allowing trading operations with the main - grid related to energy exchanges . the considered mg model has been sized for small - scale applications , such as small energy grids in rural areas or small housing units in urban areas , considered as atomic elements of a larger a multi - microgrid system . the decision making task is carried on through a fuzzy logic controller optimized by a hierarchical genetic algorithm able to tune flcs parameters ( i.e. the mf parameters and the fuzzy rule weights ) minimizing , at the same time , the number of fuzzy rules . the fuzzy rules number optimization is carried out in flcs thanks to the hierarchical structure of the adopted chromosome representation consisting of two main sections having a different semantic meaning . the control section , encoded as a binary string , controls the activation of the parametric section representing the mf parameters and the weights of fuzzy rules . the specific encoding scheme of the hga offers a higher flexibility in tuning the flc in comparison with the classic scheme which foresees a fixed and maximal number of rules due to the rigidity of the chromosome length . the proposed method for ems synthesis incorporates a learning step . in fact , once fis rules are optimized by considering training set data through an off - line procedure running on a plain workstation , the trained rules can be uploaded into an embedded system ( a microcontroller ) ready to work in real - time . from this point of view , the added complexity of the hga scheme can be well managed off - line by an inexpensive workstation ( such as an intel i7 6thh generation with 32 gb of ram ) , exploiting its power in evolving suitable rules without affecting the real time operations . note that the final aim of the hga algorithm is the reduction of the structural complexity of the rule base in the fis , dropping the number of rules and the number of antecedents in each rule . consequently , with respect to a fis resulting from a learning procedure based on a plain version of a genetic algorithm , the microcontroller is in charge to handle a lower number of rules in performing decision making , in a given time interval . moreover , note that the typical dynamics characterizing a microgrid ( at the considered abstraction level ) are slow enough to set the length of the time interval in the order of minutes ( 15 minutes , in the considered case ) . consequently , even an entry level and inexpensive microcontroller ( such as an `` arduino due '' board , based on the atmel sam3x8e arm cortex - m3 cpu ) has sufficient computational power to handle a great number of fuzzy rules in such time intervals . the fuzzy - hga paradigm is thereby evaluated for some configuration of the mg parameters , battery model parameters and genetic operator parameters with the aim to study the overall behavior of the fuzzy control system . a further comparison is performed between a classic optimization scheme within the fuzzy - ga paradigm and the adopted fuzzy - hga . the classic ga scheme foresees the optimization of the mf parameters together with the fuzzy rule weights in a fixed rule base ( rb ) whose cardinality is maximal ( grid partition ) . the fuzzy - hga algorithm outperforms the classic fuzzy - ga scheme by 67% . the hierarchical encoding method of the fis parameters and the definition of suited genetic operators explain the improvement in performance reached at the cost of a more complex design of the ga and of the encoding scheme . however , this additional computational cost at design and training stages is balanced with the possibility to obtain simpler flcs in terms of number of rules , allowing the real time implementation of the complete control system on low cost embedded electronic devices .
|
in particular this study focuses on the application of a hierarchical genetic algorithm ( hga ) for tuning the rule base ( rb ) of a fuzzy inference system ( fis ) , trying to discover a minimal fuzzy rules set in a fuzzy logic controller ( flc ) adopted to perform decision making in the microgrid .
|
bio - inspired algorithms like genetic algorithms and fuzzy inference systems ( fis ) are nowadays widely adopted as hybrid techniques in commercial and industrial environment . in this paper we present an interesting application of the fuzzy - ga paradigm to smart grids . the main aim consists in performing decision making for power flow management tasks in the proposed microgrid model equipped by renewable sources and an energy storage system , taking into account the economical profit in energy trading with the main - grid . in particular this study focuses on the application of a hierarchical genetic algorithm ( hga ) for tuning the rule base ( rb ) of a fuzzy inference system ( fis ) , trying to discover a minimal fuzzy rules set in a fuzzy logic controller ( flc ) adopted to perform decision making in the microgrid . the hga rationale focuses on a particular encoding scheme , based on control genes and parametric genes applied to the optimization of the fis parameters , allowing to perform a reduction in the structural complexity of the rb . this approach will be referred in the following as fuzzy - hga . results are compared with a simpler approach based on a classic fuzzy - ga scheme , where both fis parameters and rule weights are tuned , while the number of fuzzy rules is fixed in advance . experiments shows how the fuzzy - hga approach adopted for the synthesis of the proposed controller outperforms the classic fuzzy - ga scheme , increasing the accounting profit by 67% in the considered energy trading problem yielding at the same time a simpler rb . * _ keywords : _ * microgrid , energy management system , battery energy storage , power flow optimization , storage system management , fuzzy systems , evolutionary computation , hierarchical genetic algorithms .
|
gr-qc0610131
|
i
|
it is well known within the framework of black - hole perturbation theory @xcite that quasi - normal modes ( qnms ) ( i.e. , exponentially damped harmonic oscillations ) dominate the gravitational - wave response of a non - spherically distorted black hole if the corresponding frequencies are part of the fourier spectrum of the external source that moved the hole away from its equilibrium state . this perturbation is then radiated away in form of gravitational radiation until the black hole returns to its unperturbed , quiescent state . in practice , the qnms excitation is triggered if the frequency of the perturbing agent ( e.g. an external matter source moving close to the black hole ) is sufficiently close to the fundamental frequencies of the black hole , which then acts as an excited oscillator . as a result , the qnms and in particular the fundamental mode ( which is the one at the highest frequency and with the smallest damping time ) represent the main feature of the gravitational waves emission only for a sufficiently compact perturbation ; i.e. , for sources whose characteristic scale is comparable with the width of the peak of the potential . these results are well - known since the early studies of press @xcite ( see also vishveshwara @xcite ) , who considered the scattering of gaussian gravitational - wave packets off a schwarzschild black hole and noticed that the excitation of the qnms of the black hole is more efficient for very narrow packets . approximate relations to compute the efficiency of excitation of the various qnms were introduced by leaver @xcite and by andersson @xcite for gaussian pulses initial data with variable width . since the qnms spectrum is entirely determined by the black hole properties ( i.e. , mass , spin and charge ) , it is expected that the detection of the qnms ringdown would provide a unique opportunity to unveil the physical properties of a black hole . for this reason , the excitation of black hole qnms have been studied in various astrophysical scenarios , such as the gravitational collapse of a ( rotating ) neutron star to a black hole or the collision of two black holes . the presence of the qnms in these situations has been confirmed either through the use of perturbation theory with various degrees of sophistication @xcite , or through fully relativistic numerical simulations @xcite . however , under more general and realistic conditions , such as the excitation of the qnms by accretion of matter , and that may be encountered in gravitational stellar collapse or binary neutron star mergers , the gravitational - wave response of a black hole can be more complex . for instance , the simulations performed by @xcite showed that the gravitational - wave signal is not simply given by the superposition of exponentially - damped sinusoids at the qnms frequencies and that , in some cases , the qnm ringing is only weakly excited and analysis in the frequency domain are needed @xcite . on the other hand , `` backscattering '' effects related to the slowly - decaying features of the scattering potential and interference effects turn out to play a crucial role for the correct interpretation of the results ; a detailed discussion of these effects can be found in ref . @xcite . in the case of a schwarzschild black hole of mass @xmath0 , we recall that the appearance of qnms ringing is related to the peak of the curvature potential ( also referred to as the zerilli or regge - wheeler potential ) , that is located at @xmath1 ; the backscattering , on the other hand , is related to the fact that the curvature potential decays as @xmath2 for @xmath3 . this behavior is responsible for the late time power - law tail @xmath4 of the gravitational - wave signal , where @xmath5 refers to the radiation multipole . therefore , while early - time ringing is the result of a superposition of exponentially damped sinusoids , and is dominated by the fundamental quasi - normal mode , at later times the ringing dies out and the signal is dominated by tail effects . however , in the transition from the `` ringdown '' to the `` tail '' phase , additional oscillations appear that can not be attributed to any of the two regimes and that also seem to depend on the choice of the initial data ( see @xcite ) . as we shall show in this paper for a broad sample of initial data , there could be intermediate regimes where the `` ringdown '' and the `` tail '' terms of the potential produce competing effects , so that the qnms ringing and the backscattering effects can overlap , generating complex waveforms . these effects were first noticed in refs . @xcite , although not discussed in detail there . in a recent paper @xcite , hereafter paper i , a general analysis of the gravitational radiation emitted as a result of anisotropic accretion of matter shells onto nonrotating black holes and neutron stars was presented . that investigation made use of a procedure that combines the solution of the linearized einstein equations for the metric perturbations with fully nonlinear hydrodynamics simulations . although the study of black hole perturbations produced by infalling matter has a long history and rich literature @xcite , the approach outlined in paper i proved to be useful for a number of reasons : _ i ) _ it provided additional information on the black - hole s response to the dynamics of point - like particles in the vicinity of black holes ; _ ii ) _ it helped understanding the basic black - hole s response to extended matter perturbations ; _ iii ) _ it represented an effective way of studying black - hole physics in a linear regime without having to resort to full - scale numerical relativity simulations . one of the main results of paper i was that , in the idealized accretion processes considered , most of the energy is released at frequencies lower than that of the fundamental quasi - normal mode ( qnm ) of the black hole , the spectrum consisting of a complex pattern , mostly produced during the accretion process rather than in the ringdown phase . more precisely , the gravitational - wave emission was found to be dominated by a collection of interference `` fringes '' at frequencies of about a few hundred hz , rather than by a single monochromatic peak at the ( higher ) frequency of the fundamental mode of the black hole . moreover , the width of these fringes was found to decrease rapidly with the initial position of the matter source . these results , which were already observed in other works @xcite in the case of a point - like particle falling onto a schwarzschild black hole , also showed that the appearance of interference fringes in the energy spectra is much larger when the accreting matter is a shell of finite size and that the efficiency in gravitational - wave emission is much reduced , becoming almost two orders of magnitude smaller than in the case of point - like particles . an important feature of the calculations carried out in paper i was the minimization of the initial gravitational - wave content ; this turned out to be crucial to illustrate that the interference pattern was mainly due to the finite radial extension of the accreting source . the aim of the present paper is twofold . firstly , we intend to complete the discussion started in paper i on accreting quadrupolar shells onto a schwarzschild black hole by extending the parameter space of the initial models and by analyzing their impact on the gravitational - wave emission . in particular we study how the energy emitted in gravitational waves , and the corresponding spectra , depend on the compactness of the shells as well as on their initial locations . in doing so we show that , for a finite - size source , the ringing of the black hole is much more complex than a simple superposition of qnms and that the energy spectra ( and in particular the interference fringes ) are dependent on the choice of the initial data . secondly , we improve the astrophysical relevance of our study by analyzing the gravitational radiation produced by thick accretion disks @xcite which accrete onto the black hole on dynamical timescales . we recall , in fact , that quasi - periodic oscillations of thick accretion disks ( or tori ) orbiting around schwarzschild or kerr black holes have been recently addressed as promising sources of gravitational waves @xcite in the khz range . the paper is organized as follows : in sec . [ analytic ] we review the theory of odd- and even - parity nonspherical perturbations of schwarzschild spacetime , writing the inhomogeneous zerilli - moncrief and regge - wheeler equations in a form suitable for time - domain calculations . [ num_app ] briefly describes the numerical approach adopted for the simulations , while sec . [ results ] is devoted to the discussion of the results . finally , sec . [ conclusions ] provides a summary of the most important results and presents our conclusions . unless otherwise specified , we choose geometrized units ( @xmath6 ) , and the black hole mass @xmath0 is the unit of length .
|
the matter models considered include quadrupolar dust shells and thick accretion disks , permitting us to simulate situations which may be encountered at the end stages of stellar gravitational collapse or binary neutron star merger . we focus on the interference pattern appearing in the energy spectra of the emitted gravitational waves and on the amount of excitation of the quasi - normal modes of the accreting black hole . we show that , quite generically in the presence of accretion , the black hole ringdown is not a simple superposition of quasi - normal modes , although the fundamental mode is usually present and often dominates the gravitational - wave signal . we interpret this as due to backscattering of waves off the non - exponentially decaying part of the black hole potential and to the finite spatial extension of the accreting matter .
|
by combining the numerical solution of the nonlinear hydrodynamics equations with the solution of the linear inhomogeneous zerilli - moncrief and regge - wheeler equations we investigate the properties of the gravitational radiation emitted during the axisymmetric accretion of matter onto a schwarzschild black hole . the matter models considered include quadrupolar dust shells and thick accretion disks , permitting us to simulate situations which may be encountered at the end stages of stellar gravitational collapse or binary neutron star merger . we focus on the interference pattern appearing in the energy spectra of the emitted gravitational waves and on the amount of excitation of the quasi - normal modes of the accreting black hole . we show that , quite generically in the presence of accretion , the black hole ringdown is not a simple superposition of quasi - normal modes , although the fundamental mode is usually present and often dominates the gravitational - wave signal . we interpret this as due to backscattering of waves off the non - exponentially decaying part of the black hole potential and to the finite spatial extension of the accreting matter . our results suggest that the black hole qnm contributions to the full gravitational wave signal should be extremely small and possibly not detectable in generic astrophysical scenarios involving the accretion of extended distributions of matter .
|
gr-qc0610131
|
c
|
by performing numerical simulations that combine the solution of the nonlinear hydrodynamics equations with that of the linear inhomogeneous zerilli - moncrief and regge - wheeler equations , we have studied the features of the gravitational - wave signals generated by the accretion of matter onto a schwarzschild black hole . as extended and accreting matter - sources we have considered quadrupolar shells of dust falling radially from a finite distance , as well as geometrically thick disks undergoing either bursts of hypercritical accretion or quasi - periodic oscillations . in both cases we find that the gravitational - wave signal _ is not _ a simple superposition of the black hole qnms and that the latter can not be found in the energy spectra at times . rather , we find that quite generically the signal contains important contributions coming from radiation scattering off the tail of the curvature potential and producing a characteristic pattern of interference fringes in the energy spectra . while the relevance of this contribution differs according to the specific source considered , it is generically present as long as the source of perturbations is extended and the scattering potential does not have an exponential decay with radius . this conclusion , which was already reported in previous studies involving simpler sources , has been here confirmed unambiguously by studying the scattering off a fictitious potential , the pschl - teller potential , which however decays exponentially with radius . these generic properties of the gravitational - wave emission coming from black holes perturbed by extended sources represent important differences with respect to corresponding properties of signals produced by very compact sources , such as point - like particles . of course this conclusion prevents from the derivation of a simple and generic description of the gravitational - wave signal which would be independent of the properties of the perturbing source , but it also opens the exciting perspective of deducing many of the physical properties of the source through a careful analysis of the waveforms produced . overall , the results presented here make us confident that the black hole qnm contributions to the full gravitational - wave signal should be extremely small in generic astrophysical scenarios involving the accretion of extended distributions of matter . on the other hand , it should also be stressed that the time - domain analysis carried out here may not be the most accurate to extract the contributions coming from the perturbed black hole when these are several orders of magnitude smaller than those coming from the source itself or from the backscattering off the potential . in these cases , however , a hybrid approach such as the one proposed in ref . @xcite , combining the solution in the time - domain for the hydrodynamics with one in the frequency - domain for the perturbative equations , may be better suited . it is a pleasure to thank e. berti , h.r . beyer , s. bernuzzi , r. de pietri , i. olabarrieta , a. tartaglia and m. tiglio for useful discussions and comments . part of this work was carried out by an through visits in valencia and at the center of computation and technology ( cct ) at louisiana state university ; the support of the della riccia foundation , of cct and of ilias is gratefully acknowledged . jaf acknowledges financial support from the spanish _ ministerio de educacin y ciencia _ ( grant aya 2004 - 08067-c03 - 01 ) . the computations were performed on the cluster for numerical relativity _ `` albert '' _ at the university of parma . 200 # 1 _ gr - qc/__#1 _ # 1#2#3 phys . * # 1 * , # 2 ( # 3 ) # 1#2#3 phys . rev . d , * # 1 * , # 2 ( # 3 ) # 1#2#3 phys . b , * # 1 * , # 2 ( # 3 ) # 1#2#3 phys . reports , * # 1 * , # 2 ( # 3 ) # 1#2#3 physica , * # 1 * , # 2 ( # 3 ) # 1#2#3 j. comput . , * # 1 * , # 2 ( # 3 ) # 1#2#3 j. math . # 1 * , # 2 ( # 3 ) # 1#2#3 computer phys . , * # 1 * , # 2 ( # 3 ) # 1#2#3 class . quantum grav . , * # 1 * , # 2 ( # 3 ) # 1#2#3 computers math . , * # 1 * , # 2 ( # 3 ) # 1#2#3 math . , * # 1 * , # 2 ( # 3 ) # 1#2#3 astrophys . j. , * # 1 * , # 2 ( # 3 ) # 1#2#3 astrophys . j. lett . , * # 1 * , # 2 ( # 3 ) # 1#2#3 astrophys . j. suppl . , * # 1 * , # 2 ( # 3 ) # 1#2#3 acta astronomica , * # 1 * , # 2 ( # 3 ) # 1#2#3 sov . , * # 1 * , # 2 ( # 3 ) # 1#2#3 siam j. sci . * # 1 * , # 2 ( # 3 ) #1#2#3 astron . astrophys . , * # 1 * , # 2 ( # 3 ) # 1#2#3 mon . not . * # 1 * , # 2 ( # 3 ) # 1#2#3 proc . london , ser . a , * # 1 * , # 2 ( # 3 ) # 1#2#3 i.j.m.p . c * # 1 * , # 2 ( # 3 ) # 1#2#3 phys . lett . a * # 1 * , # 2 ( # 3 ) # 1#2#3 progr . . phys . * # 1 * , # 2 ( # 3 ) s. chandrasekhar , _ the mathematical theory of black holes _ , ( oxford university press , oxford , 1992 ) . kokkotas and b. schmidt , _ quasi - normal modes of stars and black holes _ , living rev . relativity , pub . lrr-1999 - 2 . frolov and i.d . novikov , _ black hole physics _ , kluwer academic publishers , dordrecht 1998 . w.h . press , astrophys . j. letters * 170 * , l105 ( 1971 ) . vishveshwara , nature ( london ) , * 227 * , 936 ( 1970 ) . leaver , phys . d * 34 * , 384 ( 1986 ) ; errata , phys . d * 38 * , 725 ( 1988 ) . n. andersson , phys . d * 51 * , 353 ( 1995 ) , * 52 * , 1808 ( 1995 ) , * 55 * , 468 ( 1997 ) . c.o . lousto and r.h . price , phys . d * 55 * , 2124 ( 1997 ) . cunningham , r.h . price , and v. moncrief , . e. seidel , phys . rev d * 44 * , 950 ( 1991 ) . l. baiotti , i. hawke , l. rezzolla and e. schnetter , phys . rev . lett . * 94 * , 131101 ( 2005 ) . l. baiotti and l. rezzolla , phys . . lett . * 97 * , 141101 ( 2006 ) . f. pretorius , phys . lett . * 95 * , 121101 ( 2005 ) . m. campanelli , c.o . lousto , p. marronetti , and y. zlochower , phys . lett . * 96 * 111101 ( 2006 ) . baker , j. centrella , d.i . choi , m. koppitz , and j. van meter , phys . * 96 * 111102 ( 2006 ) . p. papadopoulos and j.a . font , phys . d * 59 * , 044014 ( 1999 ) . a. nagar , g. diaz , j.a . pons , and j.a . font , phys . d * 69 * , 124028 ( 2004 ) . a. nagar , j.a . font , o. zanotti and r. de pietri , phys . d * 72 * , 024007 ( 2005 ) . v. ferrari , l. gualtieri and l. rezzolla , phys . d * 73 * 124028 ( 2006 ) . shapiro and i. wasserman , astrophys . j. * 260 * , 838 ( 1982 ) . petrich , s.l . shapiro and i. wasserman , astrophys . j. * 58 * , 297 ( 1985 ) . h. sotani and m. saijo , phys.rev . d * 74 * , 024001 ( 2006 ) . k. martel and e. poisson , phys . d * 66 * , 084001 ( 2001 ) . m. abramovicz , m. jaroszyski and m. sikora , astron . astrophys . * 63 * , 221 ( 1978 ) . j.a . font and f. daigne , mon . not . 334 * , 383 ( 2002 ) . o. zanotti , l. rezzolla , and j.a . font , mon . not . . soc . * 341 * , 832 ( 2003 ) . o. zanotti , j.a . font , l. rezzolla , and p. j. montero mon . not . . soc . * 356 * , 1371 ( 2005 ) . l. rezzolla , t. w. maccarone , si . yoshida and o. zanotti , mon . not . . soc . * 344 * , l37 ( 2003 ) . l. rezzolla , si . yoshida and o. zanotti , mon . not . . soc . * 344 * , 978 ( 2003 ) . a. nagar and l. rezzolla , class . * 22 * , r167 ( 2005 ) . k. martel and e. poisson , phys . d * 71 * , 104003 ( 2005 ) . t. regge and j.a . wheeler , phys . rev . * 108 * , 1063 ( 1957 ) . f.j . zerilli , phys . d * 2 * , 2141 ( 1970 ) . v. moncrief , ann . ( n.y . ) * 88 * , 323 ( 1974 ) . o. sarbach and m. tiglio , phys . d * 64 * , 084016 ( 2001 ) . goldberg , j. macfarlane , e.t . newman , f. rohrlich and e.c.g . sudarsahn , j. math . phys . * 8 * , 2155 ( 1967 ) . u.h . gerlach and u.k . sengupta , phys . d * 19 * , 2268 ( 1979 ) ; * 22 * , 1300 ( 1980 ) ; j.m . martn - garca and c. gundlach , phys . rev d * 59 * , 064031 ( 1999 ) ; c. gundlach and j.m . martn - garca , phys . d * 61 * , 084024 ( 2000 ) ; j.m . martn - garca and c. gundlach , phys . d * 64 * , 024012 ( 2001 ) . j.a . font , _ numerical hydrodynamics in general relativity _ , living rev . relativity , pub . lrr-2003 - 4 . lau , j. math . phys . * 46 * , 102503 ( 2005 ) . g. pschl and e. teller , z. phys . * 83 * , 143 ( 1933 ) . beyer , commun . * 204 * , 397 ( 1999 ) . v. ferrari and b. mashhoon , phys . d , * 30 * , 295(1984 ) . leaver , proc . a. , * 402 * , 285 ( 1985 ) . price , phys . d * 5 * , 2419 ( 1972 ) ; * 5 * , 2439 ( 1972 ) . e. berti , v. cardoso and c. m. will , in the proceedings of the vii international conference of the hellenic astronomical society , _ gr - qc/0601077_. a. nagar , t. damour and a. tartaglia , to appear in the special isuue of classical and quantum gravity based around the new frontiers in numerical relativity meeting , albert einstein insitutute , golm , july 17 - 21 , 2006 ; _ gr - qc/0612096_. m. davis , r. ruffini , w.h . press , and r.h . price , phys . * 27 * , 1466 ( 1971 ) ; m. davis , r. ruffini and j. tiomno , phys . d * 5 * , 2932 ( 1972 ) .
|
by combining the numerical solution of the nonlinear hydrodynamics equations with the solution of the linear inhomogeneous zerilli - moncrief and regge - wheeler equations we investigate the properties of the gravitational radiation emitted during the axisymmetric accretion of matter onto a schwarzschild black hole . our results suggest that the black hole qnm contributions to the full gravitational wave signal should be extremely small and possibly not detectable in generic astrophysical scenarios involving the accretion of extended distributions of matter .
|
by combining the numerical solution of the nonlinear hydrodynamics equations with the solution of the linear inhomogeneous zerilli - moncrief and regge - wheeler equations we investigate the properties of the gravitational radiation emitted during the axisymmetric accretion of matter onto a schwarzschild black hole . the matter models considered include quadrupolar dust shells and thick accretion disks , permitting us to simulate situations which may be encountered at the end stages of stellar gravitational collapse or binary neutron star merger . we focus on the interference pattern appearing in the energy spectra of the emitted gravitational waves and on the amount of excitation of the quasi - normal modes of the accreting black hole . we show that , quite generically in the presence of accretion , the black hole ringdown is not a simple superposition of quasi - normal modes , although the fundamental mode is usually present and often dominates the gravitational - wave signal . we interpret this as due to backscattering of waves off the non - exponentially decaying part of the black hole potential and to the finite spatial extension of the accreting matter . our results suggest that the black hole qnm contributions to the full gravitational wave signal should be extremely small and possibly not detectable in generic astrophysical scenarios involving the accretion of extended distributions of matter .
|
1302.4748
|
i
|
the pairing symmetry of the superconducting state in iron - based superconductors ( ibs ) has been one of the most debated subjects since their first discovery , in both theory and experiments.@xcite its determination is particularly challenging in the ibs for their complex electronic structure , with a strong multiband character , and a fermi surface constituted by many sheets , which can vary with doping and chemical composition . moreover , the ibs families are usually compensated metals , and thus both electrons and holes are involved in the paired state . weak - coupling rpa approaches , coupled to multiband bcs theory , yield a pairing function with global s - wave ( @xmath0 ) symmetry , but with electron and holes pockets having opposite sign . this scenario , dubbed @xmath7 , was first proposed by mazin _ _ et al.__@xcite complemented later with its variants , called `` extended @xmath7'',@xcite in the case that accidental nodes appear on a single fermi sheet without breaking the full rotational symmetry . the latest generalizations include also `` weak '' nodal lines which develop as closed loops on the 3-dimensional ( 3d ) fermi surface.@xcite a variety of experiments has been performed to probe the pairing symmetry of ibs , ranging from thermal conductivity and specific heat measurements to angle - resolved photoemission spectroscopy . while there is no doubt on the spin singlet nature of the pairing state as revealed by the knight shift,@xcite the presence of nodes and the total symmetry of the spatial part of the pairing function are controversial . in fact , the experimental outcome seems to lack universality , as it depends crucially on the `` family '' of tested compound , its doping , its isovalent substitutions , and its level of disorder . for instance , the situation is paradoxical for the `` 111 '' family , where the lifeas material shows a fully gapped superconducting order parameter , while the isovalent substitution of arsenic by phosphorus leads to thermodynamic properties compatible with a nodal pairing function.@xcite for the `` 122 '' family , there is a recent claim , supported by independent experimental probes , that the pairing in the bafe@xmath6as@xmath6 undergoes an @xmath4-to-@xmath5 symmetry change by doping with potassium,@xcite the fully substituted kfe@xmath6as@xmath6 being identified as a @xmath5-wave superconductor.@xcite in the `` 1111 '' family , the dependence of the critical temperature @xmath1 upon disorder has been studied in the lafeaso@xmath2f@xmath3 with @xmath8 . it has been found that co - doping induced disorder makes @xmath1 to fall much more slowly than mn - doping.@xcite this is certainly not compatible with a doping independent pairing function with sign changes ( either @xmath7 or @xmath5-wave ) . finally , specific heat measurements performed on the fese revealed a highly anisotropic order parameter in the `` 11 '' family,@xcite with a twofold symmetry directly observed by scanning tunneling microscopy ( stm).@xcite in this paper , we study the ibs gap structure from a different perspective , namely by looking for its _ universal _ features based on symmetry constraints induced by the 3d _ space - group _ transformations of the crystals . we prove that the ibs pairing function is a linear combination of terms having planar s- and d - wave symmetries , both fulfilling the full 3d @xmath0 representation . these properties are then verified in the fese by performing state - of - the - art quantum monte carlo ( qmc ) calculations from first - principles . our theory provides a general framework to account for the contradictory experimental outcomes . the paper is organized as it follows . in sec . [ improper_symmetry ] we present the derivation of the 2d iron lattice model consistent with the symmetries of the 3d point group of the fese . we show that it is necessary to include improper rotations and a gauge transformation for translations in order to define a 2d square lattice model consistent with the 3d structure . we study what are the implications of the symmetry constraints for the pairing structure , and show that a d@xmath9-wave channel is present beside the extended s - wave one . these predictions are verified by accurate quantum monte carlo simulations , presented in sec . [ qmc_section ] , to determine the global symmetry of the pairing function from first - principles . in sec . [ bcs_modeling ] we model the ab - initio pairing function by means of a bcs hamiltonian . this allows us to compute the bcs gap on a dense k - point grid , and study its nodal structure in the k - space . in sec . [ physical_consequences ] we relate our findings to the experimental outcome in fese , and to other experiments in different families of iron pnictides and chalcogenides .
|
by means of space - group symmetry arguments , we argue that the electronic pairing in iron - based high temperature superconductors shows a structure which is a linear combination of _ planar _ s - wave and d - wave symmetry channels , both preserving the 3-dimensional irreducible representation of the corresponding crystal point - group .
|
by means of space - group symmetry arguments , we argue that the electronic pairing in iron - based high temperature superconductors shows a structure which is a linear combination of _ planar _ s - wave and d - wave symmetry channels , both preserving the 3-dimensional irreducible representation of the corresponding crystal point - group . we demonstrate that the s- and d - wave channels are determined by the parity under reflection of the electronic orbitals through the iron planes , and by improper rotations around the iron sites . we provide evidence of these general properties by performing accurate quantum monte carlo ab - initio calculations of the pairing function , for a fese lattice with tetragonal experimental geometry at ambient pressure . we find that this picture survives even in the fese under pressure and at low temperatures , when the tetragonal point - group symmetry is slightly broken . in order to achieve a higher resolution in momentum space we introduce a bcs model that faithfully describes our qmc variational pairing function on the simulated 4x4 fese unit cell . this allows us to provide a k - resolved image of the pairing function , and show that non - isotropic contributions in the bcs gap function are related to the improper s - wave symmetry . our theory can rationalize and explain a series of contradictory experimental findings , such as the observation of twofold symmetry in the fese superconducting phase , the anomalous drop of with co - impurity in lafeasof , the-to--wave gap transition in bafeas under k doping , and the nodes appearing in the lifeas superconducting gap upon p isovalent substitution
|
gr-qc0701166
|
i
|
penrose s definition of _ asymptotic simplicity _ is an attempt to provide a characterisation of isolated systems in general relativity @xcite . it offers a framework on which diverse notions of physical interest can be defined and handled in a precise and rigorous manner . there is , however , a notorious lack of nontrivial examples where the associated formalism see e.g. @xcite has been actually been used to extract some physics . the obvious reason for this is the scarcity of radiative exact solutions to the einstein field equations other than the boost - rotation symmetric ones @xcite . a more conspicuous consideration in this regard is the fact that radiative spacetimes have to be constructed starting from an initial value problem where , say , some cauchy initial data are provided . up to fairly recently , there has not been a way of relating the properties of the initial data to the radiative properties of the development . for example , how does an initial data set have to be so that the resulting spacetime _ peels _ ? that is , the components of the weyl tensor in an adapted gauge having a distinctive decay in terms of powers of an affine parameter of the generators of outgoing light cones @xcite . another natural question on these lines would be how does the adm mass and the bondi mass relate ? similarly , is there any relation between the angular momentum defined at spatial infinity and that defined at null infinity ? which classes of initial data allow to select the poincar group out of the group of asymptotic symmetries the bms group in a canonical fashion ? arguably , there is a shortage of general results about the evolution of initial data sets for the einstein field equations near spatial infinity . this particular has made the idea of relating the structure of initial data with radiative properties a daunting endeavour . however , work by friedrich @xcite on the _ regular initial value problem at spatial infinity _ for time symmetric initial data sets has provided a tool to address questions similar to the ones raised in the first paragraph through a systematic approach . friedrich has introduced a representation of spatial infinity as a cylinder _the cylinder at spatial infinity _ , @xmath0 in stark contrast to the usual representation as a point . the cylinder at spatial infinity can be regarded as a limit set of incoming and outgoing characteristics of the einstein field equations , and as such it happens to be a _ total characteristic_. as such , it allows to transport information from the cauchy surface to null infinity without being contaminated by any sort of boundary conditions . the construction which led to the cylinder at spatial infinity allows an unfolding of the evolution process , which in turn can be analysed in all detail and to the desired order . the latter can be thought of as enabling the construction of a certain kind of asymptotic expansions which are completely determined by the cauchy data near infinity . friedrich s seminal work has been extended in several directions , automatising , in some sense , the calculation of the asymptotic expansions of the relevant field quantities @xcite and to include more general classes of initial data sets @xcite . the analysis described in these references has exhibited the existence of a hierarchy of obstructions to the smoothness of null infinity which goes beyond that arising from a mere consideration of the linear aspects of the field equations as described in e.g. @xcite . the existence of the hierarchy of obstructions has nurtured a _ rigidity conjecture _ on the smoothness of null infinity which , broadly speaking , states that the only cauchy initial data sets which have a development with a smooth null infinity . ] are those which are stationary close to infinity . similarly , a given degree of differentiability at null infinity , say @xmath1 , would only be possible if the initial data is stationary to a given order which depends on @xmath2 . the kind of initial data required can be constructed by means of the gluing techniques of @xcite . intuitively , the absence of gravitational radiation near spatial infinity which the rigidity conjecture asserts , suggests that the incoming radiation has to die off in the infinite future , and at the same time the system can not have emitted gravitational waves for all times in the infinite past . now , the calculation of the asymptotic expansions which arise from the construction of the cylinder at infinity is performed in a gauge which is adapted to a cauchy initial value problem remarkably , the gauge also allows to read off the structure and location of infinity directly from the initial data . on the other hand , the discussion of the gravitational field near null infinity is usually done using a gauge which is based on geometric structures of null hypersurfaces . having two different gauges , hampers , to a certain extent , extracting the physics of the system . it also difficults the attempts to assess the relevance of the presence of obstructions to the smoothness to null infinity in the asymptotic expansions . one of the aspects of the analysis of the asymptotic behaviour of the gravitational field via the definition of asymptotic simplicity and related constructions is that it allows to rephrase the decay of the fields in terms of differentiability at the conformal boundary . thus , it is natural to expect that the presence of obstructions to smoothness in the asymptotic expansions may result in a modified peeling behaviour this idea has already been explored in @xcite . a first study of the transformation between the gauge used in friedrich s analysis of spatial infinity and the gauge generally used to discuss null infinity has been given in @xcite . in the present article their analysis will be extended to the developments of a class of initial data sets which are not time symmetric . the consideration of initial data sets with a non - vanishing second fundamental form will allow us to examine some time asymmetric aspects of the structure of the conformal boundary of asymptotically flat spacetimes which up to now have remained _ terra incognita . _ on the other hand , the discussion shall be restricted to conformally flat initial data sets . this is done for conciseness and for the ease of calculations and presentation . previous analysis with conformally flat initial data sets see e.g. @xcite have shown that most of the crucial phenomena and structure observed in this setting admit a counterpart when discussing non - conformally flat data sets like in @xcite . furthermore , the detailed understanding of conformally flat initial data sets is of relevance in view of their use in the numerical simulation of black hole spacetimes see e.g. @xcite for a recent discussion on this . a cautionary note is , however , due : most of the conformally flat initial sets considered in the numerical simulations are boosted . this is done on physical grounds . here , initial data sets with linear momentum i.e . boosted will not be considered . the reason being that , as it will be discussed in the sequel , they are more technically involved and less smooth . the effects of this lower regularity are of interest in themselves and will be discussed elsewhere . the article is structured as follows : section [ section : basic_setup ] discusses some ideas of the basis set up and fixes some notation ; section [ section : f - gauge ] describes some aspects of the `` cylinder at spatial infinity''-formalism ; section [ section : constraints ] considers relevant aspects of the class of initial data sets to be used ; section [ section : expansions ] muses over the asymptotic expansions that can be calculated using the cylinder at spatial infinity ; section [ section : np - gauge ] goes briefly about the construction of a frame adapted to null infinity the np gauge while section [ section : relating_gauges ] discusses how it is related to the gauge used in the expansions of section [ section : expansions ] . this article revisits the analysis carried out in @xcite of `` bondi type '' systems what here is called the newman - penrose gauge and combines it with the asymptotic expansions for the development of conformally flat initial data sets which have been calculated in @xcite . in section [ section : peeling ] analysis of the implications of these expansions and the related obstructions to the smoothness of null infinity is done under some existence and regularity assumptions . in section [ section : sigma ] the machinery is used to calculate expansions of the spin coefficient @xmath3 on null infinity in the np gauge . the behaviour of this coefficient is related to various physical aspects of the theory of isolated systems . the implications of these expansions are analysed . it is shown that if the spacetime peels , then the behaviour of @xmath3 is such that it is always possible to single out the poincar group out of the asymptotic symmetry group in a canonical way . expansions of the bondi mass are also considered . in section [ section : np - constants ] the formalism is used to write down the newman - penrose constants of the developments of the class of initial data under consideration in terms of initial data quantities . these expressions allow to show that the constants at future null infinity are equal to those at past null infinity . there is much more to the aforementioned analysis than what is presented in this article . in particular , most of the calculations involved in the analysis have been carried out using _ ex profeso _ scripts in the computer algebra system maple v. a first subset of the scripts constructs the initial data ; another subset determines the coefficients of the asymptotic expansions in terms of initial data quantities ; finally a third subset calculates the transformation between gauges and allows to calculate the asymptotic expansions of np objects . in order to keep the presentation at a reasonable length , the explicit expressions of intermediate steps will be omitted , and more emphasis will be put on to the qualitative aspects of the results . some familiarity with the use of spinors in particular the space - spinor formalism of @xcite will be assumed . details of the diverse other formalisms used the cylinder at spatial infinity , the formalism of null infinity , the construction of gauges , and how to relate them is kept to the minimum necessary to motivate the analysis and to point out the many subtleties arising . in any case , the reader is referred to the various references cited for more complete details . finally , are some concluding remarks and a brief appendix containing some useful spinorial expressions .
|
certain aspects of the behaviour of the gravitational field near null and spatial infinity for the developments of asymptotically euclidean , conformally flat initial data sets are analysed . the decay of the weyl tensor for the developments of the class of initial data under consideration is analysed under some existence and regularity assumptions for the asymptotic expansions obtained using the cylinder at spatial infinity . further , the decay of the asymptotic shear on null infinity is also examined as one approaches spatial infinity . this decay is related to the possibility of selecting the poincar group out of the bms group in a canonical fashion . it is found that for the class of initial data under consideration , if the development peels , then the asymptotic shear goes to zero at spatial infinity . expansions of the bondi mass are also examined . finally , the newman - penrose constants of the spacetime are written in terms of initial data quantities and it is shown that the constants defined at future null infinity are equal to those at past null infinity . pacs : 04.20.ha , 04.20.ex , 04.20.gz
|
certain aspects of the behaviour of the gravitational field near null and spatial infinity for the developments of asymptotically euclidean , conformally flat initial data sets are analysed . ideas and results from two different approaches are combined : on the one hand the null infinity formalism related to the asymptotic characteristic initial value problem and on the other the regular cauchy initial value problem at spatial infinity which uses friedrich s representation of spatial infinity as a cylinder . the decay of the weyl tensor for the developments of the class of initial data under consideration is analysed under some existence and regularity assumptions for the asymptotic expansions obtained using the cylinder at spatial infinity . conditions on the initial data to obtain developments satisfying the peeling behaviour are identified . further , the decay of the asymptotic shear on null infinity is also examined as one approaches spatial infinity . this decay is related to the possibility of selecting the poincar group out of the bms group in a canonical fashion . it is found that for the class of initial data under consideration , if the development peels , then the asymptotic shear goes to zero at spatial infinity . expansions of the bondi mass are also examined . finally , the newman - penrose constants of the spacetime are written in terms of initial data quantities and it is shown that the constants defined at future null infinity are equal to those at past null infinity . pacs : 04.20.ha , 04.20.ex , 04.20.gz
|
1305.3530
|
i
|
the problem of checking the _ validity _ of quasiequations in finitely generated ( i.e. , generated by a finite set of finite algebras ) quasivarieties or , similarly , checking consequences from finite sets of formulas in finite - valued logics , is decidable and has been investigated extensively in the literature . in particular , uniform methods for generating proof systems to check validity such as tableaux , resolution , and multisequent calculi , have been developed , as have standard optimization techniques for these systems such as lemma generation and indexing ( see , e.g. , @xcite ) . however , checking the _ admissibility _ of quasiequations in finitely generated quasivarieties , or similarly , checking the admissibility of rules in finite - valued logics , is not so well - understood . the problem is decidable , but a naive approach leads to computationally unfeasible procedures even for small sets of small algebras . the main goal of this paper is to define uniform methods that generate computationally acceptable proof systems for checking admissibility in an arbitrary finitely generated quasivariety . intuitively , a rule is said to be admissible in a logical system if it can be added to the system without producing any new theorems . more formally , a quasiequation is admissible in a class of algebras @xmath1 if every @xmath1-unifier of the premises is a @xmath1-unifier of the conclusion , where a @xmath1-unifier of an equation @xmath2 is a substitution @xmath3 such that @xmath4 is valid in @xmath1 . admissibility plays a fundamental meta - level role in describing `` hidden properties '' of classes of algebras and logical systems . for example , establishing the completeness of a logical system with respect to some restricted class of algebras ( perhaps just one standard algebra ) often involves showing that a certain rule or quasiequation is admissible ; see , e.g. , @xcite for applications of the admissibility of rules in the context of fuzzy logics . also , the closely related problem of deciding unifiability of concepts can be a useful tool for database redundancy checking in description logics @xcite . moreover , it may be possible to automatically obtain admissible rules for classes of algebras and logics that can then be used to simplify reasoning steps or to speed up derivations for checking validity . admissibility ( in tandem with unification ) has been studied intensively in the context of intermediate and transitive modal logics and their algebras @xcite , leading in some cases to proof systems for checking admissibility @xcite . axiomatizations and characterizations have also been obtained for certain families of finite algebras and many - valued logics , in particular ukasiewicz logics ( or classes of mv - algebras ) @xcite and other fuzzy logics @xcite , fragments of the substructural logic r - mingle @xcite , and classes of de morgan algebras @xcite . however , a general theory , covering arbitrary finite algebras and finite - valued logics , has so far been lacking . the starting point for this work is the observation ( see lemma 4.1.10 of @xcite and corollary [ c : admfin ] below ) that for a finite set of finite algebras @xmath1 , admissibility in the quasivariety @xmath5 amounts to validity in the free algebra on @xmath6 generators @xmath7 , where @xmath6 is the maximum cardinality of the algebras in @xmath1 . since by birkhoff s theorem on the structure of free algebras @xcite , this algebra @xmath7 is finite , checking admissibility in @xmath5 is decidable . on the other hand , even for small @xmath6 and a small set of small algebras @xmath1 , the size of @xmath7 may be prohibitively large for checking validity . this is striking since validity and admissibility in @xmath5 may coincide , @xmath5 is then called _ structurally complete _ , or at least coincide for quasiequations with @xmath5-unifiable premises , in which case , @xmath5 is called _ almost structurally complete_. in other cases , @xmath5-admissibility may correspond to validity in other , often quite small , algebras . we provide general algorithms here that discover such algebras , or , more precisely , generate finite sets of finite algebras such that the @xmath5-admissibility of a quasiequation corresponds to validity in the quasivariety generated by these algebras . it is shown , moreover , that these are the smallest sets of algebras with this property with respect to a standard well - ordering on the multiset of their cardinalities . we proceed as follows . first , in section [ s : prelim ] , we recall some basic notions from universal algebra . then in section [ s : fingen ] , we introduce some key ideas and methods for finitely generated quasivarieties ; in particular , we apply a standard multiset well - ordering to the cardinalities of algebras in generating sets for quasivarieties , and provide an algorithm for finding the ( unique up to isomorphism ) minimal generating set of a finitely generated quasivariety . section [ s : admfree ] provides characterizations of admissibility , unifiability , structural completeness , and almost structural completeness . these characterizations are then exploited in section [ s : algorithms ] to define corresponding algorithms , and illustrated using a selection of well - known finite algebras , confirming some known results from the literature , and establishing new ones . in section [ s : logics ] , the approach is extended to finite - valued logics , where the designated values as well as the underlying finite algebra play a significant role . finally , in section [ s : concluding ] , we conclude with some remarks on future directions for this research .
|
checking the admissibility of quasiequations in a finitely generated ( i.e. , generated by a finite set of finite algebras ) quasivariety amounts to checking validity in a suitable finite free algebra of the quasivariety , and is therefore decidable . however , since free algebras may be large even for small sets of small algebras and very few generators , this naive method for checking admissibility in is not computationally feasible . in this paper , algorithms are introduced that generate a minimal ( with respect to a multiset well - ordering on their cardinalities ) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in . in particular , structural completeness ( validity and admissibility coincide ) and almost structural completeness ( validity and admissibility coincide for quasiequations with unifiable premises ) can be checked . the algorithms are illustrated with a selection of well - known finitely generated quasivarieties , and adapted to handle also admissibility of rules in finite - valued logics .
|
checking the admissibility of quasiequations in a finitely generated ( i.e. , generated by a finite set of finite algebras ) quasivariety amounts to checking validity in a suitable finite free algebra of the quasivariety , and is therefore decidable . however , since free algebras may be large even for small sets of small algebras and very few generators , this naive method for checking admissibility in is not computationally feasible . in this paper , algorithms are introduced that generate a minimal ( with respect to a multiset well - ordering on their cardinalities ) finite set of algebras such that the validity of a quasiequation in this set corresponds to admissibility of the quasiequation in . in particular , structural completeness ( validity and admissibility coincide ) and almost structural completeness ( validity and admissibility coincide for quasiequations with unifiable premises ) can be checked . the algorithms are illustrated with a selection of well - known finitely generated quasivarieties , and adapted to handle also admissibility of rules in finite - valued logics .
|
1503.07819
|
i
|
it is commonly known that variational methods are a powerful tool for studying the dynamics of various physical systems , e.g.,quantum molecular dynamics @xcite , fluid mechanics @xcite , classical motion of single particles and collective processes in plasmas @xcite , and wave propagation in both linear and nonlinear media @xcite . variational formulations are advantageous as they lead to dynamical equations in a manifestly conservative form derived from a single scalar function , a lagrangian ( or lagrangian density ) . the fact that these equations can be approximated robustly and self - consistently by approximating just one function makes the method particularly attractive for reduced calculations @xcite . however , so far , exact lagrangians have been obtained largely heuristically or _ ad hoc _ @xcite . such approaches tend to obscure the physical meaning of the results and limit their applicability , while regular ways to deduce lagrangians rigorously from first principles are yet to be found . here we show that classical variational principles ( vps ) can be deduced from quantum vps , which are well known , via formal reparameterization of the latter . such reparameterization is possible without appealing to dynamical equations and without invoking any assumptions other than classicality . this distinguishes our theory from the existing variational formulations of the quantum - classical correspondence , which are more restrictive @xcite . also as a complement to those formulations , we consider both single - particle _ and _ fluid vps and rigorously explain how they are connected through quantum vps . classical - fluid lagrangians flow as the semiclassical limit of the fundamental quantum lagrangian ( fql ) , eq . ( [ eq : gl ] ) , and the point - particle lagrangians are then yielded as corollaries for narrow _ but otherwise arbitrary _ wave packets . this approach enables the first principle classical lagrangian description of most general , relativistic vector particles ( e.g.,a dirac electron ) and , similarly , the geometrical optics ( go ) description of any vector waves , as we discuss in a companion paper @xcite . the present paper is intended as an introduction to such calculations . thus , below , we primarily focus on systematization of vps for commonly known systems , including schrdinger , pauli , and klein - gordon particles . we explicitly show how to deduce classical lagrangians for these particles and the corresponding fluids from the fql . in particular , we show that the expression for the bohm quantum potential @xcite , the so - called weizscker correction @xcite , the madelung equations @xcite , and the chen - sudan lagrangian density @xcite all emerge naturally within our unifying theory as special cases . we also obtain an alternative , manifestly lagrangian representation of takabayasi equations for a pauli particle . in contrast to the original theory @xcite , our model yields the dynamics of the spin vector @xmath0 directly from the vp and employs two , rather than three , equations ; hence , @xmath1 is ensured irrespective of initial conditions . the paper is organized as follows . in sec . [ sec : notation ] , we define the basic notation . in sec . [ sec : basic ] , we introduce the general formulas . in sec . [ sec : schrodinger ] , we discuss a schrdinger particle and the madelung equations . in sec . [ sec : pauli ] , we discuss a pauli particle and the variational formulation of takabayasi equations . in sec . [ sec : kleingordon ] , we discuss a klein - gordon particle and rederive the chen - sudan fluid lagrangian density . in sec . [ sec : conclusion ] , we summarize our main results and outline future applications of the proposed theory .
|
classical variational principles can be deduced from quantum variational principles via formal reparameterization of the latter . it is shown that such reparameterization is possible without invoking any assumptions other than classicality and without appealing to dynamical equations . as examples , first principle variational formulations of classical point - particle and cold - fluid motion are derived from their quantum counterparts for schrdinger , pauli , and klein - gordon particles .
|
classical variational principles can be deduced from quantum variational principles via formal reparameterization of the latter . it is shown that such reparameterization is possible without invoking any assumptions other than classicality and without appealing to dynamical equations . as examples , first principle variational formulations of classical point - particle and cold - fluid motion are derived from their quantum counterparts for schrdinger , pauli , and klein - gordon particles .
|
1304.5498
|
i
|
clique - width is a fundamental graph invariant that has been widely studied in combinatorics and computer science . clique - width measures in a certain sense the `` complexity '' of a graph . it is defined via a graph construction process involving four operations where only a limited number of vertex labels are available ; vertices that share the same label at a certain point of the construction process must be treated uniformly in subsequent steps . this graph composition mechanism was first considered by courcelle , engelfriet , and rozenberg @xcite and has since then been an important topic in combinatorics and computer science . graphs of small clique - width have advantageous algorithmic properties . algorithmic meta - theorems show that large classes of @xmath0-hard optimization problems and # p - hard counting problems can be solved in _ linear time _ on classes of graphs of bounded clique - width @xcite . similar results hold for the graph invariant _ treewidth _ , however , clique - width is more general in the sense that graphs of small treewidth also have small clique - width , but there are graphs of small clique - width but arbitrarily high treewidth @xcite . unlike treewidth , dense graphs ( e.g. , cliques ) can also have small clique - width . all these algorithms for graphs of small clique - width require that a certificate for the graph having small clique - width is provided . however , it seems that computing the certificate , or just deciding whether the clique - width of a graph is bounded by a given number , is a very intricate combinatorial problem . more precisely , given a graph @xmath1 and an integer @xmath2 , deciding whether the clique - width of @xmath1 is at most @xmath2 is @xmath0complete @xcite . even worse , the clique - width of a graph with @xmath3 vertices of degree greater than 2 can not be approximated by a polynomial - time algorithm with an absolute error guarantee of @xmath4 unless @xmath5 , where @xmath6 @xcite . in fact , it is even unknown whether graphs of clique - width at most @xmath7 can be recognized in polynomial time @xcite . there are approximation algorithms with an exponential error that , for fixed @xmath2 , compute @xmath8expressions for graphs of clique - width at most @xmath2 in polynomial time ( where @xmath9 by @xcite , and @xmath10 by @xcite ) . because of this intricacy of this graph invariant , the exact clique - width is not known even for very small graphs . [ [ clique - width - via - sat . ] ] clique - width via sat . + + + + + + + + + + + + + + + + + + + + + we present a new method for determining the clique - width based on a sophisticated sat encoding which entails the following ideas : 1 . _ reformulation_. the conventional construction method for determining the clique - width of a graph consists of many steps . in the worst case , the number of steps is quadratic in the number of vertices . translating this construction method into sat would result in large instances , even for small graphs . we reformulated the problem in such a way that the number of steps is less than the number of vertices . the alternative construction method allows us to compute the clique - width of much larger graphs . representative encoding_. applying the frequently - used direct encoding @xcite on the reformulation results in instances that have no arc consistency @xcite , i.e. , unit propagation may find conflicts much later than required . we developed the representative encoding that is compact and realizes arc consistency . [ [ experimental - results . ] ] experimental results . + + + + + + + + + + + + + + + + + + + + + the implementation of our method allows us for the first time to determine the exact clique - width of various graphs , including famous graphs known from the literature , as well as random graphs of various density . 1 . _ clique - width of small random graphs_. we determined experimentally how the clique - width of random graphs depends on the density . the clique - width is small for dense and sparse graphs and reaches its maximum for edge - probability @xmath11 . the larger @xmath3 , the steeper the increase towards @xmath11 . these results complement the asymptotic results of lee et al . _ smallest graphs of certain clique - width_. in general it is not known how many vertices are required to form a graph of a certain clique - width . we provide these numbers for clique - width @xmath12 . in fact , we could compute the total number of connected graphs ( modulo isomorphism ) with a certain clique - width with up to 10 vertices . for instance , there are only 7 connected graphs with 8 vertices and clique - width 5 ( modulo isomorphism ) , and no graphs with 9 vertices and clique - width 6 . there are 68 graphs with 10 vertices and clique - width 6 . the smallest one has 18 edges . clique - width of famous named graphs_. over the last 50 years , researchers in graph theory have considered a large number of special graphs . these special graphs have been used as counterexamples for conjectures or for showing the tightness of combinatorial results . we considered several prominent graphs from the literature and computed their exact clique - width . these results may be of interest for people working in combinatorics and graph theory . [ [ related - work . ] ] related work . + + + + + + + + + + + + + we are not aware of any implemented algorithms that compute the clique - width exactly or heuristically . however , algorithms have been implemented that compute upper bounds on other width - based graph invariants , including _ treewidth _ @xcite , _ branchwidth _ @xcite , _ boolean - width _ @xcite , and _ rank - width _ @xcite . samer and veith @xcite proposed a sat encoding for the exact computation of treewidth . boolean - width and rank - width can be used to approximate clique - width , however , the error can be exponential in the clique - width ; in contrast , treewidth and branchwidth can be arbitrarily far from the clique - width , hence the approximation error is unbounded @xcite . our sat encoding is based on a new characterization of clique - width that is based on partitions instead of labels . a similar partition - based characterization of clique - width , has been proposed by heggernes et al . there are two main differences to our reformulation . firstly , our characterization of clique - width uses three individual properties that can be easily expressed by clauses . secondly , our characterization admits the `` parallel '' processing of several parts of the graph that are later joined together .
|
clique - width is a graph invariant that has been widely studied in combinatorics and computer science . however , computing the clique - width of a graph is an intricate problem , the exact clique - width is not known even for very small graphs . we present a new method for computing the clique - width of graphs based on an encoding to propositional satisfiability ( sat ) which is then evaluated by a sat solver . our encoding is based on a reformulation of clique - width in terms of partitions that utilizes an efficient encoding of cardinality constraints . our sat - based method is the first to discover the exact clique - width of various small graphs , including famous graphs from the literature as well as random graphs of various density . with our method we determined the smallest graphs that require a small pre - described clique - width .
|
clique - width is a graph invariant that has been widely studied in combinatorics and computer science . however , computing the clique - width of a graph is an intricate problem , the exact clique - width is not known even for very small graphs . we present a new method for computing the clique - width of graphs based on an encoding to propositional satisfiability ( sat ) which is then evaluated by a sat solver . our encoding is based on a reformulation of clique - width in terms of partitions that utilizes an efficient encoding of cardinality constraints . our sat - based method is the first to discover the exact clique - width of various small graphs , including famous graphs from the literature as well as random graphs of various density . with our method we determined the smallest graphs that require a small pre - described clique - width .
|
astro-ph0211430
|
i
|
fluctuations in the cosmic microwave background ( cmb ) radiation can provide information about hot gas in galaxy clusters over a wide range of redshifts ( sunyaev & zeldovich 1972,1980 ( hereafter sz ) , and @xcite ) . on arcminute angular scales and smaller , the thermal sz contribution to the cmb anisotropy is expected to dominate that of the primary anisotropies ( @xcite , @xcite ( hereafter swh ) , @xcite ) . a new generation of experiments is measuring the cmb sky at these angular scales . in particular , two recent experiments , bima @xcite and cbi @xcite , both observing at frequencies around 30 ghz , have detected an excess of power in the multipole region @xmath1 , where the sz power is expected to be dominant over the cmb signal . nevertheless , at these observing frequencies , radio point sources are known to also produce a significant contribution to the power @xcite if they are not subtracted properly from the cmb maps . the reported detections of power have argued that this point - source contamination is not a problem , thus suggesting that the signal could be due to the sz effect @xcite . these arguments are based on analytical models or simulations of what we would expect to measure . thus , it would be interesting to explore , in a model - independent way , the nature of these contributions . the importance of this topic has been stressed recently by @xcite , who suggested to use a cross - correlation of cmb maps with maps of the large scale structure . this idea has been applied for this purpose to other datasets with larger angular resolutions ( e.g. @xcite , to the cobe data , or @xcite , to the tenerife data ) . here , we propose a general model - independent method to determine if the measured power excess in a _ single - frequency _ map is ( mainly ) due to point sources or sz clusters . to this end , we use the fact that for frequencies below @xmath2 ghz ( @xmath0 mm ) , the thermal sz effect produces negative features in the maps , while the point sources produce positive peaks . we illustrate this fact with figure [ typical ] , where we show two simulated one - dimensional maps , one of sz clusters observed at @xmath3 ghz , and the other one of point sources . dotted lines show the original ( without sources of any kind ) zero level of fluctuations , while dashed lines show the observed ( average ) zero level once the mean of the map has been subtracted . with the same level of fluctuations ( same rms at the observed scale ) , a power spectrum analysis is not able to distinguish these two cases , so we need to use an statistic carrying information about the sign of the subjacent signal ( e.g. the skewness ) to suggest the nature of the objects producing this excess of power . the existence of negative skewness at @xmath0 mm , while positive skewness at @xmath4 mm , is a clear prediction for sz clusters . full - width half - maximum , and no noise . dashed lines show the average ( zero ) level of fluctuations , while dotted lines show the original zero level before subtracting the mean to the map . with the same level of fluctuations , the @xmath5 analysis does not permit to distinguish between these two cases . therefore , we need to use the skewness , or to proceed with an analysis of the asymmetry of the @xmath6 curve in order to separate these two cases.,title="fig : " ] full - width half - maximum , and no noise . dashed lines show the average ( zero ) level of fluctuations , while dotted lines show the original zero level before subtracting the mean to the map . with the same level of fluctuations , the @xmath5 analysis does not permit to distinguish between these two cases . therefore , we need to use the skewness , or to proceed with an analysis of the asymmetry of the @xmath6 curve in order to separate these two cases.,title="fig : " ] we investigate here the discrimination between positive and negative sources using the probability distribution function ( pdf ) for the observed flux . from a given map , _ the pdf function can be obtained easily as an histogram of the ( normalised ) number of pixels within a given flux interval_. this tool has been widely used in radio astronomy when studying the statistical properties of a background of point sources @xcite , because in that case the shape of this function is strongly related with the statistical properties of the sources ( i.e. theirs spatial distribution ) . in this context , this function is known as the deflection probability distribution , or the @xmath6 curve . this @xmath6 formalism has been successfully applied to study the diffuse x - ray background @xcite , as well as to determine the contribution of discrete point sources to cmb maps @xcite . for the cmb , if we assume the standard inflationary scenario , then the primordial fluctuations are gaussian , so the @xmath6 itself is a gaussian , as well as for the standard instrumental noise . however , the main characteristic of this @xmath6 curve for point sources ( @xcite ) or for sz clusters ( @xcite ) is its non - gaussianity . typical curves for a @xmath6 distribution of point sources or sz clusters will exhibit long tails ( see figure [ figura_bonita ] ) . the point is that at @xmath0 mm , sources will produce a positive tail , while sz clusters will give a negative one . it is important to mention that at @xmath4 mm , both agns and sz - clusters will produce positive tails . then it is necessary to use other characteristics of both populations ( frequency spectra , etc ) to distinguish them . as an illustration , figure [ compara_pd_sz ] demonstrates p(d ) for sz sources at four frequencies , @xmath7 = 107 and 150 ghz ( where clusters are giving negative signal ) and 270 and 520 ghz ( where the signal from clusters is positive , and exactly opposite in sign to the previous cases ) . ) this curve will be explained in detail in section [ sec : general ] . ] function ( see equation ( [ s2:g(x ) ] ) ) , and four frequencies ( @xmath8 107 , 150 , 270 and 520 ghz ) where this function ( and so the flux density ) takes the same absolute value . second panel shows the p(d ) function for these four cases , using a simple truncated power - law to model the cluster source counts ( see section 4 ) , with values @xmath9 = 28 ( s/1jy)@xmath10 sr@xmath11 jy@xmath11 at 150 ghz , truncating at @xmath12 mjy , and with angular resolution @xmath13 . the p(d ) function is presented relative to its average value , so the distribution for the cases @xmath14 ghz and 150 ghz is symmetric around zero respect to the other two cases.,title="fig : " ] function ( see equation ( [ s2:g(x ) ] ) ) , and four frequencies ( @xmath8 107 , 150 , 270 and 520 ghz ) where this function ( and so the flux density ) takes the same absolute value . second panel shows the p(d ) function for these four cases , using a simple truncated power - law to model the cluster source counts ( see section 4 ) , with values @xmath9 = 28 ( s/1jy)@xmath10 sr@xmath11 jy@xmath11 at 150 ghz , truncating at @xmath12 mjy , and with angular resolution @xmath13 . the p(d ) function is presented relative to its average value , so the distribution for the cases @xmath14 ghz and 150 ghz is symmetric around zero respect to the other two cases.,title="fig : " ]
|
clusters of galaxies produce negative features at wavelengths mm in cmb maps , by means of the thermal sz effect , while point radio sources produce positive peaks . [ firstpage ] cosmology : cosmic microwave background cosmology : observations galaxies : clusters : general .
|
clusters of galaxies produce negative features at wavelengths mm in cmb maps , by means of the thermal sz effect , while point radio sources produce positive peaks . this fact implies that a distribution of unresolved sz clusters could be detected using the negative asymmetry introduced in the odd - moments of the brightness map ( skewness and higher ) , or in the probability distribution function ( pdf ) for the fluctuations , once the map has been filtered in order to remove the contribution from primordial cmb fluctuations from large scales . this property provides a consistency check to the recent detections from cbi and bima experiments of an excess of power at small angular scales , in order to confirm that they are produced by a distribution of unresolved sz clusters . however it will require at least 1.5 - 2 times more observing time than detection of corresponding power signal . this approach could also be used with the data of the planned sz experiments ( e.g. act , ami , amiba , apex , 8 m south pole telescope ) . [ firstpage ] cosmology : cosmic microwave background cosmology : observations galaxies : clusters : general .
|
astro-ph0211430
|
c
|
in this paper we have discussed five statements : * the contribution of sz clusters to the map noise at @xmath0 mm does not depend on the wavelength , and has a strong and peculiar non - gaussian negative tail in the @xmath6 function . * this contribution has characteristic negative skewness ( or bispectrum ) at @xmath0 mm . this fact can be used by current single - frequency experiments , such as cbi or bima , or by future experiments , such as act , ami , amiba , apex , or the 8-m south pole telescope , to distinguish if the detected excess of power at small angular scales is due to sz clusters . in addition , the detection of skewness only requires a factor 1.5 or 2 more integration time than the detection of an excess of power . once the skewness is detected , the @xmath6 function starts to show an asymmetry . * any multi - frequency experiment would have noise due to clusters of galaxies with @xmath6 at @xmath4 mm equal to @xmath65 at @xmath0 mm . * skewness and bispectrum will have different signs in these two spectral regions . * when dealing with real maps where primordial cmb fluctuations are present , it is necessary to use filters to remove the contribution of large angular scales . only in that case we can detect the presence of clusters / sources in the @xmath6 function .
|
this property provides a consistency check to the recent detections from cbi and bima experiments of an excess of power at small angular scales , in order to confirm that they are produced by a distribution of unresolved sz clusters . this approach could also be used with the data of the planned sz experiments ( e.g. act , ami , amiba , apex , 8 m south pole telescope ) .
|
clusters of galaxies produce negative features at wavelengths mm in cmb maps , by means of the thermal sz effect , while point radio sources produce positive peaks . this fact implies that a distribution of unresolved sz clusters could be detected using the negative asymmetry introduced in the odd - moments of the brightness map ( skewness and higher ) , or in the probability distribution function ( pdf ) for the fluctuations , once the map has been filtered in order to remove the contribution from primordial cmb fluctuations from large scales . this property provides a consistency check to the recent detections from cbi and bima experiments of an excess of power at small angular scales , in order to confirm that they are produced by a distribution of unresolved sz clusters . however it will require at least 1.5 - 2 times more observing time than detection of corresponding power signal . this approach could also be used with the data of the planned sz experiments ( e.g. act , ami , amiba , apex , 8 m south pole telescope ) . [ firstpage ] cosmology : cosmic microwave background cosmology : observations galaxies : clusters : general .
|
astro-ph0105527
|
i
|
star formation has been a long - standing target in astrophysics . the infrared protostar distribution revealed that the molecular cloud cores , which coincide with relatively high - density part ( @xmath2 ) of the molecular clouds , are the sites of star formation . the observed molecular cloud cores are divided into two categories : those observed associated with and without protostars . the molecular cloud cores without protostars are called starless cores or prestellar cores and are considered younger than the cores associated with protostars ( protostellar cores ) . from a theoretical point of view , clouds or cloud cores experience the isothermal run - away collapse first and then accretion on to the stellar core develops @xcite . prestellar cores which indicates inflow motions ( e.g. in l1544 rotation and infall velocities @xmath3 are observed by @xcite ) do indicate that they are in the dynamically contracting phase , in other words , in the run - away collapse @xcite . after the epoch when the dust thermal emissions are trapped in the central part of the cloud ( @xmath4 ) , an adiabatic core is formed and isothermal gas continues to accrete on to the core . the molecular cloud core in this phase is observed as a protostellar core . it is shown that the dynamical evolution of the cloud core is characterized by the sequence from the prestellar cores to protostellar cores . dynamical collapse of the magnetized clouds are studied by many authors @xcite . rotating clouds collapse has been attacked seriously with numerical simulations @xcite . however , a restricted number of articles are published regarding the dynamical contraction of the cloud with both rotation and magnetic fields @xcite ; for quasistatic evolution see @xcite . these researches are confined to the relatively early prestellar stage . is it sufficient to consider the effects of the rotational motion and the magnetic fields separately ? in the dynamical contraction phase , it is shown that the molecular outflow is driven by the cooperative effect of the magnetic fields and rotation motion @xcite . the toroidal magnetic fields are generated from the poloidal ones by the effect of rotation motions . the magnetic torque works only when the poloidal and toroidal magnetic fields coexist . the magnetic torque and thus magnetic angular momentum transfer along the magnetic field line is important to eject the outflow . since the outflow brings the excess angular momentum , the angular momentum that remain in the adiabatic core and thus new - born star is reduced by a factor from @xmath5 to @xmath6 from that of the parent molecular cloud core @xcite . the outflow have not observed either in the magnetized but non - rotating cloud @xcite or the rotating but non - magnetized cloud @xcite . therefore , rotation and magnetic fields are both essential to the evolution of molecular cloud cores . in the present paper , we present the dynamical contraction of the magnetized and rotating cloud . the cooperative effect of magnetic fields and the rotation motions becomes important after the adiabatic core is formed near the center of the cloud core @xcite . therefore , the evolution throughout from the prestellar to protostellar core should be studied . plan of this paper is as follows : in @xmath72 model and numerical method are described . as the initial condition , we choose a slowly rotating cloud with purely poloidal magnetic fields ( no toroidal magnetic fields ) . and we follow the evolution using magnetohydrodynamical simulations . section 3 is devoted to the numerical results . in this section we compare clouds with strong magnetic fields and those with weak magnetic fields . this shows that completely different two types of outflows are ejected in respective clouds . another comparison is made between fast rotators and slow rotators . in @xmath74 , we discuss the evolution till the second core , which becomes actually a new - born star , is formed . it is found that another outflow is found around the second core , which seems to correspond to the optical jets or high speed neutral winds . we also discuss whether the mass inflow / outflow rate and the momentum outflow rate observed in molecular bipolar outflows are explained or not .
|
the gas with excess angular momentum near the surface is finally ejected , which explains the molecular bipolar outflow . two types of outflows are observed . it is found that another outflow is ejected around the second atomic core , which seems to correspond to the optical jets or the fast neutral winds .
|
collapse of the rotating magnetized molecular cloud core is studied with the axisymmetric magnetohydrodynamical ( mhd ) simulations . due to the change of the equation of state of the interstellar gas , the molecular cloud cores experience several different phases as collapse proceeds . in the isothermal run - away collapse ( ) , a pseudo - disk is formed and it continues to contract till the opaque core is formed at the center . in this disk , a number of mhd fast and slow shock pairs appear running parallelly to the disk . after the equation of state becomes hard , an adiabatic core is formed , which is separated from the isothermal contracting pseudo - disk by the accretion shock front facing radially outwards . by the effect of the magnetic tension , the angular momentum is transferred from the disk mid - plane to the surface . the gas with excess angular momentum near the surface is finally ejected , which explains the molecular bipolar outflow . two types of outflows are observed . when the poloidal magnetic field is strong ( magnetic energy is comparable to the thermal one ) , a u - shaped outflow is formed in which fast moving gas is confined to the wall whose shape looks like a capital letter u. the other is the turbulent outflow in which magnetic field lines and velocity fields are randomly oriented . in this case , turbulent gas moves out almost perpendicularly from the disk . the continuous mass accretion leads to the quasistatic contraction of the first core . a second collapse due to dissociation of h in the first core follows . finally another quasistatic core is again formed by atomic hydrogen ( the second core ) . it is found that another outflow is ejected around the second atomic core , which seems to correspond to the optical jets or the fast neutral winds .
|
astro-ph0105527
|
r
|
in model a , we calculated the evolution with @xmath83 , @xmath84 , @xmath85 , and @xmath86 . we summarized the adopted parameters in table [ table2 ] . similar to the _ non - rotational _ magnetized cloud [ see figs . 2b and 2c of @xcite ] , the cylindrical cloud fragments into prolate spheroidal shape whose wavelength is equal to @xmath87 . this prolate spheroidal shape coincides with the structure expected from the linear stability analysis by @xcite . next , this density enhanced region begins to contract along the major axis of the cylindrical cloud , since the magnetic fields are assumed to run parallelly to the major axis . finally it forms a contracting disk ( pseudo - disk ) perpendicular to the magnetic field lines [ fig . 2d of @xcite ] . the snapshot at this stage is shown in figure [ fig1 ] . using the conversion factor shown in table[table1 ] , the epoch @xmath88 corresponds to @xmath89 from the beginning of calculation . respective panels of figure [ fig1 ] have different spatial coverage . figure [ fig1]a , which shows l1 , captures global structure of contracting disk extending horizontally which is perpendicular to the magnetic field lines . the spatial resolution of l1 is so limited that there seems no internal structures in the contracting disk . however , l5 which has 16 times finer resolution than l1 shows shock fronts facing outer directions extend parallel to the @xmath55-direction . fronts near @xmath90 are the fast - mode mhd shock front , because the magnetic fields bend toward the front passing the shock front . we can see another density jump near @xmath91 ( hereafter we will omit the sign @xmath92 and mention the upper half part of the figure since the structure is symmetric ) . the shock fronts parallel to the disk are known in non - magnetized rotating isothermal clouds @xcite . this is not due to the rotation ; multiple shock fronts are also found in the contracting magnetized cloud without rotation @xcite . however , situation becomes a bit complicated in this cloud . in the outer region @xmath93 , the magnetic field lines run almost vertically ( @xmath94 and @xmath67 ) . passing the mhd fast shock , in the intermediate region ( @xmath95 ) , the radial and toroidal components are amplified and the density increases compared to the outer region . finally after passing another front near @xmath96 , the toroidal component @xmath67 decreases . that is , since the magnetic field lines deflect departing from the front at the second front , this is the slow mhd shock . the density range captured in this panel is from @xmath97 to @xmath98 . it should be noticed that these shocks occurs in the isothermal gas . this phase is called `` run - away collapse , '' in which the central density ( @xmath99 ) increases greatly in a finite time - scale . figure [ fig1]c shows the structure captured by l10 . almost all the gas in this figure is isothermal . however , a central small part of the contracting disk @xmath100 and @xmath101 enters the adiabatic regime . at this stage , we can see another density jump is forming just outside the adiabatic part of the disk . it seems to grow into an accretion shock front , since it is known that an accretion shock forms outside the core when the adiabatic core develops @xcite . this is easily understood as follows : the adiabatic gas with specific heat ratio @xmath102 has a hydrostatic equilibrium irrespective of its mass . the scale - height in the @xmath54-direction of the core becomes larger than that of the disk . the adiabatic part of the contracting disk forms a spherical static core . figure [ fig2 ] shows the cross - cut view along two axes ( panel a : along the disk mid - plane @xmath103 and panel b : along the @xmath54-axis @xmath23 ) . the lines with 0.6066 represent the stage shown in figure [ fig1 ] . at this stage ( @xmath88 ) , inflowing gas is almost isothermal ( @xmath104 ) . in figure [ fig2]a , the radial distributions of the density , the magnetic flux density , and the radial and toroidal components of velocity are shown . we can see that density and magnetic flux density distributions are approximately expressed by power - laws as @xmath105 and @xmath106 except for the central part . at @xmath107 , the density in the core exceeds @xmath108 . and at @xmath109 , it reaches @xmath110 . at this stage , a radially outward - facing shock front is seen even inside the disk ; infall motion is abruptly decelerated and the density and magnetic flux density are amplified . this shows that a compact core is formed inside the accretion shock front . the central density increases with time and inside @xmath111 adiabatic gas ( @xmath112 ) distributes . the size of the core is equal to @xmath113 . this reduces with time since the mass of the core increases by the effect of continuous accretion . before the shock front is formed @xmath114 , the radial inflow velocity takes the maximum about @xmath115 near @xmath116 . for the larson - penston self - similar solution for the spherically symmetric dynamical collapse @xcite , this maximum inflow speed is expected equal to @xmath117 . on the other hand , it equals to @xmath118 for non - rotating isothermal _ disk _ @xcite . therefore , it is shown that the actual inflow speed ranges between the spherically symmetric self - similar solution and the axially symmetric thin disk solution . after the shock front is formed around the core , the inflow velocity takes the maximum just outside the shock front and the maximum speed increases with time . inflow motion is accelerated toward the shock front . similar acceleration is also seen in the toroidal velocity , @xmath11 . before the core formation the toroidal speed @xmath11 takes the maximum near @xmath119 . however , @xmath11 increases toward the accretion shock after core formation . at @xmath109 it reaches @xmath120 ( see also fig.1 of @xcite ) . structure seen in the cross - cut along the @xmath54-axis is more complicated ( fig.[fig2]b ) . two shock fronts mentioned earlier ( fig.[fig1]b ) correspond to the jumps near @xmath121 and @xmath122-axis it is found near @xmath123 , while departing from the @xmath54-axis ( @xmath124 ) it is found near @xmath125 . ] at @xmath88 , the density and the inflowing velocity distributions have no discontinuities besides these two shock fronts . however , at @xmath107 , the inflowing velocity distribution begins to indicate a clear discontinuity near @xmath126 . this is a newly formed shock front and propagate spatially . comparing two curves of @xmath107 and @xmath109 , it is shown that this shock front breaks into two fronts and the inner one ( @xmath126 ) is standing still , while another outer one ( @xmath127 ) is propagating outwardly . these two shock fronts are outwardly facing . thus the inwardly propagation of the inner fronts is due to the infalling gas motion . figure [ fig3 ] illustrates the structure at @xmath128 . although the gas is inflowing both inside and outside of the disk at @xmath107 ( fig.[fig2 ] ) , at this stage @xmath128 ( @xmath129 ) represents the time from the beginning of calculation but @xmath130 represents the time after the core formation . we assumed that the core consists of the gas with density @xmath112 . ] prominent outflow is formed . it is shown that the flow pattern is completely changed in @xmath131yr . outflow sweeps a sphere with radius of @xmath132 ( fig.[fig3]a ) . figure [ fig3]b indicates that the gas near the disk surface flows inwardly for @xmath133 . however , the direction of the flow is changed upwardly near @xmath134 . finally this gas is ejected . while the gas near the mid - plane of the disk ( @xmath135 ) continues to contract . this is reasonable because the total amount of angular momentum in one magnetic flux tube must be conserved in the axisymmetric ideal mhd simulation ; for the outflow gas to get angular momentum , a part of the gas in the same magnetic flux tube has to lose its angular momentum and falls further . in the acceleration process of the gas , the angular momentum is transferred from the gas near the mid - plane to the gas near the surface of the disk . considering the angular rotation speed , the angular momentum is transferred from the fast - rotating mid - plane to the slowly rotating surface gas . from figures [ fig1]c and [ fig3]a ( both show the structure captured by l10 ) , we can see the magnetic field lines run completely differently comparing before ( @xmath136 : fig.[fig1]c ) and after ( @xmath137 : fig.[fig3]a ) the adiabatic core formation . that is , in the isothermal runaway collapse phase ( fig.[fig1]c ) the magnetic fields lines run vertically , in other words , perpendicularly to the disk . in contrast , after the adiabatic core is formed , the disk continues to contract and drags the magnetic field lines inwardly . thus the angle between the magnetic field lines and the disk decreases . figure [ fig3]b is a close - up view whose spatial resolution is 4-times finer than that of figure [ fig3]a . this panel shows us that the angle between the flow and the disk is about @xmath138 . the reason why the outflow begins after the core formation is related to the angle between the magnetic field lines and the disk , @xmath139 . @xcite have pointed out that for a cold gas rotating with the keplerian speed to get angular momentum from the keplerian disk via infinitely strong magnetic fields , @xmath139 must be smaller than a critical value @xmath140 . this is understood as follows : consider the gas on one magnetic flux tube . when the magnetic flux tube rising steeply from the disk as @xmath141 , the gas has to climb the effective potential well even if it rotates with the same angular speed of the keplerian disk . on the other hand , when @xmath142 , gas can escape from the gravitational well by getting angular momentum from the disk , if the gas has the same angular speed of the keplerian disk . although the exact value of @xmath143 depends on the disk rotation speed and the disk - to - central star mass ratio , small angle is preferable to acceleration . it should be noted that this configuration is achieved only after the core formation @xcite . figure [ fig3]c shows the ratio of the toroidal magnetic component to poloidal one by contour lines , which is overlaid on the magnetic field lines . since the toroidal - to - poloidal ratio is as small as @xmath144 in the disk , the disk is poloidal - dominated . however , in the region where the gas flows outwardly the toroidal component grows and the toroidal - to - poloidal ratio reaches @xmath145 . the outflow gas is toroidal - field - dominated . the coincidence of acceleration region with the toroidal - dominant region seems to indicate that the toroidal fields play an important role to accelerate the gas . the toroidal component of the lorentz force , @xmath146 works below this toroidal - dominant region , that is , @xmath147 . this toroidal component @xmath148 accelerates the toroidal velocity @xmath11 and resultant toroidal motion amplifies the toroidal component of the magnetic fields . outflow speed exceeds the sound speed and the fastest speed reaches @xmath149 at this time . it increases with time . although the outflow seems to continue , the further evolution is hard to study , because the time - scale ( the free - fall time - scale at the central core ) becomes shorter and shorter . therefore , we study model ah with a constant polytropic index larger than that of model a. in models ah1 ( fig.[fig4]a ) and ah2 ( fig.[fig4]b ) , the polytropic indices are chosen @xmath150 and @xmath151 , respectively , for @xmath112 . ( models whose name have `` h '' have simple polytropic relation with @xmath150 or @xmath151 for @xmath112 . ) due to the hard polytropic index , the size of the adiabatic core , whose surface is determined by the jump in @xmath152 , becomes large ; for example at @xmath128 the size is equal to @xmath153 for model ah1 and @xmath154 for model ah2 while it is @xmath155 for model a. similar to model a , just outside the core , outflow begins to be accelerated . the region which the outflow sweeps expands and the surface which separates inflow and outflow forms another mhd shock front . and the expansion of the front is very similar to that of model a ( the front reaches @xmath156 at this time which is similar to model a ) . to see the similarity in more detail , we calculated the mass of the core @xmath157 for models a , ah1 , and ah2 . these are equal to @xmath158 , @xmath159 , and @xmath160 at the time @xmath128 ( @xmath129 ) for models a , ah1 and ah2 , respectively . at later epoch @xmath161 ( @xmath162yr ) , @xmath163 ( model ah1 ) and @xmath164 ( model ah2 ) . from these results , it is shown that the core mass increases with time due to the continuous accretion and the mass does not depend on the exact equation of state in the core . this is understood as follows : the core mass is determined by the accretion rate of the _ isothermal gas _ which is independent from the polytropic @xmath165 in the core . the gravity by the core has an effect on the outer inflow and outflow . since the effect depends only on its mass , the difference of the polytropic index of the core does not play an important role for the inflow and outflow . therefore , we will study this model ah to see the long time evolution of the outflow . in figure [ fig4]c , the snapshot at @xmath166 ( @xmath167 ) is plotted for model ah1 . comparing this with figure [ fig1]b for model a ( both have the same resolution but for different epochs ) , it is shown that the shock front which separates the inflow and the outflow passed the slow - mode mhd shock near @xmath168 and has just reached the outer fast - mode shock front near @xmath169 . the evolution of model ah2 is essentially the same . the maximum speed of the outflow reaches @xmath170 . this maximum speed seems smaller than that observed in the molecular outflow . since the mass accumulated in the core is only equal to @xmath171 , the outflow speed seems to be much faster than this value , when the mass has grown to typical t tauri stars . figure [ fig3]b indicates the outflow is accelerated near the core and the opening angle of the outflow in this region is wide . however , departing from the acceleration region the flow changes its direction toward the @xmath54-axis . figure [ fig4]c show that the opening angle decreases as the outflow proceeds . this indicates the flow is collimated . to see the effect of the initial rotation speed , we compare models ah1 ( @xmath172 ) , bh ( @xmath173 ) , and ch ( @xmath174 ) . these models have the same magnetic field strength , @xmath175 . in figure [ fig5 ] panels ( a - c ) , the structures at the final epoch of the isothermal run - away collapse phase are plotted for respective models . models bh and ch indicate no prominent discontinuity in l6 ( figs . [ fig5]b and c ) , while model ah1 has several shock fronts as described in @xmath7[sec3.1 ] . however , in l10 ( not shown ) , there are discontinuities near @xmath176 ( model bh ) and @xmath177 ( model ch ) as well as in model ah1 ( @xmath178 ) . comparing panels ( b ) and ( c ) , distributions of the density and magnetic field lines are similar each other . this indicates that the evolution in the isothermal phase is slightly dependent on the initial angular momentum if @xmath179 . figures [ fig5]d - f show the structure at the age @xmath180 after the core formation . in all models the outflows are formed . however , the size of the region swept by the outflow is different for each model . with increasing @xmath181 , more energetic outflow is driven . from the flow vectors , it is shown that model ah1 ( fig.[fig6]d ) forms a bit more collimated outflow than models bh and ch ( fig.[fig6]e and f ) . this seems to correspond to the differences in density distribution and magnetic field configuration . that is , in model ah1 ( also a and ah2 ) there is a relatively thick disk seen in l6 which is bounded by the shock fronts . this thick disk seems to confine the outflowing gas in model ah1 . while , in models bh and ch the disk is relatively thin , which seems to make the flow isotropic . further , the opening angle of the magnetic field lines in models bh and ch is larger than that of model ah1 . this causes the flow also open . difference between models bh and ch comes from the fact that the epochs when the outflow begins are different . since the initial angular momentum in model bh is five - times larger than that of model ch , in model bh the outflow begins earlier than model ch . at @xmath182 , however , even in model ch the top of the outflow reaches @xmath169 and the structure looks very similar to model bh at @xmath183 ( fig.[fig5]e ) . at the epoch when figure [ fig5]d - f is taken ( @xmath184^{-1/2}$ ] ) , the mass in the adiabatic core reaches @xmath185 ( model ah1 ) , @xmath186 ( model bh ) , and @xmath187 ( model ch ) . the instantaneous rate of mass accretion to the adiabatic core for each model attains @xmath188 ( model ah1 ) , @xmath189 ( model bh ) , and @xmath190 ( model ch ) , respectively . therefore , the core mass is approximately proportional to the mass accretion rate and the mass accretion rate is larger for models with smaller angular rotation speed @xmath181 . accretion rate expected from the inside - out collapse model @xcite is equal to @xmath191 . therefore , the accretion rates calculated here are @xmath192 times larger than that expected by the inside - out collapse model . consider the reason why the mass of the core decreases with increasing @xmath181 . since the gas is supplied to the core mainly through the disk , we will consider the mass inflow / outflow transported in the disk . at that time , the gas disk can be divided into three regions . outermost region is occupied with isothermal gas and the gas is contracting or inflowing ( pseudo - disk ) . therefore the inflow mass rate @xmath193 and the outflow mass rate @xmath194 in this outermost region . inside of this region , outflow is generated , although a large part of the gas is still inflowing . therefore in this middle region , the inflow rate is smaller than that of the outermost region , @xmath195 , and the excess mass is transported to the outflow , @xmath196 . innermost is the adiabatic core . since the mass accretion rate to the core is equal to the net mass inflow rate from the middle region , @xmath197 . mass inflow driven by the self - gravity is more important in a model with small @xmath181 in which the self - gravity is ineffectively counterbalanced with the centrifugal force . therefore @xmath198 becomes larger for slow rotator . this is the first effect of the rotation . furthermore , the outflow brings away appreciable amount of gas . as mentioned previously , the outflow is strongly generated in the fast rotator . thus , the mass outflow rate increases with increasing @xmath181 as @xmath199 ( model ah1 ) , @xmath200 ( model bh ) , and @xmath201 ( model ch ) . as a result , increasing @xmath181 , the portion of outflow gas to the inflow gas @xmath202 becomes large as @xmath203 40% for model ah1 , @xmath203 10% for model bh , and @xmath204 5% for model ch . these two effects works cooperatively to reduce the mass accretion rate @xmath205 to the core for the cloud with a large @xmath181 . the maximum outflow speed realized in respective figures are equal to @xmath206 ( model ah1 ) , @xmath207 ( model bh ) , and @xmath208 ( model ch ) . since the outflow is accelerated by the toroidal magnetic fields which are generated by the rotation motion , this @xmath209 increases with increasing @xmath181 . as shown in @xcite , since the excess angular momentum of the inflowing gas is effectively removed by the outflow , the total angular momentum of the first core which is defined as a gas with @xmath112 is equal to @xmath210 contained in the mass of @xmath211 ( model ah1 ) , @xmath212 in @xmath213 ( model bh ) , and @xmath214 in @xmath215 ( model ch ) . the total angular momenta contained in the first cores are only 1.1% , 0.07% , and 0.9% of the initial ones contained in respective mass @xmath216 . to see the effect of the magnetic field strength , we calculated models nh , dh and eh in which we took @xmath217 , @xmath218 , and @xmath219 . model nh has no magnetic fields . in figure [ fig6]a , a snapshot at @xmath220 captured by l8 is shown for model nh . at this stage , whole the cloud is in isothermal regime and the disk experiences the run - away collapse even if the centrifugal force may work to support the cloud . this confirms the earlier studies by @xcite and @xcite . the physical reason why the centrifugal force does not stop the contraction in the isothermal run - away collapse phase is explained in @xcite as follows : due to the the centrifugal force , the mass contained in the jeans scale ( @xmath221 ) from the center is _ decreasing _ throughout the collapse . in this sense the centrifugal force does work ! only small part of the mass that resides near the center becomes high - density . but contraction itself continues and the central density rises greatly in a finite time - scale , if the isothermal equation of state is valid . similar to the previous magnetized models , a small adiabatic core is formed first . since there is no magnetic fields , magnetic braking does not work , however , in this model . therefore , gas that accretes on the core has relatively large angular momentum in contrast to the magnetized model . as a result , a ring forms by the gas which accreted on the adiabatic core . since the specific angular momentum , @xmath222 , increases further with time , radius of the ring grows . another snapshot in panel ( b ) at @xmath223 ( @xmath224 ) shows the ring clearly . the ring seems unstable for non - axisymmetric perturbation . this may form a spiral structure similar to that found by @xcite . however this is beyond the scope of this paper . therefore , it is concluded that a rotating but non - magnetic cloud leads to a rotating ring in the adiabatic accretion stage . to see the effect of the magnetic field strength , in figure [ fig7 ] we compare models bh ( @xmath83 ) , dh(@xmath225 ) , and eh ( @xmath226 ) . all models have the same initial rotation speed @xmath227 , and polytropic gamma @xmath150 . in panels ( a)-(c ) , the structures at the epoch when an adiabatic core begins to form are plotted . comparing these panels , it is shown that in model b ( @xmath83 ) a flare - up disk is formed whose isodensity lines are departing from the disk mid - plane leaving from the center . decreasing the initial magnetic field strength , the shape of dense part of the disk becomes rounder . similar effect is already reported for non - rotating magnetized cloud collapse @xcite , that is , deceasing @xmath175 the shape of the isothermal contracting disk becomes rounder and finally reaches a sphere for @xmath217 . in panel ( d ) we plotted a snapshot for model bh at @xmath228 ( @xmath229 ) captured in l7 . this is the same as illustrated in figure [ fig5]e but captured in l7 which has twice as fine as figure [ fig5]e . figure [ fig7]d which shows the structure near the root of outflow indicates that it is very similar to that of model ah1 . for example , the outflow leaves from the disk in the direction almost parallel to the disk but it changes its direction to the @xmath54-direction . time @xmath228 is equivalent to @xmath229 . it is concluded that in a time - scale of @xmath230 , the flow pattern is completely changed from the run - away collapse to the outflow plus continuous inflow in the disk . the outflow gas flows along the surface whose shape resembles a capital letter u. the flow departs from the disk with a wide opening angle but it change its direction parallel to the @xmath54-axis . in panel ( e ) , we plotted the structure expected from a model with weaker magnetic field ( model dh : @xmath225 and @xmath227 ) . the snapshot corresponds to the epoch of @xmath231 . this corresponds to @xmath232 which is similar to the time - scale between panels ( a ) and ( d ) . in contrast to the previous model bh , the outflow gas forms a sphere and the magnetic field lines are folded inside this sphere . the magnetic field lines are folded by the pinch or hoop stress of the toroidal magnetic field . the toroidal magnetic field component is confined in the region where the adjacent poloidal magnetic field lines are running in the opposite directions . for example , the regions around @xmath233 and @xmath234 in figure [ fig7]e . in this model , the initial poloidal magnetic fields are weak compared to model bh . therefore , rotational motion amplifies the toroidal fields and their strength surpasses easily that of the poloidal ones . thus , the hoop stress by the toroidal field pinches efficiently the magnetic field lines . in the outflow acceleration region , the toroidal component is predominant over the poloidal one . magnetic field lines are pinched locally and folded . as a result , a spherical magnetic bubble is formed in this process , in which the toroidal magnetic field is predominant . toroidal component of the magnetic fields is continuously generated by the twist motion driven by the disk rotation . the disk angular momentum is transferred by this process . as a result , we do not see the ring of model nh which is supported by the centrifugal force . for the model with extremely weak field , we calculated model eh ( @xmath226 and @xmath235 ) . in panel ( f ) , we plotted the snapshot at @xmath236 ( @xmath237 ) . density distribution and magnetic field configuration show that the flow in the magnetic bubble is more complicated or turbulent in this model . the shape of the bubble is more elongated than that formed in model dh . distribution of toroidal field lines does not show any systematic pattern inside the bubble . size of the bubble both in the @xmath54- and @xmath55-directions are smaller than that of models bh and dh . it is concluded that the size of the outflow region increases with increasing the magnetic field strength @xmath175 . comparing these three models , it is shown that there are at least two types of outflows . that is , a laminar u - type flow in which fast moving gas flows along the surface whose shape resembles a capital letter u and a turbulent outflow in which the magnetic fields and the velocity are randomly oriented . the masses accumulated in @xmath238 are equal to @xmath239 for model bh , @xmath240 for model dh , and @xmath241 for model eh . this shows that the mass accretion rate , @xmath242 , is an increasing function of the initial magnetic field strength , @xmath175 . this seems strange if we remember @xmath242 is an decreasing function of the initial rotation speed , @xmath181 , since both @xmath175 and @xmath181 have an effect to counterbalance against the self - gravity . this means that the mass inflow rate in the isothermal run - away collapse region , @xmath198 , increases with increasing @xmath175 . this seems to come from a number of reasons , that is , the initial cylindrical cloud becomes massive with increasing @xmath175 . another reason is related to the characteristic wave speed in the magnetized medium . that is , the characteristic speed of the fast mode mhd wave is equal to @xmath243 in the case that the wave is propagating perpendicular to the magnetic field lines . this implies that the mass inflow rates are proportional to @xmath244 ( a : b : c = 2.83 : 1.15 : 1.02 ) . this is not inconsistent with the actual values .
|
, the angular momentum is transferred from the disk mid - plane to the surface . when the poloidal magnetic field is strong ( magnetic energy is comparable to the thermal one ) , a u - shaped outflow is formed in which fast moving gas is confined to the wall whose shape looks like a capital letter u. the other is the turbulent outflow in which magnetic field lines and velocity fields are randomly oriented . in this case , turbulent gas moves out almost perpendicularly from the disk .
|
collapse of the rotating magnetized molecular cloud core is studied with the axisymmetric magnetohydrodynamical ( mhd ) simulations . due to the change of the equation of state of the interstellar gas , the molecular cloud cores experience several different phases as collapse proceeds . in the isothermal run - away collapse ( ) , a pseudo - disk is formed and it continues to contract till the opaque core is formed at the center . in this disk , a number of mhd fast and slow shock pairs appear running parallelly to the disk . after the equation of state becomes hard , an adiabatic core is formed , which is separated from the isothermal contracting pseudo - disk by the accretion shock front facing radially outwards . by the effect of the magnetic tension , the angular momentum is transferred from the disk mid - plane to the surface . the gas with excess angular momentum near the surface is finally ejected , which explains the molecular bipolar outflow . two types of outflows are observed . when the poloidal magnetic field is strong ( magnetic energy is comparable to the thermal one ) , a u - shaped outflow is formed in which fast moving gas is confined to the wall whose shape looks like a capital letter u. the other is the turbulent outflow in which magnetic field lines and velocity fields are randomly oriented . in this case , turbulent gas moves out almost perpendicularly from the disk . the continuous mass accretion leads to the quasistatic contraction of the first core . a second collapse due to dissociation of h in the first core follows . finally another quasistatic core is again formed by atomic hydrogen ( the second core ) . it is found that another outflow is ejected around the second atomic core , which seems to correspond to the optical jets or the fast neutral winds .
|
astro-ph0105527
|
i
|
we have explored the evolution of a magnetized interstellar cloud rotating around the symmetric axis . following the change in the equation of state of the interstellar gas @xcite , the cloud experiences several phases before going to a star , that is , the isothermal run - away collapse , the slowly contracting core composed of the molecular hydrogen ( the first core ) , the second run - away collapse in the high - density gas where the dissociation of hydrogen molecules proceeds , and finally the second core which is made up of the atomic hydrogen . the magnetized cloud forms a pseudo - disk in these first and second run - away collapses , in which a supersonically contracting disk is formed and magnetic field lines are running perpendicularly to the disk . in the pseudo - disk , a number of fast- and slow - mode mhd shock pairs are formed which is extending parallelly to the disk . just after the core is formed at the center , an accretion shock front appears which surrounds the core , through which the supersonic inflow motion is decelerated . while the first and second cores are slowly contacting , the outer pseudo disk continues to contract . just outside the accretion shock front , the infall motion is accelerated and thus rotational motion becomes important from the conservation of angular momentum . by the effect of rotational motion , the toroidal magnetic fields and the poloidal currents are amplified , which bring a strong magnetic torque . the magnetic torque leads the angular momentum transfer from mid - plane to surface of the disk . this is the origin of the outflow found in star forming regions . large scale bipolar molecular outflows are made outside of the first core , while optical jets and fast neutral winds are expected to be accelerated outside of the second core . matter lost its excess angular momentum continues to contract further to form a star . this work was supported partially by the grants - in - aid ( 11640231 , 10147105 ) from mext ( the ministry of education , culture , sports , science and technology ) . numerical calculations were carried out by fujitsu vpp5000 at the astronomical data analysis center , the national astronomical observatory , an inter - university research institute of astronomy operated by mext , japan . [ [ section ] ] basu , s. , & mouschovias , t. ch . 1994 , , 432 , 720 basu , s. , & mouschovias , t. ch . 1995 , , 452 , 386 bentz , w. 1984 , , 139 , 378 berger , m.j . & colella , p. 1989 , j. comput . , 82 , 64 berger , m.j . & oliger , j. 1984 , j. comput . phys . , 53 , 484 bodenheimer , p. 1995 , , 33 , 199 bodenheimer , p. , tohline , j. e. , & black , d. c. 1980 , apj , 242 , 209 bontemps , s. , andre , p. , terebey , s. , cabrit , s. , 311 , 858 blandford , r. d. , & peyne , d. g. 1982 , , 199 , 883 ciolek , g.e . , & basu , s. 2000 , , 529 , 925 dorfi , e. 1982 , , 114 , 151 dorfi , e. 1989 , , 225 , 507 evans , c. r. , hawley , j. f. 1988 , , 332 , 659 fiedler , r. a. , & mouschovias , t. ch . 1992 , , 391 , 199 fiedler , r. a. , & mouschovias , t. ch . 1993 , , 415 , 680 gustafsson , i. 1978 , bit , 18 , 142 hayashi , c. 1980 , in `` star forming regions , '' ed . m. peimbert , & j. jugaku ( dordrecht : d.reidel ) , p.403 klein , r.i . , fisher , r.t . , mckee , c.f . , truelove , j.k . 1999 , in `` numerical astrophysics , '' ed . miyama , k. tomisaka , & t. hanawa . ( boston : kluwer academic ) , p.131 kompaneets , a.s . , soviet phys . dokl . , 5 , 46 larson , r.b . 1969 , , 145 , 271 matsumoto , t. , nakamura , f. , & hanawa , t. 1994 , , 46 , 243 matsumoto , t. , nakamura , f. , & hanawa , t. 1997 , , 478 , 569 masunaga , h. , & inutsuka , s. , 510 , 822 meijerink . j. a. , & van der vorst , h. a. 1977 , math . , 31 , 148 mouschovias , t. ch . , & morton s. a. 1991 , , 371 , 296 mouschovias , t. ch . , & morton s. a. 1992 , , 390 , 144 nakamura , f. , matsumoto , t. , hanawa , t. & tomisaka , k. 1999 , , 510 , 274 nakamura , f. , hanawa , t. & nakano , t. 1995 , , 444 , 770 nakano , t. , umebayshi , t. 1986 , , 218 , 663 narita , s. , hayashi , c. , & miyama , s. m. 1984 , prog . theor . phys . 72 , 1118 norman , m.l . , wilson , j.r . , & barton , r.t . 1980 , , 239 , 968 ogino , s. , tomisaka , k. , & nakamura , f. 1999 , , 51 , 637 ohashi , n. , lee , s.w . , wilner , d.j . , & hayashi , m. 2000 , , 518 , l41 penston , m.v . 1969 , , 144 , 425 phillips , g. l. 1986a , , 221 , 571 phillips , g. l. 1986b , , 222 , 111 saigo , k. & hanawa , t. , 493 , 342 scott , e. h. & black , d. c. 1980 , , 239 , 166 shu , f.h . 1977 , , 214 , 488 stodkiewicz , j.s . 1963 , acta astron . , 13 , 30 stone , j. m. & norman , m. l. , 80 , 753 tohline , j. e. 1982 , fundamentals of cosmic physics , 8 , 1 tomisaka , k. , ikeuchi , s. , & nakamura t. 1990 , , 362 , 202 tomisaka , k. 1995 , , 438 , 226 tomisaka , k. 1996 , , 48 , 701 tomisaka , k. 1998 , , 502 , l163 tomisaka , k. 2000 , , 528 l41 truelove , j. k. , klein , r. i. , mckee , c. f. , holliman , j. h. , ii , howell , l. h. , & greenough , j. a. 1997 , , 489 , l179 truelove , j. k. , klein , r. i. , mckee , c. f. , holliman , j. h. , ii , howell , l. h. , greenough , j. a. , & woods , d. t. 1998 , , 495 , 821 tsuribe , t. , & inutsuka , s. 1999a , , 523 , l155 tsuribe , t. , & inutsuka , s. 1999b , , 526 , 307 van leer , b. 1977 , 23 , 276 wood , d. 1982 , mnras , 199 , 331
|
collapse of the rotating magnetized molecular cloud core is studied with the axisymmetric magnetohydrodynamical ( mhd ) simulations . due to the change of the equation of state of the interstellar gas , the molecular cloud cores experience several different phases as collapse proceeds . in the isothermal run - away collapse ( ) , a pseudo - disk is formed and it continues to contract till the opaque core is formed at the center . in this disk , a number of mhd fast and slow shock pairs appear running parallelly to the disk . the continuous mass accretion leads to the quasistatic contraction of the first core . a second collapse due to dissociation of h in the first core follows .
|
collapse of the rotating magnetized molecular cloud core is studied with the axisymmetric magnetohydrodynamical ( mhd ) simulations . due to the change of the equation of state of the interstellar gas , the molecular cloud cores experience several different phases as collapse proceeds . in the isothermal run - away collapse ( ) , a pseudo - disk is formed and it continues to contract till the opaque core is formed at the center . in this disk , a number of mhd fast and slow shock pairs appear running parallelly to the disk . after the equation of state becomes hard , an adiabatic core is formed , which is separated from the isothermal contracting pseudo - disk by the accretion shock front facing radially outwards . by the effect of the magnetic tension , the angular momentum is transferred from the disk mid - plane to the surface . the gas with excess angular momentum near the surface is finally ejected , which explains the molecular bipolar outflow . two types of outflows are observed . when the poloidal magnetic field is strong ( magnetic energy is comparable to the thermal one ) , a u - shaped outflow is formed in which fast moving gas is confined to the wall whose shape looks like a capital letter u. the other is the turbulent outflow in which magnetic field lines and velocity fields are randomly oriented . in this case , turbulent gas moves out almost perpendicularly from the disk . the continuous mass accretion leads to the quasistatic contraction of the first core . a second collapse due to dissociation of h in the first core follows . finally another quasistatic core is again formed by atomic hydrogen ( the second core ) . it is found that another outflow is ejected around the second atomic core , which seems to correspond to the optical jets or the fast neutral winds .
|
1509.01985
|
c
|
the popular density - matrix functionals , the mller functional@xcite , the hartree - fock approximation and the power functional@xcite , which continuously interpolates between the other two , have been benchmarked for the hubbard dimer . the hartree - fock approximation is , for the hubbard model@xcite , analogous to hybrid density functionals@xcite , that admix a portion of exact exchange to the exchange - correlation energy . the local interaction of the hubbard model acts analogous to the range separation@xcite , which suppresses the long - ranged coulomb interaction in the fock term . in this respect , the hartree - fock approximation also captures the main effects of the lda+u method@xcite . particular emphasis has been given to left - right correlation , the dominant correlation effect for bond dissociation , which is not captured in local density functionals@xcite . left - right correlation describes that electrons localize on opposite sites of the dimer . this electron correlation , which increases with the interaction strength , avoids the energetic cost of the coulomb repulsion due to double occupancy of a site . in the hartree - fock approximation , this left - right correlation leads to an antiferromagnetic state with a spin - up electron mostly localized on one side and the spin - down electron on the other . this so - called broken - symmetry state disagrees with the exact solution , which is a singlet state , having no local moments , but nevertheless antiferromagnetic correlations similar to the broken symmetry state . more importantly , however , the antiferromagnetic transition is an abrupt one and not a continuous buildup of antiferromagnetic correlations as in the exact solution . the result is a qualitatively incorrect shape of the total energy during bond dissociation . the mller functional@xcite establishes left - right correlation in a fundamentally different manner : while the natural orbitals are mostly in the hubbard dimer exactly independent of the interaction , the occupations become fractional , which reflects the creation of electron - hole pairs that screen the interaction . one of the main successes of the mller functional besides being able to produce fractional occupations correctly , is that it captures the continuous nature of the transition to the left - right - correlated state . our calculations avoid any bias and allow for arbitrary non - collinear spin - polarized states . this strategy shall bring all potential problems to the surface , that would be present in large scale electronic structure calculations using these density - matrix functionals . our first observation is that the ground state for the mller functional , which does not have local moments , is degenerate with a one dimensional manifold of ferromagnetic states . thus the dimer has infinite magnetic susceptibility when described with the mller functional , in contrast to the vanishing zero - temperature susceptibility of the exact solution of the hubbard dimer . this large magnetic polarizability is likely to cause severe problems in extended electronic structure calculations . when turning to the power functional@xcite , we find that the system behaves analogous to the mller functional for small interactions , while it exhibits a transition to a hartree - fock - like antiferromagnetic state for large interactions . the critical interaction , where this transition occurs , drops rapidly with increasing @xmath60 from infinity in the mller functional to the hartree - fock value @xmath194 . our calculations indicate a major deficiency in the description of magnetic properties for this class of density - matrix functionals . the problems persist in modified form also for more general hamiltonians , which include off - site coulomb interactions , and for more extended systems . besides the bond - dissociation problem , we investigated the derivative discontinuity@xcite with changing the number of electrons . a balanced description of the electron affinity and ionization potential is essential for a qualitatively correct description of charge transfer . we find that the metal - like behavior of the mller functional persists : the discontinuity of the exchange - correlation energy even offsets the one of the kinetic energy . the mller functional describes the hubbard dimer with vanishing fundamental gap . the power functional inherits many of the problems of the mller functional : there is no derivative discontinuity in the entire parameter range of the power functional except for the hartree - fock limit . in the low - interaction regime the solutions are weakly ferromagnetic . like the hartree - fock approximation , the power functional exhibits an artificial abrupt magnetic transition with increasing interaction towards an antiferromagnetic configuration , albeit at a larger critical interaction . these states are intrinsically non - collinear . the absence of any derivative discontinuity also for insulating materials is expected to produce an artificial charge transfer between the constituents of large - scale electronic structure calculations . this cast severe doubt on the performance of such density - matrix functionals for complex systems . while the power functional lacks a derivative discontinuity , its chemical potential undergoes a continuous transition between two linear functions , which has been exploited to extract a band gap from data obtained further away from the integer particle number@xcite . our calculations indicate , however , that the band gap obtained from this extrapolation can be tuned by the free parameter @xmath60 of the power functional between zero and the hartree - fock result . the band gap opens in non - collinear calculations only when in the antiferromagnetic regime , while it vanishes in the mller - type regime at low interactions . the opening of a band gap obtained by the extrapolation method and its tunability are features that persist in non - magnetic calculations , while the gap opens at a larger value of the power parameter than in the magnetic calculation . these problems or signatures of them can be observed in previous calculations@xcite . the tunability of the band gap is similar to other methods such as lda+u@xcite and hybrid density functionals@xcite . however , the latter methods exhibit a true derivative discontinuity and their band gap does not shrink below the kohn - sham band gap , which is analogous to the non - interacting band gap of the hubbard dimer . approximations for ionization potentials @xcite and spectral functions@xcite have been introduced on top of rdmft . the latter method on the one hand yields spectra that agree well with experimental results for transition metal oxides@xcite for particular choices of the power functional parameter . on the other hand investigations on the hubbard dimer@xcite suggest caution and claim that the underlying physics is not correctly treated . the problems presented here demonstrate potential fundamental flaws of the class of density - matrix functionals of this study . we hope that this study provides a useful reference point for the development of new density - matrix functionals . we believe furthermore that our findings call for new approaches for the construction of density - matrix functionals that make closer contact to the many - particle description of the electronic system@xcite . 56ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrev.136.b864 [ * * , ( ) ] link:\doibase 10.1103/physrev.140.a1133 [ * * , ( ) ] link:\doibase 10.1039/b907148b [ * * , ( ) ] http://stacks.iop.org/1402-4896/2004/i=t109/a=001 [ * * , ( ) ] link:\doibase 10.1021/cr200107z [ * * , ( ) ] link:\doibase 10.1103/physrevb.30.4734 [ * * , ( ) ] link:\doibase 10.1103/physrevb.44.943 [ * * , ( ) ] \doibase doi:10.1103/revmodphys.68.13 [ * * , ( ) ] \doibase doi:10.1080/00018730701619647 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.78.865 [ * * , ( ) ] link:\doibase 10.1103/physrevb.57.6896 [ * * , ( ) ] \doibase doi:10.1088/1367 - 2630/16/9/093034 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.101.066403 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.10.159 [ * * , ( ) ] link:\doibase 10.1098/rspa.1963.0204 [ * * , ( ) ] \doibase doi:10.1143/ptp.30.275 [ * * , ( ) ] link:\doibase 10.1103/physrevb.12.2111 [ * * , ( ) ] http://www.pnas.org/content/76/12/6062.abstract [ * * , ( ) ] link:\doibase 10.1103/physrevb.84.205101 [ * * , ( ) ] link:\doibase 10.1103/physrevb.88.205139 [ * * , ( ) ] link:\doibase 10.1103/physrev.118.1417 [ * * , ( ) ] \doibase doi:10.1016/0375 - 9601(84)91034-x [ * * , ( ) ] link:\doibase 10.1103/physrevlett.81.866 [ * * , ( ) ] \doibase doi:10.1063/1.1906203 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.201103 [ * * , ( ) ] link:\doibase 10.1103/physreva.77.032509 [ * * , ( ) ] http://dx.doi.org/10.1140/epjd/e2012-30442-4 [ * * , ( ) ] link:\doibase 10.1103/physreva.79.040501 [ * * , ( ) ] http://scitation.aip.org/content/aip/journal/jcp/140/16/10.1063/1.4871875 [ * * , ( ) ] \doibase doi:10.1063/1.4926327 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.110.116403 [ * * , ( ) ] http://stacks.iop.org/0953-8984/27/i=39/a=393001 [ * * , ( ) ] link:\doibase 10.1103/physrev.97.1474 [ * * , ( ) ] link:\doibase 10.1103/revmodphys.35.668 [ * * , ( ) ] link:\doibase 10.1002/qua.560240302 [ * * , ( ) ] link:\doibase 10.1103/physreva.92.052514 [ * * , ( ) ] link:\doibase 10.1103/physrevb.54.16533 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.87.133004 [ * * , ( ) ] link:\doibase 10.1080/00268970110070243 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.55.2471 [ * * , ( ) ] \doibase doi:10.1016/0021 - 9991(77)90098 - 5 [ * * , ( ) ] link:\doibase 10.1103/physrev.101.1730 [ * * , ( ) ] \doibase doi:10.1016/0375 - 9601(72)90086 - 2 [ * * , ( ) ] link:\doibase 10.1103/physrevb.10.3626 [ * * , ( ) ] link:\doibase 10.1103/physreva.79.022504 [ * * , ( ) ] link:\doibase 10.1524/zpch.2010.6118 [ * * , ( ) ] http://stacks.iop.org/0295-5075/77/i=6/a=67003 [ * * , ( ) ] link:\doibase 10.1021/acs.jctc.5b00661 [ * * , ( ) ] , http://stacks.iop.org/1367-2630/17/i=9/a=093038 [ * * , ( ) ] link:\doibase 10.1063/1.464304 [ * * , ( ) ] link:\doibase 10.1002/qua.560340811 [ * * , ( ) ] \doibase doi:10.1063/1.1564060 [ * * , ( ) ] \doibase doi:10.1103/physrevlett.49.1691 [ * * , ( ) ] link:\doibase 10.1103/physrevb.78.201103 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.cplett.2005.06.103 [ * * , ( ) ]
|
common density - matrix functionals , the mller and the power functional , have been benchmarked for the half - filled hubbard dimer , which allows to model the bond dissociation problem and the transition from the weakly to the strongly correlated limit . the power functional actually favors the ferromagnetic state for weak interaction .
|
common density - matrix functionals , the mller and the power functional , have been benchmarked for the half - filled hubbard dimer , which allows to model the bond dissociation problem and the transition from the weakly to the strongly correlated limit . unbiased numerical calculations are combined with analytical results . despite the well known successes of the mller functional , the ground state is degenerate with a one - dimensional manifold of ferromagnetic solutions . the resulting infinite magnetic susceptibility indicates another qualitative flaw of the mller functional . the derivative discontinuity with respect to particle number is not present indicating an incorrect metal - like behavior . the power functional actually favors the ferromagnetic state for weak interaction . analogous to the hartree - fock approximation , the power functional undergoes a transition beyond a critical interaction strength , in this case however , to a non - collinear antiferromagnetic state .
|
hep-th9805123
|
i
|
the advance in the ability to analyze strong coupling effects in supersymmetric field theories @xcite led to a verification of some qualitative ideas about the behavior of gauge field theories . in particular , confinement in @xmath0 supersymmetric qcd ( sqcd ) was shown @xcite to be related , as expected , to monopole condensation . in a confining phase , the electric field is confined to flux tubes strings carrying a definite energy density ( string tension ) . electrically charged sources are connected by such strings and this leads to a confining force between them , , a force that does not vanish as the distance between them is increased . an explicit demonstration of such a situation , for @xmath0 sym theory was suggested recently , in the framework of a realization of gauge theories in m theory on @xmath4 , which is dual to type iia string theory . the 4d gauge theory is realized as the dimensionally reduced low energy effective field theory on an m5 brane wrapped on a riemann surface . when all the characteristic distances , including the radius of @xmath5 , are large with respect to the planck length , the semi - classical approximation is used @xcite . strictly speaking , the effective gauge theory is identified when the radius @xmath6 of @xmath5 is vanishingly small , corresponding to perturbative type iia string theory . as explained in @xcite , a change in @xmath6 has , in general , an effect on the world - volume field theory , so the results that one obtains for large @xmath6 are for a theory which can be different from the gauge theory one attempts to study . however , there are indications that these theories are in the same universality class and , therefore , have the same qualitative properties . a weaker expectation would be that qualitative confining features , will be shared by these theories . it is those features which we wish to study in this paper . a candidate for flux tubes in @xmath0 sym theory was suggested in @xcite : an m2 brane with one spatial dimension extended in a physical direction and the other in an internal direction , extending between points on the m5 brane . this is called _ an mqcd string _ ; it was explored further in @xcite . considering the realization of @xmath0 sqcd supersymmetric @xmath7 gauge theory with fundamental quarks the quark states were also identified as m2 branes ending on the m5 brane , these with their full extension in internal space , and it was shown that when confinement is expected , these quarks can not exist in isolation and must be either grouped in multiples of @xmath8 , forming baryons , or connected to mqcd strings , forming mesons . moreover , for the @xmath9 model , weakly broken to @xmath0 by a mass for the adjoint , the authors of @xcite identified @xmath10 types of strings with different tensions and reproduced the field theoretical results of @xcite . the string connecting two oppositely charged sources , carries flux which is determined by the sources . in the absence of dynamical matter , this flux protects the string from breaking , by charge conservation . however , when there is dynamical matter carrying an appropriate charge , a pair may be created from the vacuum , cutting the string in two . physically , this means that the potentially confining force between the external sources is screened by the dynamical matter . because of this screening , the string itself is expected to be charged only under those elements of the gauge group that act _ trivially _ on the dynamical fields . in particular , when there are no such group elements , one does not expect stable strings . instead , the force between any external sources is expected to vanish as the distance between them increases . this leads to a distinction between two physically different possibilities , depending on the algebraic structure of the matter sector . when the dynamical matter can screen any external charge , , when there are no group elements acting trivially on the dynamical fields , the forces felt by these charges are qualitatively the same as in the higgs phase and there is no actual phase boundary between the higgs and confining phases ; the corresponding branches are smoothly connected @xcite . on the other hand , when there are external charges that can not be screened by the dynamical matter , the force between them is qualitatively different in the higgs and confining branches and , therefore , these branches are expected to be separated by a boundary . to determine which of these possibilities is realized , one should find the subgroup @xmath11 of the gauge group that acts trivially on the dynamical fields . confinement and higgs phases are expected to be distinct iff @xmath11 is non - trivial . a realization of the subgroup @xmath11 was suggested in @xcite . considering @xmath0 sym theory , @xmath11 was identified as the homology group of the mqcd string and this group was , indeed , shown to be isomorphic to @xmath12 . in this work we elaborate on the geometric manifestation of the kinematic algebraic screening considerations , implied by the above identification of @xmath11 : the confining phase should be distinct from the higgs phase , when the later exists , iff there is a stable mqcd string , carrying a non - trivial @xmath11 charge . to investigate confinement and screening , it is useful to realize the external probes as dynamical , but very heavy , particles , as is done , for example , in @xcite . at energies small compared to the mass of the probes , when the string connecting them is not too long , it is energetically protected from being cut by a pair of the massive particles . therefore , such a string probes the model without the heavy particles : if in this model confinement and higgs phases are distinct , this will be manifested by the stability of the string . when the string is long enough , it should become unstable . in the present work we use m theory to study confinement and screening for systems with various local and global symmetries . we identify the mqcd string , find its conserved charge the homology group and compare to the field - theoretical expectation the subgroup @xmath11 . we then introduce external quarks and demonstrate confinement , when they are heavy and screening , when they are light . all our results are in agreement with the field theoretical analysis . the outline of this work is as follows . in section [ sec - su ] we consider @xmath7 models . we start with @xmath0 sqcd , the model considered in @xcite , as the simplest example , to explain the relevant concepts and methods . we then add an adjoint matter field with a polynomial superpotential . section [ sec - lsu ] is devoted to models with products of @xmath7 gauge factors and section [ sec - o4 ] , to models with @xmath13 and @xmath14 gauge groups . the @xmath15 and @xmath16 models are realized here by configurations that include an orientifold 4-plane , and we find ourselves on an unpaved way as far as brane dynamics is involved . guided by field theoretical expectations , we suggest some rules for m2 brane configurations in the presence of the orientifold . these rules are then used to obtain predictions for the confining behavior of all the @xmath15 and @xmath16 models considered . we end in section [ sec - disc ] with a summary and concluding remarks . in the appendix we prove some properties of m2 branes used in section [ sec - o4 ] .
|
the electric flux tubes are identified as m2 branes ending on the m5 branes and the conserved charge they carry is identified as a topological property . the group of charges carried by the flux tubes is calculated and the results agree in all cases considered with the field theoretical expectations . in particular , whenever the dynamical matter is expected to screen the confining force , this is reproduced correctly in the m theory realization .
|
confinement and screening are investigated in susy gauge theories , realized by an m5 brane configuration , extending an approach applied previously to sym theory , to other models . the electric flux tubes are identified as m2 branes ending on the m5 branes and the conserved charge they carry is identified as a topological property . the group of charges carried by the flux tubes is calculated and the results agree in all cases considered with the field theoretical expectations . in particular , whenever the dynamical matter is expected to screen the confining force , this is reproduced correctly in the m theory realization . ri-5 - 98 + cern - th-98 - 184 + hep - th/9805123 + 255= 255 by 60 255 by-60 255 by // + + shmuel elitzur , oskar pelc and eliezer rabinovici + _racah institute of physics , the hebrew university + jerusalem , 91904 , israel _ + and + _ theory division , cern + ch-1211 geneva 23 , switzerland _ + e - mail : elitzur@vms.huji.ac.il , oskar@shum.cc.huji.ac.il , + eliezer@vxcern.cern.ch +
|
hep-th9805123
|
i
|
in this work we used the realization of supersymmetric gauge theories in m theory to investigate confinement and screening . we considered the following ( @xmath0 supersymmetric ) models : * @xmath9 susy @xmath7 gauge theory with @xmath20 fundamental flavors , perturbed by a polynomial superpotential for the adjoint hypermultiplet ( @xmath0 sqcd can be seen as a special case of this family , for which the adjoint hypermultiplet is infinitely heavy ) ; * @xmath9 susy @xmath111 gauge theory with matter in fundamental and bi - fundamental representations , perturbed by masses for the adjoint hypermultiplets ; * @xmath14 gauge theory with @xmath20 fundamental hypermultiplets ; * @xmath13 gauge theory with @xmath20 vectorial hypermultiplets . for all the above models , we found results in agreement with the field theoretical expectations . starting without fundamental matter ( vectorial , for @xmath13 ) , the mqcd string ( , a candidate for the field theoretical electric flux tube ) was identified as an m2 brane ending on the m5 brane ( following @xcite ) . this string was shown to carry a topologically - conserved charge under a group isomorphic to the group @xmath11 of gauge transformations that act trivially on the fields in the field theory . this is in accord with the identification of this charge as the gauge charge that is _ not screened _ by the dynamical matter and , therefore , confined . we then introduced heavy quarks carrying unscreened charge and showed that they can not exist in isolation , but can be connected to an mqcd string , forming neutral mesons . this is a demonstration of confinement . finally , we considered higher energy scales , at which these quarks become dynamical and found that this destabilizes the mqcd string , demonstrating the screening of the charge that these quarks carry . in the @xmath13 models , we did not obtain strings carrying spinorial charge : for example , for @xmath184 , the mqcd string carries a @xmath172 charge ( which was shown to correspond to the vector representation ) , while the field theory expectation is for a charge group @xmath11 with 4 elements . the fact that the charge group was at most @xmath172 , followed from the topology of the internal space @xmath30 ( as implied by the o4 projection ) , and was independent of the m5 brane . this can serve as guidance in the search for a realization of models with spinors : the presence of heavy charge , since the vector screens the rest . ] spinors and vectors would imply the existence of a @xmath189 or @xmath198 charge group and to obtain such a charge , a different background is needed ( , larger orientifold group ) . a different configuration of ns5 branes and d4 branes would correspond to a different m5 brane in _ the same background _ , so it will not lead to the desired result . the above considerations demonstrate that the properties of the mqcd string can be used in the identification of the field theoretical model realized by an m5 configuration , or serve as a consistency check . it is , therefore , worthwhile to perform this analysis in other such realizations . * acknowledgment : * we are grateful to a. giveon , a. hanany , m. henningson , k. landsteiner , r. livne , e. lopez , y. oz and a. zaffaroni for helpful discussions . o.p . is grateful to the theory division at cern , where part of this work was performed , for hospitality . this work is supported in part by bsf american - israel bi - national science foundation , and by the israel science foundation founded by the israel academy of sciences and humanities centers of excellence program .
|
ri-5 - 98 + cern - th-98 - 184 + hep - th/9805123 + 255= 255 by 60 255 by-60 255 by // + + shmuel elitzur , oskar pelc and eliezer rabinovici + _racah institute of physics , the hebrew university + jerusalem , 91904 , israel _ + and + _ theory division , cern + ch-1211 geneva 23 , switzerland _ + e - mail : elitzur@vms.huji.ac.il , oskar@shum.cc.huji.ac.il , + eliezer@vxcern.cern.ch +
|
confinement and screening are investigated in susy gauge theories , realized by an m5 brane configuration , extending an approach applied previously to sym theory , to other models . the electric flux tubes are identified as m2 branes ending on the m5 branes and the conserved charge they carry is identified as a topological property . the group of charges carried by the flux tubes is calculated and the results agree in all cases considered with the field theoretical expectations . in particular , whenever the dynamical matter is expected to screen the confining force , this is reproduced correctly in the m theory realization . ri-5 - 98 + cern - th-98 - 184 + hep - th/9805123 + 255= 255 by 60 255 by-60 255 by // + + shmuel elitzur , oskar pelc and eliezer rabinovici + _racah institute of physics , the hebrew university + jerusalem , 91904 , israel _ + and + _ theory division , cern + ch-1211 geneva 23 , switzerland _ + e - mail : elitzur@vms.huji.ac.il , oskar@shum.cc.huji.ac.il , + eliezer@vxcern.cern.ch +
|
1406.1996
|
r
|
as pointed out in the introduction ( sec.[sec : introduction ] ) , the confinement can modify the structure of a fluid resulting in an inhomogeneous density profile . in a slit pore geometry , near the confining walls , particles form layers parallel to the walls as the temperature is decreased or the density is increased , as shown by our calculations for the density profile @xmath101 ( fig . [ fig : density ] ) . to establish the aggregation state for each layer , we compute , layer by layer , the lateral radial distribution function @xmath102 ; the 2d voronoi tessellation of each layer ; the mean square displacement ( msd ) and the survival probability ( sp ) of molecules in each layer . all these quantities together , as we discusse in the following , allow us to identify the presence and coesistence of the solid , heterogeneous fluid and homogeneous fluid . our analysis shows that the system organizes forming layers parallel to the solvophilic wall at any temperature @xmath18 and average density @xmath53 . at high @xmath18 and low @xmath53 we find only one well defined layer and no layering near the solvophobic wall , while the whole system is in the fluid state . by decreasing @xmath18 and increasing @xmath53 , the number of layers increases up to eleven . furthermore , at higher value of @xmath53 , the layers appear also near the solvophobic wall . however , for high enough @xmath18 and @xmath53 we observe that away from the walls the system is in the fluid state . nonetheless , the walls affect the density profile , changing system density and aggregation state , over an extension that in our case can be up to @xmath103 layers that depends on temperature and average density of the system . the in - layer radial distribution function @xmath104 for the @xmath105-th layer ( with @xmath106 ) is computed as @xmath107\ ] ] where @xmath108 and @xmath109 are the density of particles and the z - coordinate of the layer @xmath105 , respectively , @xmath110 is the transverse distance between two particles in the same layer ( and in internal units is @xmath111 ) . the heaviside step functions , @xmath69 , select couple of particles that lie in the layer @xmath105 of width @xmath112 . the @xmath104 is proportional to the probability of finding a molecule in the layer @xmath105 at a distance @xmath113 from a randomly chosen molecule of the same layer @xmath105 . the definition of the layer @xmath105 in which lies a particle @xmath67 is once for all established according to the value of the z - coordinate of the particle @xmath67 as : for the first layer @xmath114 , with @xmath115 , and @xmath116 for the others . this is a natural choice because particles at low temperatures or high densities tend to stratify in layers whose interdistance is approximately equal to @xmath117 . at low density ( @xmath118 , fig.[fig : gr_rho011 ] ) , we observe that the system is in a fluid state for high temperature ( @xmath119 ) in any layer . the @xmath104 of the layer @xmath120 shows a first peak around the shoulder radius ( @xmath121 ) and a second peak around the attractive well radius ( @xmath122 ) , while for the other layers only the second peak is present . we interpret this difference between the first and the other layers as the consequence of a `` templating '' effect of the solvophilic wall that at high @xmath123 is observed only on the first layer . at @xmath124 the system shows the same behavior observed for higher temperatures , except that the layer near the solvophilic wall develops patterns . this behavior is reminiscent of what has been observed in monolayers with an interparticle potential composed by a hard core and a soft repulsive shoulder @xcite . at @xmath125 the layer near the solvophilic wall is still showing patterns , while the layer @xmath126 is forming crystal patches . this is evident from the analysis of the @xmath127 that goes to zero for @xmath128 and @xmath129 at @xmath125 , consistent with an incipient triangular crystal with lattice step given by the interaction potential attractive distance @xmath130 . the other layers are in a fluid state . at @xmath131 the first layer is forming a hexagonal crystal . although the hexagonal crystal in @xmath120 does not overlap exactly with the triangular wall structure , the comparison of the @xmath132 and @xmath133 of the wall shows a strong correlation between the two structures , suggesting a `` templating '' effect . this effect due to the attraction to the solvophilic wall is so strong at @xmath131 that is forcing particles to be at their repulsive distance . the high - energy cost of the resulting honeycomb lattice forming in the layer @xmath120 is compensated by the large number of attractive interactions between the particles at @xmath120 and those of the wall ( @xmath134 ) from one hand , and between the particles themeselves at @xmath120 from the other hand . this free energy minimization process is analyzed in sec.[sec : structural ] in the discreet potential approximation to understand the stripe phase formation . the triangular structure that was incipient for @xmath126 at high @xmath123 , for @xmath131 is well defined for @xmath126 and @xmath135 , with defects in the layer @xmath135 . this triangular structure is the dual lattice of the @xmath120 hexagonal layer and its formation is the consequence of a `` molding '' effect of the layer @xmath120 onto the layer @xmath126 . note that while the wall ( @xmath134 ) layer has a templating effect on the @xmath120 layer , the @xmath120 layer has a molding effect on the @xmath126 layer . the difference between the two cases is due to the smaller density of the @xmath120 layer with respect to that of the wall . the smaller density does not allow to compensate the high energy cost of the propagation of the hexagonal crystal to the layer @xmath126 . on the other hand , the triangular crystal of the @xmath126 layer is energetically favorable , because the particles are all at the attractive distance , and at this temperature can propagate to the @xmath135 layer again with a `` templating '' effect . the layer @xmath136 is made of a few triangular crystallites immersed in the fluid , while the other layers are in a fluid state . at @xmath137 both the templating and the molding effect are stronger . in particular the template of the @xmath126 layer propagates over all the six layers that are formed at this density and temperature . at intermediate density ( @xmath138 , fig.[fig : gr_rho022 ] ) , for @xmath119 and @xmath124 we observe the same qualitative behavior as for the low density case . for @xmath125 the layer @xmath120 has less tendency to form patterns respect to the low density case , and the layer @xmath126 to order in a crystal structure . therefore , the confined system is more fluid at this density than at lower density . we understand this result as a consequence of the larger hydration at higer density . at @xmath131 the first layer has partially crystallized in the hexagonal and partially in the triangular structure following the template of the wall . therefore , the templating effect is now stronger then the corresponding case at lower density . the hexagonal crystal shows now a preferred direction of symmetry . this direction propagates to the layer @xmath126 , where we observe stripes along the preferred direction . the stripes propagate up to @xmath136 layer , while the other layers are in a fluid state . the peak of @xmath104 at @xmath139 ( that corresponds to the average second nearest neighbor distance ) is a signature of the stripe phase formation . at @xmath137 the preferred direction in the deformation of the hexagonal crystal for @xmath120 is more evident and we observe a clear stripe phase for the layers from @xmath126 to @xmath136 , with a peak of @xmath104 at @xmath139 more pronounced than the case at @xmath131 . the other layers form a triangular crystal at the attractive distance . at high density ( @xmath140 , fig.[fig : gr_rho030 ] ) , for @xmath119 we observe the same qualitative behavior as for the lower density cases . at @xmath124 we found that the only difference with the lower density case is that the layer @xmath120 is forming crystallites following the template of the wall . at @xmath125 the layer @xmath120 has a different and incipient crystal structure ( kagome lattice ) with defects that is better defined at lower @xmath18 . this is evident from the analysis of the @xmath132 that goes to zero for @xmath121 and @xmath141 at @xmath125 . the layer @xmath126 shows patterns very close to the stripe configuration . the corresponding @xmath127 goes to zero for @xmath142 and shows a peak for @xmath139 . these characteristics of the @xmath143 are typical of a stripe phase . the layer @xmath135 and @xmath136 still show patterns close to the stripe phase , but in a less pronounced way . from the layer @xmath144 to the @xmath145 the pattern is vanishing . the layer @xmath146 is showing an incipient triangular crystal with lattice step given by the interaction potential attractive distance @xmath130 . this is evident from the analysis of the @xmath147 that approaches zero for @xmath128 and @xmath129 at @xmath125 . at @xmath131 the layer @xmath120 is forming a kagome crystal with defects . the layers from @xmath126 to @xmath145 show a stripe phase and the layer @xmath146 is forming a triangular crystal with defects . at @xmath137 the kagome crystal of layer @xmath120 has no defects . the layers from @xmath126 to @xmath145 are in a stripe phase and the layer @xmath146 is forming a well defined triangular crystal . as discussed above , the layer @xmath120 close to the solvophilic wall ( @xmath134 ) is subjects to the templating effect for all densities at low temperatures . in fig.[fig : layers_gr_snap ] we compare the @xmath148 and the snapshots of the first layer for @xmath137 at several densities . in order to compare layers , that correspond to different densities , between them , we considered a portion of each layer of the same size ( @xmath48x@xmath49 of the system at @xmath140 ) . for densities @xmath149 the first layer is forming a distorted hexagonal lattice characterized by a @xmath148 with a first peak at @xmath150 and vanishing for @xmath121 and @xmath141 . by increasing @xmath53 we observe a progressive shift to higher values of @xmath113 of all the peaks of @xmath148 , but the first that , instead , is becoming more pronounced as a consequence of a better local order . for densities between @xmath151 and @xmath152 the first layer shows a polycrystal phase with coexistence of triangular and square lattices . this corresponds to an intermediate stage toward the well defined kagome lattice that is formed for @xmath153 , as showed by the splitting of the second peak of @xmath148 into two close peaks at @xmath154 and @xmath155 . in order to characterize space - dependent diffusion properties of our system , we compute the mean square displacement ( msd ) associated to each layer of the slit . we observe that , except for low temperatures , a particle can visit different layers in which the aggregation state can change from homogeneous to heterogeneous liquid and vice versa . for this reason we calculate the msd only for those particles that remain in a layer over the entire time interval under consideration and we average over all possible time interval . therefore , the msd associated to each layer @xmath105 is defined as @xmath156 where @xmath157 is the time spent in the layer @xmath105 by a particle that entered in the layer at time @xmath158 . in according to the standard definition of the msd , @xmath159 where @xmath160 is the lateral , or parallel , diffusion coefficient and @xmath161 the diffusion exponent . the value @xmath162 means that the system is arrested , as in a solid state where particles can only vibrate around theirs equilibrium positions ; for @xmath163 the system is subdiffusive corresponding in general to particles diffusing in complex structures ( with non trivial microscopic disorder ) ; @xmath164 is the standard diffusive behavior as in a normal fluid state . for @xmath165 the system is superdiffusive . on the other hand , for early times free diffusion we expect the ballistic regime with @xmath166 . our analysis ( fig.[fig : diff_layers ] ) shows that the in - layer msd has always a ballistic regime for @xmath167 . the corresponding mean displacement is approximately half particle diameter at high @xmath18 and low @xmath53 and weakly decreases for increasing @xmath53 and decreasing @xmath18 corresponding to the expected decrease of the mean free path of the particles . for @xmath168 all the layers reach the diffusive ( @xmath164 ) behavior for long times . by decreasing the temperature the behavior of the layers becomes more heterogeneous . in particular , we observe that the layer @xmath120 near to the solvophilic wall slows down in a sensible way with respect to the layers at @xmath169 and becomes arrested for @xmath170 . at these temperatures the other layers , including the one near the solvophobic wall , are diffusive at low densities . however , at @xmath140 and @xmath131 all the layers develop the plateau in the msd typical of glassy dynamics . this behavior is reminiscent of the caging effect in glasses where the plateau in the msd is followed by a diffusive regime . here , instead , at @xmath131 and @xmath140 we observe that for all the layers but the one near the solvophobic wall ( @xmath146 ) , after the plateau , the dynamics enters in a superdiffusive regime with @xmath171 . this effect is related to the presence of defects and of a nonuniform stress field , as discussed in sec.[sec : structural ] . for @xmath137 we observe that all the layers are arrested at @xmath118 . at this low density the slit is only partially filled ( fig.[fig : diff_layers ] ) . at @xmath137 and @xmath172 and @xmath138 also the layer @xmath146 near the solvophobic wall is present and it is characterized by a larger msd with respect to the other layers and by a diffusive regime at long times . the other layers have an arrested dynamics . at @xmath137 and @xmath140 all the layers from @xmath120 to @xmath145 have a superdiffusive regime at long times , while the layer @xmath146 reaches the diffusive log - time regime . however , its msd is smaller than that of the other layer for very long times ( @xmath173 ) . as the time proceeds , the average in eq . [ equ : msd ] for the msd is performed on a decreasing number of particles because some of them can leave the layer . _ a priori _ this reduction of the statistics is not homogeneous , that means that in general there can be a correlation between particles that leave the layer and theirs properties , as theirs velocity components . therefore , for @xmath174 , low enough @xmath53 , and for the most diffusive layers there is a time , @xmath175 , after which the in - layer msd ( fig.[fig : diff_layers ] ) is not well defined . to estimate @xmath175 as function of @xmath18 and @xmath53 for different layers we analyse the population relaxation of particles in each layer . in particular , we compute the survival probability ( sp ) function , @xmath176 , which is the probability that a given particle stay in the layer @xmath67 for a time interval @xmath177 . the sp can be calculated as @xmath178 where @xmath179 is the number of particle in the layer @xmath67 at time @xmath180 and @xmath181 is the number of particle that do not leave the layer @xmath67 during the time interval @xmath182 $ ] . the sp give an indication of the time interval @xmath175 over which the msd is well defined . we observe that @xmath183 has an exponential decay in our simulations ( fig.[fig : stau ] ) . we , therefore , define @xmath175 as the characteristic decay time @xmath184 . this choice is consistent with the observation that the msd in fig.[fig : diff_layers ] is well defined when @xmath185 . we observe that for @xmath186 and all the densities and for @xmath131 and @xmath187 , the sp decay is slower for the layer @xmath120 near the solvophilic wall and becomes faster for the layers away from the two walls . when the layer @xmath146 near the solvophobic wall is present , we observe that it has a decay in sp slower than those layers that are farther away from the wall . at @xmath137 for all densities , and at @xmath131 for @xmath140 , there is no decay in sp , consistent with the crystallization of the layers . the non monotonic behavior of @xmath175 is reported in fig.[fig : tau_layers ] as a function of layers for different densities and temperatures . by comparing fig.[fig : tau_layers ] and fig.[fig : density ] we observe that @xmath175 increases when the layers are more structured in the @xmath31 direction . hence , both walls facilitate the stratification of the fluid , although the structureless phobic wall does it in a less strong way with respect to the structured solvophilic wall . characteristic time decay @xmath175 as a function of fluid layer for density @xmath188 and temperature @xmath77 . for clarity , the points for @xmath124 are shifted up by 100 units.,width=377 ] in sec.[sec : msd ] , analysing the msd layer by layer , we have seen ( fig.[fig : diff_layers ] ) that for high densities and low temperatures , after a plateau , the dynamics can enter in a superdiffusive regime with @xmath171 . in this section we show how this behavior is due to the formation of liquid `` veins '' in such layers . the formation of liquid veins is of particular interest in ice during the freezing of water . recently , experiments and simulations showed the presence of liquid water between nanometer - sized ice crystal @xcite . in polycrystalline systems , the liquid is found along intergranular junctions , as grain boundaries ( see @xcite and references therein for the case of water ) . residual stress in these polycrystal structures can be localized along integranular junctions , and can results in an effective force that acts on fluid particles present in these junctions . the origin of the residual stress in our system is due to the fact that when the fluid solidifies as the temperature is decreased , the minimization process of the free energy take place locally , instead of globally . in glass forming liquids , this effect is caused by a fast cooling , while in our system it is due to the layering of the fluid caused by the confinement . in fig.[fig : snap_z1 - 2 - 3_rand1 - 2 - 3 ] we show the spatial configuration of the first three layers of the system close to the philic wall , for three different runs ( i.e. for three different realization of initial conditions ) at @xmath137 and @xmath140 . we observe that particles in the first layer ( @xmath120 ) are characterized by the same msd , while particles in other layers ( @xmath189 ) can have different msd . in particular in some configurations , we observe veins with mobility higher than the rest of the system ( fig.[fig : config_t_z2-z3 ] ) . we analyzed the trajectories of the particles of these specific realizations of the system ( fig.[fig : config_t_z2-z3 ] ) . for these cases we find that these particles with a msd higher than the majority belong to the same stripe and diffuse along the stripe itself . we observe that the majority of particles in the layer @xmath126 ( and in a less evident way for the layer @xmath135 ( fig.[fig : config_t_z2-z3]b ) , remain spatially localized during the entire simulation , except those belonging to two stripes moving in the same direction as along stripe veins ( fig.[fig : config_t_z2-z3]a ) . we observe a similar situation for the layer @xmath135 , but here all the particles are more mobile and the particles in the veins move in opposite directions . further analysis , that goes beyond the goals of the present work , is necessary to understand the effect of the vicinity of the solvophilic wall and if the veins are related to point - like defects as seems to be suggested by fig.[fig : config_t_z2-z3 ] . spatial configuration of the first three layers of the system close to the solvophilic wall , for three different runs ( i.e. for three different realization of the initial conditions ) at @xmath137 and @xmath140 . the size of circles , representing particle positions , are chosen to be equal to the particles hard core diameter @xmath3.,width=415 ] single - particle msd for layers and runs that are in fig.[fig : snap_z1 - 2 - 3_rand1 - 2 - 3 ] . for run # 1 , red and green colors are used for those particles belonging to veins performing a dynamics different from the rest of the particles in the layer.,width=226 ] in - layer trajectories for particles in the second ( a ) and third ( b ) layer of run # 1 in fig.[fig : snap_z1 - 2 - 3_rand1 - 2 - 3 ] . in red and green we show the trajectories of representative particles belonging to stripes veins . the msd of these particles is represented with the same color code in fig.[fig : diff_z1 - 2 - 3_rand1 - 2 - 3 ] . we apply periodic boundary conditions for @xmath190 where @xmath191 ( dashed lines ) and for @xmath192 where @xmath193.,title="fig:",width=94 ] in - layer trajectories for particles in the second ( a ) and third ( b ) layer of run # 1 in fig.[fig : snap_z1 - 2 - 3_rand1 - 2 - 3 ] . in red and green we show the trajectories of representative particles belonging to stripes veins . the msd of these particles is represented with the same color code in fig.[fig : diff_z1 - 2 - 3_rand1 - 2 - 3 ] . we apply periodic boundary conditions for @xmath190 where @xmath191 ( dashed lines ) and for @xmath192 where @xmath193.,title="fig:",width=94 ] our msd and sp analysis show that at low @xmath18 and high @xmath53 there are layers taht behave as a solid . however , by looking only at the msd and sp is not possible to establish if a solid layer is in an amorphous , crystal or polycrystal state @xcite . in order to better understand the structure of solid layers , we computed the standard 2d voronoi tessellation ( useful to identify defects present in the crystal structures , as vacancies , frenkel - like , dislocations and grain boundaries ) , and a modified version of it ( suitable to identify distorted crystal structures ) . with this analysis we can also disentagle the role that the three relevant length scales ( the diameter of the particles @xmath3 , the repulsive radius @xmath5 , and the attractive minimum @xmath4 ) , giving rise to two competing length scale @xmath194 and @xmath195 , play in the determination of layer s structure . indeed , the interdistance between two adjacent layers is @xmath196 , while when stripes form in a specific layer for intermediate densities , just to consider a specific case , particles within a stripe are compressed at a distance @xmath197 , while the distance between stripes depends on @xmath196 and @xmath198 , as discussed in the last part of this section . in the standard voronoi tessellation we construct polygons centered around particles forming a lattice whose edges are crossed in their middle point by the edges of the voronoi cells . this procedure garantees that each voronoi cell represent the proper volume of each particle . to better visualize the result , we represent voronoi cells having a different number of edges with different colors . to reduce the noise in our analysis we adopt also a modified version of the voronoi tessellation in which we associate a color to a polygon in according to the number of edges of the polygon that have a length @xmath199 larger than that of the average edge lenght calculated over the specific polygon itself . this procedure allows us to better visualize polycrystal structures despite the presence of small lattice deformations . we compute the voronoi tessellation for low density ( @xmath118 ) and high density ( @xmath140 ) at low temperature ( @xmath137 ) and very low temperature ( @xmath200 ) , for three different realizations of initial conditions ( figs.[fig : voronoi_rho0.11]a , b , [ fig : voronoi_rho0.30]a , b , and figs.[fig : voronoi_rho0.11_mod2]a , b , [ fig : voronoi_rho0.30_mod2]a , b in supplementary material ) . the configurations at temperature @xmath200 are obtained by annealing configurations equilibrated at @xmath137 with an annealing rate of @xmath201 . at low density ( @xmath118 ) , the first layer at @xmath137 ( fig.[fig : voronoi_rho0.11_mod2 ] ) is in a frustrated solid state that by annealing toward @xmath200 ( fig.[fig : voronoi_rho0.30_mod2 ] ) becomes a frustrated polycrystal . the very low-@xmath18 polycrystal has point and line defects as grain boundaries dividing a deformed honeycomb lattice ( the deformed green triangles ) from a stripe phase ( the stretched hexagonal cyan polygons ) . for both considered @xmath123 the other layers are organized in a triangular lattice ( where each particle is surrounded by a hexagonal cyan polygon ) . we only observe defects , such as dislocations ( run # 2 in fig.[fig : voronoi_rho0.11_mod2]a and fig.[fig : voronoi_rho0.11]a ) that are not eliminated by annealing ( run # 2 in fig.[fig : voronoi_rho0.11_mod2]b ) . at high density ( @xmath140 ) , the first layer at @xmath137 ( fig.[fig : voronoi_rho0.30_mod2]a ) is in a polycrystal state with defects . at @xmath200 ( fig.[fig : voronoi_rho0.30_mod2]b ) we observe two principal crystal grains : a triangular lattice ( cyan polygons ) and a kagome lattice with defects ( blue rhombouses ) . at @xmath137 , the layers @xmath189 present a zigzagging stripe structure with orientation and angles that can change from run to run . at @xmath200 the stripe structure of these layers becomes more regular . these observations emphasize that the increase of density induces an increase of disorder in the solid layers , propagating from the layer @xmath120 to the other layers and up to the layer @xmath146 . the formation of crystal defects during the annealing , and the fact that they are different for different initial conditions , indicate that the system can not reach easely the global minimum of the free energy landscape , corresponding to the crystal configuration , but is trapped in local minima due to the slowing down of the dynamics and the templating effect of the solvophilic wall . in particular , the mismatch of the wall structure with the bulk crystal structure induces a frustrating effect that is more evident near the wall ( in layers @xmath120 and @xmath126 ) for increasing density . to understand the formation of stripes , we follow the same approach as in refs @xcite to show that , if the principal contribution to the minimization of the free energy comes from the energetic therm , in the discreet potential approximation , under suitable conditions of density and temperature , particles organize in straight or zigzagging stripes . in the discreet potential approximation the energetic terms can be reduced to the soft core ( @xmath7 ) and the attractive well ( @xmath6 ) . as a consequence the energetic contribution coming from the interaction with particles of the adjacent layers is approximately constant when the layer density is fixed . therefore , the energetic cost of stripes formation is determined only by the contribution of the in - layer particle interactions . in particular , if a layer has a triangular structure , as the stable configuration of layers @xmath2022 , ... , 11 at low density ( fig.[fig : voronoi_rho0.11_mod2]a , b ) , then for sufficiently high density the layer will prefer to form stripes . 2d schematic representation of particles composed by an hard core ( in dark blue ) of size @xmath3 , a soft corona ( in cyan ) of size @xmath5 and an attractive external corona ( in green ) that extends uo to the potential cutoff @xmath203 . the equilateral and isosceles triangles represent the unitary cell of the triangular and straight - stripes lattice , respectively . an example of ( maximal ) zigzagging - stripes of the same density of straight - stripes is also shown.,width=264 ] consider our fluid made of particles with a hard core surrounded by a soft corona and an external attractive corona ( fig.[fig : stripes ] ) . if @xmath204 is the lattice constant of the triangular structure at the soft - corona distance , then the density of the layer is @xmath205 , where @xmath206 is the number of particles present in this layer . if we allow the triangular lattice to deform in order to minimize the energy of the layer , the new unit cell will be composed by the isosceles triangle in which one side is equal to @xmath39 and the other two are equals to @xmath40 with @xmath207 the fact that the density does nt change implies that @xmath208 . the energy per particle of the layer is @xmath209u_x+2[(\sqrt{n_l}-1)/\sqrt{n_l}]u_y$ ] . for @xmath210 it becomes @xmath211 , where @xmath212 and @xmath213 are the energy associated to the interaction between the particle along the @xmath39 and @xmath40 side of the triangle respectively . in the discreet potential approximation it is @xmath214 for @xmath215 , @xmath216 for @xmath217 and @xmath218 for @xmath219 ( note that we obtain the same result if instead of @xmath203 we consider any value between @xmath5 and @xmath203 . indeed , the present approach has been applyed to show the stability of stripes cnofiguration for a pure repulsive potential model @xcite ) . the same holds for @xmath213 substituting @xmath39 with @xmath40 . for sufficiently high density , i.e. for @xmath220 or @xmath221 , the energies per particle associated to a layer formed by equilateral or isosceles triangles are @xmath222 or @xmath223 , respectively . therefore , under these conditions , the layer will prefer to form stripes . furthermore , from geometric consideration it is possible to conclude that the zigzagging - stripe lattice can be obtained as a deformation of the straight - stripe lattice without changing the density and keeping the energies per particle @xmath224 @xcite . in general , many different zigzagging stripe lattices are possible all with comparable energy per particle ( fig.[fig : stripes ] ) . considering the stripes that form in the layers @xmath225 for system density @xmath140 and temperature @xmath137 , the resulting average in - layer density is @xmath226 , and @xmath227 . hence , from the previous equation for @xmath228 , we find @xmath229 . in view of all the approximation made , we consider this value consistent with @xmath230 of the distance between the closest particles belonging to two adjacent stripes in the same layer .
|
confinement can modify the dynamics , the thermodynamics and the structural properties of liquid water , the prototypical anomalous liquid . by considering a general anomalous liquid , suitable for globular proteins , colloids or liquid metals , we study by molecular dynamics simulations the effect of a solvophilic structured and a solvophobic unstructured wall on the phases , the crystal nucleation and the dynamics of the fluid . , the first layer induces a `` molding '' effect on the second layer determining a structure with reduced energy and density , closer to the average density of the system . this low - density , low - energy structure propagates further through the layers by templating effect and can involve all the existing layers at the lowest temperatures investigated .
|
confinement can modify the dynamics , the thermodynamics and the structural properties of liquid water , the prototypical anomalous liquid . by considering a general anomalous liquid , suitable for globular proteins , colloids or liquid metals , we study by molecular dynamics simulations the effect of a solvophilic structured and a solvophobic unstructured wall on the phases , the crystal nucleation and the dynamics of the fluid . we find that at low temperatures the large density of the solvophilic wall induces a high - density , high - energy structure in the first layer ( `` templating '' effect ) . in turn , the first layer induces a `` molding '' effect on the second layer determining a structure with reduced energy and density , closer to the average density of the system . this low - density , low - energy structure propagates further through the layers by templating effect and can involve all the existing layers at the lowest temperatures investigated . therefore , although the high - density , high - energy structure does not self - reproduce further than the first layer , the structured wall can have a long - range effect thanks to a sequence of templating , molding and templating effects through the layers . we find dynamical slowing down of the solvent near the solvophilic wall but with largely heterogeneous dynamics near the wall due to superdiffusive liquid veins within a frozen matrix of solvent . hence , the partial freezing of the first hydration layer does not correspond necessarily to an effective reduction of the channel section in terms of transport properties .
|
1406.1996
|
c
|
by considering many layers of a confined anomalous fluid @xcite we show that the effect of the structured solvophilic wall can extend up to the entire slit pore . in particular , we study structural and dynamical properties of a monocomponent anomalous liquid under confinement . the fluid has two characteristic distances and can be considered as a coarse - grained model for globular proteins @xcite , colloidal systems @xcite or , to some extent , liquid metals with water - like anomalies @xcite . we perform molecular dynamics simulations of the fluid in a slit pore with a solvophilic wall and a solvophobic wall . the solvophilic wall has structure while the solvophobic one has no structure . we observe that the molecules organize in an inhomogeneous way , forming layers that are parallel to the surfaces , with higher density near the solvophilic surface with respect to the center of the slit pore . for sufficiently high densities , for which the fluid occupy entirely the pore , we observe an increase of density also close to the solvophobic surface , but in a less prominent way . these results are consistent with experimental and theoretical works for nanoconfined fluids . at low temperature we observe coexistence between the homogeneous liquid , heterogeneous liquid and solid phase of the fluid . the influence of the structured solvophilic surface on the solid layers can extend as far as the sixth hydration layer at low @xmath18 and high @xmath53 . in particular , we find a strong correlation between the structure of the solvophilic surface and that of the first layer suggesting a `` templating '' effect . indeed , the large density of solvophilic surface particles allows the formation of a first layer at high density . the high energy cost of this first layer is compensated by the large number of attractive interactions between the particles of the first layer and those of the surface . further energy gain comes from an extra energy term due to the first in - layer particle interactions . moving further from the wall , we find that the first layer has a `` molding '' effect on the second layer . this is because the density of the first layer is smaller than that of the surface and is not high - enough to propagate its template . nevertheless , the low - density second layer is in condition to template the third layer replicating its structure and inducing a long - range effect that can , eventually , involve the whole system . from the calculation of the mean square displacement and the voronoi tessellation we conclude that at low temperature the first layer close to the solvophilic surface is a polycrystal with two competing phases that generate low - energy states with high degeneracy and very slow dynamics . at low densities the two competing phases are stripes and honeycomb lattice , while at high densities are triangular and kagome lattice . we understand this result as a consequence of the high density ( triangular ) structure of the solvophilic wall with a lattice step that corresponds to the hard repulsive distance of the solvent , and the strong wall - solvent attractive interaction . these properties of the wall generate in the first solvent layer local regions with density and energy that are higher than the average of the layer . as a consequence , other regions within the layer have density and energy below the average , giving rise to a competing crystal structure . our results remind us of some recent experiments and simulations for a thin - film of water on @xmath0 surface for which the authors found a very high density first interfacial layer for all temperatures , while they would expect , from thermodynamic arguments , a lower density liquid at supercooled conditions @xcite . in other recent experiments and simulations @xcite , the authors pointed out that it is necessary to revise the theory of heterogeneous nucleation when the crystalization induces a non - zero entropy at zero temperature and the system initially is far from equilibrium . apart from the layers close to the two surfaces , we observe that the structure of each layer mainly depends on its density . in the case of the stripe phase , using simple geometrical and energetic considerations , we find that straight and zigzagging stripes are the stable configurations for intermediate densities . furtheremore , analysing the mean square displacement layer by layer , we observe layers at high densities and low temperatures with a caging - like behavior characterized by a ballistic dynamics followed by an arrested state ( plateau ) and a superdiffusive regime with a diffusion exponent @xmath171 . our analysis shows that this behavior is due to the formation of liquid veins within the stripe phase . in particular we observe that each vein can behave differently from the others diffusing in one of the two possible directions along the stripes and having a different diffusion exponent @xmath171 . we rationalize the different possible values of @xmath161 as a consequence of the presence of residual stress that could introduce an effective force acting on the fluid . under suitable conditions , e.g. a constant effective force along the stripe , the particles in the vein could perform a biased one - dimensional random walk characterized by an exponent of the msd that approaches the ballistic value ( @xmath166 ) . the behavior of these veins can be analyzed in a more quantitative way by computing , for example , the temporal autocorrelation function , the intermediate scattering function @xcite , the relative displacement of nearest neighbors or the particles displacements following the lindemann criterion @xcite . we will present this analysis in future works . our results show that the dynamical slowing down of the anomalous solvent near the solvophilic wall does not imply by necessity the complete freezing of the first hydration layers , because at low @xmath18 and high @xmath53 we observe largely heterogeneous dynamics in three layers with the formation of liquid veins within a frozen matrix of solvent . therefore , under these considerations the partial freezing of the first hydration layer does not correspond necessarily to an effective reduction of the channel section in terms of transport properties , at variance with the conclusions of ref.@xcite .
|
we find that at low temperatures the large density of the solvophilic wall induces a high - density , high - energy structure in the first layer ( `` templating '' effect ) . in turn therefore , although the high - density , high - energy structure does not self - reproduce further than the first layer , the structured wall can have a long - range effect thanks to a sequence of templating , molding and templating effects through the layers . we find dynamical slowing down of the solvent near the solvophilic wall but with largely heterogeneous dynamics near the wall due to superdiffusive liquid veins within a frozen matrix of solvent . hence , the partial freezing of the first hydration layer does not correspond necessarily to an effective reduction of the channel section in terms of transport properties .
|
confinement can modify the dynamics , the thermodynamics and the structural properties of liquid water , the prototypical anomalous liquid . by considering a general anomalous liquid , suitable for globular proteins , colloids or liquid metals , we study by molecular dynamics simulations the effect of a solvophilic structured and a solvophobic unstructured wall on the phases , the crystal nucleation and the dynamics of the fluid . we find that at low temperatures the large density of the solvophilic wall induces a high - density , high - energy structure in the first layer ( `` templating '' effect ) . in turn , the first layer induces a `` molding '' effect on the second layer determining a structure with reduced energy and density , closer to the average density of the system . this low - density , low - energy structure propagates further through the layers by templating effect and can involve all the existing layers at the lowest temperatures investigated . therefore , although the high - density , high - energy structure does not self - reproduce further than the first layer , the structured wall can have a long - range effect thanks to a sequence of templating , molding and templating effects through the layers . we find dynamical slowing down of the solvent near the solvophilic wall but with largely heterogeneous dynamics near the wall due to superdiffusive liquid veins within a frozen matrix of solvent . hence , the partial freezing of the first hydration layer does not correspond necessarily to an effective reduction of the channel section in terms of transport properties .
|
0809.4554
|
i
|
in @xcite , dawson and perkins introduced a population dynamic model of two populations that live on a countable site space @xmath2 . the individuals migrate between sites and , at any given site , perform a critical branching process with a branching rate proportional to the local size of the population of the respective other type . more precisely , dawson and perkins considered the system of coupled stochastic differential equations ( sdes ) ( taking nonnegative values ) @xmath3 here , @xmath4 is the @xmath5-matrix of a markov chain on @xmath2 with symmetric jump kernel @xmath6 , @xmath7 is an independent family of brownian motions and @xmath8 is a parameter . dawson and perkins showed that there exists a unique weak solution of this sde taking values in a suitable subspace of @xmath9 with some growth condition . furthermore , this process is a strong markov process . while existence of a weak solution is rather standard due to the procedure proposed by shiga and shimizu @xcite , weak uniqueness was shown using a certain self - duality of the process established in @xcite . we will describe the duality in detail below , in ( [ e2.4 ] ) . a main result of dawson and perkins is a dichotomy in the long - time behavior of the solutions depending on whether @xmath10 is recurrent or transient ( assuming some mild regularity condition on @xmath11 ) . for recurrent @xmath10 ( fulfilling the regularity assumption ) , the types segregate , while for transient @xmath10 , there is coexistence of types . more precisely , let @xmath12 denote the total mass processes ( @xmath13 ) and assume that @xmath14 . then @xmath15 and @xmath16 are continuous orthogonal nonnegative @xmath17-martingales . let @xmath18 denote the almost sure limit . dawson and perkins show that @xmath19=0 $ ] if @xmath10 is recurrent and @xmath19=m_{1,0}m_{2,0}$ ] if @xmath11 is transient . furthermore , in the recurrent case , the joint distribution of @xmath20 equals @xmath21 , where , for @xmath22 , @xmath23 is the harmonic measure of planar brownian motion in @xmath24 . that is , if @xmath25 is a brownian motion in @xmath26 started at @xmath27 and @xmath28 , then @xmath23 is the probability measure on @xmath29 given by @xmath30.\ ] ] the explicit form of the densities of @xmath23 can be found in ( [ e2.5 ] ) . via the self - duality of the mutually catalytic branching process , its total mass behavior for finite initial conditions provides information on the local behavior if the initial condition is infinite and sufficiently homogeneous . for @xmath22 , let @xmath31 denote the state in @xmath9 with @xmath32 for all @xmath33 , @xmath13 . assume that @xmath34 . then @xmath35>0,\ ] ] if @xmath10 is transient , that is , types can coexist locally . on the other hand , for recurrent @xmath10 , the distribution of @xmath36 converges weakly to @xmath37 , that is , to a spatially homogeneous point @xmath38 , where @xmath39 is sampled according to the distribution @xmath23 . hence , in the recurrent case , the two types segregate locally and form clusters . the assumption that the initial point is constant can be weakened to an ergodic random initial condition ( see @xcite ) . the starting point for this work was the wish to obtain a quantitative description of the cluster growth in the recurrent case . we will only briefly describe the heuristics . dawson and perkins also constructed a version of their process in continuous space @xmath40 instead of @xmath2 as the solution of a stochastic partial differential equation @xmath41 where @xmath42 and @xmath43 are independent space time white noises and @xmath44 is the laplace operator . as @xmath44 on @xmath45 is recurrent , types also segregate here . now , due to brownian scaling , if we denote by @xmath46 the solution of ( [ e1.3 ] ) with that given value of @xmath0 , then we obtain @xmath47 = \mathbf{p}_{\underline x } [ ( y^{\gamma t}_1(r ) ) _ { r\in { \mathbb{r}}}\in \bolds{\cdot } ] .\ ] ] equation ( [ e1.4 ] ) shows that clusters of @xmath48 grow like @xmath49 and that a better understanding of the precise cluster formation can be obtained by letting @xmath1 for fixed time . hence , we aim to construct a model @xmath50 , that is , in some sense , the limit of @xmath51 as @xmath1 . in this paper , we construct @xmath50 in the simple case where @xmath2 is a singleton and where the migration between colonies is replaced by an interaction with a time - invariant mean field . this is a first step toward the investigation of the model involving infinitely many sites . we give characterizations of the process @xmath50 via an infinitesimal generator , as the solution of a well - posed martingale problem and as the limit of @xmath46 as @xmath1 . finally , we give a strong construction of the process via a time - changed planar brownian motion . this will also serve to derive path properties . in two forthcoming papers , we construct the infinite rate process on a countable site space @xmath2 via a stochastic differential equation with jump - type noise and give a characterization via a martingale problem @xcite . furthermore , we will investigate the long - time behaviour and give conditions for segregation and for coexistence of types @xcite . an alternative construction via a trotter product approach is carried out in @xcite and @xcite . we now describe the one - colony process which is the subject of investigation of this paper . assume that @xmath2 is a singleton and that immigration and emigration come from and go to some colony that is thought to be infinitely big and whose effective population size ( for immigration ) is @xmath52 . furthermore , let @xmath53 be the rate of migration . hence , we consider the solution @xmath54 of the stochastic differential equation @xmath55 this model can be thought of as a version of the model defined in ( [ e1.1 ] ) where the migration between colonies is replaced by an interaction with a time - invariant mean field @xmath56 or with an infinitely large reservoir whose types have proportions @xmath57 and @xmath58 . ( in fact , in @xcite it was shown ( proposition 1.1 ) that @xmath59 arises as the mckean vlasov limit of solutions of ( [ e1.1 ] ) with symmetric interaction on a complete graph @xmath2 . ) more formally , the interaction term @xmath60 is replaced by a drift @xmath61 . it is this simplification of the interaction that allows for a tractable exposition in this article . note that as @xmath62 , the process without drift ( @xmath63 ) converges almost surely to some random @xmath64 . hence , in the case @xmath63 , if we let @xmath65 , then the limiting process would be trivial : if it starts at @xmath64 , then it stays at @xmath27 forever . see section [ a2 ] for a more detailed description of the process @xmath66 solving ( [ e1.5 ] ) ( finite @xmath0 process ) . on a heuristic level , as the stochastic term in ( [ e1.5 ] ) defines an isotropic two - dimensional diffusion , that is , a time - transformed planar brownian motion , if we let @xmath1 , then we should end up with a process where the stochastic part is a planar brownian motion at infinite speed , stopped when it reaches the boundary of the upper - right quadrant . that is , the limiting process @xmath50 should be a markov process with values in @xmath67 . when @xmath27 is the current state and the drift moves it to @xmath68 , this point should instantaneously be replaced by a random point chosen according to @xmath69 . we will , in fact , be able to describe this infinitesimal dynamics both in terms of a martingale problem and in terms of a generator of markov transition kernels . however , we first define @xmath50 via an explicit transition semigroup and show that it is the limit of @xmath70 as @xmath1 . let @xmath71 equipped with the supremum norm @xmath72 . [ d1.1 ] let @xmath53 and @xmath52 . for @xmath73 and @xmath64 , define the stochastic kernel @xmath74 by @xmath75 define the contraction semigroup @xmath76 on @xmath77 by @xmath78 the markov process @xmath79 with state space @xmath67 , cdl ` ag paths and transition kernels @xmath80 is called the infinite rate mutually catalytic branching process ( imub ) with parameters @xmath81 . in order for this definition to make sense , in proposition [ p3.2 ] , we will show that @xmath82 is , in fact , a markov semigroup . [ p1.2 ] @xmath83 is a feller process and has the strong markov property . it is ergodic and the unique invariant measure is @xmath84 . the map @xmath85 is continuous , hence @xmath86 is also continuous , that is , @xmath83 is a feller process . since @xmath87 for @xmath64 , the semigroup @xmath88 is strongly continuous . hence , by the general theory of markov processes , there exists a cdlg version of @xmath50 that is strong markov ( see , e.g. , @xcite , chapters iii.7 and 8) . ergodicity and the explicit form of the invariant measure are trivial . [ t1.3 ] assume that @xmath89 for all @xmath90 . as @xmath1 , the finite - dimensional distributions of @xmath59 converge to those of @xmath83 . note that in theorem [ t1.3 ] , trivially , we do not have convergence in the skorohod path space , since continuous processes do not converge to discontinuous processes in that topology . in addition to the convergence of the finite - dimensional distributions , we also have convergence of the @xmath91th moments for @xmath92 [ but not for @xmath93 , of course , since for @xmath94 , the measure @xmath23 does not possess finite second moments , as can be easily derived from its density formula ( [ e2.5 ] ) ] . hence , on a suitable probability space , we have @xmath95-convergence of @xmath59 to @xmath83 . [ t1.4 ] assume that @xmath89 for all @xmath90 and let @xmath92 , @xmath73 . for every @xmath90 and @xmath13 , we have @xmath96\leq \mathbf{e}_x [ ( x^{c,\theta}_{i , t})^p ] < \infty.\ ] ] on a suitable probability space , for @xmath13 , we have @xmath97 it can be seen from the proofs of theorems [ t1.3 ] and [ t1.4 ] that the statements of these theorems also hold for @xmath98 and @xmath99 if we replace @xmath100 by a random point chosen according to @xmath101 . [ r3.3 ] while in the one - colony case considered in this paper , it is easy to explicitly write down the semigroup for the infinite rate mutually catalytic branching process @xmath83 , it is less obvious how to construct an interacting version of the process on a countable site space . one possibility is the trotter product approach that is used in @xcite and @xcite . here , we briefly sketch it for @xmath83 . in the classical setting , the trotter product approach works as follows . in order to construct a solution @xmath59 of ( [ e1.5 ] ) , in time intervals of length @xmath102 , one could alternate between a solution of the pure drift equation ( @xmath103 ) and the pure stochastic noise equation ( @xmath63 ) . as @xmath104 , this process converges to a solution of ( [ e1.5 ] ) . if we let @xmath1 , then the noise term results in an instantaneous jump to a point in @xmath67 chosen according to @xmath105 , where @xmath39 is the value of @xmath66 at the end of the preceding `` drift interval . '' more formally , let @xmath106 be an independent family of @xmath67-valued random variables with distribution @xmath107=q_x$ ] . for @xmath108 , let @xmath109 be the solution of the differential equation @xmath110 that is , @xmath111 let @xmath112 and define @xmath113 one can prove that @xmath114 converges in distribution in the skorohod topology on the space of cdlg paths to @xmath83 ( see @xcite and @xcite ) . while , in definition [ d1.1 ] , we gave an explicit formula for the transition kernels of @xmath50 , it is also interesting to characterize the process @xmath50 via its infinitesimal dynamics . in section [ a5 ] , we investigate the generator @xmath115 of the semigroup @xmath88 . for a certain class @xmath116 of smooth functions @xmath117 ( see definition [ d5.1 ] ) , we give an explicit formula for @xmath118 as an integro - differential operator . using the classical hille yoshida theorem , we show that the restricted operator @xmath119 uniquely defines @xmath120 ( theorem [ t5.3 ] ) . furthermore , we show that @xmath121 restricted to an even smaller space @xmath122 of functions that appear in the duality for @xmath50 still uniquely defines the process @xmath50 via a martingale problem ( theorem [ t5.4 ] ) . to define @xmath123 , it is crucial to study ( for suitable functions @xmath117 ) the limit @xmath124 which will also clarify the jump structure of the process @xmath50 . the description of the exact form of the operator @xmath121 and the precise statements of the theorems are a bit technical , so these are deferred to section [ a5 ] . while , for proposition [ p1.2 ] , we used general construction principles of markov processes , here , we provide an explicit strong construction of the process @xmath50 in terms of a given planar brownian motion @xmath125 . this construction also allows certain path properties to be investigated . assume @xmath126 . for @xmath127 , we write @xmath128 for the rectangular cone northeast of @xmath129 . for @xmath22 , let @xmath130 and @xmath131 for @xmath132 , we write @xmath133 if @xmath134 , that is , if @xmath135 and @xmath136 . for @xmath22 , we define the @xmath137-algebra @xmath138 in lemma [ l3.1 ] , we will show that @xmath139 is a markov process with respect to @xmath140 . let @xmath141 and @xmath142 be measurable and locally integrable . for @xmath143 , define @xmath144 [ t1.5 ] let @xmath64 and define the process @xmath145 by @xmath146 then @xmath145 is a time - inhomogeneous markov process on @xmath67 with cdlg paths and with transition probabilities @xmath147 in particular , for @xmath148 and @xmath149 , @xmath150 is an infinite rate mutually catalytic branching process with @xmath151 via a planar brownian motion . here @xmath152 for @xmath153 . ] with parameter @xmath81 , see figure [ figure1 ] . it is tempting to use this strong construction of @xmath154 in order to define an interacting version of the infinite rate mutually catalytic branching process on a countable site space @xmath2 , where @xmath155 at site @xmath33 reflects the migration from neighboring sites to @xmath156 . however , in this paper , we do not pursue this topic . rather , we use the strong construction in order to derive a path property of @xmath83 via a result of le gall and meyre @xcite on the cone points of planar brownian motion . recall that a measurable set @xmath157 is called _ polar _ for @xmath83 if for all @xmath64 , we have @xmath158=0.\ ] ] [ t1.6 ] the point @xmath159 is polar for @xmath83 . in section [ a2 ] , we give a detailed description of the duality for the process with finite branching rate . in section [ a3 ] , we establish a similar duality for the infinite rate process and use it in order to show the convergence in theorems [ t1.3 ] and [ t1.4 ] . in section [ a4 ] , we justify the strong construction of theorem [ t1.5 ] and also prove theorem [ t1.6 ] . finally , in section [ a5 ] , we describe the infinite rate process in terms of its infinitesimal dynamics and state and prove the theorem on the construction via the hille yoshida theory ( theorem [ t5.3 ] ) and via a martingale problem ( theorem [ t5.4 ] ) .
|
consider the mutually catalytic branching process with finite branching rate . we show that as , this process converges in finite - dimensional distributions ( in time ) to a certain discontinuous process . we give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem . we also give a strong construction in terms of a planar brownian motion from which we infer a path property of the process .
|
consider the mutually catalytic branching process with finite branching rate . we show that as , this process converges in finite - dimensional distributions ( in time ) to a certain discontinuous process . we give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem . we also give a strong construction in terms of a planar brownian motion from which we infer a path property of the process . this is the first paper in a series or three , wherein we also construct an interacting version of this process and study its long - time behavior . and .
|
astro-ph0703455
|
i
|
the simplest models of inflation involve only a single scalar field , the inflaton @xcite . with a suitably chosen potential , such a model can provide a simple explanation of the temperature fluctuations in the cmb at all angular scales @xcite . there are however a number of possible anomalies in the cmb power spectrum that have attracted the attention of many researchers @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . ( however see also @xcite . ) none of these anomalies is by itself statistically compelling ; however , taken together they provide a hint that these features may be significant . much discussion of anomalies involves the power spectrum at low @xmath1 , @xmath8 at large scales , where several anomalies indicate a possible spatial asymmetry in the power spectrum , most often roughly a north - south galactic coordinate asymmetry @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite , @xcite . the possibility that there may be an asymmetry in the observed cmb power spectrum was first raised by eriksen @xmath9 @xcite and hansen @xmath9 @xcite using the first year wmap data . their data analysis suggested a difference in power of roughly 20% for low @xmath1 maximized in the direction of galactic coordinates @xmath10 . interestingly no effect was seen above @xmath11 . for example , the analysis of the power spectrum in the vicinity of the first acoustic peak @xcite showed no evidence of a spatial asymmetry . at low values of @xmath1 , the cosmic variance provides an intrinsic scatter in the power spectrum data , so that even though the signal is rather large , the statistical significance of their result was below 3 standard deviations . in their three year data release @xcite , the wmap team addressed the isotropy of the power spectrum , finding a small asymmetry in a direction consistent with eriksen @xmath9 @xcite . the method introduced by the wmap team to investigate asymmetries in the cmb spectrum is to multiply an isotropic gaussian cmb field by a large scale modulation function . they test both a dipole and a quadrupole modulation and find that the significance of the signal is not statistically compelling . their analysis uses a pixel size of @xmath12 which makes their analysis sensitive up to @xmath13 . the original eriksen team has also revisited the wmap 3 year data @xcite using a statistical framework similar to the wmap team s with a modulation function . they choose a higher resolution with a pixel size of @xmath14 including multipoles up to @xmath11 in their analysis and confirm the asymmetry with a higher statistical significance than the wmap team and in consistency with their previous analysis of the first year wmap data . hansen @xmath15 @xcite and maino @xmath15 @xcite explored two different approaches to extract the cmb spectrum where wmap data itself is used for foreground removal , and both find an asymmetry of the power spectrum at largest scales consistent with previous analysis and with each other . the fact that the asymmetry does not vary when different foreground subtraction procedures are applied constitutes a strong argument against a galactic origin for the asymmetry . moreover , the asymmetry was also found in cobe data @xcite which indicates that systematics may not be the correct explanation for a large scale asymmetry in the cmb power spectrum . rth @xmath9 @xcite have also found the asymmetry in the wmap 3 year data using statistical techniques different from the ones used in previous analyses . recent analyses of wmap 5 year data show a similar anisotropy of power between the two hemispheres , but with the asymmetry possibly reaching to higher multipoles @xcite . otherwise , the nature of the asymmetry and the maximum asymmetry direction remains almost the same , and it will be interesting to see what results from planck will have to say about an asymmetry at small scales . these analyses provide motivations for the study of inflationary models that can generate a spatial asymmetry at low @xmath1 while remaining isotropic at larger values of @xmath1 . if these anomalies prove to be valid indicators of an asymmetry in the power spectrum , they can provide a direct probe of inflationary dynamics . significant work that attempts to find a solution to these anomalies has already appeared @xcite , @xcite , @xcite , @xcite in the literature . in this paper we discuss a simple situation that could lead to a spatial asymmetry in the cmb power spectrum at low values of @xmath1 within single field inflation . this involves an initial period of fast - roll expansion driven by the inflaton kinetic energy . the possibility of such an initial fast - roll period has been proposed by contaldi , peloso , kofman and linde ( cpkl ) @xcite as a mechanism to explain the lack of cmb power at low @xmath1 . this mechanism provides a suppression of the spectrum of primordial perturbations and thus of the cmb at large scales , and it has also been worked on by others @xcite . we will argue that in situations where the initial kinetic energy is significant in comparison to the potential energy , we should also expect the presence of a spatial gradient in the initial conditions of the inflaton field . we will show that even a surprisingly small value of an initial gradient of order a few percent will leave an observable spatial asymmetry in the cmb power spectrum at low @xmath1 . essentially , the power suppression in the fast - roll model occurs at scales that depend sensitively on the initial magnitude of the scalar field in the frame where the kinetic energy is uniform and isotropic . this leads to a characteristic pattern for the spatial dependence of the power spectrum . while we will provide a brief discussion of two - field models below , we here focus on the single field fast - roll option because of its simplicity and predictive power .
|
we note that in this situation it is natural that there also be a spatial gradient in the initial value of the inflaton field , and that this can provide a spatial asymmetry in the observed cmb power spectrum , manifest at low values of . mpp-2009 - 57 + * non - isotropy in the cmb power spectrum in single field inflation * + john f. donoghue , koushik dutta , andreas ross , + department of physics university of massachusetts amherst , ma 01003 , usa max - planck - institut fr physik ( werner - heisenberg - institut ) fhringer ring 6 , d-80805 mnchen , germany department of physics yale university new haven , ct 06520 , usa
|
contaldi have suggested that an initial period of kinetic energy domination in single field inflation may explain the lack of cmb power at large angular scales . we note that in this situation it is natural that there also be a spatial gradient in the initial value of the inflaton field , and that this can provide a spatial asymmetry in the observed cmb power spectrum , manifest at low values of . we investigate the nature of this asymmetry and comment on its relation to possible anomalies at low . mpp-2009 - 57 + * non - isotropy in the cmb power spectrum in single field inflation * + john f. donoghue , koushik dutta , andreas ross , + department of physics university of massachusetts amherst , ma 01003 , usa max - planck - institut fr physik ( werner - heisenberg - institut ) fhringer ring 6 , d-80805 mnchen , germany department of physics yale university new haven , ct 06520 , usa
|
astro-ph0104257
|
i
|
the last decade has witnessed an explosive growth of high - quality data for supernovae both from the space and ground observatories with spectacular results , and new perspectives for the use of sne ia as cosmological yard sticks and for constraining the physics of supernovae . one of the most important new developments in observational supernova research was to establish the long - suspected correlation between the peak brightness of sne ia and their rate of decline , @xmath1 , by means of modern ccd photometry ( phillips 1993 ) . sne ia have provided new estimates for the value of the hubble constant ( @xmath10 ) based on a purely empirical procedure ( hamuy et al . 1996ab , riess , press & kirshner , 1996 ) , and on a comparison of detailed theoretical models with observations ( hflich & khokhlov 1996 , hereafter hk96 ; nugent et al . the values obtained are in good agreement with one another . more recently , the routine successful detection of supernovae at large redshifts , z ( e.g. perlmutter et al . 1995 , 1997 ; riess et al . 1998 ; garnavich et al . 1998 ) , has provided an exciting new tool to probe cosmology . this work has provided results that are consistent with a low matter density in the universe and , most intriguing of all , yielded hints for a positive cosmological constant @xmath11 of @xmath12 . it is worth noting that the differences in the maximum magnitude between @xmath11=0 and 0.7 is @xmath13 for redshifts between 0.5 to 0.8 . these results prompted the quest for the nature of the the dark energy , i.e. cosmological equation of state . current candidates include a network of topological defects such as strings , evolving scalar fields ( i.e. quintessence ) , or the classical cosmological constant . for a recent review , see ostriker & steinhardt ( 2001 ) , and perlmutter , turner & white ( 1999b ) . to separate between the candidates by sne ia , the required accuracy has to be better than @xmath14 to @xmath15 ( albrecht & weller 2000 ) . the results on @xmath16 and future projects to measure the cosmological equation of state depend on the empirical @xmath1 which is calibrated locally . this leaves systematic effects as the main source of concern . indeed , there is already some evidence that sne ia undergo evolution . it has been argued , that the local sn ia sample covers all the possible variations that may come from different progenitors , different explosion mechanisms and environments , etc .. for local sne ia , the observational and statistical characteristics depend on their environment . they occur less often in ellipticals than in spirals , and the mean peak brightness is dimmer in ellipticals ( branch et al . 1996 ; wang , hflich & wheeler 1997 ; hamuy et al . 2000 ) . in the outer part of spirals the brightness is similar to ellipticals while , in more central regions , both intrinsically brighter and dimmer sne ia occur ( wang et al . these dependencies show us that sne ia likely depend on the underlying population and may undergo evolution . if the evolution realizes then we have to know it and take it into account going back in time . otherwise we can not _ safely _ use a local calibration . in principle , more distant sample could come from younger and more metal - poor progenitors , or the dominant explosion scenario may change . there is general agreement that sne ia result from some process of combustion of a degenerate white dwarf ( hoyle & fowler 1960 ) . within this general picture , three classes of models have been considered : ( 1 ) an explosion of a co - wd , with mass close to the chandrasekhar mass , which accretes mass through roche - lobe overflow from an evolved companion star ( whelan & iben 1973 ) . the explosion is then triggered by compressional heating near the wd center . ( 2 ) an explosion of a rotating configuration formed from the merging of two low - mass wds , caused by the loss of angular momentum due to gravitational radiation ( webbink 1984 , iben & tutukov 1984 , paczyski 1985 ) . ( 3 ) explosion of a low mass co - wd triggered by the detonation of a helium layer ( nomoto 1980 , woosley et al . 1980 , woosley & weaver 1986 ) . only the first two models appear to be viable . the third , the sub - chandrasekhar wd model , has been ruled out on the basis of predicted light curves and spectra ( hflich et al . 1996b , nugent et al . 1997 ) . delayed detonation ( dd ) models ( khokhlov 1991 , woosley & weaver 1994 , yamaoka et al . 1992 ) have been found to reproduce the optical and infrared light curves and spectra of typical sne ia reasonably well ( hflich 1995 , hereafter h95 ; hflich , khokhlov & wheeler 1995 , hereafter hkw95 ; hk96 ; fisher et al . 1998 ; nugent et al . 1997 ; wheeler et al . 1998 ; hflich et al . 2000 ; lentz et al . 2001 ; gerardy et al . this model assumes that burning starts as subsonic deflagration and then turns to a supersonic , detonative mode of burning . due to the one - dimensional nature of the model , the speed of the subsonic deflagration and the moment of the transition to a detonation are free parameters . the moment of deflagration - to - detonation transition ( ddt ) is conveniently parameterized by introducing the transition density , @xmath17 , at which ddt happens . the amount of @xmath18ni , @xmath19 , depends primarily on @xmath20 ( h95 ; hkw95 ; umeda et al . 1999 ) , and to a much lesser extent on the assumed value of the deflagration speed , initial central density of the wd , and initial chemical composition ( ratio of carbon to oxygen ) . models with smaller transition density give less nickel and hence both lower peak luminosity and lower temperatures ( hkw95 , umeda et al . in dds , almost the entire wd is burned , i.e. the total production of nuclear energy is almost constant . this and the dominance of @xmath20 for the @xmath21 production are the basis of why , to first approximation , sne ia appear to be a one - parameter family . the observed @xmath1 can be well understood as a opacity effect ( hflich et al . 1996b ) , namely , as a consequence of the rapidly dropping opacity at low temperatures ( hflich , khokhlov & mller 1993 , khokhlov , mller & hflich 1993 ) . less ni means lower temperature and , consequently , reduced mean opacities because the emissivity is shifted from the uv towards longer wavelengths with less line blocking . a more rapidly decreasing photosphere causes a faster release of the stored energy and , as a consequence , steeper declining lcs with decreasing brightness . the dd models thus give a natural and physically well - motivated origin of the @xmath1 relation of sne ia within the paradigm of thermonuclear combustion of chandrasekhar - mass co white dwarfs . nonetheless , variations of the other parameters lead to some deviation from the @xmath1 . e.g. a change of the central density results in an increased binding energy of the wd and a higher fraction of electron capture close to the center which reduce the @xmath21 production ( hflich et al . 1996b ) . because dd - models allow to reproduce the observations , we use this scenario to test the influence of the underlying stellar population on the explosion . we note that detailed analyses of observed spectra and light curves indicate that mergers and deflagration models such as w7 may contribute to the supernovae population ( hflich & khokhlov 1996 , hatano et al . in particular , the classical `` deflagration '' model w7 with its structure similar to dd models has been successfully applied to reproduce optical light curves and spectra ( e.g harkness , 1987 ) . the evidence against pure deflagration models for the majority of sneia includes ir - spectra which show signs of explosive carbon burning at high expansion velocities ( e.g. wheeler et al . 1998 ) and recent calculations for 3-d deflagration fronts by khokhlov ( 2001 ) which predict the presence of unburned and partial burned material down to the central regions . currently , pure deflagration models may be disfavored for the majority of sneia but , clearly , they can not be ruled out either . previously , hflich , wheeler & thielemann ( 1998 , hereafter hwt98 ) studied evolutionary effects induced by the progenitor . they calculate differences in the lc and nlte - spectra as a function of parameterized values of the integrated c / o ratio @xmath22 and metallicity of the exploding wd . this study showed that a change of @xmath22 alters the energetics of the explosion which results in an off - set of the brightness - decline relation . most prominently , this effect can be identified by a change in the fiducial rise - time to decline relation @xmath23 . the offset in @xmath1 is given by @xmath24 where @xmath25 is the dispersion in the rise time of the fiducial light curve . aldering et al . ( 2000 ) showed that @xmath23 are identical within @xmath26 for the local and distant sample lending strong support for the notion that we need a positive @xmath16 . a change in the metallicity z causes a change in the burning conditions at the outer layers of the wd and it alters the importance of the line blanketing in the blue to the uv . based on detailed calculations , effects of similar order have been found for both the delayed dd and the deflagration scenario ( hwt98 , lentz et al . recent studies showed the additional effect that z will influence the final structure of the progenitor and the resulting lcs ( umeda et al . 1999 , domnguez et al . 2000 , hflich et al . however , the former two studies were restricted to the progenitor evolution whereas the latter included the connection between the progenitor and the lc but it was restricted to a progenitor of @xmath27 and two metallicities , z=0.02 and 0.004 . a more comprehensive study may be useful to eliminate potential problems due to evolution of the progenitors for the determination of the cosmological equation of state , and it may provide a direct link to the progenitors of sn ia . in this work we connect @xmath0 and and the initial metallicity of the wd to the light curves and spectral properties of sne ia for the entire range of potential progenitors . in section 2 we discuss the evolutionary properties of our models . in section 3 , the results are presented for the explosion , nucleosynthesis , the light curves and spectral properties . in the final , concluding section , our model calculations are related to observations , and we discuss constraints for the progenitors and implications for the cosmological equation of state .
|
based on the structures for the wd , detailed model calculations have been performed for the hydrodynamical explosion , nucleosynthesis and light curves . the metallicity alters the isotopic composition of the outer layers of the ejecta . we use our results and recent observations to constrain the progenitors , and to discuss evolutionary effects of sne ia with redshift .
|
detailed stellar evolution calculations have been performed to quantify the influence of the main sequence mass and the metallicity z of the progenitor on the structure of the exploding wd which are thought to be the progenitors of sne ia . in particular , we study the effects of progenitors on the brightness decline relation which is a corner stone for the use of sne ia as cosmological yard - stick . both the typical and z can be expected to change as we go back in time . we consider the entire range of potential progenitors with 1.5 to 7 and metallicities between z=0.02 to . our study is based on the delayed detonation scenario with specific parameters which give a good account of typical light curves and spectra . based on the structures for the wd , detailed model calculations have been performed for the hydrodynamical explosion , nucleosynthesis and light curves . the main sequence mass has been identified as the decisive factor to change the energetics of the explosion and , consequently , dominates the variations in the rise - time to decline relation of light curves . has little effect on the color index b - v . for similar decline rates , the flux at maximum brightness relative to the flux on the radioactive tail decreases systematically with by about . this change goes along with a reduction of the photospheric expansion velocity by about 2000 km / sec . a change in the central density of the exploding wd has similar effects but produces the opposite dependency between the brightness to tail ratio and and , therefore , can be separated . the metallicity alters the isotopic composition of the outer layers of the ejecta . selective line blanketing at short wavelengths decreases with z and changes systematically the intrinsic color index b - v by up to , and it alters the fluxes in the u band and the uv . the change in b - v is critical if extinction corrections are applied . the offset in the calibration of is not monotonic in z and , in general , remains . we use our results and recent observations to constrain the progenitors , and to discuss evolutionary effects of sne ia with redshift . the narrow spread in the fiducial rise - time to decline relation in local sne ia restricts the range of main sequence masses to a factor of 2 . the upper limit of 1 day for the difference between the local and distance sample support the need for a positive cosmological constant . the size of evolutionary effects are small ( ) but are absolutely critical for the reconstruction of the cosmological equation of state .
|
astro-ph0104257
|
c
|
using a delayed detonation model and realistic structures for the exploding white dwarf , we have studied the influence of the progenitor star on the light curves and spectral properties of type ia supernovae . * stellar models : * we considered stars between 1.5 to 7 @xmath2 and metallicities between @xmath91 ( solar ) to @xmath120 which covers the full range of potential progenitors . the progenitor structures are based on detailed calculations for the stellar evolution starting at the pre - main sequence up to the thermal pulses when most of the stellar envelope is ejected and a white dwarf is formed with a mass between 0.5 and 1.0 @xmath74 . its size increases with @xmath0 and , to a lesser extend , changes with the metallicity . the subsequent accretion and burning at the surface of the wd let it grow to @xmath121 . as a final chemical structure , the wd shows a central region of reduced c abundance between 0.21 to 0.32 originating from the convective he - burning , a layer of increased c abundance from the he - shell burning , and a layer originating for the accretion phase . the mean c / o ratio decreases by about 30 % over the entire mass range . the sensitivity on the metallicity is much weaker ( @xmath122 ) , and not monotonic . * supernovae : * our study of sne ia is based on delayed detonation models because they have been found to reproduce the monochromatic light curves and and spectra of sne ia reasonably well including the brightness decline relation @xmath1 . deviation from a perfect relation are due to variations in the central density , properties of the deflagration front , and the progenitor structure . all parameters but the progenitors have been fixed to produce lcs and spectra typical for normal sne ia . in this work , rise times to maximum light are between 17.7 to 19.4 days , @xmath123 to @xmath124 , and @xmath125 to @xmath126 . differences between the models and light curves remain small because the nuclear energy production by burning carbon and oxygen to iron - group elements differs by as little as @xmath88 . the change of @xmath0 is the decisive factor to change the energetics . the @xmath87 production varies by about 14 % and the velocities of the various chemical layers differ by up to 1500 km / sec . a change in the metallicity hardly affects the overall structure of the progenitor . as already discussed in detail in hwt98 , its main effect is a change in the production of @xmath127 in the outer layers of incomplete si burning . @xmath0 alters the @xmath1 relation which may be offset by up to @xmath5 . in addition , @xmath0 changes the flux ratio between maximum light and the radioactive tail , and it alters the photospheric expansion velocities @xmath128 measured by the doppler shift of lines . e.g. a change in @xmath129 by @xmath5 is coupled to a decrease in @xmath130 at maximum light by @xmath131 . note that a change in the central density @xmath42 of the wd has a similar effect on @xmath129 but with the opposite sign for @xmath132 ( hflich , 2001 ) . in principle , this allows to decide whether differences in @xmath129 between sn with similar @xmath1 are related to a change in the progenitor or the central density at the thermonuclear runaway which is sensitive to the accretion rate . in contrast to @xmath0 , the metallicity z hardly changes the light curve shapes ( @xmath133 ) . it alters the line blocking by iron group elements at the photosphere mainly in the uv , u and b but hardly in v ( hwt98 ) . in the models presented here , b - v becomes systematically bluer with decreasing z ( up to @xmath134 ) . because b - v is the basic color index used to correct for interstellar extinction , the metallicity effect can systematically alter the estimates for the absolute brightness by up to @xmath5 . * @xmath135 : * at the example of a progenitor with @xmath136 and z=0.001 , we have tested the influence of the low nuclear rate @xmath119 on the outcome . using the lower rate suggested by caughlan & fowler ( 1988 ) instead caughlan et al . ( 1985 ) results in more energetic explosions because @xmath22 increases by a factor of @xmath137 . the rise times to maximum light are 15.3d instead of 18.0d . from detailed observations of nearby supernovae , riess et al . ( 1999 ) find the following relation between the rise time @xmath138 and the maximum brightness @xmath139}\ ] ] the theoretical lcs peak at @xmath140 and @xmath141 for @xmath142 and @xmath143 , respectively . from the empirical fit of riess , we would expect a rise time between 17.6 and 18.4 days which favors the high rate @xmath119 of caughlan et al . note that the uncertainties in absolute values for the rise times are @xmath56 1 to 2 days ( hwt98 ) . * constraints on the progenitor : * empirically , the @xmath1 has been well established with a rather small statistical error @xmath144 ( @xmath145 : riess et al . 1996 ; @xmath146 : schmidt et al . ( 1998 ) ; @xmath147 phillips 1999 ; @xmath148 riess et al . 1999 ; @xmath149 perlmutter et al . 1999a ) . from theoretical models , a spread of 0.3 to @xmath150 can be expected ( hflich et al . 1996 ) . this may imply a correlation between free model parameters , namely the properties of the progenitors , the central density or the transition density @xmath20 . in this study , we find a spread in @xmath1 of about @xmath5 for progenitors with @xmath0 between @xmath151 to @xmath152 . this may suggest a more narrow range in @xmath0 for realistic progenitors . from the fiducial rise time , progenitors with @xmath0 @xmath153 are favored . this number should be regarded as a hint because uncertainties in the lc models . another constraint can be obtained from the observed spread in the rise - time to decline relation . riess et al . ( 1999 ) find a spread in @xmath138 of @xmath154 days whereas our models show a spread of @xmath56 1.7 days for @xmath155 . to be consistent with the observations , the range of main sequence masses has to be reduced by a factor of two . this range is an upper limit because additional variations in the population of sneia such as explosion scenarios or the central density of the wd at the time of ignition will likely result in lower correlations . * the cosmological equation of state : * in the following , we want to discuss our results in context of sne ia as probes for cosmology and for the determination of the cosmological equation of state . we limit the discussion to the effects due to different progenitors . for a discussion of other systematic effects such as grey dust , gravitational lensing , the influence of a change in the importance of different possible scenarios ( e.g merger vs. @xmath121 models ) etc . we want to refer to the growing literature in this field ( e.g. schmidt et al . 1998 , hwt98 , perlmutter et al . 1999a ) . recently , there has been strong evidence for a positive cosmological constant ( e.g. perlmutter et al . 1999a , riess et al . this evidence is based on observations that sne ia in the redshift range between 0.5 to 1.2 appear to be dimmer by about @xmath156 for redshifts between 0.5 to 0.8 which is comparable to the variations produced by different progenitors . however , both the internal spread in @xmath1 ( see above ) and the similarity in @xmath1 between the local sne ia and the high - z sample ( @xmath157 , aldering et al . 2000 ) limit the likely range of models and , consequently , evolutionary effects to @xmath158 up to redshifts of @xmath159 . in addition , we do not expect a drastic change in the metallicity between local and supernovae at @xmath160 . taking the linear dependence of b - v on the metallicity ( fig . [ dep ] ) and realistic ranges for z , also reddening of non - grey dust will not change the conclusion on the need for a positive cosmological constant . as discussed in the introduction , the quest for the nature of the dark energy is one of the central questions to be addressed in future ( e.g. white 1998 , perlmutter et al . 1999b , albrecht & weller 2000 , ostriker & steinhardt 2001 ) . for @xmath161 , we can expect both very low metallicities and a significant change of the typical @xmath0 . from this study , systematic effects due to different progenitors are limited to @xmath162 . therefore , without further corrections for the progenitor evolution , some of the alternatives for the nature of the dark energy may be distinguished without correction for evolution . however , for a more detailed analysis , an accuracy of about @xmath163 ( weller & albrecht 2001 ) is required . in this paper , we have shown how a combination of spectral and lc data or different characteristics of the lc can help to achieve this goal . * limitations : * finally , we also want to mention the limitations of this study . qualitatively , our results on the z dependence agree with a previous study ( hflich et al . 2000 ) which was based on a progenitor with @xmath164 calculated by nomoto s group ( umeda et al . the relation between @xmath22 and the offset in the @xmath1 relation has been confirmed . however , the influence of z on @xmath22 was found to be about twice as large , and the central c concentrations are systematically larger . the differences point towards a general problem . the details of the central structure and evolution in the convective he burning core depend sensitively on the treatment of convection , semi - convection , overshooting , and breathing pulses ( lattanzio 1991 , schaller et al . 1991 , bressan et al . 1993 , vassiliadis & wood 1993 ) . for a detailed discussion , see domnguez et al . in particular , the central c concentration may vary between 0.1 and 0.5 . we want to note , that our value is consistent with direct measurements of the central c / o ratio found by the analysis of pulsational modes of wds ( metcalfe et al . these uncertainties will affect the efficiency to separate the contribution of @xmath0 and z , respectively , i.e. the reason for @xmath22 but not the observable relations for lcs and spectra . we provide limits on the size of evolutionary effects due to @xmath0 and metallicity of the progenitor ( @xmath9 ) . other evolutionary effects may be due to a systematic change in the dominant progenitor scenario ( e.g. mergers vs. single degenerate ) or the typical separation in binary systems which contribute to the sne ia at a given time . the latter may alter the accretion rates and , consequently , the central densities of the wd at the time of the thermonuclear runaway . we did not consider the effect of the progenitor and of its pre - conditioning just prior to the explosion on the propagation of the burning front and , in particular , on the deflagration to detonation transition or , alternatively , the phase of transition from a slow to a very fast deflagration ( hillebrandt , 1999 private communication ) . although the model parameters have been chosen to allow for a representation of typical " sne ia , more comprehensive studies and detailed fitting of actual observations are needed e.g. to detangle effects due to a change in the ignition density vs. the progentor . in particular , observations of local sne ia have to be employed to narrow down and test for the proposed range of flux ratio between maximum and tail , and its relation to the expansion velocities . * acknowledgments : * this work has been supported by nasa grant nag5 - 7937 , the murst italian grant cofin2000 , by the mec spanish grant pb96 - 1428 , by the andalusian grant fqm-108 and it is part of the italy - spain integrated action ( murst - mec agreement ) hi1998 . the calculations for the explosion and light curves were done on a beowulf - cluster financed by the john w. cox - fund of the department of astronomy at the university of texas . albrecht a. , weller j. 2000 , american astronomical society , 197 , 61 aldering g. , knop , p. , nugent p. 2000 , 119 , 2110 becker s.a . , iben i. jr . 1979 , 232 , 831 becker s.a . , iben i. jr . 1980 , 273 , 111 branch d. , romanishing w. , baron e. 1996 , 465 , 73 & e467 , 473 bressan a. , fagotto f. , bertelli g. , chiosi c. 1993 , a&as 100 , 647 buchmann l. 1996 , 468 , l127 buchmann l. 1997 , 479 , l153 caputo f. , castellani , v. , chieffi a. , pulone l. , tornamb a. 1989 , 340 , 241 caughlan g.r , fowler a.m. , harris m.j . , zimmerman b.a . 1985 , at . data nucl . dat a tables 32 , 197 caughlan g.r , fowler a.m. 1988 , atomic data nucl . table , 40 , 283 chieffi a. , dominguez i. , limongi m. , straniero o. 2001 , apj , in press chieffi a. , straniero o. 1989 , 71 , 47 chieffi a. , limongi m. , straniero o. 1998 , 502 , 737 collela , p. , & woodward , p.r . 1984 , j.comp.phys . , 54 , 174 domnguez i. , chieffi a. , limongi m. , straniero o. 1999 , 524 , 226 domnguez i. , & hflich p. 2000 , apj 528 , 854 domnguez i. , hflich p. , straniero o. , wheeler c.j . 2000 , in : _ nuclei in the cosmos _ , eds . n. pranzos & s. harissulos , edition frontiers , paris , p. 259 domnguez i. 1994 , in : abstracts of the conferences on thermonuclear supernovae , eds . p. ruiz - lapuente , r. canal & j. isern . domnguez i. 1991 , phd thesis , university of barcelona , isbn 84.7875.736.8 ezer d. , cameron a. 1972 , apss 14 , 399 fisher a. , branch d. , hflich p. , khokhlov a. 1998 , 494 , 47 garnavich p.m. et al . 1998 , apj 509 , 74 gerardy , c.l , hflich p. , fesen r. , & wheeler j.c . 2001 , apj in preparation hamuy m. , phillips m.m , schommer r.a . suntzeff n.b , maza j. , aviles a. 1996a , 112 , 2391 hamuy m. , phillips m.m . , suntzeff n.b . , schommer r.a . , maza j. , aviles a. 1996b , 112 , 2398 hamuy m. et al . 2000 , 120 , 1479 harkness r. 1987 , in : 13th texas symposium on relativistic astrophysics , world scientific publishing co. , p. 413 hatano k. , branch d. , lentz e.j . , baron e. , filippenko a.v . , garnavich p. 2000 , apj 543l , 49 hillebrandt w. 1999 , private communication at the aspen workshop on sne ia hflich p. 2001 , , in preparation hflich p. , nomoto , k. , umeda h. , wheeler j.c . 2000 , 528 , 590 hflich p. , wheeler j.c . , thielemann f.k 1998 , 495 , 617 [ hwt98 ] hflich p. , khokhlov a. 1996 , 457 , 500 [ hk96 ] hflich , p. , khokhlov a. , wheeler j.c . , phillips m.m . , suntzeff n.b . , hamuy m. 1996a , apj 472l , 81 hflich , p. ; dominik , c. ; khokhlov , a. ; mller , e. ; wheeler , j.c . 1996b , @xmath165 texas symposium on relativistic astrophysics , annals of the new york academy of science 759 , 348 hflich , p. 1995 , 443 , 89 [ h95 ] hflich p. , khokhlov a. , wheeler j.c . 1995 , 444 , 211 [ hkw95 ] hflich p. , khokhlov a. , wheeler j.c . 1993 , a&a , 270 , 223 hoyle p. , fowler w.a . 1960 , 132 , 565 iben i.j , tutukov a.v . 1984 , apjs 54 , 335 iben i.j 1972 , 178 , 433 ivanov v.d . , hamuy , m. , pinto , p.a . , 2000 , 542 , 588 iwamoto , k. , brachwitz , f. , nomoto , k. , kishimoto , n. , umeda , h. , hix w.r . , thielemann f.k . 1999 , apjs 125 , 439 khokhlov a. 2001 , , in press & astro - ph/0008463 khokhlov a. 1995 , 449 , 695 khokhlov , a. , mller , e. , hflich p. 1993 , a&a 270 , 223 khokhlov a. 1991 , 245 , 114 lattanzio j. 1991 , apjs 76 , 215 lentz e.j . , baron e. , branch d. , hauschildt p. , nugent p.e . 2000 , apj 530 , 966l metcalfe t.s . , winget d.e . , charbonneau p. 2001 , , submitted niemeyer , j. c. , & hillebrandt , w. 1995 , apj 452 , 779 nomoto , k. 1980 , apj 248 , 798 nomoto k. , thielemann f .- k . , yokoi k. 1984 , 286 , 644 nugent , p. , baron , e. , hauschildt , p. , & branch . , d. 1997a , 485 , 812 ostriker p. , steinhardt p.j . 2001 , scientific american , jan . 2001 , p. 47 paczyski b. 1985 , in _ cataclysmic variables and low - mass x - ray binaries _ , eds . lamb & j. patterson , ( dordrecht : reidel ) p. 1 perlmutter , s. , et al . 1999a , apj 517 , 565 perlmutter s. , turner m.s . , white m. 1999b , physical review letters 83 , 670 perlmutter c. et al . 1997 , 483 , 565 perlmutter c. et al . 1995 , apj 440 , l95 phillips , m.m . 1993 , 413 , 105 phillips , m.m . , lira p. , sunzeff n.b . , schommer r.a . , hamuy m. , maza j. 1999 , 118 , 1766 renzini a. , fusi pecci f. 1988 , 26 199 riess a.g . , press w.h . , kirshner r.p . 1995 , 438 , l17 riess a.g . , press w.h . , kirshner r.p . 1996 , 473 , 88 riess a.g . , et al . 1998 , 116 , 1009 riess a.g . , et al . 1999 , 118 , 2675 saha a. et al . 1997 , apj 486 , 1 schaller g. , shaerer d. , meynet g. , maeder a. 1992 , a&as 96 , 269 schmidt , b. p. , et al . 1998 , apj 507 , 46 straniero , o. chieffi , a. and limongi m. 1997 , 490 , 425 thielemann f.k . , nomoto k. , hashimoto m. 1996 , 460 , 408 tornamb a. , chieffi . a. 1986 , , 220 , 529 umeda , h. , nomoto , k. 1999 , 513 , 861 vassiliadis e. , wood p.r . 1993 , 413 , 641 wang l. , hflich p. , wheeler j.c . 1997 , 483 , 29 webbink , r. f. 1984 , apj 277 , 355 weller j. , albrecht a. 2001 , _ opportunities for future supernova studies of cosmic acceleration _ , astro - ph/0008314 whelan j. , iben i.jr . 1973 , apj 186 , 1007 wheeler j. c. , hflich p. , harkness r. p. , spyromilio j. 1998 , apj 496 , 908 white m. 1998 , apj 506 , 495 weidemann v. 1987 , a&a 188 , 74 woosley s.e . , weaver t.a . 1994 , 423 , 371 woosley s.e . , weaver t.a . 1986 , 24 , 205 woosley s. e. , weaver t.a . , taam r.e . 1980 , in : type i supernovae , ed . j.c.wheeler , austin , u.texas , p.96 yamaoka h. , nomoto k. , shigeyama t. , & thielemann f. 1992 , apj 393 , 55 llcccccc z=0.02 & 1p5z22 & 1.5 & 0.55 & 0.21 & 0.27 & 0.75 & 0.589 y=0.28 & 3p0z22 & 3.0 & 0.57 & 0.21 & 0.28 & 0.76 & 0.584 & 5p0z22 & 5.0 & 0.87 & 0.29 & 0.46 & 0.72 & 0.561 & 7p0z22 & 7.0 & 0.99 & 0.28 & 0.70 & 0.60 & 0.516 & & & & & & & z=10@xmath166 & 1p5z13 & 1.5 & 0.59 & 0.24 & 0.31 & 0.76 & 0.587 y=0.23 & 3p0z13 & 3.0 & 0.77 & 0.26 & 0.39 & 0.74 & 0.567 y=0.23 & 5p0z13 & 5.0 & 0.90 & 0.29 & 0.58 & 0.66 & 0.541 & 6p0z13 & 6.0 & 0.98 & 0.29 & 0.71 & 0.60 & 0.522 & & & & & & & & low rate & & & & & & & 3p0z13lr & 3.0 & 0.76 & 0.51 & 0.38 & 1.22 & 0.620 & & & & & & & z=10@xmath167 & 3p0z14 & 3.0 & 0.80 & 0.27 & 0.41 & 0.73 & 0.568y=0.23 & 5p0z14 & 5.0 & 0.90 & 0.29 & 0.58 & 0.65 & 0.541 & 6p0z14 & 6.0 & 0.99 & 0.28 & 0.72 & 0.59 & 0.511 & & & & & & & z=10@xmath73 & 5p0z00 & 5.0 & 0.89 & 0.32 & 0.49 & 0.70 & 0.549 y=0.23 & 7p0z00 & 7.0 & 0.99 & 0.31 & 0.59 & 0.62 & 0.525 ccccccc he & c & o & ne & na & mg & si 6.62e-04 & 1.19e-02 & 9.18e-02 & 5.34e-03 & 4.91e-05 & 1.82e-02 & 2.61e-01 & & & & & & p & s & cl & ar & k & ca & sc 2.51e-05 & 1.59e-01 & 4.00e-06 & 3.28e-02 & 1.99e-06 & 3.44e-02 & 1.02e-08 & & & & & & ti & v & cr & mn & fe & co & ni 1.61e-05 & 9.07e-04 & 6.56e-04 & 2.68e-02 & 6.57e-01 & 6.15e-03 & 6.47e-02
|
our study is based on the delayed detonation scenario with specific parameters which give a good account of typical light curves and spectra . a change in the central density of the exploding wd has similar effects but produces the opposite dependency between the brightness to tail ratio and and , therefore , can be separated . the narrow spread in the fiducial rise - time to decline relation in local sne ia restricts the range of main sequence masses to a factor of 2 . the upper limit of 1 day for the difference between the local and distance sample support the need for a positive cosmological constant . the size of evolutionary effects are small ( ) but are absolutely critical for the reconstruction of the cosmological equation of state .
|
detailed stellar evolution calculations have been performed to quantify the influence of the main sequence mass and the metallicity z of the progenitor on the structure of the exploding wd which are thought to be the progenitors of sne ia . in particular , we study the effects of progenitors on the brightness decline relation which is a corner stone for the use of sne ia as cosmological yard - stick . both the typical and z can be expected to change as we go back in time . we consider the entire range of potential progenitors with 1.5 to 7 and metallicities between z=0.02 to . our study is based on the delayed detonation scenario with specific parameters which give a good account of typical light curves and spectra . based on the structures for the wd , detailed model calculations have been performed for the hydrodynamical explosion , nucleosynthesis and light curves . the main sequence mass has been identified as the decisive factor to change the energetics of the explosion and , consequently , dominates the variations in the rise - time to decline relation of light curves . has little effect on the color index b - v . for similar decline rates , the flux at maximum brightness relative to the flux on the radioactive tail decreases systematically with by about . this change goes along with a reduction of the photospheric expansion velocity by about 2000 km / sec . a change in the central density of the exploding wd has similar effects but produces the opposite dependency between the brightness to tail ratio and and , therefore , can be separated . the metallicity alters the isotopic composition of the outer layers of the ejecta . selective line blanketing at short wavelengths decreases with z and changes systematically the intrinsic color index b - v by up to , and it alters the fluxes in the u band and the uv . the change in b - v is critical if extinction corrections are applied . the offset in the calibration of is not monotonic in z and , in general , remains . we use our results and recent observations to constrain the progenitors , and to discuss evolutionary effects of sne ia with redshift . the narrow spread in the fiducial rise - time to decline relation in local sne ia restricts the range of main sequence masses to a factor of 2 . the upper limit of 1 day for the difference between the local and distance sample support the need for a positive cosmological constant . the size of evolutionary effects are small ( ) but are absolutely critical for the reconstruction of the cosmological equation of state .
|
cond-mat0412123
|
i
|
the casimir force @xcite between uncharged metallic plates attracts considerable attention as a macroscopic manifestation of the quantum vacuum @xcite . with the development of microtechnologies that routinely allow control of the separation between bodies smaller than 1 @xmath0 the force became a subject of systematic experimental investigation . modern high precision experiments were made using different techniques such as torsion pendulum @xcite , atomic force microscope ( afm ) @xcite , microelectromechanical systems ( mems ) @xcite and different geometrical configurations : sphere - plate @xcite , plate - plate @xcite , crossed cylinders @xcite . in most cases the bodies were covered with gold evaporated or sputter deposited to the thickness of 100 - 200 nm . in the only plate - plate configuration experiment @xcite the bodies were covered with chromium . one of the bodies was covered with copper in the mems experiment @xcite . the root mean square ( rms ) roughness of deposited metal films was varied from 30 - 40 nm as in the mems experiments @xcite to nearly atomically flat surfaces as in the crossed cylinders experiment @xcite . relatively low precision , 15% , in the force measurement was reached for the plate - plate configuration @xcite because of the parallelism problem . in the torsion pendulum experiment @xcite the force was measured with the accuracy of 5% . in the experiments @xcite it was found with 1% precision . in the most precise up to date experiment @xcite the improvement was achieved due to the use of the dynamical method @xcite . additionally the change in the resonance frequency of the mechanical oscillator was measured using the phase jump instead of the resonance behavior of the amplitude . in this way the force between gold and copper covered bodies was found with a relative accuracy of 0.25% . to draw any conclusion from the experiments one has to predict the force theoretically with the precision comparable with the experimental errors . it is a real challenge to the theory because the force is material dependent ; it is very difficult to fix the material properties since different uncontrolled factors are involved . in its original form , the casimir force @xcite @xmath1 was calculated between the ideal metals . it depends only on the fundamental constants and the distance between the plates @xmath2 . the force between real materials was found for the first time by lifshitz @xcite . the material properties enter in the lifshitz formula via the dielectric function @xmath3 at imaginary frequencies @xmath4 . correction to the expression ( [ fc ] ) is significant at the separations between bodies smaller than 1 @xmath0 . to calculate the force the lifshitz formula is used with the optical data taken from the handbooks @xcite . the data are available only up to some low - frequency cutoff @xmath5 . for good metals such as @xmath6 @xmath7 @xmath8 the data can be extrapolated to the lower frequencies @xmath9 with the drude dielectric function @xmath10 which includes two parameters : the plasma frequency @xmath11 and the relaxation frequency @xmath12 . these parameters can be extracted from the optical data at the lowest accessible frequencies . the exact values of the drude parameters are very important for the precise evaluation of the force . when two plates are separated by the distance @xmath2 , one can introduce a characteristic imaginary frequency @xmath13 of electromagnetic field fluctuations in the gap . the important fluctuations are those that have frequency @xmath14 . for @xmath15 the characteristic frequency is in the near infrared region . this qualitative consideration is often continued to the real frequency domain @xcite with the real characteristic frequency @xmath16 . because @xmath17 is in the near infrared it is stated that the low - frequency behavior of the dielectric function is not significant for the casimir force . it is not difficult to see that this consideration is wrong @xcite . the force depends on the dielectric function at imaginary frequencies @xmath3 . with the help of the dispersion relation @xmath18 can be expressed via the observable function @xmath19 : @xmath20 because for metals @xmath21 is large at low frequencies , the main contribution in the integral in eq . ( [ k - k ] ) is given by the low frequencies even if @xmath22 is in the visible range . the characteristic frequency is well defined on the imaginary axis @xmath23 but it is wrong to introduce it in the real frequency domain . this aspect is carefully discussed in this paper . two different procedures to get the drude parameters were discussed in the literature . in the first approach @xcite the plasma frequency was fixed independently on the optical data assuming that each atom gives one conduction electron ( for @xmath24 ) with the effective mass equal to the mass of free electron . the optical data at the lowest frequencies were used then to find @xmath25 with the help of eq . ( [ drude ] ) . in this way the parameters @xmath26 and @xmath27 have been found . in the second approach @xcite no assumptions were made and both of the parameters were extracted from the low - frequency optical data by fitting them with eq . ( [ drude ] ) . when the best data from ref . @xcite were used the result @xcite was close to that found by the first approach , but using different sources for the optical data collected in ref . @xcite an appreciable difference was found @xcite . this difference was attributed to the defects in the metallic films which appear as the result of the deposition process . it was indicated that the density of the deposited films is typically smaller and the resistivity is larger than the corresponding values for the bulk material . however , it was difficult to make definite conclusions without analysis of significant amount of the data on the deposited metallic films . in this paper this analysis is performed and it is proven that there is a genuine sample dependence of optical properties of gold . first of all it means that the gold films prepared in different ways will have different drude parameters . the change in the optical properties will lead to the sample dependence of the casimir force , which is carefully investigated in this paper . the main ideas were outlined before @xcite with the conclusion that the absolute value of the casimir force between gold films is of about 2% smaller than between bulk material at the smallest separations investigated in the experiments @xcite . here we come to the same conclusion but present much more details and facts . the paper is organized as follows . in sec . [ sec2 ] it is explained qualitatively and quantitatively why the exact values of the drude parameters are crucial for the precise calculation of the casimir force . existing optical data for gold are reviewed and analyzed in sec . it is unambiguously demonstrated that there is the sample dependence of the optical properties of gold films . in sec . [ sec4 ] the dielectric function of the perfect gold single crystal is discussed and the upper limit on the casimir force is deduced . in sec . [ sec5 ] a simple two parameter model is proposed allowing us to take into account the main reasons for the sample dependence of the casimir force : voids and grains in the films . the volume fraction of voids and the mean grain size are bounded from the available optical data and the smallest sample correction to the force is estimated . in sec . [ sec6 ] necessary experimental information on the films that has to be known for the precise evaluation of the force is briefly discussed . comparison between the afm experiment and prediction of this paper is given . our conclusions are summarized in the last section .
|
it is argued that the precise values of the drude parameters are crucial for accurate evaluation of the force . the dielectric function of perfect single crystalline gold is discussed . it is used to establish the upper limit on the absolute value of the force . it is demonstrated that the force between films is smaller than that between bulk samples mainly due to the presence of voids and electron scattering on the grain boundaries in the films . the minimal reduction of the force is estimated as 2% for the smallest separations investigated in the most precise experiments .
|
difference between bulk material and deposited film is shown to have an appreciable influence on the casimir force . analysis of the optical data on gold films unambiguously demonstrates the sample dependence : the dielectric functions of the films deposited in different conditions are different on the level that can not be ignored in high precision prediction of the force . it is argued that the precise values of the drude parameters are crucial for accurate evaluation of the force . the dielectric function of perfect single crystalline gold is discussed . it is used to establish the upper limit on the absolute value of the force . it is demonstrated that the force between films is smaller than that between bulk samples mainly due to the presence of voids and electron scattering on the grain boundaries in the films . the minimal reduction of the force is estimated as 2% for the smallest separations investigated in the most precise experiments . the other sample effects can reduce the force further but the correction is expected to be smaller than 1% .
|
1310.2926
|
i
|
distance covariance ( dcov ) and distance correlation characterize multivariate independence for random vectors in arbitrary , not necessarily equal dimension . in this work we focus on the open problem of _ partial distance correlation_. our intermediate results include methods for applying distance correlation to dissimilarity matrices . the distance covariance , denoted @xmath0 , of two random vectors @xmath1 and @xmath2 characterizes independence ; that is , @xmath3 with equality to zero if and only if @xmath1 and @xmath2 are independent . this coefficient is defined by a weighted @xmath4 norm measuring the distance between the joint characteristic function ( c.f . ) @xmath5 of @xmath1 and @xmath2 , and the product @xmath6 of the marginal c.f.s of @xmath1 and @xmath2 . if @xmath1 and @xmath2 take values in @xmath7 and @xmath8 , respectively , @xmath0 is the non - negative square root of @xmath9 where @xmath10 . the integral exists provided that @xmath1 and @xmath2 have finite first moments . note that feuerverger ( 1993 ) @xcite proposed a bivariate test based on this idea and applied the same weight function @xmath11 , where it may have first appeared . the following identity is established in szkely and rizzo ( * ? ? ? * theorem 8 , p. 1250 ) . let @xmath12 , @xmath13 , and @xmath14 be independent and identically distributed ( iid ) , each with joint distribution @xmath12 . then @xmath15 provided that @xmath1 and @xmath2 have finite first moments . in section [ s4 ] an alternate version of ( [ edcov ] ) is defined for @xmath1 and @xmath2 taking values in a separable hilbert space . that definition and intermediate results lead to the definition of partial distance covariance . we summarize a few key properties and computing formulas below for easy reference . the distance correlation ( dcor ) @xmath16 is a standardized coefficient , @xmath17 , that also characterizes independence : @xmath18 for more details see @xcite and @xcite . properties , extensions , and applications of distance correlation have been discussed in the recent literature ; see e.g. dueck et al . @xcite , lyons @xcite , kong et al . @xcite , and li , zhong , and zhu @xcite . however , there is considerable interest among researchers and statisticians on the open problem of a suitable definition and supporting theory of partial distance correlation . among the many potential application areas of partial distance correlation are variable selection ( see example [ ex7 ] ) and graphical models ; see e.g. wermuth and cox @xcite for an example of work that motivated the question in that context . in this work we introduce the definition of partial distance covariance ( pdcov ) and partial distance correlation ( pdcor ) statistics and population coefficients . first , let us see why it is not straightforward to generalize distance correlation to partial distance correlation in a meaningful way that preserves the essential properties one would require , and allows for interpretation and inference . one could try to follow the definitions of the classical partial covariance and partial correlation that are based on orthogonal projections in a euclidean space , but there is a serious difficulty . orthogonality in case of partial distance covariance and partial distance correlation means independence , but when we compute the orthogonal projection of a random variable onto the condition variable , the `` remainder '' in the difference is typically not independent of the condition . alternately , the form of sample distance covariance ( definition [ defvn ] ) may suggest an inner product , so one might think of working in the hilbert space of double centered distance matrices ( [ dcenter ] ) , where the inner product is the squared distance covariance statistic ( [ e : anxy ] ) . here we are facing another problem : what would the projections represent ? the difference @xmath19 of double centered distance matrices is typically not a double centered distance matrix of any sample . this does not affect formal computations , but if we can not interpret our formulas in terms of samples then inference becomes impossible . to overcome these difficulties while preserving the essential properties of distance covariance , we finally arrived at an elegant solution which starts with defining an alternate type of double centering called `` @xmath20-centering '' ( see definition [ def : tildea ] and proposition [ p : unbiased ] below ) . the corresponding inner product is an unbiased estimator of squared population distance covariance . in the hilbert space of `` @xmath20-centered '' matrices , all linear combinations , and in particular projections , are zero diagonal @xmath20-centered matrices . we prove a representation theorem that connects the orthogonal projections to random samples in euclidean space . methods for inference are outlined and implemented , including methods for non - euclidean dissimilarities . definitions and background for dcov and dcor are summarized in section 2 . the partial distance correlation statistic is introduced in section 3 , and section 4 covers the population partial distance covariance and inference for a test of the hypothesis of zero partial distance correlation . examples and applications are given in section 5 , followed by a summary in section 6 . appendix a contains proofs of statements .
|
distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension . properties , extensions , and applications of distance correlation have been discussed in the recent literature , but the problem of defining the partial distance correlation has remained an open question of considerable interest . the problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space . for the definition of partial distance correlation we introduce a new hilbert space where the squared distance covariance is the inner product . we define the partial distance correlation statistics with the help of this hilbert space , and develop and implement a test for zero partial distance correlation .
|
distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension . properties , extensions , and applications of distance correlation have been discussed in the recent literature , but the problem of defining the partial distance correlation has remained an open question of considerable interest . the problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space . for the definition of partial distance correlation we introduce a new hilbert space where the squared distance covariance is the inner product . we define the partial distance correlation statistics with the help of this hilbert space , and develop and implement a test for zero partial distance correlation . our intermediate results provide an unbiased estimator of squared distance covariance , and a neat solution to the problem of distance correlation for dissimilarities rather than distances .
|
1103.3268
|
i
|
directly extending our previous publication ( ymm09 ) , we continue to study massless particles moving under the gravitational influence of density fluctuations due to saturated magneto - rotational turbulence in a local , isothermal , keplerian gas disk . we include linearized vertical gravity from the host star and thus vertical stratification of the gas disk . for comparison , the conditions in the mid - plane of the vertically stratified disks are exactly the same as those in the unstratified disks of ymm09 . in order to accurately measure the gravitational effect of the turbulent gas , we separate the gas density @xmath8 into two components : the basic state @xmath37 for the vertical hydrostatic equilibrium ( equation ( [ e : rho_0 ] ) ) and the density deviation @xmath194 from this basic state . we use the exact gravitational acceleration due to the basic state ( equation ( [ e : g_0 ] ) ) and only solve the poisson equation for the gravitational potential due to the density deviation ( equation ( [ e : poisson ] ) ) . we emphasize that since the poisson equation is linear in density , this approach does not assume small density fluctuations . furthermore , we implement isolated boundary conditions in the vertical direction and thus any density fluctuation outside the vertical computational domain is neglected . by imposing a weak , uniform external magnetic field , we maintain a constant level of saturated magneto - rotational turbulence in the disk mid - plane . several turbulence properties demonstrate convergence with both resolution up to 64 points per disk scale height @xmath22 and horizontal box size up to 16@xmath22 . the @xcite @xmath81 parameter in the mid - plane of our models is controlled at the level of @xmath14810@xmath195 . however , even though the properties of the turbulent gas appear numerically convergent , the dynamics of massless particles moving under the gravity of this turbulent gas does not converge with the horizontal box size @xmath1 . the larger the horizontal box size , the stronger the gravitational effect of the gas on the particles . specifically , the evolution of the orbital radius , the eccentricity , and the horizontal velocity dispersion of the particles is roughly linearly dependent on @xmath1 up to 16@xmath22 . this trend was also found in our unstratified models ( ymm09 ) , and we find consistency between the unstratified models and the mid - plane of the stratified models . in contrast to the horizontal components of the particle movement , we find that the evolution of the inclination and the vertical velocity dispersion is not significantly affected by @xmath1 . the dependence of particle dynamics on the horizontal box size can be traced back to the density structure of the gas . consistent with the large - box models studied by @xcite , the longest wavelength fourier mode dominates the density spectrum along the radial direction in our models . furthermore , we find that the longest wavelength fourier mode in the azimuthal direction also strongly influences the particle dynamics , leading to the diffusive evolution of both the orbital radius and the eccentricity of the particles . the spectral amplitudes of these longest wavelength modes are roughly constant against the horizontal box size . using a simple single - mode analysis , we show that the linear dependence of the particle response is a natural outcome of these findings for the density spectrum . correlation of particle dynamics with box size poses a major difficulty for the interpretation of local - shearing - box models involving gravitational physics of magneto - rotational turbulence . we can nevertheless conjecture that @xmath185 , where @xmath2 is the distance of the box center to the host star , might be a natural scale of choice for a local model to approach reality . if this conjecture holds , we find that our previous conclusions in ymm09 on the unimportance of radial diffusive migration for protoplanets as well as the survivability of kilometer - sized planetesimals under collisional destruction may still be valid . ultimately , high - resolution global disk models and detailed comparisons with large - box local models might be necessary to settle this issue . we thank the anonymous referee for critical comments that significantly improved the rigor and clarity of this paper , and thank charles gammie , anders johansen , roman rafikov , clment baruteau , martin pessah , richard nelson , and oliver gressel for their insightful discussion on this research . this research was supported in part by the perimeter institute for theoretical physics . resources supporting this work were provided by the nasa high - end computing ( hec ) program through the nasa advanced supercomputing ( nas ) division at ames research center . partial support of this work was provided by the nasa origins of solar systems program under grant nnx07ai74 g . rcccccc 2@xmath882@xmath884 & 16 & 0.9@xmath1960.2 & 5.3@xmath1962.6 & 1.9@xmath1960.8 & 0.5@xmath1960.3 & 1.2@xmath1960.4 + 4@xmath884@xmath884 & 16 & 1.4@xmath1960.2 & 5.9@xmath1961.8 & 2.4@xmath1960.7 & 0.7@xmath1960.2 & 1.5@xmath1960.4 + 8@xmath888@xmath884 & 16 & 1.6@xmath1960.3 & 5.2@xmath1960.8 & 2.2@xmath1960.3 & 0.7@xmath1960.1 & 1.4@xmath1960.2 + 16@xmath8816@xmath884 & 16 & 1.7@xmath1960.3 & 5.1@xmath1960.4 & 2.2@xmath1960.2 & 0.7@xmath1960.1 & 1.4@xmath1960.1 + 2@xmath882@xmath884 & 32 & 1.0@xmath1960.3 & 7.6@xmath1963.4 & 2.8@xmath1961.0 & 0.6@xmath1960.3 & 1.6@xmath1960.5 + 4@xmath884@xmath884 & 32 & 1.2@xmath1960.3 & 6.3@xmath1961.8 & 2.6@xmath1960.6 & 0.7@xmath1960.2 & 1.6@xmath1960.4 + 8@xmath888@xmath884 & 32 & 1.5@xmath1960.4 & 5.1@xmath1960.5 & 2.3@xmath1960.2 & 0.6@xmath1960.1 & 1.4@xmath1960.1 + 16@xmath8816@xmath884 & 32 & 1.3@xmath1960.1 & 5.1@xmath1960.4 & 2.3@xmath1960.1 & 0.6@xmath1960.1 & 1.4@xmath1960.1 + 2@xmath882@xmath884 & 64 & 1.6@xmath1960.6 & 17.1@xmath1969.4 & 5.5@xmath1963.0 & 0.9@xmath1960.6 & 3.0@xmath1961.5 + 4@xmath884@xmath884 & 64 & 1.4@xmath1960.3 & 8.1@xmath1961.6 & 3.4@xmath1960.7 & 0.8@xmath1960.2 & 2.0@xmath1960.4 + 8@xmath888@xmath884 & 64 & 1.6@xmath1960.4 & 7.0@xmath1960.9 & 3.0@xmath1960.4 & 0.6@xmath1960.1 & 1.7@xmath1960.2 rcccccccc 2@xmath882@xmath884 & 16 & 2.4(-4 ) & 2.4(-4 ) & 2.0(-4 ) & 3.6(-4 ) & 2.1(-4 ) & 8.3(-5 ) & 1.2(-4 ) + 4@xmath884@xmath884 & 16 & 4.4(-4 ) & 3.5(-4 ) & 2.9(-4 ) & 5.0(-4 ) & 3.0(-4 ) & 2.9(-4 ) & 1.7(-4 ) + 8@xmath888@xmath884 & 16 & 1.3(-3 ) & 7.4(-4 ) & 2.8(-4 ) & 1.1(-3 ) & 2.9(-4 ) & 2.6(-3 ) & 3.7(-4 ) + 16@xmath8816@xmath884 & 16 & 3.0(-3 ) & 1.6(-3 ) & 2.4(-4 ) & 2.4(-3 ) & 3.3(-4 ) & 1.4(-2 ) & 8.0(-4 ) + 2@xmath882@xmath884 & 32 & 2.6(-4 ) & 2.7(-4 ) & 2.6(-4 ) & 3.8(-4 ) & 2.2(-4 ) & 1.0(-4 ) & 1.3(-4 ) + 4@xmath884@xmath884 & 32 & 3.7(-4 ) & 3.0(-4 ) & 2.5(-4 ) & 4.3(-4 ) & 2.3(-4 ) & 2.1(-4 ) & 1.5(-4 ) + 8@xmath888@xmath884 & 32 & 1.1(-3 ) & 6.4(-4 ) & 1.7(-4 ) & 9.3(-4 ) & 2.4(-4 ) & 1.7(-3 ) & 3.2(-4 ) + 16@xmath8816@xmath884 & 32 & 2.3(-3 ) & 1.3(-3 ) & 2.2(-4 ) & 1.9(-3 ) & 3.0(-4 ) & 7.8(-3 ) & 6.4(-4 ) + 2@xmath882@xmath884 & 64 & 4.0(-4 ) & 4.1(-4 ) & 4.5(-4 ) & 6.1(-4 ) & 4.7(-4 ) & 2.4(-4 ) & 2.0(-4 ) + 4@xmath884@xmath884 & 64 & 4.2(-4 ) & 3.3(-4 ) & 2.6(-4 ) & 4.7(-4 ) & 2.8(-4 ) & 2.7(-4 ) & 1.6(-4 ) + 8@xmath888@xmath884 & 64 & 1.1(-3 ) & 6.6(-4 ) & 2.3(-4 ) & 9.6(-4 ) & 3.0(-4 ) & 1.9(-4 ) & 3.2(-4 ) rccccc 2@xmath882@xmath884 & 16 & 1.0@xmath1960.4 & 2.9@xmath1960.8 & 0.016 & 0.020 + 4@xmath884@xmath884 & 16 & 3.6@xmath1961.4 & 5.0@xmath1960.9 & 0.019 & 0.014 + 8@xmath888@xmath884 & 16 & 4.4@xmath1961.4 & 8.0@xmath1961.6 & 0.067 & 0.067 + 16@xmath8816@xmath884 & 16 & 6.0@xmath1962.2 & 13.1@xmath1962.8 & 0.130 & 0.120 + 2@xmath882@xmath884 & 32 & 1.0@xmath1960.3 & 2.7@xmath1960.7 & 0.023 & 0.029 + 4@xmath884@xmath884 & 32 & 2.8@xmath1961.1 & 4.2@xmath1960.8 & 0.020 & 0.014 + 8@xmath888@xmath884 & 32 & 6.4@xmath1962.9 & 6.3@xmath1961.3 & 0.070 & 0.065 + 16@xmath8816@xmath884 & 32 & 4.5@xmath1961.3 & 10.4@xmath1962.1 & 0.119 & 0.112 + 2@xmath882@xmath884 & 64 & 1.3@xmath1960.6 & 3.5@xmath1961.0 & 0.033 & 0.022 + 4@xmath884@xmath884 & 64 & 3.6@xmath1961.5 & 4.7@xmath1960.9 & 0.020 & 0.022 + 8@xmath888@xmath884 & 64 & 6.1@xmath1963.0 & 6.7@xmath1961.5 & 0.068 & 0.070
|
due to the gravitational influence of density fluctuations driven by magneto - rotational instability in the gas disk , planetesimals and protoplanets undergo diffusive radial migration as well as changes in other orbital properties . this correlation indicates that caution should be exercised when interpreting local - shearing - box models involving gravitational physics of magneto - rotational turbulence .
|
due to the gravitational influence of density fluctuations driven by magneto - rotational instability in the gas disk , planetesimals and protoplanets undergo diffusive radial migration as well as changes in other orbital properties . the magnitude of the effect on particle orbits can have important consequences for planet formation scenarios . we use the local - shearing - box approximation to simulate an ideal , isothermal , magnetized gas disk with vertical density stratification and simultaneously evolve numerous massless particles moving under the gravitational field of the gas and the host star . we measure the evolution of the particle orbital properties , including mean radius , eccentricity , inclination , and velocity dispersion , and its dependence on the disk properties and the particle initial conditions . although the results converge with resolution for fixed box dimensions , we find the response of the particles to the gravity of the turbulent gas correlates with the horizontal box size , up to 16 disk scale heights . this correlation indicates that caution should be exercised when interpreting local - shearing - box models involving gravitational physics of magneto - rotational turbulence . based on heuristic arguments , nevertheless , the criterion , where is the horizontal box size and is the distance to the host star , is proposed to possibly circumvent this conundrum . if this criterion holds , we can still conclude that magneto - rotational turbulence seems likely to be ineffective at driving either diffusive migration or collisional erosion under most circumstances .
|
1309.0526
|
i
|
the chemical abundance of galaxies is an integrated information on the sequential process of star formation , stellar death , and ejection of heavy elements lasting over a galactic age under an individual specific galaxy environment . therefore , an unusual observed chemical feature detected in some galaxies , that is , the feature which is theoretically hard to understand , will tell us some physical key process that drives their chemical evolution and we have overlooked . the low n abundance in h ii regions and very young stellar populations of the magellanic clouds ( mcs ) is one of the long - standing puzzling problem that remains unsolved . the relative n / o ratio in the large mc ( lmc ) by many studies thus far leads to a mean @xmath0 ( n / o ) @xmath3@xmath4 @xmath5 ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , while @xmath3@xmath5 @xmath6 for the small mc ( smc ) @xcite . since the solar n / o ratio is @xmath7 @xcite , it turns out that the mcs exibit lower n / o ratio by @xmath8@xmath9@xmath10 dex than the sun . to assess the origin of low n / o , two observational facts are worth being highlighted . first , the observed n / o ratios in the mcs are broadly equivalent to an average n / o ratio of extremely metal - poor ( emp ) stars ( [ fe / h]@xmath11 ) in the galactic halo @xcite . since the chemical composition of emp stars in the galactic halo can be explained in the context of sne - ii nucleosynthesis in massive stellar progenitors ( @xmath12 10 ) ( e.g. , * ? ? ? * ) , it can be interpreted that n released from asymptotic giant branch ( agb ) stars with a time delay , which has lifted the n abundance by @xmath8 0.6 dex in the solar vicinity , is missing in the present - day interstellar matter ( ism ) of the mcs . secondly , the n - deficient feature is in common with many other dwarf irregular galaxies ( dirrs ) in the local universe , as represented by the well - known plateau of the n / o ratio ( @xmath1 , e.g. , * ? ? ? * ) for extragalactic h ii regions at 12 + @xmath0 ( o / h ) 8.0 . this fact may suggest that the mechanism to lower the n abundance is associated with the common properties characteristic to these dirrs . moreover , neutral gas in bursting dirrs ( i.e. , blue compact dwarf ( bcd ) galaxies ) that accounts for a much larger mass in galaxies than ionized one also exhibit low n / o ratios ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? this argument holds for the mcs , as implied from low n / o of young mc stars ( e.g. , * ? ? ? * ; * ? ? ? * ) . theoretically , it is a hard task to predict such a low n / o ratio for dirrs covering a wide metallicity range up to 12 + @xmath0 ( o / h ) @xmath8 8.4 for the lmc as not a passing point but a kind of terminus of galaxy evolution . @xcite show that a low n / o ratio can be identical to the values at the early phase ( @xmath8 a first 1 - 2 gyr ) of predicted evolutionary tracks ( see also * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? on the other hand , @xcite find that a low n / o ratio can be reproduced as a present - day value as long as 12 + @xmath0 ( o / h ) 8.0 , utilizing a very low n yield in massive stars . it should be here of note that a low n / o ratio ( @xmath13 ) extends to 12 + @xmath0 ( o / h ) @xmath3 8.6 - 8.7 , i.e. , a solar metallicity , for dirrs in the virgo cluster @xcite . some models adopting starbursts with burst duration of 0.01 gyr and/or 0.5 gyr yield a low n / o ratio ( * ? ? ? * see also lanfranchi & matteucci 2003 ; henry et al . 2006 ) . in modeling , however , we should be mindful of the star formation history of individual dirrs . many studies clearly reveal that the mcs undergo a continuous star formation over more than 10 gyr ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? in addition , since a deep image provided by _ hubble space telescope _ ( hst ) in particular leads to a color magnitude diagram reaching a very faint magnitude , accurate star formation histories of other dirrs in local volume ( @xmath14 5 mpc ) are now accessible ( e.g. , * ? ? ? indeed , as discussed in the next section , dirrs exhibiting low n / o ratios have a long - term star formation similar to the mcs . the production site of n in agb stars is inclined to massive ones such as 4 in which a hot bottom burning operates at the bottom of an envelope ( e.g. , * ? ? ? * ) . accordingly , agb stars release n with a time delay of @xmath80.1 gyr from their steller birth . this fact implies that the n - rich agb ejecta are likely to interact with those of type ia supernovae ( sne ia ) . recent results regarding the delay time distribution ( dtd ) of sne ia yielded by the studies on the sn ia rate in external galaxies dramatically shorten the sn ia s delay time , compared with its conventional timescale of @xmath81 gyr @xcite . @xcite claim that about 50 % of sne ia explode soon after their stellar birth , and further works reveal that the dtd is proportional to @xmath15 with its peak at around 0.1 gyr extending to @xmath8 10 gyr @xcite . thus , the current view on sne ia is organized such that a majority of sne ia explode promptly after the bursting explosions of sne ii ( prompt sne ia ) , and the rest gradually emerge with a long interval of gyrs ( slow sne ia ) . therefore , the timing of n ejection from agb stars is broadly equivalently to that of the explosion of prompt sne ia . @xcite discuss the origin of the different [ cr , mn , ni / fe ] features between the milky way and the lmc , and conclude that their difference can be nicely explained if the ejecta of prompt sne ia escape from the gravitational potential of the lmc . they also show that the predicted evolution of [ @xmath16-elements / fe ] under the process of chemical enrichment by massive stars together with slow sne ia is in good agreement with the observed trend of the lmc . their finding suggests that the total number of sne ia contributing to the chemical enrichment in the lmc is smaller than that in the milky way . this prediction is fully compatible with the implication from the observed high [ ba / fe ] ratios in the lmc as well as in the fornax dwarf spheroidal ( dsph ) galaxy , which demands about @xmath81/3 of the sn ia rate of the milky way @xcite . in addition , recent observed finding of the chemical feature mainly enriched by massive stars , i.e. , enhanced [ @xmath16/fe ] ratios , of the neutral medium in bcd galaxies @xcite may be suggestive of a preferential removal of sn ia ejecta . this line of evidence suggests that the energetic ejecta of prompt sne ia entrain massive stellar agb ejecta out of the galaxy potential of dwarf galaxies owing to the coincidental occurrence timing between the two . therefore , the question is raised of what determines the fate of agb ejecta associated with the bursting prompt sn ia event . it is no doubt that one primary factor is a galaxy potential . the shallow potential is necessary to strip off agb ejecta but not a sufficient condition . for instance , ic 1613 and ngc 5152 , which are much less luminous galaxies than the smc , show a signature of agb enrichment in hii regions as their elevated n / o ratios @xcite . in search for the second key factor , some witness might be obtainable from a well - studied evolution of the mcs . the mcs have experienced close encounters with the milky way with the orbital period of @xmath8 2.5 gyr @xcite . such interactions cause the mcs to lose a part of dark and stellar halo @xcite . this process is likely to cause the joint stripping of agb and prompt sne ia ejecta . on the other hand , if a dirr is isolated for its overall evolution , agb ejecta seem hard to be stripped since the prompt sne ia are unlikely to retain a sufficient energy to push them out of a galaxy potential . this view is supported by the observations that the speed of sn - driven outflows detected in many dwarf galaxies does not exceed the escape velocity of the host galaxy ( e.g. , * ? ? ? * ; * ? ? ? this paper is organized as follows . we start with a brief review on the properties of dirrs with stress on interactions , classifying them into two groups in terms of the n / o ratio ( 2 ) . then , it is followed by modeling the chemical evolutions of n / o for some dirrs including the lmc ( 3 ) . in 4 , we perform numerical simulations for tracing the fate of agb ejecta in a dirr analogous to the lmc under the interactions with the milky way to investigate the role of an interaction on the stripping of agb ejecta from dirrs .
|
dwarf irregular galaxies ( dirrs ) including the magellanic clouds in the local universe , in many cases , exhibit an unusually low n / o abundance ratio ( this ratio is broadly equivalent to the average level of extremely metal - poor stars in the galactic halo , suggesting that n released from asymptotic giant branch ( agb ) stars is missing in the present - day interstellar matter of these dirrs . furthermore , we perform n - body + hydrodynamical simulations to trace the fate of agb ejecta inside a dirr orbiting the milky way , and confirm that a tidal interaction is responsible for the efficient stripping of agb ejecta from dirrs . [ firstpage ] galaxies : abundances galaxies : dwarf galaxies : evolution galaxies : irregular .
|
dwarf irregular galaxies ( dirrs ) including the magellanic clouds in the local universe , in many cases , exhibit an unusually low n / o abundance ratio ( n / o ) in h ii regions as compared with the solar value ( ) . this ratio is broadly equivalent to the average level of extremely metal - poor stars in the galactic halo , suggesting that n released from asymptotic giant branch ( agb ) stars is missing in the present - day interstellar matter of these dirrs . we find evidence for past tidal interactions in the properties of individual dirrs exhibiting low n / o ratios , while a clear signature of interactions is unseen for dirrs with high n / o ratios . accordingly , we propose that the ejecta of massive agb stars that correspond to a major production site of n can be stripped from dirrs that have undergone a strong interaction with a luminous galaxy . the physical process of its stripping is made up of two stages : ( i ) the ejecta of massive agb stars in a dirr are first merged with those of the bursting prompt sne ia and pushed up together to the galaxy halo , and ( ii ) subsequently through tidal interactions with a luminous galaxy , these ejecta are stripped from a dwarf galaxy s potential well . our new chemical evolution models with stripping of agb ejecta succeed in reproducing the observed low n / o ratio . furthermore , we perform n - body + hydrodynamical simulations to trace the fate of agb ejecta inside a dirr orbiting the milky way , and confirm that a tidal interaction is responsible for the efficient stripping of agb ejecta from dirrs . [ firstpage ] galaxies : abundances galaxies : dwarf galaxies : evolution galaxies : irregular .
|
1309.0526
|
c
|
our theoretical scheme is constructed under a hypothesis that preferential loss of massive agb ejecta together with those of prompt sne ia occurs in dirrs . this view is favoured by hydrodynamical simulations in the starburst model @xcite . in this model , sne ii explode in dense ism where most of energy from sne ii is dissipated away by radiative cooling owing to the high - density environment ( e.g. , 97% ; * ? ? ? * ) , and the rest of energy is used for the formation of expanding shells , which is responsible for the formation of hi hole or very diffuse ism . afterward , sne ia explode and agb ejecta are released in an environment with little ism where these ejecta can be expelled from the disk driven by sne ia s explosion energy . we suppose that an efficient joint mass loss from prompt sne ia and massive agb stars is realized even under a continuous star formation by regarding it as a sequence of small - scale starbursts . in other words , global continuous star formation is an ensemble of localized small bursts . in each local region , stars are intermittently formed at intervals of a few @xmath102 yrs . in localized small bursts of star formation , the ambient ism is first thermalized by sn ii explosions so as to suppress further star formation within @xmath103yrs , then followed by feedback of prompt sne ia , and finally the ism can cool down without a major heat source . such an interval of star formation is supported by the discussion on the turbulent ism following sn explosions . @xcite discuss the mixing timescales of sn ii ejecta and quote a shortest mixing timescale of @xmath104yr for the ambient ism . it might predict an unacceptably large scatter in stellar abundances due to the incomplete chemical homogeneity of the ism unless we adopt the interval of star formation more than a few @xmath102yr . this scheme will be validated by performing highly - resolved hydrodynamical simulations applied to local regions where individual stars and ejecta can be decomposed , extracted from an entire galaxy ( bekki & tsujimoto , in preparation ) . we believe that such future numerical simulations can shed light on the connection between the star formation and the removal efficiencies among the different heavy - element ejectors . observationally , there is evidence for bursting star formation even in the milky way s past over most of cosmic time . @xcite find that the local disk within @xmath105 pc has experienced a complex star formation history , using a chromospheric age distribution of dwarf stars . we will see that the lmc has a much more complex one with the strong sfr bumpiness on account of its small mass if we can have a close look at stellar properties of the individual lmc stars , and thereby deduce the localized , not averaged , star formation history . we do not object to removal of sn ii ejecta from a galaxy potential of dirrs . it is reasonable to anticipate that the ejecta of sne ii is in part removed from dwarf galaxies . as supporting evidence , galactic winds from dwarf starburst galaxy , ngc 1569 , exhibit an enhancement of @xmath16-elements relative to fe which is indicative of a sn ii - like feature @xcite . note that this galaxy shows a marginally low ( n / o ) ratio of @xmath106 with 12+@xmath0 ( o / h ) = 8.19 @xcite . additionally , two dwarf starburst galaxies ( ngc 4449 , he 2 - 10 ) are likely to retain the hot gas with an enhanced @xmath16/fe while a non - enhanced feature is seen for two galaxies ( ngc 3077 , ngc 4214 ) @xcite . as already discussed , chemical evolution models for dirrs including the lmc support a steeper imf . this required reduction of sn ii enrichment can be identical to the outcome of a partial removal triggered by galactic winds associated with sne ii . this angle is supported by the observational studies on the imf of the mcs claiming a salpeter - like imf ( e.g. , * ? ? ? an imf slope of @xmath70=@xmath6 adopted in the present study indeed broadly corresponds to a 50% removal of sn ii ejecta . it turns out that the differences in removal efficiencies among snii , sn ia , and agb ejecta may be quite moderate . recall here that we assume a removal of subset ( not all ) of both agb ( only massive ones ) and sn ia ( only prompt ones ) ejecta . these similar efficiencies are rather favored by previous works @xcite . in any event , the degree of escape of sn ii ejecta is of secondary importance for low n / o dirrs focused in this study : the low n / o ratios for dirrs such as the lmc mimic the effects of sn ii nucleosynthesis , though the underlying cause is the preferential stripping of n - rich agb ejecta . on the other hand , the n / o evolution of dirrs that can confine agb ejecta is largely influenced by a degree of sn ii removal since the n / o track goes up by the combined effects of n enrichment by agb stars and less o contribution by sne ii . the loss of agb ejecta from dirrs is restricted to those originated from massive ( 4 ) agb stars since prompt sne ia play a key role of their thermalization and eventual removal . therefore , in our scenario , @xmath107-process elements such as ba synthesized in agb stars with a mass of 1.5 - 3@xcite are hard to suffer a removal from dirrs . on the contrary , the observed results present that stars belonging to the lmc as well as to some dsph galaxies exhibit unusual high ba abundances @xcite . on the other hand , since the production site of c is inclined to a more massive star , i.e. , a 2 -4 agb star @xcite , a partial removal of c ejected form agb stars is expected . indeed , the mcs show the deficiency of c in h ii regions but to a lesser extent than n ( e.g. , * ? ? ? * ) . the n / o ratio as diagnosis of a galaxy environment as we propose can be applied to discussion on the debatable motion of the lmc . recent measurement of the proper motion of the lmc using hst suggests that the lmc is now in the middle of a first infall to the milky way @xcite . their canonical orbit implies that the lmc first entered within 100 kpc of the milky way only a few @xmath102 yr ago . here we calculate the n / o evolution in the lmc for the case analogous to this first infall model . we assume that n enrichment by massive agb stars ceases from 1.5 gyr ago corresponding the accretion epoch reaching a virial radius ( 300 kpc ) of the milky way . as shown in figure 6 , the recent accretion model does not give the sufficient time to lower the n / o ratio down to the present value ( dashed line ) , while the periodic orbit model which has the orbital period of @xmath108 gyr within a virial radius matches the observation ( solid line ) , as already discussed in $ 4 . these results suggest that the lmc has approached the milky way at least more than a few gyr ago . this approach can be strengthened by acquiring stellar n / o for unevolved stars in the lmc . the n / o ratios for stars with different metallicities will give information on how the n / o ratio changes with metallicity , which acts as an indicator of age . this n / o - age relation thus obtained will enable us to pinpoint an accretion epoch of the lmc , though it is currently a formidable task to measure the n abundance for dim unevolved stars in local group . in future , the same method will be also applied to infer the epoch of accretions of dsph galaxies onto the milky way . in cluster of galaxies , dwarf galaxies may be inclined to exhibit low n / o owing to an efficient stripping of agb ejecta by strong tide as well as ram pressure of hot intra - cluster medium . in the virgo cluster , @xmath870% of observed 21 dirrs exhibit low n / o ratios @xcite . this percentage derived from a small database , in fact , seems not so high since 50 dirrs in the local volume listed by @xcite involves @xmath8 80% low n / o dirrs ( however , @xmath109 shows @xmath0 ( n / o)@xmath110 for 12+@xmath0 ( o / h)@xmath148.2 in extragalactic h ii region data compiled by pettini et al . in addition , among the member dirrs in the virgo , we do not see a clear correlation between a low - n / o cluster member and its location relative to the centre of the cluster ( i.e. , m 87 ) . given the absence of a clear correlation between them , a scenario involving ram stripping of cold galactic gas may lead to an increased n / o ratio ( e.g. , * ? ? ? * ; * ? ? ? * ) , contrary to our main thesis . this ram - stripping process results in a suppression of further star formation and production of o , followed by a delayed ejection of n. it is indeed different from our proposed stripping of halo gas , where star formation should be negligible . more data for the reliable statistical analysis will be surely demanded to validate how the n / o ratio is associated with a galaxy environment . in local group , we predict that n / o ratios in dsph galaxies are essentially low as in the mcs because of their much closer distance to a luminous galaxy , i.e. , the milky way or m 31 than dirrs . in our proposed stripping scenario , severe truncation of recycling of agb ejecta due to tidal stripping of gas from dirrs is responsible for the origin of low n / o . on the other hand , @xcite propose an alternative infall scenario in which the observed unusually low n / o in the young populations of the lmc is due to the infall of gas with very low n / o from the smc in the predicted scheme of a gas - transfer from the smc to the lmc ( e.g. , * ? ? ? it is possible to consider that similar dilution occurred in some dirrs by an infall of the objects analogous to h i high velocity clouds ( hvcs ) that surround the milky way . galactic hvcs are observed to have low [ n / h ] ranging from @xmath111 - @xmath112 , giving @xmath0 ( n / o)@xmath113 @xcite.this infall scenario predicts a low n / o ratio only for young stars and h ii regions . therefore , a way to distinguish between the two scenarios is to measure the n / o ratio for intermediate - age stars in dirrs .
|
accordingly , we propose that the ejecta of massive agb stars that correspond to a major production site of n can be stripped from dirrs that have undergone a strong interaction with a luminous galaxy . the physical process of its stripping is made up of two stages : ( i ) the ejecta of massive agb stars in a dirr are first merged with those of the bursting prompt sne ia and pushed up together to the galaxy halo , and ( ii ) subsequently through tidal interactions with a luminous galaxy , these ejecta are stripped from a dwarf galaxy s potential well . our new chemical evolution models with stripping of agb ejecta succeed in reproducing the observed low n / o ratio .
|
dwarf irregular galaxies ( dirrs ) including the magellanic clouds in the local universe , in many cases , exhibit an unusually low n / o abundance ratio ( n / o ) in h ii regions as compared with the solar value ( ) . this ratio is broadly equivalent to the average level of extremely metal - poor stars in the galactic halo , suggesting that n released from asymptotic giant branch ( agb ) stars is missing in the present - day interstellar matter of these dirrs . we find evidence for past tidal interactions in the properties of individual dirrs exhibiting low n / o ratios , while a clear signature of interactions is unseen for dirrs with high n / o ratios . accordingly , we propose that the ejecta of massive agb stars that correspond to a major production site of n can be stripped from dirrs that have undergone a strong interaction with a luminous galaxy . the physical process of its stripping is made up of two stages : ( i ) the ejecta of massive agb stars in a dirr are first merged with those of the bursting prompt sne ia and pushed up together to the galaxy halo , and ( ii ) subsequently through tidal interactions with a luminous galaxy , these ejecta are stripped from a dwarf galaxy s potential well . our new chemical evolution models with stripping of agb ejecta succeed in reproducing the observed low n / o ratio . furthermore , we perform n - body + hydrodynamical simulations to trace the fate of agb ejecta inside a dirr orbiting the milky way , and confirm that a tidal interaction is responsible for the efficient stripping of agb ejecta from dirrs . [ firstpage ] galaxies : abundances galaxies : dwarf galaxies : evolution galaxies : irregular .
|
astro-ph0511284
|
i
|
the cumulative stellar mass loss from the stars in an early - type galaxy is typically 0.1 - 3 m@xmath6 yr@xmath7 ( e.g. , @xcite ) , which , integrated over a hubble time , is comparable to the gaseous mass of a spiral galaxy . the absence of a massive disk of cool atomic and molecular gas in early - type galaxies indicates a different life cycle for the gas , which may be divided into distinct stages . ignoring accretion onto a galaxy , the first stage is mass loss from the stars , a process that can be measured by detecting the infrared signature associated with the stellar winds of red giants . the measurement of this process successfully detects the infrared emission near 10 @xmath11 m and yields a value for the mass loss rate that is approximately the value predicted from theoretical stellar evolution models @xcite . the stellar ejecta will not remain in orbit around its star because the expanding ejecta will eventually collide with the ejecta from other stars , undergoing shocks that convert their random orbital motion to thermal energy ( e.g. , @xcite ) . this process heats the gas to 10@xmath12 -10@xmath13 k , and if there were no additional heating or cooling , the gas would be bound to the galaxy and have the same spatial distribution as the stars . however , the radiative cooling time for the gas is less than a hubble time , so the gas will evolve with time , and this is the basis of the cooling flow model . in the absence of a heating mechanism , radiative cooling drains energy from the gas most rapidly at small radii , causing a loss of buoyancy and a subsequent inflow of gas . then , the rate at which gas cools and flows inward ( @xmath10 ) is proportional to two observed quantities , the energy loss rate ( @xmath14 ) divided by the thermal energy per gram ( @xmath15 ) , or @xmath10 @xmath16 @xmath14/@xmath17 , typically 0.03 - 3 m@xmath6 yr@xmath7 for ellipticals ( e.g. , @xcite ) . the location at which the gas cools ( to 10@xmath3 k or cooler ) depends upon a free parameter in this _ standard _ cooling flow picture . picture does not include heating effects by supernovae , which will make profound changes , since the net cooling rate is given by @xmath10 @xmath16 ( @xmath14-_h_)/@xmath17 , where _ h _ is a heating component . in principle , the rate at which cool gas is produced can be reduced to zero . if one uses the recent values for the supernova rate in early - type galaxies ( @xcite ; a factor of three lower than the older tammann and sandage rates ; @xcite ) , the characteristic temperature of the ism would be 1@xmath010@xmath13 k , which is about the escape temperature for a typical elliptical galaxy ( @xmath18 @xmath2 0.3@xmath19 ) . more detailed hydrodynamic calculations @xcite show that galactic winds ( or partial galactic winds ) will play an important role in the evolution of the hot gas even if the supernova heating rate is lower than that implied by @xcite . a galactic wind carries away nearly all of its energy rather than radiating it , substantially lowering the x - ray luminosity . not only is a galactic wind important for understanding the x - ray emission and the ism in the galaxy , it has implications for the surrounding intergalactic medium , as this is the primary way that it becomes polluted with metals . the x - ray emission surveys of early - type galaxies show that there can be a wide range in the mass and luminosity of x - ray emitting gas for galaxies of similar optical properties ( e.g. , @xcite ) . this variation , along with the trend of rapidly decreasing @xmath14 with decreasing optical luminosity @xmath18 , can be understood if galactic winds play a role in some systems ( other systems may have accretion from their surrounding region ; review by @xcite ) . therefore , the prediction is that the systems with large x - ray gas masses and short cooling times ( high @xmath14 ) should have cooling flows while the x - ray poor systems have partial or total galactic winds , so the cooling flow will be weak or absent in those systems . because the heating rate is not well known , it is difficult to determine the net cooling rate through x - ray luminosity measurements . alternatively , if gas is cooling to lower temperatures , emission lines will be produced that are indicative of those lower temperatures . the best lines for this test are from ovi because this ionization state dominates the total radiative cooling as the gas passes through the 2 - 4@xmath010@xmath8 k range @xcite . the primary cooling lines come from the doublet at @xmath20@xmath201032 , 1038 , which is accessible with the _ far ultraviolet spectroscopic explorer _ ( _ fuse _ ) . calculations show that there is a linear relationship between the line luminosity and the cooling rate , which is insensitive to the metallicity of the gas or whether the gas is out of collisional ionization equilibrium @xcite . at 10@xmath21 k , the radiative cooling rate per unit volume is significantly higher than at the temperature of the x - ray emitting gas ( 10@xmath12 - 10@xmath13 k ) , so the relative contribution of a heating mechanism is greatly reduced and can be ignored . therefore , the luminosity of the ovi doublet should be a direct measure of the gas cooling rate , @xmath22 . previously , we reported upon ovi observations of two early - type galaxies , ngc 1404 and ngc 4636 @xcite , two of the x - ray luminous galaxies widely believed to host cooling flows . for ngc 1404 , ovi was not detected and the upper limit is several times below the standard cooling flow prediction , based on _ rosat _ data . however , ovi emission was detected from ngc 4636 , and the luminosity of these lines corresponds to a cooling rate of 0.4 m@xmath6 yr@xmath7 . this is less than the total rate from the cooling flow model of 2 m@xmath6 yr@xmath7 , but the _ fuse _ aperture ( a 30@xmath23 square aperture , or an effective radius of about 17@xmath23 ) only takes in a part of the galaxy . correcting for the flux that falls outside the aperture is a model - dependent procedure , but if one uses a model with distributed mass drop out ( _ q_=1 from @xcite ) , the corrected ovi luminosity approximately equals the cooling flow prediction . following on this work , we began a ovi emission line survey of an unbiased sample of nearby early - type galaxies . the basic observations define the emission line characteristics of the sample , and permit us to test a few predictions of the model . one would expect that the x - ray faint galaxies would be very weak ovi emitters , if most of the thermal energy is being carried away in galactic winds . secondly , the galaxies with significant hot gas masses should usually possess ovi emission .
|
early - type galaxies often contain a hot x - ray emitting interstellar medium ( 3 - 8 k ) with an apparent radiative cooling time much less than a hubble time . we report on a study of an unbiased sample of 24 galaxies , obtaining _ far ultraviolet spectroscopic explorer _ there is a correlation between and , although there is significant dispersion about the relationship , where the ovi is brighter or dimmer than expected by a factor of three or more .
|
early - type galaxies often contain a hot x - ray emitting interstellar medium ( 3 - 8 k ) with an apparent radiative cooling time much less than a hubble time . if unopposed by a heating mechanism , the gas will radiatively cool to temperatures 10 k at a rate proportional to/ , typically 0.03 - 1 m yr . we can test if gas is cooling through the 3 k range by observing the ovi doublet , whose luminosity is proportional to the cooling rate . here we report on a study of an unbiased sample of 24 galaxies , obtaining _ far ultraviolet spectroscopic explorer _ spectra to complement the x - ray data of _ rosat _ and _ chandra_. the ovi line emission was detected in about 40% of the galaxies and at a luminosity level similar to the prediction from the cooling flow model . there is a correlation between and , although there is significant dispersion about the relationship , where the ovi is brighter or dimmer than expected by a factor of three or more . if the cooling flow picture is to be retained , this dispersion requires that cooling flows be time - dependent , as might occur by the activity of an agn . however , of detected objects , those with the highest or lowest values of/ are not systematically hot or cool , as one might predict from agn heating .
|
cond-mat9806107
|
i
|
understanding of the physical origins and systematics underlying the variations of materials properties with size , form of aggregation , and dimensionality are some of the main challenges in modern materials research , of ever increasing importance in the face of the accelerated trend toward miniaturization of electronic and mechanical devices . @xcite interestingly , it has emerged that concepts and methodologies developed in the context of isolated gas - phase clusters and atomic nuclei are often most useful for investigations of finite - size solid - state structures . in particular , it has been shown most recently @xcite through first - principles molecular dynamics simulations that as metallic ( sodium ) nanowires are stretched to just a few atoms in diameter , the reduced dimensions , increased surface - to - volume ratio , and impoverished atomic environment , lead to formation of structures , made of the metal atoms in the neck , which can be described in terms of those observed in small gas - phase sodium clusters ; hence they were termed @xcite as supported _ cluster - derived structures ( cds)_. the above prediction of the occurrence of `` magic - number '' cds s in nanowires , due to characteristics of electronic cohesion and atomic bonding in such structures of reduced dimensions , are directly correlated with the energetics of metal clusters , where magic - number sequences of cluster sizes , shapes and structural motifs due to electronic and/or geometric shell effects , have been long predicted and observed . @xcite these results lead one directly to conclude that other properties of nanowires , derived from their energetics , may be described using methodologies developed previously in the context of clusters . indeed , in a previous letter , @xcite we showed that certain aspects of the mechanical response ( i.e. , elongation force ) and electronic transport ( e.g. , quantized conductance ) in metallic nanowires can be analyzed using the local - density - approximation ( lda ) -based shell correction method ( scm ) , developed and applied previously in studies of metal clusters . @xcite specifically , we showed that in a jellium - modelled , volume - conserving , and uniform in shape nanowire , variations of the total energy ( particularly terms associated with electronic subband corrections ) upon elongation of the wire lead to _ self - selection _ of a sequence of stable `` magic '' wire configurations ( mwc s , specified by a sequence of the wire s radii ) , with the force required to elongate the wire from one configuration to the next exhibiting an oscillatory behavior . moreover , we showed that due to the quantized nature of electronic states in such wires , the electronic conductance varies in a quantized step - wise manner ( in units of the conductance quantum @xmath2 ) , correlated with the transitions between mwc s and the above - mentioned force oscillations . in this paper , we expand our lda - based treatment to wires of variable shape , that is allowing for a constricted region . from this investigation , we conclude that the above self - selection principles and the direct correlations between the oscillatory patterns in the energetic stability , forces , and stepwise variations of the quantized conductance maintain for the variable - shaped wire as well , with the finding that underlying these oscillatory patterns and correlations are the contributions from the narrowmost region of the wire . furthermore , this finding is analyzed and corroborated through a semiclassical analysis . prior to introducing the model studied in this paper , it is appropriate to briefly describe certain previous theoretical and experimental investigations , which form the background and motivation for this study . atomistic descriptions , based on realistic interatomic interactions , and/or first - principles modelling and simulations played an essential role in discovering the formation of nanowires , @xcite and in predicting and elucidating the microscopic mechanisms underlying their mechanical , spectral , electronic and transport properties . these predictions @xcite [ particularly those pertaining to generation of nanowires through separation of the contact between two materials bodies ; size - dependent evolution of the wire s mechanical response to elongation transforming from multiple - slips for wider wires to a succession of stress accumulation and fast relief stages leading to a sequence of structural instabilities and order - disorder transformations localized in the neck region when its diameter shrinks to about 15 ; consequent oscillations of the elongation force and the calculated high value of the resolved yield stress ( @xmath3 4 gpa for au nanowires ; which is over an order of magnitude that of the bulk ) , as well as anticipated electronic quantization effects on transport properties @xcite ] have been corroborated in a number of experiments using scanning tunneling and force microscopy , @xcite break junctions , @xcite and pin - plate techniques @xcite at ambient environments , as well as under ultrahigh vacuum and/or cryogenic conditions . particularly pertinent to our current study are experimental observations of the oscillatory behavior of the elongation forces and the correlations between the changes in the conductance and the force oscillations ; see especially the simultaneous measurements of force and conductance in gold nanowires in ref . , where in addition the predicted `` ideal '' value of the critical yield stress has also been measured ( see also ref . ) . the lda - jellium - based model introduced in our previous paper @xcite and extended to generalized wire shapes herein , while providing an appropriate solution within the model s assumptions ( see section ii ) , is devoid by construction of atomic crystallographic structure and does not address issues pertaining to nanowire formation methods , atomistic configurations , and mechanical response modes [ e.g. , plastic deformation mechanisms , interplanar slip , ordering and disordering mechanisms ( see detailed descriptions in refs . and , and a discussion of conductance dips in ref . ) , defects , mechanical reversibility , @xcite and roughening of the wires s morphology during elongation @xcite ] , nor does it consider the effects of the above on the electron spectrum , transport properties , and dynamics . @xcite nevertheless , as shown below , the model offers a useful framework for linking investigations of solid - state structures of reduced dimensions ( e.g. , nanowires ) with methodologies developed in cluster physics , as well as highlighting certain nanowire phenomena of mesoscopic origins and their analogies to clusters . in this context , we note that several other treatments related to certain of the issues in this paper , but employing free - electron models , have been pursued most recently . @xcite in both of these treatments an infinite confining potential on the surface of the wire is assumed and only the contribution from the kinetic energy of the electrons to the total energy is considered , neglecting the exchange - correlation and hartree terms , and electrostatic interactions due to the positive ionic ( jellium ) background . a comprehensive discussion of the limitations of such free - electron models in the context of calculations of electronic structure and energetics ( e.g. , surface energies ) of metal surfaces can be found in ref . . in section ii.a . , we outline the lda - based shell correction method , describe the jellium model for variable - shaped nanowires , and derive expressions for the energetics of such nanowires ( density of states , energy , and force ) . numerical results pertaining to energetics , force , and electronic conductance , calculated as a function of elongation for variable - shaped sodium nanowires , are given in section ii.b . , including a discussion on the main finding that the contribution from the narrowmost part of the constriction underlies the properties of these quantities and the correlations between them . these correlations between the energetic and transport properties and their dependence on the narrowmost part of the nanowire are further analyzed in section iii , using a semiclassical treatment . we summarize our results in section iv .
|
b * 101 * , 5780 ( 1997 ) ] to wires containing a variable - shaped constricted region . these subbands are the analogs of shells in finite - size , zero - dimensional fermionic systems , such as metal clusters , atomic nuclei , andhe clusters , where magic numbers are known to occur . these variations in the energy result in oscillations in the force required to elongate the wire and are directly correlated with the stepwise variations of the conductance of the nanowire in units of . energetics , forces , and quantized conductance in jellium - modeled metallic nanowires +
|
energetics and quantized conductance in jellium - modeled nanowires are investigated using the local - density - functional - based shell correction method , extending our previous study of uniform - in - shape wires [ c. yannouleas and u. landman , j. phys . chem . b * 101 * , 5780 ( 1997 ) ] to wires containing a variable - shaped constricted region . the energetics of the wire ( sodium ) as a function of the length of the volume - conserving , adiabatically shaped constriction , or equivalently its minimum width , leads to formation of self - selecting magic wire configurations , i.e. , a discrete configurational sequence of enhanced stability , originating from quantization of the electronic spectrum , namely , formation of transverse subbands due to the reduced lateral dimensions of the wire . these subbands are the analogs of shells in finite - size , zero - dimensional fermionic systems , such as metal clusters , atomic nuclei , andhe clusters , where magic numbers are known to occur . these variations in the energy result in oscillations in the force required to elongate the wire and are directly correlated with the stepwise variations of the conductance of the nanowire in units of . the oscillatory patterns in the energetics and forces , and the correlated stepwise variation in the conductance are shown , numerically and through a semiclassical analysis , to be dominated by the quantized spectrum of the transverse states at the narrowmost part of the constriction in the wire . energetics , forces , and quantized conductance in jellium - modeled metallic nanowires +
|
cond-mat9806107
|
c
|
in this paper , we extended our investigations @xcite of energetics , conductance , and mesoscopic forces in a jellium modelled nanowire ( sodium ) using the local - density - functional - based shell correction method to variable - shaped wires , i.e. , containing a constricted region modeled here by a parabolic dependence of the cross - sectional radii in the constriction on @xmath4 ( see fig . 1 ) . the results shown above , particularly , the oscillations in the total energy of the wire as a function of the length of the variable - shaped constricted region ( and correspondingly its narrowmost width ) , the consequent oscillations in the elongation force , the corresponding discrete sequence of magic wire configurations , and the direct correlation between these oscillations and the stepwise quantized conductance of the nanowires , originate from quantization of the electronic states ( i.e. , formation of subbands ) due to the reduced lateral ( transverse ) dimension of the nanowires . these results are in correspondence with our earlier lda - scm investigation of jellium - modeled uniform nanowires . @xcite moreover , in the current study of a wire with a variable ( adiabatic ) shaped constriction , we found that the oscillatory behavior of the energetic and transport properties are governed by the subband quantization spectrum ( termed here electronic shells ) at the narrowmost part of the constriction . this characteristic is supported and corroborated by our semiclassical analysis ( section iii ) . we reiterate here that such oscillatory behavior , as well as the appearance of `` magic numbers '' and `` magic configurations '' of enhanced stability , are a general characteristic of finite - size fermionic systems and are in direct analogy with those found in simple - metal clusters ( as well as in @xmath0he clusters @xcite and atomic nuclei@xcite ) , where electronic shell effects on the energetics @xcite ( and most recently shape dynamics @xcite of jellium modelled clusters driven by forces obtained from shell - corrected energetics ) have been studied for over a decade . while these calculations provide a useful and instructive framework , we remark that they are not a substitute for theories where the atomistic nature and specific atomic arrangements are included @xcite in evaluation of the energetics ( and dynamics ) of these systems ( see in particular refs . , where first - principles molecular - dynamics simulations of electronic spectra , geometrical structure , atomic dynamics , electronic transport and fluctuations in sodium nanowires have been discussed ) . indeed , the atomistic structural characteristics of nanowires @xcite ( including the occurrence of cluster - derived structures of particular geometries@xcite ) , which may be observed through the use of high resolution microscopy , @xcite influence the electronic spectrum and transport characteristics , as well as the energetics of nanowires and their mechanical properties and response mechanisms . in particular , the mechanical response of materials involves structural changes through displacement and discrete rearrangenent of the atoms . the mechanisms , pathways , and rates of such structural transformations are dependent on the arrangements and coordinations of atoms , the magnitude of structural transformation barriers , and the local shape of the wire , as well as possible dependency on the history of the material and the conditions of the experiment ( i.e. , fast versus slow extensions ) . further evidence for the discrete atomistic nature of the structural transformations is provided by the shape of the force variations ( compare the calculated fig . 3(b ) in ref . and fig . 3 in ref . with the measurements shown in figs . 1 and 2 in ref . ) , and the interlayer spacing period of the force oscillations when the wire narrows . while such issues are not addressed by models which do not include the atomistic nature of the material , the mesoscopic ( in a sense universal ) phenomena described by our model are of interest , and may guide further research in the area of finite - size systems in the nanoscale regime . such further investigations include the occurrence of magic configurations ( i.e. , sequences of enhanced stability specified by number of particles , size , thickness or shape ) in clusters , dots , wires , and thin films of normal , as well as superconducting metals , and the effect of magnetic fields which can influence the energetics in such systems ( e.g. , leading to magnetostriction effects ) through variations of the subband spectra , in analogy with magnetotransport phenomena in nanowires @xcite . several directions for improving the model ( while remaining within a jellium framework ) are possible . these include : ( i ) consideration of more complex shapes . for example , in our current model the elongation is distributed over the entire constriction throughout the process , while a more realistic description should include a gradual concentration of the elongation , and consequent shape variation , to the narrower part of the constriction as found through molecular - dynamics simulations ; @xcite ( ii ) use of a stabilized - jellium description @xcite of the energetics of the nanowire in order to give it certain elements of mechanical stability . in this context , note also that from the total energy shown in fig . 4(c ) , and the corresponding total force [ fig . 5(c ) ] , it is evident that in our current model , except for the region of large elongation close to the breaking point ( i.e. , @xmath165 ) , the wire is unstable against spontaneous collapse ( that is shortening ) , i.e. , there are no energetic barriers against such process , while both experiments @xcite and md simulations @xcite show that compression of such wires requires the application of an external force . improvements of the model in these directions are most desirable in light of the aforementioned experimental @xcite and md - simulations @xcite observations that the total oscillating forces for elongation and compression of nanowires are of opposite signs ( i.e. , negative and positive , respectively ) , while our current ( equilibrium model ) is limited to certain aspects of the tensile part of an elongation - compression cycle ; ( iii ) inclusion of bias voltage effects in calculations of the energetics and conductance of nanowires . @xcite while such effects may be expected to have little influence ( particularly on the energetics ) at small voltages , they could be of significance at larger ones . work in these directions is in progress in our laboratory . this research was supported by a grant from the u.s . department of energy ( grant no . fg05 - 86er45234 ) and the afosr . useful comments by w.d . luedtke are gratefully acknowledged . calculations were performed at the georgia institute of technology center for computational materials science .
|
energetics and quantized conductance in jellium - modeled nanowires are investigated using the local - density - functional - based shell correction method , extending our previous study of uniform - in - shape wires [ c. yannouleas and u. landman , j. phys . the energetics of the wire ( sodium ) as a function of the length of the volume - conserving , adiabatically shaped constriction , or equivalently its minimum width , leads to formation of self - selecting magic wire configurations , i.e. , a discrete configurational sequence of enhanced stability , originating from quantization of the electronic spectrum , namely , formation of transverse subbands due to the reduced lateral dimensions of the wire .
|
energetics and quantized conductance in jellium - modeled nanowires are investigated using the local - density - functional - based shell correction method , extending our previous study of uniform - in - shape wires [ c. yannouleas and u. landman , j. phys . chem . b * 101 * , 5780 ( 1997 ) ] to wires containing a variable - shaped constricted region . the energetics of the wire ( sodium ) as a function of the length of the volume - conserving , adiabatically shaped constriction , or equivalently its minimum width , leads to formation of self - selecting magic wire configurations , i.e. , a discrete configurational sequence of enhanced stability , originating from quantization of the electronic spectrum , namely , formation of transverse subbands due to the reduced lateral dimensions of the wire . these subbands are the analogs of shells in finite - size , zero - dimensional fermionic systems , such as metal clusters , atomic nuclei , andhe clusters , where magic numbers are known to occur . these variations in the energy result in oscillations in the force required to elongate the wire and are directly correlated with the stepwise variations of the conductance of the nanowire in units of . the oscillatory patterns in the energetics and forces , and the correlated stepwise variation in the conductance are shown , numerically and through a semiclassical analysis , to be dominated by the quantized spectrum of the transverse states at the narrowmost part of the constriction in the wire . energetics , forces , and quantized conductance in jellium - modeled metallic nanowires +
|
0803.0295
|
i
|
the optical and near - infrared spectral energy distributions of very low mass stars and brown dwarfs late - type m , l and t dwarfs are distinctly non - blackbody . overlapping molecular bands and strong line emission produce a rich array of spectral diagnostics for classification and characterization of physical properties . considerable effort is now being devoted toward decrypting the spectral fingerprints of late - type dwarfs to determine masses , ages , metallicities and other fundamental parameters ( e.g. , @xcite ) . in some cases , spectral peculiarities arise when an observed source is in fact an unresolved multiple system , with components of different masses , effective temperatures , and other spectral properties . while several classes of stellar multiples are recognized on the basis of their unusual spectral or photometric properties ( u geminorum stars , m dwarf + white dwarf systems , etc . ) , identifying such cases amongst late - type dwarfs is complicated by the influence of other physical effects . delineation of spectral peculiarities that arise purely from multiplicity as opposed to other physical effects is essential if we hope to unambiguously characterize the physical properties of the lowest luminosity stars and brown dwarfs . very low mass multiple systems are important in their own right , as they enable mass and occasionally radius measurements ( e.g. , @xcite ) , provide constraints for star / brown dwarf formation scenarios ( e.g. , @xcite ) and facilitate detailed studies of atmospheric properties ( e.g. , @xcite ) . of the roughly 90 very low mass multiple systems currently known , the majority have been identified through high angular resolution imaging , using the _ hubble space telescope _ ( _ hst _ ; e.g. , @xcite ) , ground - based adaptive optics systems ( e.g. , @xcite ) and more recently aperture masking interferometry ( e.g. , @xcite ) . however , as the vast majority of very low mass binaries have small separations ( @xmath390% have @xmath4 @xmath5 20 au ; @xcite ) , expanding the population of known binaries to greater distances requires either finer angular sampling or the identification of systems that are unresolved . the frequency of nearby , tightly - bound binaries is also essential for a complete assessment of the overall very low mass dwarf binary fraction , since imaging studies provide only a lower limit to this fundamental statistic . such systems are also more likely to eclipse , enabling radius measurements and fundamental tests of evolutionary models ( e.g. , @xcite ) . while searches for radial velocity variability via high resolution spectroscopy can be useful in this regime ( e.g. , @xcite ) , in many cases very low luminosity and/or distant late - type dwarfs are simply too faint to be followed up in this manner . recently , @xcite demonstrated that in certain cases the presence of an unresolved companion can be inferred directly from the morphology of a source s low - resolution near - infrared spectrum . in particular , it was shown that the spectrum of the peculiar l dwarf sdss j080531.84 + 481233.0 ( hereafter sdss j0805 + 4812 ; @xcite ) , which has highly discrepant optical and near - infrared spectral classifications , could be accurately reproduced as a combination of `` normal '' l4.5 + t5 components . indeed , the binary hypothesis provides a far simpler and more consistent explanation for the unusual optical , near - infrared and mid - infrared properties of sdss j0805 + 4812 than other alternatives ( e.g. , @xcite ) . the identification of unresolved multiples like sdss j0805 + 4812 by low - resolution near - infrared spectroscopy is a potential boon for low - mass multiplicity studies , as this method is not subject to the same physical or projected separation limitations inherent to high - resolution imaging and spectroscopic techniques . this article reports the discovery of a second unresolved very low mass binary system , 2mass j03202839@xmath00446358 ( hereafter 2mass j0320@xmath00446 ) , identified by the morphology of its low - resolution , near - infrared spectrum . the spectroscopic observations leading to this conclusion are described in @xmath6 2 , as are laser guide star adaptive optics ( lgs ao ) imaging observations aimed at searching for a faint companion . analysis of the spectral data using the binary template matching technique described in @xcite is presented in @xmath6 3 . @xmath6 4 discusses the viability of 2mass j0320@xmath00446 being a binary , with specific comparison to the known m dwarf + t dwarf system scr 1845@xmath06357 . we also constrain the projected separation of the 2mass j0320@xmath00446 system based on our imaging observations , and discuss overall limitations on the variety of unresolved m dwarf + t dwarf binaries that can be identified from composite near - infrared spectroscopy . conclusions are summarized in @xmath6 5 .
|
evidence is presented that 2mass j03202839 , a late - type dwarf with discrepant optical ( m8 : ) and near - infrared ( l1 ) spectral types , is an as - yet unresolved stellar / brown dwarf binary with late - type m dwarf and t dwarf components . this conclusion is based on low - resolution , near - infrared spectroscopy that reveals a subtle but distinctive absorption feature at 1.6 . 2mass j0320 is the second very low mass binary to be identified from unresolved , low - resolution , near - infrared spectroscopy , a technique that complements traditional high resolution imaging and spectroscopic methods .
|
evidence is presented that 2mass j03202839 , a late - type dwarf with discrepant optical ( m8 : ) and near - infrared ( l1 ) spectral types , is an as - yet unresolved stellar / brown dwarf binary with late - type m dwarf and t dwarf components . this conclusion is based on low - resolution , near - infrared spectroscopy that reveals a subtle but distinctive absorption feature at 1.6 . the feature , which is also present in the combined light spectrum of the m8.5 + t6 binary scr 1845 , arises from the combination of feh absorption from an m8.5 primary and pseudo - continuum flux from a t5 secondary , as ascertained from binary spectral templates constructed from empirical data . the binary templates provide a far superior match to the overall near - infrared spectral energy distribution of 2mass j0320 than any single comparison spectra . laser guide star adaptive optics ( lgs ao ) imaging observations , including the first application of lgs ao aperture mask interferometry , fail to resolve a faint companion , restricting the projected separation of the system to less than 8.3 au at the time of observation . 2mass j0320 is the second very low mass binary to be identified from unresolved , low - resolution , near - infrared spectroscopy , a technique that complements traditional high resolution imaging and spectroscopic methods .
|
0803.0295
|
c
|
we have found that subtle peculiarities observed in the near - infrared spectrum of 2mass j0320@xmath00446 , in particular a characteristic bowl - shaped dip at 1.6 @xmath1 , indicate the presence of a mid - type t dwarf companion . this companion is unresolved in lgs ao imaging observations ( including the first application of aperture mask interferometry with lgs ao ) , indicating a maximum projected separation of 8.3 au at the time of observations . the binary scenario not only provides a simple and straightforward explanation for the 1.6 @xmath1 feature also present in the composite spectrum of the known m8.5 + t6 binary scr 1845@xmath06357but also resolves the discrepancy between the optical and near - infrared classifications of 2mass j0320@xmath00446 . furthermore , empirical binary templates composed of `` normal '' m dwarf plus t dwarf pairs provide a far superior match to the overall near - infrared spectral energy distribution of 2mass j0320@xmath00446 than any single comparison source . the hypothesis that 2mass j0320@xmath00446 is an unresolved binary is therefore compelling , and could potentially be verified through radial velocity monitoring observations . in addition , we estimate that roughly 1 - 5% of all late - type m dwarfs may harbor a mid - type t dwarf companion that could similarly be identified and characterized using low resolution near - infrared spectroscopy and binary spectral template analysis . the authors acknowledge telescope operator paul sears and instrument specialist john rayner at irtf , and al conrad , randy campbell , jason mcilroy , and gary punawai at keck , for their assistance during the observations . we also thank markus kasper for providing the spectral data for scr 1845@xmath06357 and sandy leggett , dagny looper and kevin luhman for providing a portion of the spex prism spectra used in the binary spectral template analysis . our anonymous referee provided a helpful and very prompt critique of the original manuscript . mcl and tjd acknowledge support for this work from nsf grant ast-0507833 and an alfred p. sloan research fellowship . this publication makes use of data from the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center , and funded by the national aeronautics and space administration and the national science foundation . 2mass data were obtained from the nasa / ipac infrared science archive , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administration . this research has benefitted from the m , l , and t dwarf compendium housed at dwarfarchives.org and maintained by chris gelino , davy kirkpatrick , and adam burgasser ; the vlm binaries archive maintained by nick siegler at http://www.vlmbinaries.org ; and the spex prism spectral libraries , maintained by adam burgasser at http://www.browndwarfs.org/spexprism . the authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of mauna kea has always had within the indigenous hawaiian community . we are most fortunate to have the opportunity to conduct observations from this mountain . burgasser , a. j. , reid , i. n. , siegler , n. , close , l. m. , allen , p. , lowrance , p. j. , & gizis , j. e. 2007b , in planets and protostars v , eds . b. reipurth , d. jewitt and k. keil ( univ . arizona press : tucson ) , p. 427 wilson , j. c. , miller , n. a. , gizis , j. e. , skrutskie , m. f. , houck , j. r. , kirkpatrick , j. d. , burgasser , a. j. , & monet , d. g. 2003 , in brown dwarfs ( iau symp . 211 ) , ed . e. martn ( san francisco : asp ) , p. 197 llllcl sdss j0000 + 2554 & j00001354 + 2554180 & & t4.5 & 15.06@xmath20.04 & * 1*;2 + 2mass j0034 + 0523 & j00345157 + 0523050 & & t6.5 & 15.54@xmath20.05 & * 3*;1 + 2mass j0036 + 1821 & j00361617 + 1821104 & l3.5 & l4@xmath21 & 12.47@xmath20.03 & * 4*;2,5,6 + hd 3651b & j0039191 + 211516 & & t7.5 & 16.16@xmath20.03 & * 7*;8,9,10 + 2mass j0050@xmath03322 & j00501994@xmath03322402 & & t7 & 15.93@xmath20.07 & * 11*;1,12 + 2mass j0103 + 1935 & j01033203 + 1935361 & l6 & & 16.29@xmath20.08 & * 13*;6 + 2mass j0117@xmath03403 & j01174748@xmath03403258 & l2 : & & 15.18@xmath20.04 & * 56*;14 + sdss j0119 + 2403 & j01191207 + 2403317 & & t2 & 17.02@xmath20.18 & * 15 * + ipms 0136 + 0933 & j01365662 + 0933473 & & t2.5 & 13.46@xmath20.03 & * 4*;16 + 2mass j0144@xmath00716 & j01443536@xmath00716142 & l5 & & 14.19@xmath20.03 & * 4*;17 + sdss j0151 + 1244 & j01514155 + 1244300 & & t1 & 16.57@xmath20.13 & * 3*;1,18 + 2mass j0205 + 1251 & j02050344 + 1251422 & l5 & & 15.68@xmath20.06 & * 19*;6 + sdss j0207 + 0000 & j02074284 + 0000564 & & t4.5 & 16.80@xmath20.16 & * 1*;18 + 2mass j0208 + 2542 & j02081833 + 2542533 & l1 & & 13.99@xmath20.03 & * 4*;6 + sips j0227@xmath01624 & j02271036@xmath01624479 & l1 & & 13.57@xmath20.02 & * 4*;20 + 2mass j0228 + 2537 & j02281101 + 2537380 & l0 : & l0 & 13.84@xmath20.03 & * 4*;14,21 + gj 1048b & j02355993@xmath02331205 & l1 & l1 & & * 4*;22 + 2mass j0241@xmath01241 & j02415367@xmath01241069 & l2 : & & 15.61@xmath20.07 & * 56*;14 + 2mass j0243@xmath02453 & j02431371@xmath02453298 & & t6 & 15.38@xmath20.05 & * 3*;1,23 + sdss j0247@xmath01631 & j02474978@xmath01631132 & & t2@xmath21.5 & 17.19@xmath20.18 & * 15 * + so 0253 + 1625 & j02530084 + 1652532 & m7 & & 8.39@xmath20.03 & * 4*;24,25 + denis j0255@xmath04700 & j02550357@xmath04700509 & l8 & l9 & 13.25@xmath20.03 & * 1*;26,27 + 2mass j0310 + 1648 & j03105986 + 1648155 & l8 & l9 & 16.03@xmath20.08 & * 28*;1,6 + sdss j0325 + 0425 & j03255322 + 0425406 & & t5.5 & 16.25@xmath20.14 & * 15 * + 2mass j0328 + 2302 & j03284265 + 2302051 & l8 & l9.5 & 16.69@xmath20.14 & * 4*;2,6 + lp 944@xmath020 & j03393521@xmath03525440 & m9 & & 10.73@xmath20.02 & * 4 * + 2mass j0345 + 2540 & j03454316 + 2540233 & l0 & l1@xmath21 & 14.00@xmath20.03 & * 29*;2,30,31 + sdss j0351 + 4810 & j03510423 + 4810477 & & t1@xmath21.5 & 16.47@xmath20.13 & * 15 * + 2mass j0407 + 1514 & j04070885 + 1514565 & & t5 & 16.06@xmath20.09 & * 3*;1 + 2mass j0415@xmath00935 & j04151954@xmath00935066 & t8 & t8 & 15.70@xmath20.06 & * 3*;1,23,32 + 2mass j0439@xmath02353 & j04390101@xmath02353083 & l6.5 & & 14.41@xmath20.03 & * 28*;14 + 2mass j0510@xmath04208 & j05103520@xmath04208140 & & t5 & 16.22@xmath20.09 & * 33 * + 2mass j0516@xmath00445 & j05160945@xmath00445499 & & t5.5 & 15.98@xmath20.08 & * 4*;1,34 + 2mass j0559@xmath01404 & j05591914@xmath01404488 & t5 & t4.5 & 13.80@xmath20.02 & * 1*;32,35 + 2mass j0602 + 4043 & j06020638 + 4043588 & & t4.5 & 15.54@xmath20.07 & * 33 * + lehpm 2@xmath0461 & j06590991@xmath04746532 & m6.5 & m7 & 13.64@xmath20.03 & * 4*;36,37 + 2mass j0727 + 1710 & j07271824 + 1710012 & t8 & t7 & 15.60@xmath20.06 & * 11*;23,32 + 2mass j0729@xmath03954 & j07290002@xmath03954043 & & t8 & 15.92@xmath20.08 & * 33 * + 2mass j0755 + 2212 & j07554795 + 2212169 & t6 & t5 & 15.73@xmath20.06 & * 1*;23,32 + sdss j0758 + 3247 & j07584037 + 3247245 & & t2 & 14.95@xmath20.04 & * 4*;1,2 + ssspm 0829@xmath01309 & j08283419@xmath01309198 & l2 & & 12.80@xmath20.03 & * 38*;39,40 + sdss j0830 + 4828 & j08300825 + 4828482 & l8 & l9@xmath21 & 15.44@xmath20.05 & * 4*;18,27 + sdss j0837@xmath00000 & j08371718@xmath00000179 & t0@xmath22 & t1 & 17.10@xmath20.21 & * 33*;1,32,41 + 2mass j0847@xmath01532 & j08472872@xmath01532372 & l2 & & 13.51@xmath20.03 & * 42*;14 + sdss j0858 + 3256 & j08583467 + 3256275 & & t1 & 16.45@xmath20.12 & * 15 * + sdss j0909 + 6525 & j09090085 + 6525275 & & t1.5 & 16.03@xmath20.09 & * 15 * + 2mass j0939@xmath02448 & j09393548@xmath02448279 & & t8 & 15.98@xmath20.11 & * 1*;12 + 2mass j0949@xmath01545 & j09490860@xmath01545485 & & t2 & 16.15@xmath20.12 & * 1*;12 + 2mass j1007@xmath04555 & j10073369@xmath04555147 & & t5 & 15.65@xmath20.07 & * 33 * + 2mass j1010@xmath00406 & j10101480@xmath00406499 & l6 & & 15.51@xmath20.06 & * 19 * + hd 89744b & j10221489 + 4114266 & l0 & l ( early ) & 14.90@xmath20.04 & * 4*;43 + sdss j1039 + 3256 & j10393137 + 3256263 & & t1 & 16.41@xmath20.15 & * 15 * + 2mass j1047 + 2124 & j10475385 + 2124234 & t7 & t6.5 & 15.82@xmath20.06 & * 4*;1,32,44 + sdss j1048 + 0111 & j10484281 + 0111580 & l1 & l4 & 12.92@xmath20.02 & * 4*;45,46 + sdss j1052 + 4422 & j10521350 + 4422559 & & t0.5@xmath21 & 15.96@xmath20.10 & * 4*;15 + wolf 359 & j10562886 + 0700527 & m6 & & 7.09@xmath20.02 & * 4 * + 2mass j1104 + 1959 & j11040127 + 1959217 & l4 & & 14.38@xmath20.03 & * 3*;14 + 2mass j1106 + 2754 & j11061197 + 2754225 & & t2.5 & 14.82@xmath20.04 & * 33 * + sdss j1110 + 0116 & j11101001 + 0116130 & & t5.5 & 16.34@xmath20.12 & * 11*;1,18 + 2mass j1114@xmath02618 & j11145133@xmath02618235 & & t7.5 & 15.86@xmath20.08 & * 11*;1,12 + 2mass j1122@xmath03512 & j11220826@xmath03512363 & & t2 & 15.02@xmath20.04 & * 1*;12 + 2mass j1124 + 3808 & j11240487 + 3808054 & m8.5 & & 12.71@xmath20.02 & * 3*;14 + sdss j1206 + 2813 & j12060248 + 2813293 & & t3 & 16.54@xmath20.11 & * 15 * + sdss j1207 + 0244 & j12074717 + 0244249 & l8 & t0 & 15.58@xmath20.07 & * 33*;1,45 + 2mass j1209@xmath01004 & j12095613@xmath01004008 & & t3 & 15.91@xmath20.07 & * 3*;1,27 + sdss j1214 + 6316 & j12144089 + 6316434 & & t3.5@xmath21 & 16.59@xmath20.12 & * 15 * + 2mass j1217@xmath00311 & j12171110@xmath00311131 & t7 & t7.5 & 15.86@xmath20.06 & * 11*;1,32,44 + 2mass j1221 + 0257 & j12212770 + 0257198 & l0 & & 13.17@xmath20.02 & * 4*;47 + 2mass j1231 + 0847 & j12314753 + 0847331 & & t5.5 & 15.57@xmath20.07 & * 3*;1 + 2mass j1237 + 6526 & j12373919 + 6526148 & t7 & t6.5 & 16.05@xmath20.09 & * 48*;1,32,44 + sdss j1254@xmath00122 & j12545393@xmath00122474 & t2 & t2 & 14.89@xmath20.04 & * 3*;1,32,44 + 2mass j1324 + 6358 & j13243559 + 6358284 & & t2 & 15.60@xmath20.07 & * 33 * + sdss j1346@xmath00031 & j13464634@xmath00031501 & t7 & t6.5 & 16.00@xmath20.10 & * 11*;1,32,49 + sdss j1358 + 3747 & j13585269 + 3747137 & & t4.5@xmath21 & 16.46@xmath20.09 & * 15 * + 2mass j1404@xmath03159 & j14044941@xmath03159329 & & t2.5 & 15.60@xmath20.06 & * 33 * + lhs 2924 & j14284323 + 3310391 & m9 & & 11.99@xmath20.02 & * 29 * + sdss j1435 + 1129 & j14355323 + 1129485 & & t2@xmath21 & 17.14@xmath20.23 & * 15 * + 2mass j1439 + 1929 & j14392836 + 1929149 & l1 & & 12.76@xmath20.02 & * 3*;31 + sdss j1439 + 3042 & j14394595 + 3042212 & & t2.5 & 17.22@xmath20.23 & * 15 * + gliese 570d & j14571496@xmath02121477 & t7 & t7.5 & 15.32@xmath20.05 & * 3*;1,32,50 + 2mass j1503 + 2525 & j15031961 + 2525196 & t6 & t5 & 13.94@xmath20.02 & * 3*;1,32,51 + 2mass j1506 + 1321 & j15065441 + 1321060 & l3 & & 13.37@xmath20.02 & * 28*;52 + 2mass j1507@xmath01627 & j15074769@xmath01627386 & l5 & l5.5 & 12.83@xmath20.03 & * 28*;2,5,6 + sdss j1511 + 0607 & j15111466 + 0607431 & & t0@xmath22 & 16.02@xmath20.08 & * 15 * + 2mass j1526 + 2043 & j15261405 + 2043414 & l7 & & 15.59@xmath20.06 & * 3*;6 + 2mass j1546@xmath03325 & j15462718@xmath03325111 & & t5.5 & 15.63@xmath20.05 & * 4*;1,23 + 2mass j1615 + 1340 & j16150413 + 1340079 & & t6 & 16.35@xmath20.09 & * 33 * + sdss j1624 + 0029 & j16241436 + 0029158 & & t6 & 15.49@xmath20.05 & * 11*;1,53 + 2mass j1632 + 1904 & j16322911 + 1904407 & l8 & l8 & 15.87@xmath20.07 & * 28*;1,31 + 2mass j1645@xmath01319 & j16452211@xmath01319516 & l1.5 & & 12.45@xmath20.03 & * 4*;54 + vb 8 & j16553529@xmath00823401 & m7 & & 9.78@xmath20.03 & * 4 * + sdss j1750 + 4222 & j17502385 + 4222373 & & t2 & 16.47@xmath20.10 & * 1*;2 + sdss j1750 + 1759 & j17503293 + 1759042 & & t3.5 & 16.34@xmath20.10 & * 3*;1,18 + 2mass j1754 + 1649 & j17545447 + 1649196 & & t5 & 15.81@xmath20.07 & * 4 * + sdss j1758 + 4633 & j17580545 + 4633099 & & t6.5 & 16.15@xmath20.09 & * 11*;1,2 + 2mass j1807 + 5015 & j18071593 + 5015316 & l1.5 & l1 & 12.93@xmath20.02 & * 4*;14,21 + 2mass j1828@xmath04849 & j18283572@xmath04849046 & & t5.5 & 15.18@xmath20.06 & * 3*;1 + 2mass j1901 + 4718 & j19010601 + 4718136 & & t5 & 15.86@xmath20.07 & * 3*;1 + vb 10 & j19165762 + 0509021 & m8 & & 9.91@xmath20.03 & * 3 * + 2mass j2002@xmath00521 & j20025073@xmath00521524 & l6 & & 15.32@xmath20.05 & * 4*;55 + sdss j2028 + 0052 & j20282035 + 0052265 & l3 & & 14.30@xmath20.04 & * 3*;45 + lhs 3566 & j20392378@xmath02926335 & m6 & & 11.36@xmath20.03 & * 3 * + 2mass j2049@xmath01944 & j20491972@xmath01944324 & m7.5 & & 12.85@xmath20.02 & * 3 * + sdss j2052@xmath01609 & j20523515@xmath01609308 & & t1@xmath21 & 16.33@xmath20.12 & * 4,15 * + 2mass j2057@xmath00252 & j20575409@xmath00252302 & l1.5 & l1.5 & 13.12@xmath20.02 & * 3*;14,46 + 2mass j2107@xmath00307 & j21073169@xmath00307337 & l0 & & 14.20@xmath20.03 & * 3*;14 + sdss j2124 + 0100 & j21241387 + 0059599 & & t5 & 16.03@xmath20.07 & * 15*;1,2 + 2mass j2132 + 1341 & j21321145 + 1341584 & l6 & & 15.80@xmath20.06 & * 59*;55 + 2mass j2139 + 0220 & j21392676 + 0220226 & & t1.5 & 15.26@xmath20.05 & * 1*;56 + hn peg b & j21442847 + 1446077 & & t2.5 & 15.86@xmath20.03 & * 10 * + 2mass j2151@xmath02441 & j21512543@xmath02441000 & l3 & & 15.75@xmath20.08 & * 56*;55,57 + 2mass j2151@xmath04853 & j21513839@xmath04853542 & & t4 & 15.73@xmath20.07 & * 4*;1,58 + 2mass j2154 + 5942 & j21543318 + 5942187 & & t6 & 15.66@xmath20.07 & * 33 * + 2mass j2212 + 1641 & j22120345 + 1641093 & m5 & & 11.43@xmath20.03 & * 3 * + 2mass j2228@xmath04310 & j22282889@xmath04310262 & & t6 & 15.66@xmath20.07 & * 3*;1,34 + 2mass j2234 + 2359 & j22341394 + 2359559 & m9.5 & & 13.15@xmath20.02 & * 3 * + sdss j2249 + 0044 & j22495345 + 0044046 & l3 & l5@xmath21.5 & 16.59@xmath20.13 & * 4*;2,18,45 + 2mass j2254 + 3123 & j22541892 + 3123498 & & t4 & 15.26@xmath20.05 & * 3*;1,23 + 2mass j2331@xmath04718 & j23312378@xmath04718274 & & t5 & 15.66@xmath20.07 & * 3*;1 + 2mass j2339 + 1352 & j23391025 + 1352284 & & t5 & 16.24@xmath20.11 & * 1*;23 + lehpm 1@xmath06333 & j23515012@xmath02537386 & m8 & m8 & 12.47@xmath20.03 & * 4*;36,40,55 + lehpm 1@xmath06443 & j23540928@xmath03316266 & m8.5 & m8 & 13.05@xmath20.02 & * 4*;36,40 + 2mass j2356@xmath01553 & j23565477@xmath01553111 & & t5.5 & 15.82@xmath20.06 & * 1*;23 + lccc spectral type & m8.5@xmath20.3 & t5@xmath20.9 & + @xmath68 ( mag ) & 13.25@xmath20.03 & 16.4@xmath20.4 & 3.1@xmath20.4 + @xmath69 ( mag ) & 12.61@xmath20.03 & 16.4@xmath20.5 & 3.8@xmath20.5 + @xmath70 ( mag ) & 12.13@xmath20.03 & 16.5@xmath20.6 & 4.3@xmath20.6 + @xmath59 & -3.48@xmath20.10 & -5.0@xmath20.3 & 1.5@xmath20.3 + @xmath71 ( @xmath32 yr@xmath10 ) & 0.562@xmath20.005 & & + @xmath72 ( @xmath11 ) & 205.9@xmath20.5 & & + @xmath73 ( pc ) & 25@xmath23 & & + @xmath13 ( km s@xmath10 ) & 67@xmath28 & & + @xmath4 ( au ) & @xmath58.3 ( @xmath50@xmath933 ) & & + mass ( m@xmath66 ) at 1 gyr & 0.081 & 0.035 & 0.44 + mass ( m@xmath66 ) at 5 gyr & 0.086 & 0.068 & 0.79 + mass ( m@xmath66 ) at 10 gyr & 0.086 & 0.074 & 0.86 +
|
the feature , which is also present in the combined light spectrum of the m8.5 + t6 binary scr 1845 , arises from the combination of feh absorption from an m8.5 primary and pseudo - continuum flux from a t5 secondary , as ascertained from binary spectral templates constructed from empirical data . the binary templates provide a far superior match to the overall near - infrared spectral energy distribution of 2mass j0320 than any single comparison spectra . laser guide star adaptive optics ( lgs ao ) imaging observations , including the first application of lgs ao aperture mask interferometry , fail to resolve a faint companion , restricting the projected separation of the system to less than 8.3 au at the time of observation .
|
evidence is presented that 2mass j03202839 , a late - type dwarf with discrepant optical ( m8 : ) and near - infrared ( l1 ) spectral types , is an as - yet unresolved stellar / brown dwarf binary with late - type m dwarf and t dwarf components . this conclusion is based on low - resolution , near - infrared spectroscopy that reveals a subtle but distinctive absorption feature at 1.6 . the feature , which is also present in the combined light spectrum of the m8.5 + t6 binary scr 1845 , arises from the combination of feh absorption from an m8.5 primary and pseudo - continuum flux from a t5 secondary , as ascertained from binary spectral templates constructed from empirical data . the binary templates provide a far superior match to the overall near - infrared spectral energy distribution of 2mass j0320 than any single comparison spectra . laser guide star adaptive optics ( lgs ao ) imaging observations , including the first application of lgs ao aperture mask interferometry , fail to resolve a faint companion , restricting the projected separation of the system to less than 8.3 au at the time of observation . 2mass j0320 is the second very low mass binary to be identified from unresolved , low - resolution , near - infrared spectroscopy , a technique that complements traditional high resolution imaging and spectroscopic methods .
|
1701.04490
|
r
|
we implement all algorithms and perform all calculations using the open - source plane - wave density - functional theory software , jdftx.@xcite all calculations in this work employ the pbe@xcite exchange - correlation functional with gbrv ultrasoft pseudopotentials@xcite at a kinetic energy cutoff of 20 hartrees for kohn - sham orbitals and 100 hartrees for the charge density . the metal surface calculations use inversion - symmetric slabs of at least five layers , with at least 15 @xmath153 vacuum separation and truncated coulomb potentials@xcite to minimize interactions with periodic images . for brillouin zone integration , we use a fermi smearing of @xmath154 hartrees and a monkhorst - pack @xmath155-point mesh along the periodic directions with the number of @xmath155-points chosen such that the effective supercell is larger than 30 in each direction . we use the candle solvation model to describe the effect of liquid water and debye screening due to 1 m electrolyte , which we showed recently to most accurately capture the solvation of highly - charged negative and positive solutes.@xcite we emphasize that the methods and algorithms described above do not rely on specific choices for the pseudopotential , exchange - correlation functional , @xmath155-mesh or solvation model ; we keep these computational parameters constant here for consistency . . upper panel shows the convergence of grand free energy @xmath156 on a logarithmic scale , and lower panel shows that of electron number @xmath78 . best convergence is obtained with @xmath157 , the inverse debye - screening length , which we use as the default value henceforth . results shown here are for a five - layer ( 5ml ) cu(111 ) slab solvated in 1 m candle aqueous electrolyte , with potential fixed to 1v she ( @xmath158 ) , starting from a converged neutral calculation of the same slab in vacuum . [ fig : scf - qkappa ] ] the self - consistent field ( gc - scf ) algorithm summarized in section [ sec : algoscf ] depends on several parameters that control its iterative convergence . all these parameters are common to the conventional fixed - charge version ( scf ) and the fixed - potential variant ( gc - scf ) introduced here , except for the low - frequency cutoff wavevector @xmath122 that is necessary to allow the net electron number to change in the fixed - potential case . figure [ fig : scf - qkappa ] compares the dependence of gc - scf convergence on @xmath122 for a prototypical calculation of an electrochemical system : a cu(111 ) surface treated using a five - layer inversion - symmetric slab surrounded by 1 m aqueous non - adsorbing electrolyte treated using the candle solvation model,@xcite at a fixed potential of @xmath158 ( 1v she ) . ( the remaining gc - scf parameters are set to their default values which we discuss below . ) the best convergence is obtained with @xmath159 which corresponds to the debye screening length of the electrolyte ( [ eqn : qkappaopt ] ) . for small @xmath122 , including the conventional case of @xmath160 , the number of electrons does not respond sufficiently quickly , stalling at about 0.1 electrons from the converged value , correspondingly with the free energy stalling at about 0.1 @xmath161 ( @xmath162 ev ) away from the converged value . larger @xmath122 causes the electron number to change too rapidly , hindering convergence and eventually leading to a divergence as seen for the case of @xmath163 . an issue remains in the convergence independent of @xmath122 : after initial convergence , the free energy oscillates at the @xmath164 level , while the electron number oscillates at the @xmath165 level . . good convergence is observed near the typical recommended value @xmath166 ( @xmath167 @xmath168 ) . convergence is relatively insensitive to @xmath105 near this value , but becomes unstable for small @xmath105 approaching the low - frequency cutoff @xmath122 . system and remaining details are identical to figure [ fig : scf - qkappa ] . [ fig : scf - qkerker ] ] keeping @xmath122 at this optimum value given by ( [ eqn : qkappaopt ] ) , we next examine the dependence of gc - scf convergence on the remaining algorithm parameters for the same example system . figure [ fig : scf - qkerker ] shows the dependence on the kerker mixing wavevector @xmath105 , which helps stabilize the gc - scf algorithm against long - wavelength charge oscillations . optimal convergence is obtained for the typical recommended value@xcite of @xmath169 ( @xmath167 @xmath168 ) . as expected , convergence is relatively insensitive to the exact choice of @xmath105 , as long as @xmath105 does not become so small that convergence is ruined by charge sloshing . notice that the final convergence beyond the @xmath164 and @xmath165 electron level remains an issue that is not resolved for any choice of @xmath105 . . convergence is relatively insensitive to @xmath106 , and we set @xmath170 henceforth . system and remaining details are identical to figure [ fig : scf - qkappa ] . [ fig : scf - qmetric ] ] next , figure [ fig : scf - qmetric ] shows the variation of gc - scf convergence with the wavevector @xmath106 controlling the reciprocal - space metric used by the pulay algorithm . the convergence is entirely insensitive to this choice , and we henceforth set @xmath170 ( the recommended value@xcite ) . again , the final convergence issue remains unaffected by the choice of @xmath106 . . convergence is relatively insensitive to @xmath0 , except that it slows down for small @xmath0 we henceforth set @xmath171 which nominally exhibits the best convergence . system and remaining details are identical to figure [ fig : scf - qkappa ] . [ fig : scf - mixfraction ] ] finally , figure [ fig : scf - mixfraction ] compares the dependence of gc - scf convergence on the maximum kerker mixing fraction @xmath0 , which effectively controls what fraction of the new electron density is mixed into the current value . we find nominally best convergence for @xmath171 , but the performance of other values is not much worse . smaller values of @xmath0 lead to greater stability initially , but marginally slower convergence later on , while larger values of @xmath0 lead to greater oscillations initially , but faster convergence later on . regardless , as before , good convergence is obtained until the free energy reaches the @xmath164 level , but continues to oscillate at that level beyond that point . , and with vacuum spacing ranging from 15 @xmath172 to 35 @xmath172 with the 5ml slab . convergence slows only marginally with increasing system size , with either vacuum spacing or layer count . each calculations is for solvated cu(111 ) charged to 1v she in candle electrolyte , starting from the state of the corresponding converged neutral vacuum calculation . [ fig : scf - size ] ] this final convergence issue may not affect practical calculations where relevant energy differences are at the @xmath173 level or higher . however , smooth exponential convergence to the final answer is desirable as this makes it easier to determine when the target accuracy has been reached . unfortunately , no combination of gc - scf parameters achieves uniformly smooth convergence for the fixed - potential case . on the other hand , with our @xmath122 modification , the gc - scf algorithm at least converges to the @xmath164 level independent of system size ( figure [ fig : scf - size ] ) , with the number of cycles required for convergence remaining mostly unchanged with increasing number of cu(111 ) layers , and with increasing thickness of the solvent region . , the preconditioning scale factor for subspace rotations generated by the auxiliary hamiltonian . near - optimal convergence is obtained when @xmath143 is automatically adjusted using the heuristic given by ( [ eqn : kadjust ] ) , which we use by default henceforth . calculations are for solvated 5ml cu(111 ) at 1v she , starting from the corresponding neutral vacuum calculation , exactly as in figures [ fig : scf - qkappa]-[fig : scf - mixfraction ] . note the smooth exponential convergence ( without oscillations in electron number and free energy ) here , in contrast to the gc - scf case . [ fig : aux - k ] ] the grand - canonical auxiliary hamiltonian ( gc - auxh ) approach discussed in section [ sec : algoaux ] , directly minimizes the total free energy of the system without assuming any models for physical properties of the system ( such as dielectric response models that are built into the scf mixing schemes ) . this algorithm contains a single parameter @xmath143 , which weights the relative contributions of the kohn - sham orbital and subspace hamiltonian degrees of freedom in the conjugate - gradients search direction for free energy minimization . figure [ fig : aux - k ] shows the dependence of iterative convergence of the gc - auxh algorithm on this preconditioning parameter @xmath143 for the same cu(111 ) test problem considered above . if the preconditioning factor @xmath143 is held fixed , the rate of convergence is sensitive to the choice of @xmath143 , with the optimum choice being @xmath174 for this system . with the preconditioner auto - adjusted using the heuristic given by ( [ eqn : kadjust ] ) , we find that indeed the convergence picks up from that of the sub - optimal @xmath175 towards that of the optimal value . more importantly , we observe smooth exponential convergence of both the free energy and the electron number , in contrast to our experiences with the gc - scf method . for the gc - scf method ) . convergence is invariant with vacuum spacing , and slows down only marginally with number of layers . [ fig : aux - size ] ] figure [ fig : aux - size ] further shows that this smooth convergence sustains with changing system size . in particular , the convergence is virtually unchanged with the thickness of the solvent regions , but slows down slightly with increasing number of copper layers in the surface slabs . having analyzed and optimized the convergence of the gc - scf and gc - auxh methods , we now compare the performance of these algorithms for a few different cases . in this comparison , we also include the present state of the art : the ` loop ' method which uses a secant method to adjust the number of electrons in an outer loop to match the specified electron chemical potential.@xcite for a fair comparison , we use the fixed - charge scf method in the inner loops , because it achieves the fastest convergence ; this algorithm works equally well with the auxh method in the inner loop , but is then marginally slower . also , we now compare the wall time between algorithms , because there is no straightforward correspondence between gc - scf cycles and conjugate - gradient iterations of the gc - auxh method . free energy accuracy in half the wall time of the loop method , but the gc - auxh method is the clear winner with smooth exponential convergence . calculations here are for the 5ml cu(111 ) slab at 1v she , as before . timings are measured on a single 32-core nersc cori node in all cases . [ fig : compare - cu5 ] ] first , figure [ fig : compare - cu5 ] compares the performance of the algorithms for the 5-layer cu(111 ) slab used in all the tests so far . the spikes in the free energy seen in the loop method are the points where the electron number changes in the outer loop and a new scf convergence at fixed charge begins . both the gc - scf and gc - auxh methods are quite competitive , cutting the time to convergence within @xmath164 in half compared to the loop method . given the smooth convergence beyond @xmath164 however , the gc - auxh method is preferable over gc - scf for fixed potential calculations . , but for a 5ml pt(111 ) slab at 0v she ( @xmath176 ) . platinum is neutral at @xmath177v she , so this corresponds to negatively charging the slab , which activates candle s asymmetry correction . additionally , platinum has @xmath178 bands crossing the fermi level in contrast to copper which has occupied @xmath178 bands . this calculation therefore explores a more complicated charge vs. potential landscape , causing deviations from exponential convergence , but the relative performance of the algorithms remains similar . [ fig : compare - pt5 ] ] next , we compare these algorithms for more complex text cases . figure [ fig : compare - pt5 ] compares the convergence for a five - layer pt(111 ) slab fixed to a potential of @xmath176 ( 0v she ) . at this potential , the surface of pt(111 ) charges negatively , and the candle solvation model brings the cavity closer to the electrons to capture the more effective solvation of negative charges by liquid water . additionally , the dielectric response of platinum is more complex than copper due to the partially filled @xmath178 shell . both of these factors make this system harder to converge than the previous test case , and therefore the convergence of the gc - auxh method is no longer clearly a single exponential . despite this , both the direct grand - canonical methods converge faster than the loop method , with the gc - auxh method eking out an advantage in final convergence as before . and [ fig : compare - pt5 ] , but for a 5ml pt(111 ) slab decorated with a @xmath179 partial monolayer of adsorbed chloride anions ( 1/3 coverage ) at 0v she . the increased complexity slows down the convergence of the gc - scf method , which also slows down the loop over fixed - charge scf calculations , while the gc - auxh method continues to exhibit rapid near - exponential convergence with no charge oscillations , beating the loop method by a factor of 4 in time . all subsequent calculations of complex metal surfaces with adsorbates therefore use the gc - auxh method . [ fig : compare - pt5_cl ] ] finally , figure [ fig : compare - pt5_cl ] compares the convergence for chloride anions adsorbed at one - third monolayer coverage , in a @xmath179 supercell of a five - layer pt(111 ) slab . despite the increased complexity , the direct grand - canonical methods exhibit the best convergence , edging out the loop method by a factor of four in wall time now , again with smoothest convergence for the gc - auxh method . due to the systematic convergence advantage of the gc - auxh method , we use it for all remaining calculations in this work and recommend it as the default general purpose algorithm for converging electrochemical calculations . in the theory section and all calculations so far , we used fermi smearing where the electron occupations are given by the fermi function ( [ eqn : fermioccupations ] ) . practical @xmath155-point meshes typically require the use of a temperature @xmath2 substantially higher than room temperature ( we used 0.01 @xmath161 which is approximately ten times higher ) , which could result in inaccurate free energies . such errors can be reduced substantially by changing the functional form of the occupations and the electronic entropy , with the caveat that the smearing width @xmath2 no longer corresponds to an electron temperature . common modifications include gaussian smearing , where the fermi functions are replaced with error functions , and cold smearing,@xcite where the functional form is chosen to cancel the lowest order variation of the free energy with @xmath2 . ( see ref . for details . ) , the gc - auxh method with cold smearing converges marginally slower than gaussian or fermi smearing , but still faster than when using any of the smearing methods with the smaller width of 0.001 @xmath161 . [ fig : smearing ] ] figure [ fig : smearing ] compares the performance of the preferred gc - auxh method for various smearing methods . the insets show the variation of the converged free energy and electron number with smearing width @xmath2 . gaussian smearing reduces the coefficient of the quadratic @xmath2 dependence compared to fermi smearing , while cold smearing cancels the quadratic dependence altogether , by design . the variation of electron number with smearing width is also reduced by cold smearing , but to a lesser extent . notice that the use of cold smearing marginally slows down the iterative convergence of the gc - auxh method . however , using cold smearing at a high width of 0.01 @xmath161 is still faster than using any smearing method with the lower width 0.001 @xmath161 that is close to room temperature ( because of the far denser brillouin zone sampling required for smaller widths ) . therefore , it is still advantageous to use cold smearing at elevated smearing widths , and so we use cold smearing with a width of 0.01 @xmath161 for the final demonstration below . the application of sufficiently negative ( reductive ) potentials on an electrode immersed in a solution containing metal ions , reduces those ions and results in bulk electro - deposition of metal on the surface . additionally , for many pairs of metals , a single monolayer of one metal deposits on a surface of the other at an under potential , that is , at a potential less favorable than for bulk deposition . this phenomenon of under - potential deposition ( upd ) has several technological applications since it enables precise synthesis of heterogeneous metal interfaces . it also serves as an archetype for fundamental studies of electrochemical processes ( see ref . for an extensive review ) , which makes it a perfect example for demonstrating our grand - canonical density - functional theory method . the basic reason for underpotential deposition is that the heterogeneous binding between the two metals is stronger than the homogeneous binding of the depositing metal to itself . indeed , metal pairs that exhibit underpotential deposition also display analogous phenomena in vapor adsorption.@xcite . however , the process in solution is far more complicated and highly sensitive to the composition of the solution because of competing adsorbates,@xcite as well as to the structure of the electrode surface.@xcite the upd of copper on pt(111 ) in the presence of chloride anions is particularly interesting and the subject of considerable debate in the literature . voltammetry for this system@xcite exhibits two well - separated under - potential peaks , as shown in the background of figure [ fig : upd ] . certain leed and in situ x - ray scattering studies of this system@xcite find evidence of a @xmath180 bilayer of copper and chloride ions co - adsorbed on the surface at potentials between the two peaks , suggesting that one peak corresponds to a formation of a partial layer , and the second peak , to the formation of the full monolayer . in contrast , other studies@xcite do not find this signature and propose that the additional peak arises from adsorption and desorption of chloride ions alone . to address this debate , we perform grand - canonical density functional theory calculations of various configurations of copper and chlorine adsorbed on a 5-layer pt(111 ) slab in @xmath179 and @xmath180 supercells . we determine the most stable configurations at each potential(@xmath23 ) and the potentials at which transitions between configurations occur by comparing their grand free energies . the relevant grand free energy of a configuration @xmath38 containing @xmath181 platinum atoms in the slab with @xmath182 copper and @xmath183 chlorine atoms adsorbed at the surface within the calculation cell is @xmath184 normalized by the number of surface platinum atoms @xmath185 in order to correctly compare energies of calculations in different supercells . ( @xmath186 for the unit cell , 6 for the @xmath179 supercell and 8 for the @xmath180 supercell , accounting for the top and bottom surfaces in the inversion - symmetric setup . ) since no light atoms are present , we safely neglect changes in vibrational contributions to the free energy between adsorbate configurations . above , @xmath187 is the free energy of adsorbate configuration @xmath38 calculated by fixed - potential dft , which is grand canonical with respect to the electrons at chemical potential @xmath23 ( related to the electrode potential by ( [ eqn : mucalibration ] ) as discussed at the end of section [ sec : theoryjdft ] ) . then , ( [ eqn : freeenergyupd ] ) above calculates the free energy @xmath188 which is additionally grand canonical with respect to all relevant atoms with chemical potentials @xmath189 , @xmath190 and @xmath191 . several conventions are possible in defining the electron - grand - canonical free energy @xmath156 , depending on what electron number we subtract : change from neutral value , total electron number , or number of valence electrons in pseudopotential dft calculations . the atom chemical potentials would then respectively correspond to neutral atoms , bare nuclei or pseudo - nuclei ( nuclei + core electrons in pseudopotential ) . the full grand canonical free energy @xmath192 does not depend on this choice . in our jdftx implementation , we choose the last option above ( number of valence electrons and correspondingly atom chemical potentials of the pseudo - nuclei ) . the bulk of the platinum electrode sets the pt chemical potential , @xmath193 , where @xmath194 is the dft energy of a bulk fcc pt calculation with a single atom in the unit cell , and @xmath195 is the number of valence electrons in that calculation . the second term here implements the electron counting convention discussed above . next , copper ions in solution set @xmath190 , but directly calculating the free energy of such ions using solvation models is error - prone.@xcite so , instead , we use the dft calculated energy , @xmath196 , of a bulk fcc cu calculation ( containing @xmath197 valence electrons ) , and relate it to the free energy of the ion via the experimentally - determined standard reduction potential @xmath198 v she.@xcite this yields @xmath199,\end{gathered}\ ] ] where the second term accounts for the change from cu(s ) to @xmath200 ions , and the final term accounts for change in ionic concentration from the standard value of 1 mol / liter to the current value of [ cu@xmath201 ( in mol / liter ) . similarly , chlorine ions in solution set @xmath191 , but to minimize dft errors , we connect to the dft calculated energy , @xmath202 , of an isolated chlorine atom ( containing @xmath203 valence electrons ) , via the experimentally - determined atomization energy @xmath204 kj / mol,@xcite gas - phase entropy @xmath205 j / mol - k,@xcite and reduction potential @xmath206 v she.@xcite specifically , @xmath207,\end{gathered}\ ] ] where the second term accounts for the change from atomic to gas - phase chlorine , the third term for the change to chloride ions , and the final term for the change in chloride ion concentration to [ cl@xmath208 ( in mol / liter ) . mol / liter cu@xmath209 ions and 10@xmath210 mol / liter cl@xmath211 ions . calculated free energies for various adsorbate configurations as a function of electrode potential are shown as calculated by explicit fixed - potential solvated calculations in ( a ) , and from vacuum calculations in ( b ) , with the experimental voltammogram@xcite shown for comparison . solid red lines indicate no copper , purple dotted lines indicate @xmath180 partial copper monolayer and blue dot - dash lines indicate full copper monolayer . thickness of the lines indicate cl coverage , ranging from no cl ( thinnest ) , @xmath180 ( 1/4 ) coverage ( intermediate ) to @xmath179 ( 1/3 ) coverage ( thickest ) . prominent configurations are sketched in ( c - f ) . vacuum calculations in ( b ) predict only a single voltammetric peak in disagreement with experiment . explicit potential - dependent solvated calculations in ( a ) predict two peaks in qualitative agreement with experiment ( @xmath212 v accuracy ) , with the second peak due to chloride desorption.@xcite the partial cu monolayer ( d ) proposed by some@xcite is not predicted to be the most stable configuration at any relevant potential . [ fig : upd ] ] figure [ fig : upd](a ) shows the calculated grand free energies as a function of electrode potential for a number of cu and cl adsorbate configurations on the surface of pt(111 ) . at high potentials , the most stable ( lowest free energy ) configuration is 1/4 cl coverage ( figure [ fig : upd](f ) ) , which transitions to a clean pt surface ( figure [ fig : upd](e ) ) at a potential of 0.55 v she . upon further lowering the potential , the stable configuration transitions to a full monolayer of copper with 1/3 cl coverage ( figure [ fig : upd](c ) ) at a potential of 0.46 v experimentally , the two voltammogram peaks are at approximately @xmath213 and @xmath214 v she , averaging over the forward and reverse direction sweeps . therefore chlorine desorption and full - copper - monolayer formation are plausible explanations@xcite of the two peaks , with our first - principles predictions reproducing well the peak spacing ( 0.09 ev versus 0.12 ev in experiment ) , and placing the absolute locations of the peaks to within 0.07 ev . the partial 2x2 monolayer of copper ( figure [ fig : upd](d ) ) proposed by others@xcite as the reason for the second peak is not the most stable configuration in our calculations at any potential , lying a singinificant 0.3 ev above the other phases at relevant potentials . for comparison , figure [ fig : upd](b ) shows the analogous results that would be obtained using only conventional vacuum calculations . in the above formalism , this corresponds to assuming @xmath215 , where @xmath216 is the helmholtz energy from a neutral vacuum dft calculation of configuration @xmath38 containing @xmath217 valence electrons . this approximation results in a single transition directly from the cl - covered pt surface to the one with a copper monolayer , predicting a single voltammogram peak in disagreement with experiment . accurate predictions for electrochemical systems therefore require treating charged configurations stabilized by the electrolyte at relevant electron potentials , now easily accomplished with the methods and algorithms introduced in this work .
|
both methods substantially improve performance compared to a sequence of conventional fixed - number calculations targeting the desired potential , with the gc - auxh method additionally exhibiting reliable and smooth exponential convergence of the grand free energy . enables theoretical elucidation of reaction mechanisms at complex catalyst surfaces , making it now possible to design efficient heterogeneous catalysts for various industrial applications from first principles , for example for high - temperature gas - phase transformation of hydrocarbons to a variety of valuable chemical products. the extension of this predictive power to electrocatalysis would be highly valuable for an even broader class of technological problems , including a cornerstone of future technology for renewable energy : converting solar energy to chemical fuels by electrochemical water splitting and carbon dioxide reduction. accurately describing electrochemical phenomena , however , presents two additional challenges . sections [ sec : resultsscf ] and [ sec : resultsaux ] establish the algorithm parameter(s ) that optimize the iterative convergence of the gc - scf and gc - auxh methods respectively , while section [ sec : algocompare ] compares the performance of these algorithms for a number of prototypical electrochemical systems .
|
first - principles calculations combining density - functional theory and continuum solvation models enable realistic theoretical modeling and design of electrochemical systems . when a reaction proceeds in such systems , the number of electrons in the portion of the system treated quantum mechanically changes continuously , with a balancing charge appearing in the continuum electrolyte . a grand - canonical ensemble of electrons at a chemical potential set by the electrode potential is therefore the ideal description of such systems that directly mimics the experimental condition . we present two distinct algorithms , a self - consistent field method ( gc - scf ) and a direct variational free energy minimization method using auxiliary hamiltonians ( gc - auxh ) , to solve the kohn - sham equations of electronic density - functional theory directly in the grand canonical ensemble at fixed potential . both methods substantially improve performance compared to a sequence of conventional fixed - number calculations targeting the desired potential , with the gc - auxh method additionally exhibiting reliable and smooth exponential convergence of the grand free energy . finally , we apply grand - canonical dft to the under - potential deposition of copper on platinum from chloride - containing electrolytes and show that chloride desorption , not partial copper monolayer formation , is responsible for the second voltammetric peak . density - functional theory ( dft ) enables theoretical elucidation of reaction mechanisms at complex catalyst surfaces , making it now possible to design efficient heterogeneous catalysts for various industrial applications from first principles , for example for high - temperature gas - phase transformation of hydrocarbons to a variety of valuable chemical products. the extension of this predictive power to electrocatalysis would be highly valuable for an even broader class of technological problems , including a cornerstone of future technology for renewable energy : converting solar energy to chemical fuels by electrochemical water splitting and carbon dioxide reduction. accurately describing electrochemical phenomena , however , presents two additional challenges . first , the electrolyte , typically consisting of ions in a liquid solvent , strongly affects the energetics of structures and reactions at the interface . treating liquids directly in dft requires expensive molecular dynamics to sample the thermodynamic phase space of atomic configurations . historically , a number of continuum solvation models that empirically capture liquid effects have enabled theoretical design of liquid - phase catalysts. more recently , empirical solvation models suitable for solid - liquid interfaces, joint density - functional theory ( jdft ) for efficiently treating liquids with atomic - scale structure, and minimally - empirical solvation models derived from jdft, have made great strides towards reliable yet efficient treatment of electrochemical systems . second , electrons can flow in and out of the electrode as electrochemical reactions proceed . changes in electronic charge of electrode surfaces and adsorbates can be especially important because the electrolyte stabilizes charged configurations with a counter charge from the ionic response . for example , reduction of formic acid on platinum at experimentally relevant potentials is dominated by formate ions rather than neutral molecules at the surface. proton adsorption on stepped and polycrystalline surfaces involves displacing oxidatively - adsorbed water at relevant potentials , resulting in non - integer charge transfers and an anomalous ph dependence deviating from the nernst equation. accounting for the electrolyte response using our solvation models, and adjusting the electron number to match experimentally relevant electrode potentials , realistic predictions of electrochemical reaction mechanisms have now become possible. in particular , application of this methodology to the reduction of co on cu(111 ) predicts onset potentials for methane and ethene formation with 0.05 v accuracy in comparison to experiment , for a wide range of ph varying from 1 to 12. however , conventional dft software and algorithms are optimized for solving the quantum - mechanical problem at fixed electron number , requiring extra work ( both manual and computational ) to calculate properties for a specified electrode potential . this paper introduces algorithms for grand canonical dft , where electron number adjusts automatically to target a specified electron chemical potential ( related to electrode potential ) , thereby enabling efficient and intuitive first - principles treatment of electrochemical phenomena . section [ sec : theory ] summarizes the theoretical background of first - principles electrochemistry using jdft and continuum solvation models , and sets up the fundamental basis of grand - canonical dft . then , section [ sec : algorithms ] introduces the modifications necessary to make two distinct classes of dft algorithms , the self - consistent field ( gc - scf ) method and the variational free energy minimization using auxiliary hamiltonians ( gc - auxh ) , directly converge the grand free energy of electrons at fixed potential . sections [ sec : resultsscf ] and [ sec : resultsaux ] establish the algorithm parameter(s ) that optimize the iterative convergence of the gc - scf and gc - auxh methods respectively , while section [ sec : algocompare ] compares the performance of these algorithms for a number of prototypical electrochemical systems . finally , section [ sec : upd ] demonstrates the utility of grand canonical dft by solving an electrochemical mystery : the identity of the second voltammetric peak in the under - potential deposition ( upd ) of copper on platinum in chloride - containing electrolytes .
|
1701.04490
|
c
|
this work introduces algorithms for directly converging dft calculations in the grand - canonical ensemble of electrons , where the number of electrons adjusts to maintain the system at constant electron chemical potential , while ionic response in a continuum solvation model of electrolyte keeps the system neutral . we show that , with appropriate modifications , grand canonical versions of both the self - consistent field ( gc - scf ) method as well as direct free - energy minimization with auxiliary hamiltonians ( gc - auxh ) method are able to rapidly converge the grand free energy of electrons . this substantially improves upon the current state of the art of running an outer loop over conventional fixed - charge dft calculations . with detailed tests of the convergence of all these algorithms , we show that the gc - auxh method is the most suitable default choice exhibiting smooth exponential convergence to the minimum . grand - canonical dft directly mimics the experimental condition in electrochemical systems , where electrode potential sets the chemical potential of electrons , and the number of electrons at the electrode surface ( including adsorbates in the electrochemical interface ) changes continuously in response . describing this change in charge at the surface plays an important role in accurately modeling several electrochemical phenomena.@xcite here , we showcase the new algorithms by analyzing the under - potential deposition ( upd ) of copper on platinum in an electrolyte containing chloride ions . we resolve an old debate about the identity of a second under - potential peak , showing that partial copper monolayers are not plausible and that the second peak is due to desorption of chloride ions . we expect the new methods presented here to substantially advance the realistic treatment of electrochemical phenomena in first principles calculations . rs and wag acknowledge support from the joint center for artificial photosynthesis ( jcap ) , a doe energy innovation hub , supported through the office of science of the u.s . department of energy under award number de - sc0004993 . rs and taa acknowledge support from the energy materials center at cornell ( emc@xmath218 ) , an energy frontier research center funded by the u.s . department of energy , office of science , office of basic energy sciences under award number de - sc0001086 . calculations in this work used the national energy research scientific computing center ( nersc ) , a doe office of science user facility supported by the office of science of the u.s . department of energy under contract no . de - ac02 - 05ch11231 . we thank kendra letchworth - weaver , kathleen schwarz , yuan ping , hai xiao , tao cheng , robert smith nielsen and jason goodpaster for insightful discussions . 49ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop `` , '' ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) , ed . , @noop _ _ ( , ) @noop _ _ ,
|
when a reaction proceeds in such systems , the number of electrons in the portion of the system treated quantum mechanically changes continuously , with a balancing charge appearing in the continuum electrolyte . a grand - canonical ensemble of electrons at a chemical potential set by the electrode potential is therefore the ideal description of such systems that directly mimics the experimental condition . we present two distinct algorithms , a self - consistent field method ( gc - scf ) and a direct variational free energy minimization method using auxiliary hamiltonians ( gc - auxh ) , to solve the kohn - sham equations of electronic density - functional theory directly in the grand canonical ensemble at fixed potential . finally , we apply grand - canonical dft to the under - potential deposition of copper on platinum from chloride - containing electrolytes and show that chloride desorption , not partial copper monolayer formation , is responsible for the second voltammetric peak . density - functional theory ( dft ) changes in electronic charge of electrode surfaces and adsorbates can be especially important because the electrolyte stabilizes charged configurations with a counter charge from the ionic response . this paper introduces algorithms for grand canonical dft , where electron number adjusts automatically to target a specified electron chemical potential ( related to electrode potential ) , thereby enabling efficient and intuitive first - principles treatment of electrochemical phenomena . then , section [ sec : algorithms ] introduces the modifications necessary to make two distinct classes of dft algorithms , the self - consistent field ( gc - scf ) method and the variational free energy minimization using auxiliary hamiltonians ( gc - auxh ) , directly converge the grand free energy of electrons at fixed potential . finally , section [ sec : upd ] demonstrates the utility of grand canonical dft by solving an electrochemical mystery : the identity of the second voltammetric peak in the under - potential deposition ( upd ) of copper on platinum in chloride - containing electrolytes .
|
first - principles calculations combining density - functional theory and continuum solvation models enable realistic theoretical modeling and design of electrochemical systems . when a reaction proceeds in such systems , the number of electrons in the portion of the system treated quantum mechanically changes continuously , with a balancing charge appearing in the continuum electrolyte . a grand - canonical ensemble of electrons at a chemical potential set by the electrode potential is therefore the ideal description of such systems that directly mimics the experimental condition . we present two distinct algorithms , a self - consistent field method ( gc - scf ) and a direct variational free energy minimization method using auxiliary hamiltonians ( gc - auxh ) , to solve the kohn - sham equations of electronic density - functional theory directly in the grand canonical ensemble at fixed potential . both methods substantially improve performance compared to a sequence of conventional fixed - number calculations targeting the desired potential , with the gc - auxh method additionally exhibiting reliable and smooth exponential convergence of the grand free energy . finally , we apply grand - canonical dft to the under - potential deposition of copper on platinum from chloride - containing electrolytes and show that chloride desorption , not partial copper monolayer formation , is responsible for the second voltammetric peak . density - functional theory ( dft ) enables theoretical elucidation of reaction mechanisms at complex catalyst surfaces , making it now possible to design efficient heterogeneous catalysts for various industrial applications from first principles , for example for high - temperature gas - phase transformation of hydrocarbons to a variety of valuable chemical products. the extension of this predictive power to electrocatalysis would be highly valuable for an even broader class of technological problems , including a cornerstone of future technology for renewable energy : converting solar energy to chemical fuels by electrochemical water splitting and carbon dioxide reduction. accurately describing electrochemical phenomena , however , presents two additional challenges . first , the electrolyte , typically consisting of ions in a liquid solvent , strongly affects the energetics of structures and reactions at the interface . treating liquids directly in dft requires expensive molecular dynamics to sample the thermodynamic phase space of atomic configurations . historically , a number of continuum solvation models that empirically capture liquid effects have enabled theoretical design of liquid - phase catalysts. more recently , empirical solvation models suitable for solid - liquid interfaces, joint density - functional theory ( jdft ) for efficiently treating liquids with atomic - scale structure, and minimally - empirical solvation models derived from jdft, have made great strides towards reliable yet efficient treatment of electrochemical systems . second , electrons can flow in and out of the electrode as electrochemical reactions proceed . changes in electronic charge of electrode surfaces and adsorbates can be especially important because the electrolyte stabilizes charged configurations with a counter charge from the ionic response . for example , reduction of formic acid on platinum at experimentally relevant potentials is dominated by formate ions rather than neutral molecules at the surface. proton adsorption on stepped and polycrystalline surfaces involves displacing oxidatively - adsorbed water at relevant potentials , resulting in non - integer charge transfers and an anomalous ph dependence deviating from the nernst equation. accounting for the electrolyte response using our solvation models, and adjusting the electron number to match experimentally relevant electrode potentials , realistic predictions of electrochemical reaction mechanisms have now become possible. in particular , application of this methodology to the reduction of co on cu(111 ) predicts onset potentials for methane and ethene formation with 0.05 v accuracy in comparison to experiment , for a wide range of ph varying from 1 to 12. however , conventional dft software and algorithms are optimized for solving the quantum - mechanical problem at fixed electron number , requiring extra work ( both manual and computational ) to calculate properties for a specified electrode potential . this paper introduces algorithms for grand canonical dft , where electron number adjusts automatically to target a specified electron chemical potential ( related to electrode potential ) , thereby enabling efficient and intuitive first - principles treatment of electrochemical phenomena . section [ sec : theory ] summarizes the theoretical background of first - principles electrochemistry using jdft and continuum solvation models , and sets up the fundamental basis of grand - canonical dft . then , section [ sec : algorithms ] introduces the modifications necessary to make two distinct classes of dft algorithms , the self - consistent field ( gc - scf ) method and the variational free energy minimization using auxiliary hamiltonians ( gc - auxh ) , directly converge the grand free energy of electrons at fixed potential . sections [ sec : resultsscf ] and [ sec : resultsaux ] establish the algorithm parameter(s ) that optimize the iterative convergence of the gc - scf and gc - auxh methods respectively , while section [ sec : algocompare ] compares the performance of these algorithms for a number of prototypical electrochemical systems . finally , section [ sec : upd ] demonstrates the utility of grand canonical dft by solving an electrochemical mystery : the identity of the second voltammetric peak in the under - potential deposition ( upd ) of copper on platinum in chloride - containing electrolytes .
|
1606.09532
|
i
|
let @xmath6 be an odd prime , and @xmath5 be an algebraically closed field of characteristic @xmath6 . for @xmath7 an integer , we consider the symplectic group @xmath4 , thought of as an algebraic group of rank @xmath8 . it is well - known that the classification ( due to chevalley ) of rational simple @xmath4-modules is the same as in characteristic zero ( see jantzen ( * ? ? ? * ii.2 ) ) . more precisely , for every dominant weight @xmath9 there is a simple module @xmath3 , and these exhaust all isomorphism classes of simple modules . here the set of dominant weights is the same as in characteristic zero : @xmath9 is dominant iff it is a linear combination of the fundamental weights @xmath10 ( @xmath11 ) with nonnegative integer coefficients . on the other hand , while the dimensions of simple @xmath12-modules can be computed from the weyl character formula , it seems that explicit dimension formulae for the modules @xmath13 for @xmath14 are quite rare , except in rather special situations . we refer to @xcite for a survey . for fundamental weights , premet and suprunenko @xcite gave an algorithm to compute the dimensions of @xmath15 by reducing the problem to known properties of symmetric group representations . later , gow @xcite gave an explicit construction of @xmath15 for the last @xmath2 fundamental weights ( that is : @xmath10 where @xmath16 ) which allowed him to obtain a recursive formula for their dimensions . even later , foulle @xcite obtained a dimension formula for all fundamental weights . as for other weights , it is known that for weights @xmath9 in the fundamental alcove the dimension of @xmath3 is the same as the dimension of @xmath17 ( the corresponding simple module in characteristic zero ) , and can thus be computed by the weyl character formula . but for weights outside the fundamental alcove , no general dimension formula is known . a conjectural formula by lusztig for primes in a certain range was shown to hold for @xmath18 by andersen - jantzen - soergel @xcite but was recently shown not to hold for all @xmath6 in the hoped - for range by williamson @xcite . in this paper , we show that topological quantum field theory ( tqft ) can give new information about the dimensions of some of these simple modules . specifically , we show that for every prime @xmath0 and in every rank @xmath1 , there is a family of @xmath2 dominant weights @xmath19 , lying outside of the fundamental alcove except for one weight in rank @xmath20 , for which we can express the dimension of @xmath13 by formulae similar to the verlinde formula in tqft . we found this family as a byproduct of integral @xmath21-tqft @xcite , an integral refinement of the witten - reshetikhin - turaev tqft associated to @xmath21 . more precisely , we use integral @xmath21-tqft in what we call the ` equal characteristic case ' which we studied in @xcite . the family of weights @xmath19 we found together with our formulae for @xmath22 is given in the following theorem [ 1.1 ] . we can also compute the weight space decomposition of @xmath13 for these weights @xmath19 ; this will be given in theorem [ 1.8 ] . we follow the notation of ( * ? ? ? * planche iii ) , where the fundamental weights @xmath23 are expressed in the usual basis @xmath24 ( @xmath25 ) of weights of the maximal torus as @xmath26 . let @xmath0 be prime and put @xmath27 . for rank @xmath1 , consider the following @xmath2 dominant weights for the symplectic group @xmath4 : @xmath28 put @xmath29 in case @xmath30 and @xmath31 and @xmath32 in case @xmath33 and @xmath34 . then @xmath35 @xmath36 @xmath37 and @xmath38 is the same @xmath38 used in the definition of @xmath9 , except in case @xmath30 and @xmath34 , where we put @xmath39 . in case @xmath40 in rank @xmath20 , @xmath41 should be interpreted as zero . formula ( [ verl ] ) is an instance of the famous verlinde formula in tqft . formula ( [ d - verl ] ) appeared first in @xcite . note that the difference between the two formulae is that certain sines in ( [ verl ] ) have become cosines in ( [ d - verl ] ) , and the overall prefactor is different . for fixed @xmath8 , both @xmath42 and @xmath43 can be expressed as polynomials in @xmath6 and @xmath38 . see @xcite for more information and further references . in appendix [ app - b ] , we give explicit polynomial expressions for the dimensions of our @xmath13 in rank @xmath44 . when @xmath45 , the list above produces ( in order ) the fundamental weights @xmath46 . these are exactly the weights considered by gow @xcite . for @xmath47 , our weights are different from those of gow . it is intriguing that both gow s and our family of weights have @xmath2 elements . can one find similar verlinde - like dimension formulae for other families of dominant weights ? in @xcite , we answered this question affirmatively for the @xmath2 fundamental weights considered by gow . but @xcite was based on gow s recursion formula , not on tqft as in the present paper . on the other hand , @xmath21-tqft is just one of the simplest tqfts within the family of witten - reshetikhin - turaev tqfts , and it is conceivable that other integral tqfts ( _ e.g. _ @xcite ) might produce more families of weights @xmath19 where the methods of the present paper could be applied . a difficulty here is that integral tqft as we need it in this paper is so far not developed for other tqfts . throughout the paper , we assume @xmath0 and we use the notation @xmath27 . the construction of the modules @xmath3 goes as follows . for @xmath48 and @xmath49 , we construct certain simple modules which we denote by @xmath50 . note that there are @xmath2 choices of pairs @xmath51 . the construction of the modules @xmath50 is based on results from integral tqft obtained in @xcite . from the tqft description , we shall compute the dimension and weight space decomposition of @xmath50 . in particular , we shall compute the highest weight occuring in @xmath50 , thereby identifying @xmath50 with one of the @xmath13 in theorem [ 1.1 ] . , width=172 ] here is the construction of @xmath50 . we give a description which can be read without any knowledge of tqft . consider the graph @xmath52 depicted in figure [ lol ] which we call a lollipop tree . it has @xmath53 trivalent vertices and one univalent vertex which in the figure is labelled @xmath54 . the @xmath55-valent ` corner ' vertex to the left of the figure should be ignored , and the two edges meeting there are to be considered a single edge . thus , @xmath52 has @xmath56 edges , @xmath8 of which are loop edges . the edges incident to a loop edge are called stick edges , and we refer to a loop edge together with its stick edge as a lollipop . a @xmath6-color is an integer @xmath57 . a @xmath6-coloring of @xmath52 is an assignment of @xmath6-colors to the edges of @xmath52 . a @xmath6-coloring is _ admissible _ if whenever @xmath58 , @xmath59 and @xmath60 are the colors of edges which meet at a vertex , then @xmath61 admissibility at the trivalent vertex of the @xmath58-th lollipop implies that the stick edge has to receive an even color , which we denote by @xmath62 , and the loop edge has to receive a color of the form @xmath63 , with @xmath64 . we denote the colors of the remaining edges by @xmath65 as in figure [ lol2 ] , and we write an admissible @xmath6-coloring as @xmath66 . , width=172 ] a @xmath6-coloring is of type @xmath51 if the color @xmath54 is assigned to the edge incident with the univalent vertex and if @xmath67 a @xmath6-coloring is _ small _ if the colors @xmath63 of the loop edges satisfy @xmath68 let @xmath69 denote the set of small admissible @xmath6-colorings of @xmath52 of type @xmath51 . let @xmath70 denote the finite field with @xmath6 elements , and let @xmath71 be the @xmath70-vector space with basis @xmath69 . there is an irreducible representation of the finite symplectic group @xmath72 on @xmath71 . this will be proved in section [ sec3 ] . steinberg s restriction theorem ( see _ e.g. _ @xcite ) implies that there is a unique simple @xmath4-module @xmath50 characterized by the following two properties : \(i ) the restriction of @xmath50 to the finite group @xmath72 is @xmath73 . \(ii ) @xmath50 has @xmath6-restricted highest weight . we recall that a dominant weight @xmath74 is _ @xmath6-restricted _ if , for each @xmath75 , we have @xmath76 . part ( i ) of the following theorem says that the @xmath50 are precisely the simple modules @xmath3 listed in theorem [ 1.1 ] . part ( ii ) gives the weight space decomposition and thus determines the formal character of these modules . to state the result , let @xmath77 be the multiset of weights occuring in @xmath50 . ( by a multiset , we mean a set with multiplicities . ) \(i ) the @xmath4-module @xmath50 is isomorphic to @xmath3 where the highest weight @xmath78 is given by @xmath79 \(ii ) we have @xmath80 where the weight of a coloring @xmath66 is @xmath81 in case @xmath82 , the highest weight corresponds to the @xmath6-coloring @xmath83 where all edges are colored zero . indeed , formula ( [ defw ] ) gives @xmath84 in the other cases , the coloring @xmath83 is not allowed as it is not of type @xmath51 for @xmath85 . we shall describe the colorings corresponding to the highest weights in case @xmath86 in section [ sec4 ] . the @xmath45 case of theorem [ 1.8 ] answers affirmatively the question raised in ( * ? ? ? * p. 257 ( after theorem 8.1 ) ) ( see also ( * ? ? ? * ( after corollary 3 ) ) ) . the remainder of this paper is organized as follows . in section [ sec2 ] , we formulate two results ( lemma [ elb1 ] and lemma [ elb2 ] ) about the @xmath72-modules @xmath71 . in section [ sec3 ] , we review the construction of @xmath71 and the proof of theorem [ 1.7 ] , and then prove lemma [ elb1 ] and lemma [ elb2 ] using further arguments from tqft . in section [ sec4 ] , we prove theorems [ 1.1 ] and [ 1.8 ] . the only results from tqft that will be used in the proof of these two theorems are those stated in section [ sec2 ] . finally , in section [ sec5 ] , we make a few further comments and discuss the rank 3 case as an example . * acknowledgements . * we thank henning h. andersen for helpful discussions . he suggested checking our results against the jantzen sum formula in the rank @xmath87 case ( see section [ sec5 ] ) and showed us how to do it . g. m. thanks the mathematics department of louisiana state university , baton rouge , the centre for quantum geometry of moduli spaces , aarhus , denmark , and the max planck institute for mathematics , bonn , germany , for hospitality while part of this paper was written . p. g. also thanks the max planck institute for mathematics for hospitality . last but not least , we thank the referee for his insightful comments .
|
for a prime , and an integer , we use topological quantum field theory ( tqft ) to study a family of highest weight modules for the symplectic group where is an algebraically closed field of characteristic . this permits explicit formulae for the dimension and the formal character of for these highest weights .
|
for a prime , and an integer , we use topological quantum field theory ( tqft ) to study a family of highest weight modules for the symplectic group where is an algebraically closed field of characteristic . this permits explicit formulae for the dimension and the formal character of for these highest weights .
|
1211.5426
|
i
|
the famous lacunary fourier series @xmath13 was proposed by riemann in the 1850 s as an example of continuous but nowhere differentiable function . since then , this series has drawn much attention from many mathematicians ( amongst them , hardy and littlewood ) , and its complete local study was finally achieved by gerver in @xcite and jaffard in @xcite . in particular , its local regularity at a point @xmath14 depends on the diophantine type of @xmath14 , and it is differentiable only at rationals @xmath15 where @xmath16 and @xmath17 are both odd . in this article , we study the series defined for @xmath18 and @xmath19 by @xmath20 and we denote by @xmath21 its @xmath22-th partial sum . both are periodic functions of period @xmath23 in @xmath14 and @xmath24 in @xmath25 . for @xmath26 and @xmath27 the imaginary part of is . for any fixed @xmath25 , if @xmath28 , @xmath4 is in @xmath29 and it converges almost everywhere by carleson s theorem . it is not everywhere convergent however . one of the aim of this paper is to understand better the convergence of @xmath30 especially when @xmath27 . we set @xmath31 , @xmath32 , resp . @xmath33 if @xmath34 , resp . @xmath35 , and @xmath36 . we define @xmath37 with @xmath38 . we denote by @xmath39 and @xmath40 the integer part and fractional part respectively of a real number @xmath14 . for @xmath34 , @xmath41 and @xmath42 , we set @xmath43 this function is well - defined and if @xmath44 , the second integral is equal to @xmath45 ( because the series vanishes ) . we then define a function @xmath46 as follows : @xmath47 for simplicity , given a function @xmath48 , we will write @xmath49 for @xmath50 . the function @xmath46 will be particularly important in this paper . our first result is a consequence of the celebrated `` approximate functional equation for the theta series '' of hardy and littlewood ( proposition [ prop : h - l ] in section [ sec : hl ] ) , which corresponds exactly to the case @xmath44 in our theorem [ theo:1 ] below ; see @xcite where many references and historical notes are given . [ theo:4 ] let @xmath14 be an irrational number in @xmath51 whose ( regular ) continued fraction is denoted by @xmath52 , and let @xmath41 . @xmath53 if @xmath54 and @xmath55 then @xmath30 is absolutely convergent . @xmath56 if @xmath12 and @xmath57 then @xmath58 is absolutely convergent . conditions and hold for lebesgue - almost all @xmath14 . it is possible to prove a quantitative version of therorem [ theo:4 ] . we need to introduce more notations . we introduce the two transformations @xmath59-\frac tx \right\}.\ ] ] then , for all @xmath14 satisfying at least the conditions or , it can be proved ( and this would add nearly ten more pages to the paper . ] ) that @xmath60 where @xmath61 if @xmath62 is even , @xmath63 if @xmath62 is odd , and @xmath64 eq . holds very generally , the right - hand side converges quickly and the appearence of gauss transform @xmath65 is a nice feature . but this is at the cost of the simultaneous appearance of the operator @xmath66 and this makes looks very complicated , even when @xmath27 because @xmath67 in general . however , the underlying modular nature of @xmath30 implies that the transformation of @xmath2\setminus\{0\}$ ] given by @xmath68 is more natural than gauss in this specific study , and in particular it leads to another expression ( i.e , below ) for @xmath30 which is formally similar to but simpler . the comparison of both approaches is one of our motivations . our next theorem below explains what we mean by `` the modular nature of @xmath30 '' and the subsequent theorems are devoted to convergence conditions of @xmath30 ( mainly when @xmath27 ) in terms of series defined by the operator @xmath69 , as well as their relations with theorem [ theo:4 ] . [ theo:1 ] @xmath53 for any @xmath70\setminus \{0\}$ ] , @xmath41 , @xmath42 , we have the estimate @xmath71 when @xmath22 tends to infinity . the implicit constant depends on @xmath72 and @xmath25 , but not on @xmath14 . @xmath56 when @xmath73 , the function @xmath74 is continuous on @xmath75 , differentiable at any rational number @xmath15 with @xmath76 both odd , and @xmath77 are bounded on @xmath78 . @xmath79 when @xmath80 , the function @xmath74 is differentiable on @xmath81 and continuous at @xmath45 . the function @xmath82 is the same as the one used by cellarosi @xcite and fedotov - klopp @xcite . when @xmath80 , the proof of item @xmath56 yields only that @xmath8 is bounded around @xmath45 . see figures [ fig3 ] and [ fig4 ] for an illustration of theorem [ theo:1 ] . as @xmath83 , the left hand side of tends to @xmath46 when @xmath34 . the resulting `` modular '' equation @xmath84 holds a priori at least _ almost everywhere _ for @xmath85 for any fixed @xmath86 and @xmath87 , and theorem [ theo:1 ] shows in which sense we can say it holds _ everywhere_. for other examples of this phenomenon , see @xcite for instance . of course , if @xmath80 , holds for all @xmath14 . in fact , we will obtain a more precise estimate for the error term in , uniform in @xmath88\setminus\{0\}$ ] , @xmath89 and @xmath41 : @xmath90 where the constants in the @xmath91 on the right - hand side depend now at most on @xmath72 and are effective . we need this refinement to prove theorem [ theo:2 ] below . theorem [ theo:1 ] will be used to get informations of the convergence of @xmath92 in terms of the diophantine properties of @xmath14 . in the sequel @xmath93 denotes the @xmath94-th iterate of @xmath14 by @xmath69 . by @xmath23-periodicity of @xmath69 , eq . can be rewritten as follows ( when @xmath27 ) : for any @xmath70\setminus \{0\}$ ] , @xmath95 and integer @xmath89 , @xmath96 ( when @xmath27 , the error term @xmath97 in is absorbed by the error term in , for some constant that depends only on @xmath72 ; this is enough for the application we have in mind . ) the second sum @xmath98 in involves less terms than the first one for any irrational number in @xmath99 , because @xmath100 . hence for any fixed @xmath22 and @xmath14 , we can iterate because @xmath101 . after a finite number of steps ( say @xmath102 , which depends on @xmath14 ) , we get an empty sum @xmath103 on the right hand side together with a finite sum defined in terms of iterates of @xmath74 and a quantity we expect to be an error term ( i.e. , that tends to @xmath45 as @xmath22 tends to infinity under suitable condition on @xmath14 ) . we prove the following result . [ theo:2 ] let @xmath104 be an irrational number . + @xmath53 if @xmath105 and if @xmath106 then @xmath92 is also convergent and the following identity holds : @xmath107 @xmath56 if @xmath108 then @xmath109 is also convergent and the following identity holds : @xmath110 the series in , and do not converge everywhere . it was an open question whether such series converge lebesgue - almost everywhere . note that the results in the cited papers of cellarosi @xcite , kraaikamp - lopes @xcite , schweiger @xcite , and sinai @xcite give estimates for the average behavior of @xmath111 when @xmath62 tends to infinity , but these estimates are not sharp enough ( is ergodic with respect to a measure @xmath112 supported on @xmath2 $ ] but , in contrast with the ergodic theory of gauss transformation @xmath65 , the measure @xmath112 is infinite . as a consequence , the analogue of birkoff s ergodic theorem is not known and one must content with `` convergence in probability '' results ( see @xcite ) , which are not well suited to our study . ] ) to guarantee the almost everywhere convergence . it would be interesting to know if @xmath109 converges under the assumption of convergence of only . we now precise the diophantine content of theorem [ theo:2 ] . [ theo:3 ] let @xmath113 , @xmath114 , and set @xmath115 . @xmath53 if @xmath116 , then the series @xmath117 converges if @xmath118 @xmath56 if @xmath119 , then the series converges if @xmath120 @xmath79 if @xmath121 , then and impose the convergence of . @xmath122 if @xmath123 and @xmath124 then the series @xmath125 converges . the condition can be simplified according to the values of @xmath126 and @xmath127 , in terms of the irrationality exponent @xmath128 of an irrational @xmath129 , defined as @xmath130 it is well - known that @xmath131 for almost every real numbers @xmath14 . when @xmath12 , choosing @xmath132 , @xmath133 , one sees that implies . when @xmath134 , putting @xmath135 and @xmath136 , a simple computation shows that @xmath137 in theorem [ theo:3 ] . [ coro:4 ] @xmath53 if @xmath138 and @xmath139 is convergent , then the identity holds true . the series converges for every @xmath14 such that @xmath140 . @xmath56 if @xmath141 converges , then the equality holds true . the series converges for every @xmath14 such that @xmath142 . corollary [ coro:4 ] is not entirely satisfying , since the series @xmath92 converges ( even absolutely ) when a weaker condition on the standard convergents of @xmath14 holds ( theorem [ theo:4 ] , conditions and ) . the main reason for this discrepency is the factor @xmath143 : it is present in but not in , respectively in but not in - . our next result shows that the role of this factor is very important , even though its modulus is @xmath24 . we explain after the theorem why we are not able to keep track of it in our proof of theorem [ theo:2 ] . [ theo:5 ] @xmath53 let @xmath144 be a bounded function , differentiable at @xmath145 and @xmath146 ( in particular , if @xmath147 ) . then for any @xmath113 and any irrational number @xmath85 , the series @xmath148 converges . @xmath56 for any @xmath149 , any @xmath150 and any irrational number @xmath85 , the series @xmath151 converges if @xmath152 @xmath79 for any @xmath149 and any irrational number @xmath85 , the series @xmath153 converges if @xmath154 these convergence properties are essentially optimal . they are very different from the absolute convergence properties ( which are also essentially optimal ) , as stated in theorem [ theo:2 ] . in fact , in our proof of theorem [ theo:2 ] , only one technical detail impedes us to prove that formulas and hold true when conditions and are satisfied ( and not only when the more constraining conditions in @xmath53 and @xmath56 of corollary [ coro:4 ] hold ) . indeed , we show that the convergence of the series and is equivalent to the convergence of three auxiliary simpler series ( see equations and ) . for two of these series , their convergence follows from theorem [ theo:5 ] , and the convergence conditions are optimal ( i.e. , when conditions and are satisfied ) . for the third series , which contains heuristically a sort of `` error '' term , we do not have an estimate precise enough to apply theorem [ theo:5 ] , and we can only use theorem [ theo:3 ] and the conditions ensuring absolute convergence of the series and , which are stronger . nevertheless , we do believe that conditions and imply the identities and respectively . for @xmath25 not necessarily an integer , the iteration of leads to an identity , for which we have to introduce the following sequences of operators : @xmath155 then , for any fixed @xmath156 $ ] and @xmath95 , the following identity ( but , again , we skip the details to shorten the length of the paper . ] ) holds for almost every @xmath14 : @xmath157 this new representation of @xmath30 is similar to identity displayed after theorem [ theo:4 ] . for @xmath14 unspecified , eq . is simpler than when @xmath27 , because @xmath158 for all @xmath62 while neither @xmath159 nor @xmath160 necessarily vanish finally , it is easy to see that when @xmath14 has _ even _ partial quotients and @xmath27 , then the summands of and are equal . the simplicity of ( relatively to ) when @xmath27 was our motivation to make the detailed study of various series defined in term of the operator @xmath69 , for which apparently nothing was done in the direction of our results . finally , when @xmath80 , the series @xmath30 obviously converges absolutely for any real numbers @xmath14 and @xmath25 . it turns out that identities and hold for any @xmath25 and any irrational number @xmath14 , and with minor modification for any rational number @xmath14 as well . on the one hand , this is not difficult to prove for , whose right hand side converges very quickly . on the other hand , the convergence of the right - hand side of for all irrational @xmath14 is a consequence of theorem [ theo:5]@xmath53 applied with @xmath135 because for @xmath80 , @xmath74 is bounded on @xmath2 $ ] and differentiable at @xmath161 by theorem [ theo:1]@xmath79 . we note that @xmath69 is closely related to the theta group , a subgroup of @xmath162 of the matrices @xmath163 with @xmath164 , @xmath165 ; see @xcite . this relation has been used in the papers @xcite to study the ( non)-derivability of riemann series @xmath166 , culminating with jaffard s determination of its spectrum of singularities @xcite . it would be very interesting to know if jaffard s results can be recovered by a direct study of in the case @xmath26 and @xmath27 , which reads @xmath167 where @xmath168 is differentiable on @xmath2\setminus\{0\}$ ] , and continuous at 0 . the paper is organized as follows . we start by recalling some facts on regular and even continued fractions in section [ sec:1 ] . then , in section [ sec : hl ] , we prove theorem [ theo:4 ] . section [ sec : newthm ] , which is rather long , contains the proof of the approximate functional equation for @xmath169 ( part @xmath53 of theorem [ theo:1 ] ) . the second part @xmath56 of theorem [ theo:1 ] , i.e. the regularity properties of the functions @xmath8 , is dealt with in section [ sec:2 ] . theorem [ theo:2 ] and the diophantine identities and are proven in section [ sec:4 ] . finally , the standard and absolute convergence properties of the series and ( theorems [ theo:5 ] and [ theo:3 ] ) are studied in sections [ sec:10_2 ] and [ sec:10_1 ] .
|
for any $ ] , the series converges almost everywhere on $ ] by a result of hardy - littlewood concerning the growth of the sums , but not everywhere . however , there does not yet exist an intrinsic description of the set of convergence for . in this paper , we define in terms of even continued fractions a subset of points of $ ] of full measure where the series converges . as an intermediate step , we prove that , for , the sequence of functions converges when to a function continuous on\setminus\{0\}$ ] with ( at most ) a singularity at of type or a logarithmic singularity ( ) . we provide an explicit expression for and the error term . finally , we study thoroughly the convergence properties of certain series defined in term of the convergents of the even continued fraction of an irrational number .
|
for any $ ] , the series converges almost everywhere on $ ] by a result of hardy - littlewood concerning the growth of the sums , but not everywhere . however , there does not yet exist an intrinsic description of the set of convergence for . in this paper , we define in terms of even continued fractions a subset of points of $ ] of full measure where the series converges . as an intermediate step , we prove that , for , the sequence of functions converges when to a function continuous on\setminus\{0\}$ ] with ( at most ) a singularity at of type or a logarithmic singularity ( ) . we provide an explicit expression for and the error term . finally , we study thoroughly the convergence properties of certain series defined in term of the convergents of the even continued fraction of an irrational number .
|
0802.1352
|
i
|
stable commutator length is a numerical invariant of elements in the commutator subgroup of a group . it is intimately related to ( two - dimensional ) bounded cohomology , and appears in many areas of low - dimensional topology and dynamics , from the milnor - wood inequality , to the @xmath2 conjecture . however , although a great deal of work has gone into estimating or bounding stable commutator length in many contexts , there are very few known nontrivial examples of finitely presented groups in which it can be calculated exactly , and virtually no cases where the range of stable commutator length on a given group can be understood arithmetically . the most significant results of this paper are as follows : 1 . we show that stable commutator length in free groups takes on only rational values , and give an explicit algorithm to compute the value on any given element . this is the first example of a group with infinite dimensional second bounded cohomology group @xmath3 in which stable commutator length can be calculated exactly . we show how to extend stable commutator length to a ( pseudo)-norm on the vector space @xmath0 of homogenized real ( group ) @xmath1-cochains that are boundaries of @xmath4-cochains . in the case of a free group , this is a genuine norm . we show that the intersection of the unit ball in this norm with any finite dimensional rational subspace of @xmath0 of a free group is a finite - sided rational polyhedron . this invites comparison with the thurston norm on the @xmath4-dimensional homology of a @xmath5-manifold @xcite , although the relationship between the two cases is subtle and deserves further investigation . we give examples of explicit elements in the commutator subgroup of @xmath6 ( the free group of rank @xmath4 ) for which the stable commutator length is not in @xmath7 . this answers in the negative a well - known question of bavard @xcite . we now elaborate on these points in turn . let @xmath8 be a group . for @xmath9 $ ] , the _ commutator length _ of @xmath10 , denoted @xmath11 , is the smallest number of commutators in @xmath8 whose product is equal to @xmath10 . the _ stable commutator length _ of @xmath10 , denoted @xmath12 , is the limit of @xmath13 as @xmath14 . in geometric language , ( see e.g. gromov @xcite ) @xmath15 is sometimes called `` filling genus '' , and @xmath16 is called `` stable filling genus '' . this quantity is intimately related , by bavard duality and an exact sequence ( see theorem [ duality_theorem ] and proposition [ exact_sequence_prop ] ) to the second bounded cohomology @xmath3 of @xmath8 , with its banach norm . despite a considerable amount of research , there are very few known examples of groups @xmath8 in which @xmath16 can be calculated exactly ( except when it vanishes identically ) . this is partly because the groups @xmath3 , when nontrivial , tend to be very large in general : when @xmath8 is word - hyperbolic , @xmath3 is not merely infinite dimensional , but is not even separable as a banach space . calculating @xmath16 is tantamount to solving an extremal problem in @xmath3 . ( technically , one solves the extremal problem in the space of _ homogeneous quasimorphisms _ @xmath17 , see definition [ quasimorphism_definition ] . the spaces @xmath17 and @xmath3 are both banach spaces , and are related by the coboundary @xmath18 , which is fredholm when @xmath8 is finitely presented . ) gromov ( @xcite 6.@xmath19 ) asked whether @xmath16 is always rational in finitely presented groups . the answer to gromov s question is known to be _ no _ : dongping zhuang gave the first examples in @xcite . these examples occur in generalized stein - thompson groups of pl homeomorphisms of the circle , where one can show that @xmath3 is actually finite dimensional , and everything can be calculated explicitly . in this paper we show that _ @xmath16 is rational _ in free groups ( and some closely related groups ) , and moreover we give an explicit algorithm to compute the value of @xmath16 on any element . if @xmath20 are elements in @xmath8 , define @xmath21 to be the smallest number of commutators in @xmath8 whose product is equal to the product of conjugates of the @xmath22 . let @xmath23 denote the limit of @xmath24 as @xmath14 . this function can be extended by linearity and continuity in a unique way to a pseudo - norm on @xmath25 , the vector space of real group @xmath1-chains on @xmath8 that are in the image of the boundary map @xmath26 . this function vanishes identically on the subspace @xmath27 of @xmath25 spanned by terms of the form @xmath28 and @xmath29 for @xmath30 and @xmath31 , and descends to a pseudo - norm on the quotient @xmath32 , or @xmath0 for short . when @xmath8 is hyperbolic , @xmath16 is a genuine norm on @xmath0 . we show that in a free group , this @xmath16 norm is _ piecewise rational linear _ ( denoted pql ) on finite dimensional rational subspaces of @xmath0 . so for any finite set of elements @xmath33 , there is a uniform upper bound on the denominators of the values of @xmath16 on integral linear chains @xmath34 in @xmath0 . one should compare the @xmath16 norm with the gromov - thurston norm @xcite , which is a norm on @xmath35 of an irreducible , atoroidal @xmath5-manifold , and whose most significant feature is that it is a piecewise rational linear function . in thurston s definition ( in which one restricts to embedded surfaces ) this is straightforward to show . in gromov s definition ( in terms of chains , or immersed surfaces ) this is a very deep theorem , whose proof depends on the full power of gabai s theory of sutured hierarchies @xcite , and taut foliations . in fact , it is reasonable to think of the @xmath16 norm as a relative gromov - thurston norm , with gromov s definition . our proof of rationality is conceptually close in some ways to an argument due to oertel @xcite insofar as both proofs reduce the problem of calculating the norm to a linear programming problem in the vector space of weights carried by a finite constructible branched surface . however , there are crucial differences between the two cases . in oertel s case , the branched surface might have complicated branch locus , but it comes with an embedding in a @xmath5-manifold . in our case , the branch locus is simple , but the branched surface is merely _ immersed _ in a @xmath5-manifold . it is intriguing to try to find a natural generalization of both theories . in his seminal paper @xcite on stable commutator length , bavard asked whether stable commutator length in free groups takes values in @xmath7 . there were several pieces of direct and indirect evidence for this conjecture . firstly , where certain ( geometric or homological ) methods for estimating stable commutator length in free groups give exact answers , these answers in every case confirm bavard s guess . secondly , in the ( analogous ) context of @xmath5-manifold topology , one knows that the gromov norm of an integral @xmath4-dimensional homology class is in @xmath36 ( the factor of @xmath37 arises because gromov norm counts triangles , whereas stable commutator norm counts handles ) . it was generally felt that bavard s conjecture was eminently plausible , and it is therefore surprising that our algorithm produces many elements whose stable commutator length is not in @xmath7 . in fact experiments suggest that arbitrarily large denominators occur , with arbitrary prime factors . in view of these examples , the fact that stable commutator length is rational in free groups is seen to be a more delicate and subtle fact than one might have imagined , and stable commutator length to be a richer invariant than previous work has suggested . the organization of this paper is as follows . in [ background_section ] we state definitions and sketch proofs of background results which pertain to stable commutator length in groups in general . in [ free_group_section ] we specialize to the case of free groups . the purpose of this section is to state and prove the `` rationality theorem '' , whose precise statement is the following : let @xmath38 be a free group . 1 . @xmath39 for all @xmath40 $ ] . every @xmath41 $ ] rationally bounds an extremal surface ( in fact , every rational chain @xmath42 in @xmath0 rationally bounds an extremal surface ) 3 . the function @xmath16 is piecewise rational linear on @xmath0 . 4 . there is an algorithm to calculate @xmath16 on any finite dimensional rational subspace of @xmath0 . similar rationality results hold for stable commutator length in virtually free groups , and fundamental groups of noncompact seifert - fibered @xmath5-manifolds . finally , in [ algorithm_section ] we explicitly describe an algorithm for computing the stable commutator length in free groups , and discuss a simple example that answers bavard s question in the negative .
|
for any group , there is a natural ( pseudo-)norm on the vector space of real homogenized ( group )-boundaries , called the _ stable commutator length _ norm . this norm is closely related to , and can be thought of as a relative version of , the gromov ( pseudo)-norm on ( ordinary ) homology . we show that for a free group , the unit ball of this pseudo - norm is a rational polyhedron . it follows that stable commutator length in free groups takes on only rational values . moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group . the proof of these facts yields an algorithm to compute stable commutator length in free groups . using this algorithm , we answer a well - known question of bavard in the negative , constructing explicit examples of elements in free groups whose stable commutator length is not a half - integer . [ section ] [ theorem]lemma [ theorem]proposition [ theorem]corollary [ theorem]conjecture [ theorem]question [ theorem]problem [ theorem]definition [ theorem]construction [ theorem]notation [ theorem]remark [ theorem]example p
|
for any group , there is a natural ( pseudo-)norm on the vector space of real homogenized ( group )-boundaries , called the _ stable commutator length _ norm . this norm is closely related to , and can be thought of as a relative version of , the gromov ( pseudo)-norm on ( ordinary ) homology . we show that for a free group , the unit ball of this pseudo - norm is a rational polyhedron . it follows that stable commutator length in free groups takes on only rational values . moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group . the proof of these facts yields an algorithm to compute stable commutator length in free groups . using this algorithm , we answer a well - known question of bavard in the negative , constructing explicit examples of elements in free groups whose stable commutator length is not a half - integer . [ section ] [ theorem]lemma [ theorem]proposition [ theorem]corollary [ theorem]conjecture [ theorem]question [ theorem]problem [ theorem]definition [ theorem]construction [ theorem]notation [ theorem]remark [ theorem]example p
|
0909.0286
|
i
|
particle - based numerical methods are routinely used in plasma physics calculations @xcite . in many cases these methods are more efficient and simpler to implement than the corresponding continuum eulerian methods . however , particle methods face the well known statistical sampling limitation of attempting to simulate a physical system containing @xmath0 particles using @xmath1 computational particles . particle methods do not seek to reproduce the exact individual behavior of the particles , but rather to approximate statistical macroscopic quantities like density , current , and temperature . these quantities are determined from the particle distribution function . therefore , a problem of relevance for the success of particle - based simulations is the reconstruction of the particle distribution function from discrete particle data . the difference between the distribution function reconstructed from a simulation using @xmath2 particles and the exact distribution function gives rise to a discretization error generically known as `` particle noise '' due to its random - like character . understanding and reducing this error is a complex problem of importance in the validation and verification of particle codes , see for example refs . @xcite and references therein for a discussion in the context of gyrokinetic calculations . one obvious way to reduce particle noise is by increasing the number of computational particles . however , the unfavorable scaling of the error with the number of particles , @xmath3 @xcite , puts a severe limitation on this straightforward approach . this has motivated the development of various noise reduction techniques including finite size particles ( fsp ) @xcite , monte - carlo methods @xcite , fourier - filtering @xcite , coarse - graining @xcite , krook operators @xcite , smooth interpolation @xcite , low noise collision operators @xcite , and proper orthogonal decomposition ( pod ) methods @xcite among others . in the present paper we propose a wavelet - based method for noise reduction in the reconstruction of particle distribution functions from particle simulation data . the method , known as wavelet based density estimation ( wbde ) , was originally introduced in ref . @xcite in the context of statistics to estimate probability densities given a finite number of independent measurements . however , to our knowledge , this method has not been applied before to particle - base computations . wbde , as used here , is based on truncations of the wavelet representation of the dirac delta function associated with each particle . the method yields almost optimal results for functions with unknown local smoothness without compromising computational efficiency , assuming that the particles coordinates are statistically independent . as a first step in the application of the wbde method to plasma particle simulations , we limit attention to `` passive denoising '' . that is the wbde method is treated as a post - processing technique applied to independently generated particle data . the problem of `` active denoising '' , e.g. the application of wbde methods in the evaluation of self - consistent fields in particle in cell simulations , will not be addressed . this simplification will allow us to assess the efficiency of the proposed noise reduction method in a simple setting . another simplification pertains the dimensionality . here , for the sake of simplicity , we limit attention to the reconstruction and denoising problem in two dimensions . however , the extension of the wbde method to higher dimensions is in principle straightforward . collisions , or the absence of them , play an important role in plasma transport problems . particle methods handle the collisional and non - collisional parts of the dynamics differently . fokker - planck - type collision operators are typically introduced in particle methods using langevin - type stochastic differential equations . on the other hand , the non - collisional part of the dynamics is described using deterministic ordinary differential equations . collisional dominated problems tend to washout small scale structures whereas collisionless problems typically develop fine scale filamentary structures in phase space . therefore , it is important to test the dependence of the efficiency of denoising reconstruction methods on the level of collisionality . here we test the wbde method in strongly collisional , weakly collisional and collisionless regimes . for the strongly collisional regime we consider particle data of force - free collisional relaxation involving energy and pinch - angle scattering . the weakly collisional regime is illustrated using guiding - center particle data of a magnetically confined plasma in toroidal geometry . the collisionless regime is studied using particle in cell ( pic ) data corresponding to bump - on - tail and two streams instabilities in the vlasov - poisson system . beyond the role of collisions , the data sets that we are considering open the possibility of exploring the role of external and self - consistent fields in the reconstruction of the particle density . in the collisional relaxation problem no forces act on the particles , in the guiding - center problem particles interact with an external magnetic field , and in the vlasov - poisson problem particle interactions are incorporated through a self - consistent electrostatic mean field . one of the goals of this paper is to compare the wbde method with the proper orthogonal decomposition ( pod ) density reconstruction method proposed in ref . @xcite . the rest of the paper is organized as follows . in sect . ii we review the main properties of kernel density estimation ( kde ) and show its relationship with finite size particles ( fsp ) . we then review basic notions on orthogonal wavelet and multiresolution analysis and outline a step by step algorithm for wbde . also , for completeness , in this section we include a brief description of the pod reconstruction method proposed in ref . @xcite . section iii discusses applications of the wbde method and the comparison with the pod method . we start by post - processing a simulation of plasma relaxation by random collisions against a background thermostat . we then turn to a @xmath4 monte - carlo simulation in toroidal geometry , whose phase space has been reduced to two dimensions . finally , we analyze the results of particle - in - cell ( pic ) simulations of a 1d vlasov - poisson plasma . the conclusions are presented in sec . iv .
|
the method , known as wavelet based density estimation ( wbde ) , was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements . the method is compared with a recently proposed proper orthogonal decomposition based method , and it is tested with three particle data sets that involve different levels of collisionality and interaction with external and self - consistent fields .
|
for given computational resources , the accuracy of plasma simulations using particles is mainly held back by the noise due to limited statistical sampling in the reconstruction of the particle distribution function . a method based on wavelet analysis is proposed and tested to reduce this noise . the method , known as wavelet based density estimation ( wbde ) , was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements . its novel application to plasma simulations can be viewed as a natural extension of the finite size particles ( fsp ) approach , with the advantage of estimating more accurately distribution functions that have localized sharp features . the proposed method preserves the moments of the particle distribution function to a good level of accuracy , has no constraints on the dimensionality of the system , does not require an a priori selection of a global smoothing scale , and its able to adapt locally to the smoothness of the density based on the given discrete particle data . most importantly , the computational cost of the denoising stage is of the same order as one time step of a fsp simulation . the method is compared with a recently proposed proper orthogonal decomposition based method , and it is tested with three particle data sets that involve different levels of collisionality and interaction with external and self - consistent fields .
|
0909.0286
|
m
|
this section presents the wavelet - based density estimation ( wbde ) algorithm . we start by reviewing basic ideas on kernel density estimation ( kde ) which is closely related to the use of finite size particles ( fsp ) in pic simulations . following this , we we give a brief introduction to wavelet analysis and discuss the wbde algorithm . for completeness , we also include a brief summary of the pod approach . given a sequence of independent and identically distributed measurements , the nonparametric density estimation problem consists in finding the underlying probability density function ( pdf ) , with no a priori assumptions on its functional form . here we discuss general ideas on this difficult problem for which a variety of statistical methods have been developed . further details can be found in the statistics literature , e.g. ref . @xcite . consider a number @xmath2 of statistically independent particles with phase space coordinates @xmath5 distributed in @xmath6 according to a pdf @xmath7 . this data can come from a pic or a monte - carlo , full @xmath7 or @xmath4 simulation . formally , the sample pdf can be written as @xmath8 where @xmath9 is the dirac distribution . because of its lack of smoothness , eq . ( [ dirac_estimate ] ) is far from the actual distribution @xmath7 according to most reasonable definitions of the error . moreover , the dependence of @xmath10 on the statistical fluctuations in @xmath11 can lead to an artificial increase of the collisionality of the plasma . the simplest method to introduce some smoothness in @xmath10 is to use a histogram . consider a tiling of the phase space by a cartesian grid with @xmath12 cells . let @xmath13 denote the set of all cells with characteristic function @xmath14 defined as @xmath15 if @xmath16 and @xmath17 otherwise . then the histogram corresponding to the tiling is @xmath18 which can also be viewed as the orthogonal projection of @xmath10 on the space spanned by the @xmath14 . the main difference between @xmath19 and @xmath20 is that the latter can not vary at scales finer than the grid scale which is of order @xmath21 . by choosing @xmath22 small enough , it is therefore possible to reduce the variance of @xmath20 to very low levels , but the estimate then becomes more and more biased towards a piecewise continuous function , which is not smooth enough to be the true density . histograms correspond to the nearest grid point ( ngp ) charge assignment scheme used in the early days of plasma physics computations @xcite . one of the most popular methods to achieve higher level of smoothness is kernel density estimation ( kde ) @xcite . given @xmath5 , the kernel estimate of @xmath7 is defined as @xmath23 where the smoothing kernel @xmath24 is a positive definite , normalized , @xmath25 , function . equation ( [ kernel_estimate ] ) corresponds to the convolution of @xmath24 with the dirac delta measure corresponding to each particle . a typical example is the gaussian kernel @xmath26 where the so - called `` bandwidth '' , or smoothing scale , @xmath27 , is a free parameter . the optimal smoothing scale depends on how the error is measured . for example , in the one dimensional case , to minimize the mean @xmath28-error between the estimate and the true density , the smoothing volume @xmath29 should scale like @xmath30 , and the resulting error scales like @xmath31 @xcite . as in the case of histograms , the choice of @xmath27 relies on a trade - off between variance and bias . in the context of plasma physics simulations the kernel @xmath24 corresponds to the charge assignment function @xcite . a significant effort has been devoted in the choice of the function @xmath24 since it has a strong impact on computational efficiency and on the conservation of global quantities . concerning @xmath27 , it has been shown that it should not be much larger than the debye length @xmath32 of the plasma to obtain a realistic and stable simulation @xcite . given a certain amount of computational resources , the general tendency has thus been to reduce @xmath27 as far as possible in order to fit more debye lengths inside the simulation domain , which means that the effort has been concentrated on reducing the bias term in the error . since the force fields depend on @xmath7 through integral equations , like the poisson equation , that tend to reduce the high wavenumber noise , we do not expect the disastrous scaling @xmath33 , which would mean @xmath34 in @xmath35 dimensions , to hold . nevertheless , the problem remains that if we want to preserve high resolution features of @xmath7 or of the electromagnetic fields , we need to reduce @xmath27 , and therefore greatly increase the number of particles to prevent the simulation from drowning into noise . bandwidth selection has long been recognized as the central issue in kernel density estimation @xcite . we are not aware of a theoretical or numerical prediction of the optimal value of @xmath27 taking into account the noise term . to bypass this difficulty , it is possible to use new statistical methods which do not force us to choose a global smoothing parameter . instead , they adapt locally to the behavior of the density @xmath7 based on the available data . wavelet based - density estimation , which we will introduce in the next two sections , is one of these methods . wavelets are a standard mathematical tool to analyze and compute non stationary signals . here we recall basic concepts and definitions . further details can be found in ref . @xcite and references therein . the construction takes place in the hilbert space @xmath36 of square integrable functions . an orthonormal family @xmath37 is called a wavelet family when its members are dilations and translations of a fixed function @xmath38 called the mother wavelet : @xmath39 where @xmath40 indexes the scale of the wavelets and @xmath41 their positions , and @xmath38 satisfies @xmath42 . in the following we shall always assume that @xmath38 has compact support of length @xmath43 . the coefficients @xmath44 of a function @xmath7 for this family are denoted by @xmath45 . these coefficients describe the fluctuations of @xmath7 at scale @xmath46 around position @xmath47 . large values of @xmath40 correspond to fine scales , and small values to coarse scales . some members of the commonly used daubechies 6 wavelet family are shown in the left panel of fig . it can be shown that the orthogonal complement in @xmath36 of the linear space spanned by the wavelets is itself orthogonally spanned by the translates of a function @xmath48 , called the scaling function . defining @xmath49 and the scaling coefficients @xmath50 , one thus has the reconstruction formula : @xmath51 the first sum on the right hand side of eq . ( [ wavelet_reconstruction ] ) is a smooth approximation of @xmath7 at the coarse scale , @xmath52 , and the second sum corresponds to the addition of details at successively finer scales . if the wavelet @xmath38 has @xmath53 vanishing moments : @xmath54 for @xmath55 , and if @xmath7 is locally @xmath56 times continuously differentiable around some point @xmath57 , then a key property of the wavelet expansion is that the coefficients located near @xmath57 decay when @xmath58 like @xmath59 @xcite . hence , localized singularities or sharp features in @xmath7 affect only a finite number of wavelet coefficients within each scale . another important consequence of ( [ vanishing_moments ] ) of special relevance to particle methods is that for @xmath55 , the moments @xmath60 of the particle distribution function depend only on its scaling coefficients , and not on its wavelet coefficients . if the scaling coefficients @xmath61 at a certain scale @xmath62 are known , all the wavelet coefficients at coarser scales ( @xmath63 ) can be computed using the fast wavelet transform ( fwt ) algorithm @xcite . we shall address the issue of computing the scaling coefficients themselves in section [ critical_discussion ] . the generalization to @xmath35 dimensions involves tensor products of wavelets and scaling functions at the same scale . for example , given a wavelet basis on @xmath64 , a wavelet basis on @xmath65 can be constructed in the following way : @xmath66 where we refer to the exponent @xmath67 as the direction of the wavelets . this name is easily understood by looking at different wavelets shown in fig . [ daubechies_wavelets_1d ] ( right ) . the corresponding scaling functions are simply given by @xmath68 . wavelets on @xmath6 are constructed exactly in the same way , but this time using @xmath69 directions . to lighten the notation we write the @xmath35-dimensional analog of eq . ( [ wavelet_reconstruction ] ) as @xmath70 where @xmath71 is a multi - index , with the integer @xmath40 denoting the scale and the integer vector @xmath72 denoting the position of the wavelet . the wavelet multiresolution reconstruction formula in eq . ( [ wavelet_reconstruction ] ) involves an infinite sum over the position index @xmath41 . one way of dealing with this sum is to determine a priori the non - zero coefficients in eq . ( [ wavelet_reconstruction ] ) , and work only with these coefficients , but still retaining the full wavelet basis on @xmath6 as presented above . another alternative , which we have chosen because it is easier to implement , is to periodize the wavelet transform on a bounded domain @xcite . assuming that the coordinates have been rescaled so that all the particles lie in @xmath73^d$ ] , we replace the wavelets and scaling functions by their periodized counterparts : @xmath74 throughout this paper we will consider only periodic wavelets . for the sake of completeness we mention a third alternative which is technically more complicated . it consists in constructing a wavelet basis on a bounded interval @xcite . the advantage of this approach is that it does not introduce artificially large wavelet coefficients at the boundaries for functions @xmath7 that are not periodic . the multiscale nature of wavelets allows them to adapt locally to the smoothness of the analyzed function @xcite . this fundamental property has triggered their use in a variety of problems . one of their most fruitful applications has been the denoising of intermittent signals @xcite . the practical success of wavelet thresholding to reduce noise relies on the observation that the expansion of signals in a wavelet basis is typically sparse . sparsity means that the interesting features of the signal are well summarized by a small fraction of large wavelet coefficients . on the contrary , the variance of the noise is spread over all the coefficients appearing in eq . ( [ wavelet_reconstruction_2d ] ) . although the few large coefficients are of course also affected by noise , curing the noise in the small coefficients is already a very good improvement . the original setting of this technique , hereafter referred to as global wavelet shrinkage , requires the noise to be additive , stationary , gaussian and white . it found a first application in plasma physics in ref . @xcite , where coherent bursts were extracted out of plasma density signals . since ref . @xcite , wavelet denoising has been extended to a number of more general situations , like non - gaussian or correlated additive noise , or to denoise the spectra of locally stationary time series @xcite . in particular , the same ideas were developed in ref . @xcite to propose a wavelet - based density estimation ( wbde ) method based on independent observations . at this point we would like to stress that wbde assumes nothing about the gaussianity of the noise or whether or not it is stationary . in fact , under the independence hypothesis which is admittedly quite strong the statistical properties of the noise are entirely determined by standard probability theory . we refer to ref . @xcite for a review on the applications of wavelets in statistics . in ref . @xcite , global wavelet shrinkage was applied directly to the charge density of a 2d pic code , in a case were the statistical fluctuations were quasi gaussian and stationary . in particular , an iterative algorithm @xcite , which crucially relies on the stationnarity hypothesis , was used to determine the level of fluctuations . however , in the next section we will show an example where the noise is clearly non - stationary , and this procedure fails . let us now describe the wbde method as we have generalized it to several dimensions . the first step is to expand the sample particle distribution function , @xmath10 , in eq . ( [ dirac_estimate ] ) in a wavelet basis according to eq . ( [ wavelet_reconstruction_2d ] ) with the wavelet coefficients @xmath75 since this reconstruction is exact , keeping all the wavelet coefficients does not improve the smoothness of @xmath10 . the simple and yet efficient remedy consists in keeping only a subset of the wavelet coefficients in eq . ( [ wavelet_reconstruction_2d ] ) . a straightforward prescription would be to discard all the wavelet coefficients at scales finer than a cut - off scale @xmath76 . this approach corresponds to a generalization of the histogram method in eq . ( [ histogram_estimate ] ) with @xmath77 . because the characteristic functions @xmath14 of the cells in a dyadic grid are the scaling functions associated with the haar wavelet family , eqs . ( [ wavelet_reconstruction_2d ] ) and ( [ histogram_estimate ] ) are in fact equivalent for this wavelet family . accordingly , like in the histogram case , we would have to choose @xmath76 quite low to obtain a stable estimate , at the risk of losing some sharp features of @xmath7 . better results can be obtained by keeping some wavelet coefficients down to a much finer scale @xmath78 . however , to prevent that statistical fluctuations contaminate the estimate , only those coefficients whose modulus are above a certain threshold should be kept . we are thus naturally led to a nonlinear thresholding procedure . in the one dimensional case , values of @xmath62 , @xmath76 , and of the threshold within each scale that yield theoretically optimal results have been given in ref . this reference discusses the precise smoothness requirements on @xmath7 , which can accommodate well localized singularities , like shocks and filamentary structures known to arise in collisionless plasma simulations . there remains the question of how to compute the @xmath79 based on the positions of the particles . although more accurate methods based on ( [ empirical_wavelet_coefficients ] ) may be developed in the future , our present approximation relies on the computation of a histogram , which creates errors of order @xmath21 . the complete procedure is described in the following * wavelet - based density estimation * algorithm : 1 . [ histogram_approx ] construct a histogram @xmath20 of the particle data with @xmath80 cells in each direction , 2 . [ scaling_function_approx ] approximate the scaling coefficients at the finest scale @xmath81 by : @xmath82 3 . compute all the needed wavelet coefficients using the fwt algorithm , 4 . keep all the coefficients for scales coarser than @xmath76 , defined by @xmath83 where @xmath84 is the order of regularity of the wavelet ( 1 in our case ) , 5 . discard all the coefficients for scales strictly finer than @xmath62 defined by @xmath85 , 6 . [ thresholding_function ] for scales @xmath40 in between @xmath76 and @xmath62 , keep only the wavelet coefficients @xmath86 such that @xmath87 where @xmath88 is a constant that must in principle depend on the smoothness of @xmath7 and on the wavelet family @xcite . in the following , except otherwise indicated , @xmath89 . for the wavelet bases we used orthonormal daubechies wavelets with 6 vanishing moments and thus support of size @xmath90 @xcite . in our case , @xmath91 , which means that the wavelets have a first derivative but no second derivative , and the size of the wavelets at scale @xmath76 for @xmath92 is roughly @xmath93 . since @xmath94 , it follows from the definition at stage 5 of the algorithm that the size of the wavelets at scale @xmath62 is orders of magnitude smaller than that . using the adaptive properties of wavelets , we are thus able to detect small scale structures of @xmath7 without compromising the stability of the estimate . note that the error at stage [ scaling_function_approx ] could be reduced by using coiflets @xcite instead of daubechies wavelets , but the gain would be negligible compared to the error made at stage [ histogram_approx ] . we will denote the wbde estimate of @xmath7 as @xmath95 . in the one - dimensional case , @xmath96 where @xmath97 is the thresholding function as defined by stage [ thresholding_function ] of the algorithm : @xmath98 if @xmath99 and @xmath100 otherwise . finally , let us propose two methods for applying wbde to postprocess @xmath4 simulations . recall that the lagrangian equations involved in the @xmath4 schemes are identical to their full @xmath7 counterparts . the only difficulty introduced by the @xmath4 method lies in the evaluation of phase space integrals of the form @xmath101 , where @xmath102 is a function on phase space and @xmath103 is a known reference distribution function . in these integrals , @xmath104 should be replaced by @xmath4 , which is in turn written as a product @xmath105 , where @xmath106 is a `` weighting '' function . numerically , @xmath106 is known via its values at particles positions , @xmath107 , and the usual expression for @xmath108 is thus @xmath109 . we can not apply wbde directly to @xmath4 , since this function is not a density function.an elegant approach would be to first apply wbde to the unweighted distribution @xmath10 to determine the set of statistically significant wavelet coefficients , and to include the weights only in the final reconstruction ( [ explicit_wavelet_estimate ] ) of @xmath95 . a simpler approach , which we will illustrate in section [ delta_5d_example ] , consists in renormalizing @xmath4 , so that @xmath110 , and treat it like a density . in this section we discuss how the wbde method handles two issues of direct relevance to plasma simulations : conservation of moments and computational efficiency . as mentioned before , due to the vanishing moments of the wavelets in eq . ( [ vanishing_moments ] ) , the moments up to order @xmath53 of the particle distribution distribution are solely determined by its scaling function coefficients . as a consequence , we expect the thresholding procedure to conserve these moments , in the sense that @xmath111 for @xmath112 and for all @xmath113 . this conservation holds up to round - off error if the wavelet coefficients can be computed exactly . due to the type of wavelets that we have used , we were not able to achieve this in the results presented here . there remains a small error related to stages 1 and 2 of the algorithm , namely the construction of @xmath20 and the approximation of the scaling function coefficients by eq . ( [ scaling_function_approximation ] ) . they are both of order @xmath21 . we will present numerical examples of the moments of @xmath95 in the next section . conservation of moments is closely related to a peculiarity of the denoised distribution function resulting from the wbde algorithm : it is not necessarily everywhere positive . indeed , wavelets are oscillating functions by definition , and removing wavelet coefficients therefore can not preserve positivity in general . further studies are needed to assess if this creates numerical instabilities when @xmath95 is used in the computation of self - consistent fields . the same issue was discussed in ref . @xcite where a kernel with two vanishing moments was used to linearly smooth the distribution function . the fact that this kernel is not everywhere positive was not considered harmful in this reference . we acknowledge that it may render the resampling of new particles from @xmath95 , if it is needed in the future , more difficult . there are ways of forcing @xmath95 to be positive , for example by applying the method to @xmath114 and then taking the square of the resulting estimate , but this implies the loss of the moment conservation , and we have not pursued in this direction . daubechies 6 wavelet family . left , bold red : scaling function @xmath48 at scale @xmath115 . left , bold blue : wavelet @xmath38 at scale @xmath115 . left , thin black , from left to right : wavelets at scales 6 , 7 , 8 and 9 . right : ( a ) 2d scaling function @xmath116 . ( b ) first 2d wavelet @xmath117 . ( c ) second 2d wavelet @xmath118 . ( d ) third 2d wavelet @xmath119 . , title="fig : " ] daubechies 6 wavelet family . left , bold red : scaling function @xmath48 at scale @xmath115 . left , bold blue : wavelet @xmath38 at scale @xmath115 . left , thin black , from left to right : wavelets at scales 6 , 7 , 8 and 9 . right : ( a ) 2d scaling function @xmath116 . ( b ) first 2d wavelet @xmath117 . ( c ) second 2d wavelet @xmath118 . ( d ) third 2d wavelet @xmath119 . , title="fig : " ] the number of arithmetic operations to perform a fast wavelet transform from scale @xmath120 to scale @xmath52 with the fwt in @xmath35 dimensions is @xmath121 , where @xmath43 is the length of the wavelet filter ( 12 for the daubechies filter that we are using ) . the definitions of @xmath62 and @xmath76 imply that @xmath122 scales like @xmath123 . the cost of the binning stage of order @xmath124 , so that the total cost for computing @xmath95 is @xmath125 , not larger than the cost of one time step during the simulation that produced the data . the amount of memory needed to store the wavelet coefficients during the denoising procedure is proportional to @xmath12 , which should at least scale like @xmath126 , and therefore also like @xmath2 . if one wishes to use a finer grid to ensure high accuracy conservation of moments , the storage requirements grow like @xmath12 . thanks to optimized in - place algorithms , the amount of additional memory needed during the computation does not exceed @xmath127 . another consequence of using the fwt algorithm is that @xmath22 must be an integer multiple of @xmath128 . for comparison purposes , let us recall that most algorithms to compute the pod have a complexity proportional to @xmath129 when @xmath130 . to conclude this subsection , fig . [ example_1d ] presents an example of the reconstruction of a 1d discontinuous density that illustrates the difference between the kde and wbde methods . the probability density function is uniform on the interval @xmath131 $ ] and the estimates were computed on @xmath132 $ ] to include the discontinuities . the sample size was @xmath133 , and the binning used @xmath134 cells to compute the scaling function coefficients . for this 1d case the value @xmath135 was used to determine the thresholds ( step [ thresholding_function ] of the algorithm ) . the kde estimate is computed using a gaussian kernel with smoothing scale @xmath136 @xcite . the relative mean squared errors associated with the kde and wbde estimates are respectively @xmath137 and @xmath138 . the error in the kde estimate comes mostly from the smoothing of the discontinuities . the better performance of wbde stems from the much sharper representation of these discontinuities . it is also observed that the wbde estimate is not everywhere positive . the approximate conservation of moments is demonstrated on table [ example_1d_moments ] . note that the error on all these moments for @xmath95 could be made arbitrary low by increasing @xmath22 . the overshoots could also be mitigated by using nearly shift invariant wavelets @xcite . estimation of the density of a sample of size @xmath133 drawn uniformly in @xmath139 $ ] , using gaussian kernels ( left ) or wavelets ( right ) . the discontinuous analytical density is plotted with a dashed line in the two cases . , title="fig : " ] estimation of the density of a sample of size @xmath133 drawn uniformly in @xmath139 $ ] , using gaussian kernels ( left ) or wavelets ( right ) . the discontinuous analytical density is plotted with a dashed line in the two cases . , title="fig : " ] for completeness , in this subsection we present a brief review of the pod density reconstruction method . for the sake of comparison with the wbde method , we limit attention to the time independent case . further details , including the reconstruction of time dependent densities using pod methods can be found in ref . @xcite . the first step in the pod method is to construct the histogram @xmath20 from the particle data . this density is represented by an @xmath140 matrix @xmath141 containing the fraction of particles with coordinates @xmath142 such that @xmath143 and @xmath144 . in two dimensions , the pod method is based on the singular value decomposition of the histogram . according to the svd theorem @xcite , the matrix @xmath145 can always be factorized as @xmath146 where @xmath147 and @xmath148 are @xmath149 and @xmath150 orthogonal matrices , @xmath151 , and @xmath152 is a diagonal matrix , @xmath153 , such that @xmath154 . with @xmath155 . in vector form , the decomposition can be expressed as [ svd_vector ] _ ij= _ k=1^n w_k u^(k)_i v^(k)_j , where the @xmath156-dimensional vectors , @xmath157 , and the @xmath158-dimensional vectors , @xmath159 , are the orthonormal pod modes and correspond to the columns of the matrices @xmath147 and @xmath148 respectively . given the decomposition in eq . ( [ svd_vector ] ) , we define the rank-@xmath160 approximation of @xmath145 as [ lr_svd ] ^(r)_ij= _ k=1^r w_k u^(k)_i v^(k)_j , where @xmath161 , and define the corresponding rank-@xmath160 reconstruction error as [ nre ] e(r ) = || -^(r ) ||^2 = _ i = r+1^n w_i^2 , where @xmath162 is the frobenius norm . since @xmath163 , we define @xmath164 . the key property of the pod is that the approximation in eq . ( [ lr_svd ] ) is optimal in the sense that e(r ) = min \ { ||-g||^2 | rank ( g ) = r . } . that is , of all the possible rank-@xmath160 cartesian product approximations of @xmath145 , @xmath165 is the closest to @xmath145 in the frobenius norm . the svd spectrum , @xmath166 , of noise free coherent signals decays very rapidly after a few modes , but the spectrum of noise dominated signals is relatively flat and decays very slowly . when a coherent signal is contaminated with low level noise , the svd spectrum exhibits an initial rapid decay followed by a weakly decaying spectrum known as the noisy plateau . in the pod method the denoised density is defined as the truncation @xmath167 , where @xmath168 corresponds to the rank where the noisy plateau starts . in general it is difficult to provide a precise a priori estimate of @xmath168 , and this is one of the potential limitations of the pod method . one possible quantitative criterion used in ref . @xcite is to consider the relative decay of the spectrum , @xmath169 , for @xmath170 , and define @xmath168 by the condition @xmath171 where @xmath172 is a predetermined threshold . l*6cr & @xmath173 & @xmath174 & @xmath175 & @xmath176 + @xmath177 & @xmath178 & @xmath179 & @xmath180 & @xmath181 + @xmath95 & @xmath182 & @xmath183 & @xmath184 & @xmath185 +
|
its novel application to plasma simulations can be viewed as a natural extension of the finite size particles ( fsp ) approach , with the advantage of estimating more accurately distribution functions that have localized sharp features . most importantly , the computational cost of the denoising stage is of the same order as one time step of a fsp simulation .
|
for given computational resources , the accuracy of plasma simulations using particles is mainly held back by the noise due to limited statistical sampling in the reconstruction of the particle distribution function . a method based on wavelet analysis is proposed and tested to reduce this noise . the method , known as wavelet based density estimation ( wbde ) , was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements . its novel application to plasma simulations can be viewed as a natural extension of the finite size particles ( fsp ) approach , with the advantage of estimating more accurately distribution functions that have localized sharp features . the proposed method preserves the moments of the particle distribution function to a good level of accuracy , has no constraints on the dimensionality of the system , does not require an a priori selection of a global smoothing scale , and its able to adapt locally to the smoothness of the density based on the given discrete particle data . most importantly , the computational cost of the denoising stage is of the same order as one time step of a fsp simulation . the method is compared with a recently proposed proper orthogonal decomposition based method , and it is tested with three particle data sets that involve different levels of collisionality and interaction with external and self - consistent fields .
|
0909.0286
|
i
|
wavelet based density estimation was investigated as a post - processing tool to reduce the noise in the reconstruction of particle distribution functions starting from discrete particle data . this is a problem of direct relevance to particle - based transport calculations in plasma physics and related fields . in particular , particle methods present many advantages over continuum methods , but have the potential drawback of introducing noise due to statistical sampling . in the context of particle in cell methods this problem is typically approached using finite size particles . however , this approach , which is closely related to the kernel density estimation method in statistics , requires the choice of a smoothing scale , @xmath27 , ( e.g. , the standard deviation for gaussian shape functions ) whose optimal value is not known a priori . a small @xmath27 is desirable to fit as many debye wavelengths as possible , whereas a large @xmath27 would lead to smoother distributions . this situation results from the compromise between bias and variance in statistical estimation . to address this problem we proposed a wavelet based density estimation ( wbde ) method that does not require an a priori selection of a global smoothing scale and that its able to adapt locally to the smoothness of the density based on the given discrete data . the wbde was introduced in statistics @xcite . in this paper we extended the method to higher dimension and applied it for the first time to particle - based calculations . the resulting method exploits the multiresolution properties of wavelets , has very weak dependence on adjustable parameters , and relies mostly on the raw data to separate the relevant information from the noise . as a first example , we analyzed a plasma collisional relaxation problem modeled by stochastic differential equations . thanks to the sparsity of the wavelet expansion of the distribution function , we have been able to extract the information out of the statistical fluctuations by nonlinear thresholding of the wavelet coefficients . at late times , when the particle distribution approaches a maxwellian state , we have been able to quantify the difference between the denoised particle distribution function and its analytical counterpart , thus demonstrating the improvement with respect to the raw histogram . the pod - smoothed and wavelet - smoothed particle distribution functions were shown to be roughly equivalent in this respect . these results were then extended to a more complex situation simulated with a @xmath4 code . finally , we have turned to the vlasov - poisson problem , which includes interactions between particles via the self - consistent electric field . the pod and wbde methods were shown to yield quantitatively close results in terms of mean squared error for a particle distribution function resulting from nonlinear saturation after occurrence of a bump - on - tail instability . we have then studied the denoising algorithm during nonlinear evolution after the two - streams instability starting from two counter - streaming cold electron beams . this initial condition violates the decorrelation hypothesis underlying the wbde algorithm , and thus offers a good way to test its robustness regarding this aspect . the wbde method was shown to yield qualitatively good results without changing the threshold values . one limitation of the present work comes from the way denoising quality is measured . we have considered the quadratic error on the distribution function @xmath7 as a first indicator of the quality of our denoising methods . however , it may be more relevant to compute the error on the force fields , which determine the evolution of the simulated plasma . these forces depend on @xmath7 through integrals , and statistical analysis of the estimation of @xmath7 using weak norms , like was done in @xcite in the deterministic case , could therefore be of great help to obtain threshold parameters more efficient than those considered in this study . the computational cost of our method scales linearly with the number of particles and with the grid resolution . therefore , wbde is an excellent candidate to be performed at each time step during the course of a simulation . once the wavelet expansion of the denoised particle distribution function is known , it is possible to continue using the wavelet representation to solve the poisson equation @xcite and to compute the forces . the moment conservation properties that we have demonstrated in this paper should mitigate the unavoidable dissipative effects implied by the smoothing stage . in ref . @xcite , a dissipative term was introduced in a global pic code to avoid unlimited growth of particle weights in @xmath4 codes , and this was shown to improve long time convergence of the simulations . it would be of interest to assess if the nonlinear dissipation operator corresponding to wbde has the same effect . we thank d. spong for providing the delta5d monte - carlo guiding center simulation data in fig.6 , originally published in ref . we also thank xavier garbet for his comments on the paper and for pointing out several key references . mf and ks acknowledge financial support by anr under contract m2tfp , mthodes multichelles pour la turbulence dans les fluides et les plasmas . dcn and gch acknowledge support from the oak ridge national laboratory , managed by ut - battelle , llc , for the u.s . department of energy under contract de - ac05 - 00or22725 . dcn also gratefully acknowledges the support and hospitality of the cole centrale de marseille for the three , one month visiting positions during the elaboration of this work . this work , supported by the european communities under the contract of association between euratom , cea and the french research federation for fusion studies , was carried out within the framework of the european fusion development agreement . the views and opinions expressed herein do not necessarily reflect those of the european commission .
|
the proposed method preserves the moments of the particle distribution function to a good level of accuracy , has no constraints on the dimensionality of the system , does not require an a priori selection of a global smoothing scale , and its able to adapt locally to the smoothness of the density based on the given discrete particle data .
|
for given computational resources , the accuracy of plasma simulations using particles is mainly held back by the noise due to limited statistical sampling in the reconstruction of the particle distribution function . a method based on wavelet analysis is proposed and tested to reduce this noise . the method , known as wavelet based density estimation ( wbde ) , was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements . its novel application to plasma simulations can be viewed as a natural extension of the finite size particles ( fsp ) approach , with the advantage of estimating more accurately distribution functions that have localized sharp features . the proposed method preserves the moments of the particle distribution function to a good level of accuracy , has no constraints on the dimensionality of the system , does not require an a priori selection of a global smoothing scale , and its able to adapt locally to the smoothness of the density based on the given discrete particle data . most importantly , the computational cost of the denoising stage is of the same order as one time step of a fsp simulation . the method is compared with a recently proposed proper orthogonal decomposition based method , and it is tested with three particle data sets that involve different levels of collisionality and interaction with external and self - consistent fields .
|
1602.07406
|
c
|
this work has presented a theoretical framework of stochastic weak passivity serving for stabilizing the stochastic differential systems with nonvanishing noise . the main contributions include : i ) deriving the necessary conditions to say a stochastic system stochastically passive or the sufficient conditions that a stochastic system must lose stochastic passivity ; ii ) proving that it is impossible for some stochastic systems to be stabilized in probability ; iii ) defining a new concept of stochastic weak passivity to serve for those systems losing stochastic passivity , which captures the stochastic passivity of the system not in the whole state space but only outside a ball centered around the desired state ; iv ) associating stochastic weak passivity to asymptotic weak stability of systems , and further providing the sufficient conditions for global and local asymptotic weak stabilization of nonlinear stochastic differential systems by means of negative feedback laws . the stochastic weak passivity provides an alternative way to stabilize the transition measure as well as capturing the ergodicity of the stochastic differential systems with nonvanishing noise . however , there is still a large room for this method to be improved or expanded . an important issue is that the whole theoretical framework works under the assumption that the stochastic term @xmath55 is a standard wiener process . the motivation of making such an assumption is that the current concept is developed based on the stochastic passivity . for the latter , the noise term is assumed as a standard wiener process . however , the standard wiener process is just a kind of ideal noise , and is used mainly for the simplicity of analysis . as far as many practical systems are concerned , this ideal noise is not accurate enough to represent the internal modeling uncertainty . therefore , it is interesting but challenging to use other stochastic processes instead of the standard wiener process for developing stochastic weak passivity . towards this task , the infinitesimal generator @xmath429}$ ] of eq . needs to be redefined accordingly . in addition , other possible points of future research include : i ) weakening the condition of nonsigularity of the diffusion matrix @xmath430 ; ii ) applying the stochastic weak passivity theory to some special stochastic differential systems , such as stochastic affine systems , thermodynamic process systems , and drive the development of these fields in control techniques ; iii ) developing the determinist version of stochastic weak passivity .
|
the stochastic asymptotic weak stability is proposed in this paper which suggests the transition measure of the state to be convergent and the ergodicity . by defining stochastic weak passivity that admits stochastic passivity only outside a ball centered around the desired state but not in the whole state space , we develop stochastic weak passivity theorems to ensure that the stochastic systems with nonvanishing noise can be globally locally stabilized in weak sense through negative feedback law . stochastic differential systems , transition measure , ergodicity , stochastic weak passivity , asymptotic weak stability , stabilization 60h10 , 62e20 , 70k20 , 93c10 , 93d15 , 93e15
|
for stochastic systems with nonvanishing noise , i.e. , at the desired state the noise port does not vanish , it is impossible to achieve the global stability of the desired state in the sense of probability . this bad property also leads to the loss of stochastic passivity at the desired state if a radially unbounded lyapunov function is expected as the storage function . to characterize a certain ( globally ) stable behavior for such a class of systems , the stochastic asymptotic weak stability is proposed in this paper which suggests the transition measure of the state to be convergent and the ergodicity . by defining stochastic weak passivity that admits stochastic passivity only outside a ball centered around the desired state but not in the whole state space , we develop stochastic weak passivity theorems to ensure that the stochastic systems with nonvanishing noise can be globally locally stabilized in weak sense through negative feedback law . applications are shown to stochastic linear systems and a nonlinear process system , and some simulation are made on the latter further . stochastic differential systems , transition measure , ergodicity , stochastic weak passivity , asymptotic weak stability , stabilization 60h10 , 62e20 , 70k20 , 93c10 , 93d15 , 93e15
|
1302.4692
|
i
|
let @xmath0 be a closed ( i.e. compact , without boundary ) manifold of dimension two , different from the 2-sphere , equipped with an orientation 2-form @xmath3 . if @xmath4 and @xmath5 are two @xmath6 closed curves on @xmath0 which intersect transversally , we call algebraic intersection of @xmath7 and @xmath5 the number @xmath8 where * @xmath9 denotes the tangent vector to @xmath4 at @xmath10 * the sum is taken over all pairs of parameter values @xmath11 such that @xmath12 . it is classical that this number only depends on the homology classes of @xmath4 and @xmath5 . we denote it by @xmath13,\left[\beta\right])$ ] . the map @xmath1 extends by linearity to a symplectic ( i.e bilinear , antisymmetric , nondegenerate ) form on the first homology @xmath14 of @xmath0 . the central question in this paper is _ when @xmath0 is endowed with a riemannian metric @xmath2 , how much can two curves of a given length intersect ? _ this amounts to evaluating the norm of the bilinear form @xmath1 with respect to a certain norm on @xmath14 , called the stable norm . informally speaking the stable norm measures the size , relative to the metric @xmath2 , of a homology or cohomology class . various equivalent definitions exist , see @xcite . we shall use that of @xcite : for @xmath15 and a vector @xmath16 , we denote by @xmath17 its riemannian norm . the _ comass _ of a differential one - form on @xmath0 is given by @xmath18 equation ( [ comass ] ) defines a norm on the space @xmath19 of smooth 1-forms on @xmath0 . we get a norm on the first cohomology of @xmath0 by taking the infimum of the comass over all smooth closed 1-forms in a given cohomology class : @xmath20=c \}.\ ] ] the norm @xmath21 is called the stable norm on @xmath22 . we denote in the same way the dual norm on @xmath14 . we say a homology class @xmath23 is integer if @xmath24 is the image in @xmath25 of an element of @xmath26 . when @xmath0 is an orientable surface of genus @xmath27 , and the homology class @xmath24 is integer , the stable norm has a nice expression , see @xcite : @xmath28 is the minimum of all sums @xmath29 , where * the index @xmath30 ranges over @xmath31 * @xmath32 denotes the length with respect to @xmath2 * the @xmath33 are integer numbers * the @xmath34 are pairwise disjoint simple closed geodesics * @xmath35 $ ] . the norm of the bilinear form @xmath36 with respect to the stable norm on @xmath37 is then defined as : @xmath38 observe that in the above expression , the supremum is actually a maximum , since the function @xmath39 is zero - homogeneous , so it is actually defined on the projectivized of @xmath14 , which is compact . when there is no ambiguity on @xmath0 and @xmath2 , we shall sometimes abbreviate the notation @xmath40 to @xmath41 . while , from a geometrical standpoint , the stable norm is the most natural norm on @xmath14 , from the complex analysis viewpoint , the most natural norm is the @xmath42-norm . for any differential one - form @xmath43 and for @xmath44 we denote by @xmath45 the norm , with respect to the metric @xmath2 , of the corresponding linear form on @xmath46 . then we define the @xmath42-norm of @xmath43 by the formula @xmath47 where @xmath48 denotes the volume element of the metric @xmath2 . we define the @xmath42-norm of a cohomology class @xmath49 as @xmath50 , over all 1-forms @xmath51 . it is a remarkable fact ( see @xcite ) that this infimum is actually a minimum , and is achieved by the unique harmonic 1-form in the cohomology class @xmath49 . the norm on @xmath37 dual to the @xmath42-norm on @xmath22 will also be called @xmath42-norm , and will be denoted by the same symbol . the original motivation for this article was to compare the stable norm and the @xmath42-norm . this is done in section [ comparison ] , and our result is : [ compa1 ] let * @xmath52 be a closed , oriented surface equipped with a riemannian metric * @xmath53 be the total volume of @xmath52 . then for all @xmath54 , we have @xmath55 this theorem was originally proved as equation ( 4.8 ) of @xcite , see also @xcite . in section [ section compa ] we give a short and simple proof . the first inequality , which is a straightforward consequence of the cauchy - schwarz inequality , has been extended to higher dimensions in @xcite . it is also used in @xcite . now that we ve been introduced to the number @xmath40 , we want to know more about it . a trivial , but nice observation , is that theorem [ compa1 ] entails @xmath56 the first question that comes to mind is [ question 1 ] is the lower bound of equation ( [ poincare - vol - eq ] ) optimal ? if not , what is the best possible lower bound ? is it realized by some surfaces , and if so , how to characterize such surfaces ? such as it is , question [ question 1 ] is readily answered by @xcite , and the answer is that @xmath57 if and only if @xmath0 is the two - torus and the metric @xmath2 is flat . the `` if '' part may be checked by elementary calculations and we leave it as an exercise . the `` only if '' holds because , by @xcite , if the stable norm and the @xmath42-norm are proportional , then each harmonic 1-form has constant norm . then proposition 6.2 of @xcite implies that @xmath52 is a flat torus . this answer to question [ question 1 ] prompts new questions : [ question 2 ] if we fix a genus @xmath58 for @xmath0 , what is the optimal lower bound ? is it realized by some riemannian metrics ? if so , are those metrics of constant curvature ? another obvious question is [ question 3 ] does @xmath40 have an upper bound involving known geometric quantities such as the length of a homological systole ( the length of a shortest , non - separating closed geodesic ) ? the best we can do about questions [ question 2 ] , [ question 3 ] is summed up in corollary [ bounds in arbitrary curvature ] which we restate here for the commodity of the reader : let * @xmath0 be a closed , oriented surface of genus @xmath59 * @xmath2 be a riemannian metric on @xmath0 * @xmath60 be the diameter of @xmath52 * @xmath61 be the length of a homological systole of @xmath52 . then we have @xmath62 in section [ section constant curvature ] we specialize to metrics of constant negative curvature , and we obtain the [ encadrement courbure -1 ] let * @xmath0 be a closed , oriented surface of genus @xmath58 * @xmath2 be a riemannian metric of constant curvature @xmath63 on @xmath0 * @xmath64 be the length of a homological systole of @xmath52 . then there exist positive numbers @xmath65 and @xmath66 , which depend only on the genus @xmath27 of @xmath0 , such that when @xmath64 is small enough , @xmath67 it would be interesting to know if there is a more precise asymptotic estimate for the behaviour of @xmath40 when @xmath2 tends to infinity in the moduli space @xmath68 of surfaces of genus @xmath27 and curvature @xmath63 . at least we know that @xmath40 does not have a maximum in @xmath68 , but the following question remains : does @xmath40 have a minimum when @xmath52 ranges over the moduli space @xmath68 of surfaces of genus @xmath27 ? if so , which surfaces realize the minimum ? there is still an obvious question that we havent addressed : given a surface @xmath52 , by which homology classes is @xmath40 realized , as the maximum in equation ( [ def k ] ) ? when is it realized by ( the homology classes of ) simple closed geodesics ? in the case of flat tori , it can be checked by elementary calculations that for almost every flat torus ( with respect to lebesgue measure on the moduli space of flat tori ) , @xmath40 is not realized by the homology classes of simple closed geodesics . in the case of surfaces of constant negative curvature , we propose the following conjecture , inspired by theorem 10.7 of @xcite : for any @xmath69 , for almost every @xmath52 in @xmath68 , @xmath40 is realized by the homology classes of simple closed geodesics .
|
given a closed , oriented surface , the algebraic intersection of closed curves induces a symplectic form on the first homology group of . if is equipped with a riemannian metric , the first homology group of inherits a norm , called the stable norm . we study the norm of the bilinear form , with respect to the stable norm .
|
given a closed , oriented surface , the algebraic intersection of closed curves induces a symplectic form on the first homology group of . if is equipped with a riemannian metric , the first homology group of inherits a norm , called the stable norm . we study the norm of the bilinear form , with respect to the stable norm .
|
1303.5180
|
i
|
in this note we study the problem concerning the optimality of the aew in the regression model with random design . to formulate the problem , we need to introduce several definitions . let @xmath7 and @xmath8 be two measure spaces , and set @xmath9 and @xmath10 to be @xmath11 i.i.d . random variables with values in @xmath7 . from a statistical standpoint , @xmath12 is the set of given data at our disposal . the _ risk _ of a measurable real - valued function @xmath13 defined on @xmath8 is given by @xmath14 where @xmath15 is a non - negative function , called the _ loss function _ and @xmath16 is the set of all real - valued measurable functions defined on @xmath8 . if @xmath17 is a statistic constructed using the data @xmath18 , then the risk of @xmath17 is the random variable @xmath19.\ ] ] throughout this article , we restrict our attention to functions @xmath13 , loss functions @xmath20 , and random variables @xmath9 for which @xmath21 almost surely . ( note that some results have been obtained in the same setup for unbounded loss functions in @xcite , and @xcite . ) the loss function on which we focus throughout most of the article is the quadratic loss function , defined when @xmath22 by @xmath23 . in the aggregation framework , one is given a finite set @xmath0 of real - valued functions defined on @xmath8 , usually called a _ dictionary_. the problem of _ aggregation _ ( see , e.g. , @xcite , and @xcite ) is to construct a procedure , usually called an _ aggregation procedure _ , that produces a function with a risk as close as possible to the risk of the best element in @xmath0 . keeping this in mind , one can define the _ optimal rate of aggregation _ @xcite , which is the smallest price , as a function of the cardinality of the dictionary @xmath24 and the sample size @xmath25 , that one has to pay to construct a function with a risk as close as possible to that of the best element in the dictionary . we recall the definition for the `` expectation case ; '' a similar definition for the `` probability case '' can be formulated as well ( see , e.g. , @xcite ) . [ def : definition - optimality ] let @xmath26 . we say that @xmath27 is an optimal rate of aggregation in expectation when there exist two positive constants , @xmath28 and @xmath5 , depending only on @xmath29 , for which the following holds for any @xmath30 and @xmath31 : 1 . there exists an aggregation procedure @xmath32 such that for any dictionary @xmath0 of cardinality @xmath24 and any random variable @xmath9 satisfying @xmath21 almost surely for all @xmath33 , one has @xmath34 2 . for any aggregation procedure @xmath35 , there exists a dictionary @xmath0 of cardinality @xmath24 and a random variable @xmath9 such that @xmath21 almost surely for all @xmath33 and @xmath36 in our setup , one can show ( cf . @xcite ) that in general , an optimal rate of aggregation ( in the sense of @xcite [ optimality in expectation ] and of @xcite [ optimality in probability ] ) is lower - bounded by @xmath37 . thus , procedures satisfying an exact oracle inequality like ( [ eq : exact - oracle - inequality])that is , an oracle inequality with a factor of 1 in front of @xmath38with a residual term of @xmath39 are said to be optimal . only a few aggregation procedures have been shown to achieve this optimal rate , including the exponential aggregating schemes of @xcite , the the `` empirical star algorithm '' in @xcite , and the `` preselection / convexification algorithm '' in @xcite . for a survey on optimal aggregation procedures , see the hdr dissertation of j .- y . audibert . our main focus here is on the problem of the optimality of the aggregation procedure with exponential weights ( aew ) . this procedure originate from the thermodynamic standpoint of learning theory ( see @xcite for the state of the art in this direction ) . aew can be viewed as a relaxed version of the trivial aggregation scheme , which is to minimize the empirical risk @xmath40 in the dictionary @xmath0 . a procedure that minimizes ( [ eq : r - n ] ) is called _ empirical risk minimization _ ( erm ) . it is well known that erm generally can not achieve the optimal rate of @xmath41 , unless one assumes that the given class @xmath0 has certain geometric properties , which we discuss below ( see also @xcite ) . to have any chance of obtaining better rates , one has to consider aggregation procedures that take values in larger sets than @xmath0 . the most natural set is the convex hull of @xmath0 . aew is a very popular candidate for the optimal procedure , and it was one of the first procedures to be studied in the context of the aggregation framework @xcite . it is defined by the following convex sum : @xmath42 for the dictionary @xmath43 . the parameter @xmath2 is called the _ temperature_. can be seen as a gibbs measure with temperature @xmath44 on the dictionary @xmath0 . ] thus far , there have been three main results concerning the optimality of the aew . the first of these is that the progressive mixture rule is optimal in expectation for @xmath44 larger than some parameters of the model ( see @xcite and @xcite ) , and under certain convexity assumption on the loss function @xmath20 . this procedure is defined by @xmath45 where @xmath46 is the function generated by aew ( with a common temperature parameter @xmath44 ) associated with the dictionary @xmath0 and constructed using only the first @xmath47 observations @xmath48 . ( see @xcite for more details and for other procedures related to the progressive mixture rule . ) second , the optimality in expectation of aew was obtained by @xcite for the regression model @xmath49 with a deterministic design @xmath50 with respect to the risk @xmath51 ( with its empirical version being @xmath52 ) . that is , it was shown that for @xmath53 , where @xmath54 is the variance of the noise @xmath55 , @xmath56 finally , @xcite , and @xcite proved that in the high - temperature regime , aew can achieve the optimal rate @xmath37 under the bernstein assumption , recalled below in definition [ def : bernstein ] in expectation and in high probability . this result is discussedin more detail later . despite the long history of aew , the literature contains no results on the optimality ( or suboptimality ) of aew in the regression model with random design in the general case ( when the dictionary does not necessarily satisfy the bernstein condition ) . in this article , we address this issue and complement the results ( assuming the bernstein condition ) of @xcite for the low - temperature regime by proving the following : 1 . aew is suboptimal for low temperatures @xmath4 ( where @xmath5 is an absolute positive constant ) , both in expectation and in probability , for the quadratic loss function and a dictionary of cardinality @xmath57 ( theorem [ tha ] ) . aew is suboptimal in probability for some large dictionaries ( of cardinality @xmath58 ) and small temperatures @xmath59 ( theorem [ thb ] ) . 3 . aew achieves the optimal rate @xmath37 for low temperatures under the bernstein condition on the dictionary ( theorem [ thc ] ) . together with the high - temperature results of @xcite and @xcite , this proves that the temperature parameter has almost no impact ( as long as @xmath60 ) on the performance of the aew under this condition , with a residual term of the order of @xmath61 for every @xmath2 . [ tha]there exist absolute constants @xmath62 for which the following holds . for any integer @xmath63 , there are random variables @xmath64 and a dictionary @xmath65 such that @xmath66 almost surely for @xmath67 , for which the quadratic risk of the aew satisfies the following : 1 . if @xmath4 and @xmath25 is odd , then @xmath68 2 . if @xmath69 , then , with probability greater than @xmath70 , @xmath71 theorem [ tha ] proves that aew is suboptimal in expectation in the low - temperature regime and suboptimal in probability in both the low- and high - temperature regimes , since it is possible to construct procedures that achieve the rate @xmath72 with high probability @xcite and in expectation @xcite in the same setup as for theorem [ tha ] . it should be noted that the problem of the optimality in probability of the progressive mixture rule ( and other related procedures ) was studied by @xcite , who proved that , for a loss function @xmath20 satisfying some convexity and regularity assumption ( e.g. , the quadratic loss used in theorem [ tha ] ) , the progressive mixture rule @xmath73 defined in ( [ eq : progressive - mixture ] ) satisfies that for any temperature parameter , with probability greater than an absolute constant @xmath74 , @xmath75 in addition , it is important to observe that the suboptimality in probability does not imply suboptimality in expectation for the aggregation problem , or vice versa . this property of the aggregation problem was first noted by @xcite , who found the progressive mixture rule ( and other related aggregation procedures ) to be suboptimal in probability for dictionaries of cardinality two but , on the other hand , to be optimal in expectation ( @xcite and @xcite ) . this peculiar property of the problem of aggregation comes from the fact that an aggregate @xmath17 is not restricted to the set @xmath0 , which allows @xmath76 to take negative values . @xcite showed that for the progressive mixture rule @xmath73 , these negative values do compensate on average for larger values , but there is still an event of constant probability on which @xmath77 takes values greater than @xmath78 . the proof of theorem [ tha ] shows that a dictionary consisting of two functions is sufficient to yield a lower bound in expectation in the low - temperature regime and in probability in both the small temperature regime , @xmath79 , and the large temperature regime , @xmath80 . in the following theorem , we study the behavior of aew for larger dictionaries . to the best of our knowledge , negative results on the behavior of exponential weights based aggregation procedures are not known for dictionaries with more than two functions , and we show that the behavior of the aew deteriorates in some sense as the cardinality of the dictionary increases . [ thb ] there exist an integer @xmath81 and absolute constants @xmath5 and @xmath82 for which the following holds . for every @xmath83 , there are random variables @xmath64 and a dictionary @xmath43 of cardinality , @xmath84 , for which the quadratic loss function of any element in @xmath0 is bounded by @xmath57 almost surely , and for every @xmath85 , if @xmath86 , then with probability at least @xmath87 , @xmath88 moreover , if @xmath89 denotes the optimal function in @xmath0 with respect to the quadratic loss ( the oracle ) , then there exists @xmath90 with an excess risk greater than @xmath91 and for which the weight of @xmath92 in the aew procedure satisfies @xmath93 theorem [ thb ] implies that the aew procedure might cause the weights to concentrate around a `` bad '' element in the dictionary ( i.e. , an element whose risk is larger than the best in the class by at least @xmath94 ) with high probability . in particular , theorem [ thb ] provides additional evidence that the aew procedure is suboptimal for low temperatures . the analysis of the behavior of aew for a dictionary of cardinality larger than two is considerably harder than in the two - function case and requires some results on rearrangement of independent random variables that are almost gaussian ( see proposition [ prop : shahar - proposition ] below ) . fortunately , not all is lost as far as optimality results for aew go . indeed , we show that under some geometric condition , aew can be optimal and in fact can even adapt to the `` real complexity '' of the dictionary . intuitively , a good aggregation scheme should be able to ignore the elements in the dictionary whose risk is far from the optimal risk in @xmath0 , or at least the impact of such elements on the function produced by the aggregation procedure should be small . thus , a good procedure is one with a residual term of the order of @xmath95 , where @xmath96 is a complexity measure that is determined only by the richness of the set of `` almost minimizers '' in the dictionary . this leads to the following question : [ qu : adaptivitytocomplexityofdictionary ] is it possible to construct an aggregation procedure that adapts to the real complexity of the dictionary ? this question was first addressed by the pac - bayesian approach . @xcite and @xcite showed that in the high - temperature regime , aew satisfies the requirements of question [ qu : adaptivitytocomplexityofdictionary ] , assuming that the class has a geometric property , called the bernstein condition . [ def : bernstein ] we say that a function class @xmath0 is a @xmath97-bernstein class ( @xmath98 and @xmath99 ) with respect to @xmath9 if every @xmath100 satisfies @xmath101 and @xmath102 there are many natural situations in which the bernstein condition is satisfied . for instance , when @xmath20 is the quadratic loss function and the regression function is assumed to belong to @xmath0 , the excess loss function class @xmath103 satisfies the bernstein condition with @xmath104 , where @xmath105 is the minimizer of the risk in the class @xmath0 . another generic example is when the target function @xmath106 is far from the set of targets with `` multiple minimizers '' in @xmath0 and @xmath107 satisfies the bernstein condition with @xmath104 . ( see @xcite for an exact formulation of this statement and related results . ) the bernstein condition is very natural in the context of erm because it has two consequences : that the empirical excess risk has better concentration properties around the excess risk , and that the complexity of the subset of @xmath0 consisting of almost minimizers is smaller under this assumption . consequently , if the class @xmath107 is a @xmath97-bernstein class for @xmath108 , then the erm algorithm can achieve fast rates ( see , e.g. , @xcite and references therein ) . as the results below show , the same is true for aew . indeed , under a bernstein assumption , @xcite and @xcite proved that if @xmath109 is a convex risk function and if @xmath0 is such that @xmath110 almost surely for any @xmath33 , then for every @xmath111 and @xmath112 , with probability greater than @xmath113 , @xmath114 although the pac - bayesian approach can not be used to obtain ( [ eq : pac - bound ] ) in the low - temperature regime ( @xmath115 ) , such a result is not surprising . indeed , because fast error rates for the erm are expected when the underlying excess loss functions class satisfies the bernstein condition , and because aew converges to the erm when the temperature @xmath44 tends to 0 , it is likely that for `` small values '' of @xmath44 , aew inherits some of the properties of erm , such as fast rates under a bernstein condition . we show this in theorem [ thc ] , proving that aew answers question [ qu : adaptivitytocomplexityofdictionary ] for low temperatures under the bernstein condition . before formulating theorem [ thc ] , we introduce the following measure of complexity . for every @xmath116 , let @xmath117 where @xmath118 denotes the cardinality of the set @xmath119 . observe that @xmath120 is a weighted sum of the number of elements in @xmath0 that assigns smaller and smaller weights to functions with a relatively large excess risk . [ thc ] there exist absolute constants @xmath28 , @xmath121 , and @xmath122 for which the following holds . let @xmath0 be a class of functions bounded by @xmath29 such that the excess loss class @xmath107 is a @xmath123-bernstein class with respect to @xmath9 . if the risk function @xmath109 is convex and if @xmath124 , then for every @xmath125 , with probability at least @xmath113 , the function @xmath126 produced by the aew algorithm satisfies @xmath127 where @xmath128 . in particular , @xmath129 in other words , the scaling factor @xmath130 that we use is proportional to @xmath131 , and if the class is regular ( in the sense that the complexity of @xmath0 is well spread and not concentrated just around one point ) , then @xmath132 is roughly the cardinality of the elements in @xmath0 with risk at most latexmath:[$\sim\!(b+b)(\log observe that for every @xmath116 , @xmath134 for a suitable absolute constant @xmath135 . thus , if @xmath44 is reasonably small ( below a level proportional to @xmath136 ) , then the resulting aggregation rate is the optimal one , proportional to @xmath137 with probability @xmath113 , and proportional to @xmath138 in expectation . thus , theorem [ thc ] indeed gives a positive answer to question [ qu : adaptivitytocomplexityofdictionary ] in the presence of a bernstein condition and for low temperatures . although the residual terms in theorem [ thc ] and in ( [ eq : pac - bound ] ) are not the same , they are comparable . indeed , the contribution of each element in @xmath0 in the residual term depends exponentially on its excess risk . theorem [ thc ] together with the results for high temperatures from @xcite and @xcite show that the aew is an optimal aggregation procedure under the bernstein condition as long as @xmath60 when @xmath24 and @xmath25 tend to infinity . in general , the residual term obtained is on the order of @xmath139 , and it can be proven that the optimal rate of aggregation under the bernstein condition is proportional to @xmath37 using the classical tools in @xcite . finally , a word about the organization of the article . in the next section we present some comments about our results . the proofs of the three theorems follow in the subsequent sections . throughout , we denote absolute constants or constants that depend on other parameters by @xmath5 , @xmath82 , etc . ( of course , we specify when a constant is absolute and when it depends on other parameters . ) the values of constants may change from line to line . we write @xmath140 if there are absolute constants @xmath135 and @xmath141 such that @xmath142 , and write @xmath143 if @xmath144 .
|
given a finite class of functions , the problem of aggregation is to construct a procedure with a risk as close as possible to the risk of the best element in the class . a classical procedure ( pac - bayesian statistical learning theory ( 2004 ) paris 6 , _ statistical learning theory and stochastic optimization _ ( 2001 ) springer , _ ann . statist . _ * 28 * ( 2000 ) 7587 ) is the aggregate with exponential weights ( aew ) , defined by where is called the temperature parameter and is an empirical risk . in this article , we study the optimality of the aew in the regression model with random design and in the low - temperature regime . we prove three properties of aew . first , we show that aew is a suboptimal aggregation procedure in expectation with respect to the quadratic risk when , where is an absolute positive constant ( the low - temperature regime ) , and that it is suboptimal in probability even for high temperatures . second , we show that as the cardinality of the dictionary grows , the behavior of aew might deteriorate , namely , that in the low - temperature regime it might concentrate with high probability around elements in the dictionary with risk greater than the risk of the best function in the dictionary by at least an order of . this result holds for small values of the temperature parameter , thus complementing an analogous result for high temperatures .
|
given a finite class of functions , the problem of aggregation is to construct a procedure with a risk as close as possible to the risk of the best element in the class . a classical procedure ( pac - bayesian statistical learning theory ( 2004 ) paris 6 , _ statistical learning theory and stochastic optimization _ ( 2001 ) springer , _ ann . statist . _ * 28 * ( 2000 ) 7587 ) is the aggregate with exponential weights ( aew ) , defined by where is called the temperature parameter and is an empirical risk . in this article , we study the optimality of the aew in the regression model with random design and in the low - temperature regime . we prove three properties of aew . first , we show that aew is a suboptimal aggregation procedure in expectation with respect to the quadratic risk when , where is an absolute positive constant ( the low - temperature regime ) , and that it is suboptimal in probability even for high temperatures . second , we show that as the cardinality of the dictionary grows , the behavior of aew might deteriorate , namely , that in the low - temperature regime it might concentrate with high probability around elements in the dictionary with risk greater than the risk of the best function in the dictionary by at least an order of . third , we prove that if a geometric condition on the dictionary ( the so - called `` bernstein condition '' ) is assumed , then aew is indeed optimal both in high probability and in expectation in the low - temperature regime . moreover , under that assumption , the complexity term is essentially the logarithm of the cardinality of the set of `` almost minimizers '' rather than the logarithm of the cardinality of the entire dictionary . this result holds for small values of the temperature parameter , thus complementing an analogous result for high temperatures .
|
math0010271
|
i
|
in this work we continue the investigation begun in @xcite of the spectral relationships within an @xmath0-tuple of self adjoint operators in a finite von neumann algebra as reflected in the geometry of the corresponding spectral scale . let us begin by introducing some notation and reviewing some of the results in @xcite . * the following notation will apply throughout the rest of the paper without explicit reference . * let @xmath2 denote a finite von neumann algebra equipped with a faithful normal tracial state @xmath3 , let @xmath0 denote a positive integer , and let @xmath13 denote an @xmath0tuple of self adjoint operators in @xmath2 . we define a map @xmath4 from @xmath2 to @xmath14 by the formula @xmath15 and write @xmath16 , where @xmath17 . since @xmath3 is normal and @xmath18 is weak * compact and convex , it follows that @xmath8 is a compact , convex subset of @xmath14 . we call @xmath8 the * spectral scale * of the @xmath10 s * relative to @xmath3*. we use parentheses [ @xmath19 to denote vectors in @xmath5 and angle brackets [ @xmath20 to denote inner products . 1 . we let @xmath12 denote the von neumann subalgebra of @xmath2 generated by @xmath13 and the identity . if @xmath21 and @xmath22 are projections in @xmath2 and @xmath23 , then the * order interval * that they determine is @xmath24 = \{a\in m^+_1 : p \le a \le q\}.\ ] ] 3 . for each nonzero vector @xmath25 we write @xmath26 4 . by a * spectral pair * we mean a pair of the form @xmath27 , where @xmath28 is a real number and @xmath29 is a nonzero vector in @xmath5 . our standard way of embedding @xmath5 into @xmath14 is @xmath30 . if @xmath27 is a spectral pair , then @xmath31 and @xmath32 denote the spectral projections of @xmath33 determined by the intervals @xmath34 $ ] and @xmath35 . we call @xmath36 the * spectral interval projections * determined by @xmath28 and @xmath37 . 6 . if @xmath27 is a spectral pair and @xmath38 is a real number , then @xmath39 denotes the hyperplane in @xmath14 defined by the formula @xmath40 and we write @xmath41for the half space defined by the inequality @xmath42 7 . if @xmath43 , then the * isotrace slice * of @xmath8 at @xmath28 is by definition @xmath44 the notion of an isotrace slice may seem artificial , but , as we shall show in a forthcoming paper , for @xmath45 it essentially contains the more familiar concept of the numerical range of the operator @xmath46 . isotrace slices are particularly useful when there are two operators so that @xmath47 as will be seen in 6 of this paper . as shown in @xcite , the geometry ( i.e. the facial structure ) of the spectral scale @xmath8 contains information about the spectrum and spectral projections of real linear combinations of the @xmath13 . in this paper we complete this part of the theory by giving an exact description of an arbitrary face of @xmath8 in terms of the @xmath10 s . in @xcite this was done only for the case @xmath48 , and it is useful to begin by presenting a description of these results . assume @xmath49 and @xmath50 . thus , @xmath8 is a convex , compact subset of the plane so that a face in @xmath8 is either an extreme point or a line segment . also , for each real @xmath28 write @xmath51 , ( resp . , @xmath52 ) for the spectral projection of @xmath53 corresponding to the interval @xmath54 $ ] ( resp . , @xmath55 ) . finally write @xmath56 for the spectrum of @xmath53 . the results obtained in ( * ? ? ? * theorems 1.5 and 1.6 ) may be summarized as follows . 1 . @xmath8 lies between the lines @xmath57 and @xmath58 and has `` sharp points '' at @xmath59 and @xmath60 . thus , the boundary of @xmath8 is divided into an upper and lower boundary . the spectrum of @xmath53 is exactly the set of slopes of the tangent lines to the lower ( or upper ) boundary of @xmath8 . a point @xmath61 in @xmath8 is an extreme point on the lower boundary if and only if @xmath61 has the form @xmath62 for some @xmath63 . extreme points on the upper boundary have the form @xmath64 . 4 . the line segments in the lower ( or upper ) boundary of @xmath8 are in one - to - one correspondence with the eigenvalues of @xmath53 . if @xmath28 is an eigenvalue for @xmath53 so that @xmath65 and @xmath66 is the line segment on the lower boundary of @xmath8 that it determines , then @xmath67)$ ] . the corners in the lower ( or upper ) boundary of the spectral scale are in one - to - one correspondence with gaps in the spectrum of the operator @xmath53 . let us now review the basic facts obtained in @xcite for the higher dimensional case . since we shall refer to these facts repeatedly , it is convenient to give them local numbers . the following is a restatement of the results in theorems 2.3 and 2.4 in @xcite . the following statements hold . 1 . if @xmath61 is an extreme point of @xmath8 , then there is a projection @xmath21 in @xmath12 such that @xmath68 further , @xmath69 and @xmath70 are extreme points of @xmath8 for every spectral pair @xmath27 . we have @xmath71 . the hyperplane @xmath72 is a hyperplane of support for @xmath8 with @xmath73 if and only if @xmath74 in this case we have @xmath75 . 3 . if @xmath76 , then @xmath77 is a face in @xmath8 . moreover , @xmath78\ ] ] and @xmath79).\ ] ] recall that a face @xmath66 of a convex set @xmath80 is said to be an * exposed face * if there is a hyperplane of support @xmath81 for @xmath80 such that @xmath82 . if the exposed face @xmath66 is a single point @xmath61 , and so necessarily an extreme point of @xmath8 , we call @xmath61 an * exposed point*. ( see 2 for a more detailed review of these notions ) . although it was not specifically noted in @xcite , theorem 0.2 allows the complete classification of the exposed faces in the spectral scale which we record below . the following statements hold . 1 . @xmath66 is an exposed face in @xmath8 if and only if there is a spectral pair @xmath27 such that @xmath83).\ ] ] 2 . if @xmath61 is an extreme point in the spectral scale , then @xmath61 is an exposed point if and only if there is a spectral pair @xmath27 such that @xmath84 and @xmath85 . if @xmath66 is a face that is not an extreme point , then @xmath66 is an exposed face if and only if there is a spectral pair @xmath27 such that @xmath28 is an eigenvalue for @xmath37 . thus , in this case @xmath86 and @xmath87 . the first conclusion follows immediately from parts @xmath88 and @xmath89 of theorem 0.2 . suppose @xmath61 is an exposed point so that by theorem 0.2 and part 1 of this corollary we have @xmath90 ) = \psi({p_{s,{\mathbf t}}^{\pm}}),\ ] ] where @xmath27 is a spectral pair . write @xmath91 and @xmath92 if @xmath93 , then there is a nontrivial projection @xmath94 in @xmath95 such that @xmath96 . but in this case , we have @xmath97 so that @xmath98 and @xmath99 ) \ne \{{\mathbf x}\}$ ] . as this is a contradiction , we get @xmath84 , as desired . conversely , if @xmath27 is a spectral pair such that @xmath84 and @xmath85 , then @xmath100)\ ] ] is a singleton and so this face is an exposed point . finally , by part 1 and the fact that @xmath4 is faithful , we have that @xmath101 exactly when the face @xmath102)$ ] has positive dimension . the results to be presented below concern the following topics . @xmath103 a general analysis of a face of a compact , convex set in @xmath5 . ( 2 and 5 ) . @xmath103 a complete description of the facial structure of the spectral scale in the general case . ( 3 ) . @xmath103 an analysis of the `` corners '' of the spectral scale in higher dimensions . ( 4 ) . @xmath103 applications of the results in 4 to show how geometric properties of @xmath8 imply the existence of central projections in @xmath12 . ( 5 and 6 ) . let us now describe our results in more detail . although we defined exposed faces of convex compact subsets in @xmath5 using hyperplanes of support , there is also an equivalent formulation in terms of linear functionals . from this point of view , a face @xmath66 in the ( convex , compact ) set @xmath80 is said to be * exposed * if there is linear functional @xmath104 and a scalar @xmath105 such that @xmath106 for each @xmath61 in @xmath66 and @xmath107 for each @xmath61 in @xmath108 . in this case , we say that @xmath104 * exposes * @xmath66 . since the former definition is more geometric it fits better with the emphasis of this paper , but it will sometimes be convenient to use the latter , more algebraic , definition . observe that not every face need be exposed . for example , it is easy to construct a convex set in two dimensions that has a face in its boundary of dimension one such that the end points of this line segment ( which are extreme points ) are not exposed . we wish to make distinctions among exposed faces as follows . the exposed face @xmath66 is said to be have * degree * @xmath109 if @xmath110 is the cardinality of the largest linearly independent set of linear functionals that expose @xmath66 . in terms of hyperplanes , this means that there are @xmath110 hyperplanes of support for @xmath66 such that their normal vectors are linearly independent . we study such faces in 4 . in 3 we show that a general face of the spectral scale provides spectral information for linear combinations of the defining @xmath10 s and certain cut downs of these operators . if @xmath66 is not exposed , then a condition similar to that of corollary 0.1 ( 1 ) holds , except that it is first necessary to cut down by a certain spectral projection . this analysis allows us to complete the characterization of the extreme points of the spectral scale which was begun in @xcite . generalized `` corners '' are studied in 4 . a corner of a convex planar set is a point on the boundary that admits more than one tangent line of support . the natural generalization of this notion in higher dimensions is a face that admits more than one hyperplane of support , i.e. , a face of degree @xmath111 for some @xmath112 . we call such faces * sharp faces * since they generalize sharp ( i.e. , nondifferentiable ) corners on the boundary of a 2 dimensional convex set . our main result on this topic can be described as follows . in two dimensions , a corner of a convex set admits precisely two tangent lines of support with linearly independent normal vectors . in higher dimensions , there can be much wider variation and a new phenomenon occurs . specifically , in the case of a spectral scale formed by a single operator , a tangent line of support for a face of the spectral scale is determined by real numbers @xmath28 and @xmath38 , where @xmath113 . in the general case where there are @xmath0 @xmath10 s , the hyperplanes of the support for a sharp face are determined by a sequence @xmath114 of spectral pairs and real numbers @xmath115 . for example , suppose @xmath66 is a sharp face which is contained in hyperplanes of the form @xmath116 and @xmath117 , where the spectral pairs @xmath118 and @xmath119 are linearly independent . if @xmath120 and @xmath121 , then , just as in two dimensions , the interval @xmath122 lies in a gap in the spectrum of @xmath37 . on the other hand if @xmath123 is a linearly independent set , then something new occurs . in this case , there is a projection @xmath124 that commutes with @xmath125 and @xmath126 and such that @xmath127 if there are @xmath111 hyperplanes containing @xmath66 , with linearly independent vectors @xmath128 , then the projection @xmath124 commutes with each @xmath129 and the compression of each @xmath129 to @xmath124 is a scalar . next , we observe in 5 that if @xmath130 for the sharp face @xmath66 of the previous paragraph , then the projection @xmath124 commutes with each @xmath10 and so @xmath124 is a nontrivial central projection in @xmath12 . put colloquially , if we can `` wobble '' a sharp face in all @xmath0 `` @xmath29directions '' , then the center of @xmath12 is not trivial . 6 is devoted to showing that if @xmath8 has a countable number of extreme points , then @xmath12 is abelian .
|
given an-tuple of self - adjoint operators in a finite von neumann algebra and a faithful , normal tracial state on , we define a map from to by . the image of the positive part of the unit ball under is called the * spectral scale * of relative to and is denoted by . in a previous paper with nik weaver we showed that the geometry of reflects spectral data for real linear combinations of the operators \{}. for example , we showed that an exposed face in is determined by a certain pair of spectral projections of a real linear combination of the s . in the present paper we extend this study to faces that are not exposed . in order to do this we need to introduce a recursive method for describing faces of compact convex sets in . using this new method , we completely describe the structure of arbitrary faces of in terms of and . we also study faces of convex , compact sets that are exposed by more than one hyperplane of support ( we call these * sharp faces * ) .
|
given an-tuple of self - adjoint operators in a finite von neumann algebra and a faithful , normal tracial state on , we define a map from to by . the image of the positive part of the unit ball under is called the * spectral scale * of relative to and is denoted by . in a previous paper with nik weaver we showed that the geometry of reflects spectral data for real linear combinations of the operators \{}. for example , we showed that an exposed face in is determined by a certain pair of spectral projections of a real linear combination of the s . in the present paper we extend this study to faces that are not exposed . in order to do this we need to introduce a recursive method for describing faces of compact convex sets in . using this new method , we completely describe the structure of arbitrary faces of in terms of and . we also study faces of convex , compact sets that are exposed by more than one hyperplane of support ( we call these * sharp faces * ) . when such faces appear on , they signal the existence of commutativity among linear combinations of the operators \{}. although many of the conclusions of this study involve too much notation to fit nicely in an abstract , there are two results that give their flavor very well . theorem 6.1 : if the set of extreme points of is countable , then is abelian . corollary 5.6 : has a finite number of extreme points if and only if is abelian and has finite dimension .
|
astro-ph9901242
|
i
|
in this work , we have addressed two essential features of the x - ray temperatures derived by davis & white ( 1996 ) for an optically complete sample of elliptical galaxies : ( 1 ) the x - ray emitting gas is always hotter than the stars and , typically , twice as hot ( @xmath298 ) ; ( 2 ) the gas / stellar temperature ratio tends to be higher for galaxies with lower velocity dispersions ( the `` @xmath0@xmath1 relation '' ) . we have constructed physically plausible models of the mass distribution in bright elliptical galaxies in an effort to constrain their average dark matter properties by matching these observations . the stellar models ( described in 3 ) are fully consistent with the fundamental plane scaling relations , and are designed to either conform to the latest published _ hst _ results on the structure of the centers of elliptical galaxies or to follow the hernquist ( 1990 ) approximation to a de vaucouleurs profile ( see figure 2 ) . we have made other plausible , but conservative , assumptions , such as ( 1 ) maximizing the non - gravitational ( _ i.e _ pressure ) contribution to the gas temperature by allowing the gas and dark matter distributions to extend to infinity and ( 2 ) assuming that the stellar orbits are isotropic for @xmath299 . a basic and general result of our calculations is that , in the absence of dark matter , @xmath300 ( see figure 3 ) . since @xmath139 is observed , a convincing case is made that dark matter is an extremely common , if not ubiquitous , constituent of elliptical galaxies . although x - ray ( and other ) observations have been used to infer the presence of dark matter in individual cases , we have shown that dark halos are generic to luminous , nearby elliptical galaxies . furthermore , the dark matter / stellar temperature ratios ( derived at the peak of their velocity dispersion distributions , assuming isotropic orbits ) are greater than one for models with @xmath297 . thus , the observation that the temperature of the extended hot gas exceeds the central stellar temperature is a reflection of the fact that the dark matter is dynamically hotter " than the stars , as suggested by davis & white ( 1996 ) . in 4 we described how @xmath9 varies as functions of the two relative parameters of the `` universal '' dark matter distribution ( equation 8) the ratio of dark matter to stellar scale lengths , @xmath301 , and the ratio of dark - to - luminous matter within @xmath264 , @xmath302_{r_{\rm max}}$ ] ( see figures 4a - b and 5 ) . because we attempt to match only the single global observable @xmath9 , there is an allowed range in the details of the dark matter spatial distribution ; however , we have derived absolute limits on the dark matter parameters required to obtain any particular value of @xmath9 ( see figures 6a - b ) . the observations do not require that dark matter dominate the inner luminous regions of elliptical galaxies : more than half of the mass within @xmath5 is baryonic for models with @xmath303 if @xmath158 . the most natural explanation of the tendency for galaxies with lower stellar temperatures to have larger gas - to - stellar temperature ratios is that @xmath132_{r_{\rm max}}$ ] decreases with @xmath27 in such a way that , on average , the total mass - to - light ratio inside @xmath59 is nearly independent of optical luminosity . this ratio , @xmath304m@xmath13l@xmath14 , is exactly what is predicted for mass models of elliptical galaxies designed to explain the gravitational shear of background field galaxies . if one specifies a scaling relation for the dark halo concentration , one can extend the dark matter distribution out to the virial radius and calculate the total baryon fraction and mass - to - light ratio . when we attempt to embed our models within the cdm theory of hierarchical halo formation , the implied dark matter scaling badly fails to reproduce the observed @xmath0@xmath1 relation unless smaller galaxies lose an increasingly larger fraction of their initial baryonic content ( see figure 7b ) , such that the average @xmath2 galaxy has lost most of its initial baryonic mass . alternatively , the global dark - to - luminous mass ratio could be constant if the dark halo concentration declines much more steeply with virial mass than cdm models predict , so that the decrease in @xmath132_{r_{\rm max}}$ ] for larger systems is a result of a more diffuse dark matter halo rather than a less massive ( relative to the stars ) halo . in this latter scenario , dark matter may become increasingly important inside @xmath5 as @xmath27 decreases , becoming dominant for @xmath305 . this deviation from cdm predictions of dark halo scaling could conceivably be due to a relatively flat primordial fluctuation spectrum on mass scales @xmath306m@xmath18 or to the effects on the dark matter density profile of the evolution of the baryonic component ; to date , large scale structure numerical simulations that can resolve halos on galactic scales _ and _ include a dissipational component have not been attempted . in this paper we have shown that the observed relationship between optical velocity dispersions and x - ray temperatures in giant elliptical galaxies implies that they have dark matter halos with @xmath12m@xmath13l@xmath14 within @xmath8 . in the future , we plan to fully utilize the available x - ray and optical imaging and spectroscopic data for our sample on a case - by - case basis in order derive the mass distributions in individual galaxies in the highest possible detail and to investigate the scatter in the @xmath0@xmath1 relationship . for galaxies with x - ray temperature profiles , we will be able to constrain the detailed form of the dark matter distribution as well as its integrated properties . we thank lars hernquist for reminding us of the consistency considerations of ciotti & pelligrini ( 1992 ) , and richard mushotzky for feedback on the original manuscript . comments from an anonymous referee led to significant improvements in the quality of this paper . r.e.w . acknowledges partial support from nasa grants nag 5 - 1718 and nag 5 - 1973 .
|
given the recently deduced relationship between x - ray temperatures and stellar velocity dispersions ( the `` relation '' ) in an optically complete sample of elliptical galaxies ( ) , we demonstrate that ellipticals contain substantial amounts of dark matter _ in general_. we present constraints on the dark matter scale length and on the dark - to - luminous mass ratio within the optical half - light radius and within the entire galaxy . we also confirm the prediction of davis & white ( 1996 ) that the dark matter is characterized by velocity dispersions that are greater than those of the luminous stars : .
|
given the recently deduced relationship between x - ray temperatures and stellar velocity dispersions ( the `` relation '' ) in an optically complete sample of elliptical galaxies ( ) , we demonstrate that ellipticals contain substantial amounts of dark matter _ in general_. we present constraints on the dark matter scale length and on the dark - to - luminous mass ratio within the optical half - light radius and within the entire galaxy . for example , we find that minimum values of dark matter core radii scale as kpc and that the minimum dark matter mass fraction is% within one optical effective radius and is% within , depending on the stellar density profile and observed value of . we also confirm the prediction of davis & white ( 1996 ) that the dark matter is characterized by velocity dispersions that are greater than those of the luminous stars : . the relation implies a nearly constant mass - to - light ratio within six half - light radii :ml . this conflicts with the simplest extension of cdm theories of large scale structure formation to galactic scales ; we consider a couple of modifications which can better account for the observed relation .
|
1101.5641
|
i
|
graph pebbling is like a number of network models , including network flow , transportation , and supply chain , in that one must move some commodity from a set of sources to a set of sinks optimally according to certain constraints . network flow constraints restrict flow along edges and conserve flow through vertices , and the goal is to maximize the amount of commodity reaching the sinks . the transportation model includes per unit costs along edges and aims to minimize the total cost of shipments that satisfy the source supplies and sink demands . at its simplest , the supply chain model ignores transportation costs while seeking to satisfy demands with minimum inventory . the graph pebbling model introduced by chung @xcite also tries to meet demands with minimum inventory , but constrains movement across an edge by the loss of the commodity itself , much like an oil tanker using up the fuel it transports , not unlike heat or other energy dissipating during transfer . specifically , a _ configuration _ @xmath4 of pebbles on the vertices of a connected graph @xmath5 is a function @xmath6 ( the nonnegative integers ) , so that @xmath7 counts the number of pebbles placed on the vertex @xmath8 . we write @xmath9 for the _ size _ @xmath10 of @xmath4 ; i.e. the number of pebbles in the configuration . a _ pebbling step _ from a vertex @xmath11 to one of its neighbors @xmath8 reduces @xmath12 by two and increases @xmath7 by one ( so that one can think of it as moving one pebble at the _ cost _ of another as toll ) . given two configurations @xmath4 and @xmath13 we say that @xmath4 is @xmath13-_solvable _ if some sequence of pebbling steps converts @xmath4 to @xmath13 . in this paper we study the traditional case in which the target distribution consists of a single pebble at some _ root _ vertex @xmath14 ( one can peruse @xcite for a wide array of variations on this theme ) . we are concerned with determining @xmath15 , the minimum number @xmath0 of pebbles so that every configuration of size @xmath0 is @xmath14-solvable . then the _ pebbling number _ of @xmath5 equals @xmath16 . alternatively , @xmath17 is one more than the maximum @xmath18 such that there is some root @xmath14 and some size @xmath18 configuration @xmath4 so that @xmath4 does not solve @xmath14 . the primary focus of this paper is to exploit this duality with newly discovered algebraic constraints . given a graph @xmath5 , configuration @xmath4 , and root @xmath14 , one can ask how difficult it is to determine if @xmath4 solves @xmath14 . in @xcite it was determined that this problem is -hard . subsequently , @xcite proved that the problem is -complete , with @xcite showing further that answering the question `` is @xmath19 ? '' is @xmath2-complete ( and hence both -hard and co - hard , and therefore in neither nor co unless @xmath20 co ) . finding classes of graphs on which we can answer more quickly is therefore relevent , and there is some evidence that one can be successful in this direction . besides what we share in this introduction , we show later that many graphs can have very short certificates that @xmath19 . the @xmath14-unsolvable configuration with one pebble on every vertex other than the root @xmath14 shows that @xmath21 , where @xmath22 denotes the number of vertices of @xmath5 . in @xcite it is proved that graphs of diameter two satisfy @xmath23 , with a characterization separating the two classes ( _ class 0 _ means @xmath24 and _ class 1 _ means @xmath25 ) given in @xcite . one of the consequences of this is that 3-connected diameter two graphs are class 0 . as an extension it is proved in @xcite that @xmath26-connected diameter @xmath27 graphs are also class 0 , and they use this result to show that almost every graph with significantly more than @xmath28 ( for any fixed @xmath27 ) edges is class 0 . consequently , it is a very ( asymptotically ) small collection of graphs that cause all the problems . knowing the pebbling number of a graph and actually solving a particular configuration are two different things , as even a configuration that is known to be solvable ( say , one of size equal to the pebbling number ) can be difficult to solve . evidence that most configurations are not so difficult , though , comes in the following form . the work of @xcite shows that every infinite graph sequence @xmath29 has a _ pebbling threshold _ @xmath30 , which yields the property that almost every configuration @xmath31 on @xmath32 of size @xmath33 is solvable ( and almost every configuration of size @xmath34 is not ) . in papers such as @xcite we find that @xmath35 is significantly smaller than @xmath36 for example , @xmath37 as opposed to @xmath38 for the complete graph @xmath39 , and roughly @xmath40 as opposed to @xmath41 for the path @xmath42 . moreover , the proof techniques reveal that almost all of these solvable configurations can be solved _ greedily _ , meaning that every pebbling step reduces the distance of the pebble to the root . so the hardness of the problem stems from a rare collection of configurations . with these results as backdrop , @xcite presents a polynomial algorithm for determining the solvability of a configuration on diameter two graphs of connectivity some fixed @xmath43 . furthermore , @xcite contains an algorithm that calculates pebbling numbers , and is able to complete the task for every graph on at most 9 vertices . also , the proof in @xcite that the @xmath27-dimensional cube is of class 0 is a polynomial algorithm ( actually bounded by its number of edges @xmath44 ) . along these lines , our main objective is to develop algorithmic tools that will in a reasonable amount of time yield good upper bounds on @xmath17 for much larger graphs , and in particular decide in some cases whether or not a graph is of class 0 . this latter determination is motivated most by the following conjecture of graham in @xcite . for graphs @xmath5 and @xmath45 , let @xmath46 denote the _ cartesian product _ whose vertices are @xmath47 , with edges @xmath48 whenever @xmath49 in @xmath5 and @xmath50 whenever @xmath51 in @xmath45 . [ graham ] * ( graham ) * every pair of graphs @xmath5 and @xmath45 satisfy @xmath52 . the conjecture has been verified for many graphs ; see @xcite for the most recent work . however , as noted in @xcite , there is good reason to suspect that @xmath53 might be a counterexample to this conjecture , if one exists , where @xmath54 is the _ lemke _ graph of figure [ lemke ] . since @xmath54 is class 0 , graham s conjecture requires that @xmath53 is also , but it is a formiddable challenge to compute the pebbling number of a graph on 64 vertices . one hopes that graph structure and symmetry will be of use , but purely graphical methods have failed to date . the methods of this paper represent the first strides toward the computational resolution of the @xmath55 question , `` is @xmath56 ? '' . certainly , these methods alone will not suffice in theorem [ lem2 ] in fact , for one root @xmath14 we show @xmath57 . ] , but if they produce a decent upper bound then the methods of @xcite might be able to finish the job . the main tool we develop is the weight function lemma [ wfl ] . this lemma allows us to define a ( very large ) integer linear optimization problem that yields an upper bound on the pebbling number . this has several important consequences , including the following . 1 . the pebbling numbers of reasonably small graphs often can be computed easily . moreover , it is frequently the case that the fractional relaxation suffices for the task , allowing the computation for somewhat larger graphs . 2 . it is also common that only a small portion of the constraints are required , expanding the pool of computable graphs even more . , with graphs on @xmath58 and @xmath59 vertices . ] one can restrict the types of constraints to greedy , bounded depth , and so on , with great success , seemingly because of the comments above . potentially , this allows one to begin to catalog special classes of graphs such as class 0 , ( semi-)greedy , and tree - solvable . the dual solutions often yield very short certificates of the results , in most cases quadratic in the number of vertices , and usually at most the number of vertices times the degree of the root . these certificates are remarkably simple compared to the usual solvability arguments that chase pebbles all over the graph in a barrage of cases . one can sometimes find such certificates for infinite families of graphs by hand , without resorting to machine for more than the smallest one or two of its members . this was our approach in section [ classes ] , for example . our method gives trivial proofs of 1 . @xmath60 and @xmath61 , which we write as @xmath62 , and 2 . @xmath63 is class 0 for @xmath64 , where @xmath65 denotes the @xmath66 _ graph power _ of @xmath5 ( as opposed to the _ cartesian _ power @xmath67 ) . this answers a question of @xcite , who defined the _ pebbling exponent _ of @xmath5 minimum such @xmath68 for which @xmath69 . thus @xmath70 ( see theorem [ pebbexpo ] ) , which is fairly close to the obvious lower bound of @xmath71 . in this paper we apply the weight function lemma to several specific graphs , including the petersen , lemke , @xmath3 weak bruhat , lemke squared , and two random graphs , as well as to a number of infinite families of graphs , such as trees , cycles , graph powers of cycles , cubes , and some generalized petersen and coxeter graphs .
|
graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves . a pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint ; the other pebble is lost in transit as a toll . it has been shown that deciding whether a supply can meet a demand on a graph is -complete . the pebbling number of a graph is the smallest such that every supply of pebbles can satisfy every demand of one pebble . deciding if the pebbling number is at most is-complete . in this paper we apply the weight function lemma to several specific graphs , including the petersen , lemke , weak bruhat , lemke squared , and two random graphs , as well as to a number of infinite families of graphs , such as trees , cycles , graph powers of cycles , cubes , and some generalized petersen and coxeter graphs . this partly answers a question of pachter , et al .
|
graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves . a pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint ; the other pebble is lost in transit as a toll . it has been shown that deciding whether a supply can meet a demand on a graph is -complete . the pebbling number of a graph is the smallest such that every supply of pebbles can satisfy every demand of one pebble . deciding if the pebbling number is at most is-complete . in this paper we develop a tool , called the weight function lemma , for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization . with this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms , and much more quickly as well . we also obtain results for many families of graphs , in many cases by hand , with much simpler and remarkably shorter proofs than given in previously existing arguments ( certificates typically of size at most the number of vertices times the maximum degree ) , especially for highly symmetric graphs . here we apply the weight function lemma to several specific graphs , including the petersen , lemke , weak bruhat , lemke squared , and two random graphs , as well as to a number of infinite families of graphs , such as trees , cycles , graph powers of cycles , cubes , and some generalized petersen and coxeter graphs . this partly answers a question of pachter , et al . , by computing the pebbling exponent of cycles to within an asymptotically small range . it is conceivable that this method yields an approximation algorithm for graph pebbling .
|
1310.2871
|
i
|
it has long been understood that , in general relativity gravitational phenomena are the physical manifestation of spacetime curvature . nevertheless , gravitational wave detectors , which make physical measurements , are typically described as responding to spacetime metric perturbations , which are coordinate gauge dependent and correspondingly unphysical quantities . just as early attempts to understand gravitational waves in terms of metric perturbations led to confusion regarding whether such waves existed or how they might be generated , so attempts to describe how gravitational wave detectors respond to metric perturbations lead to wooly statements and , sometimes , outright misconceptions @xcite . by way of contrast , gravitational waves described as spacetime curvature perturbations are , in a well - defined sense @xcite , physically unambiguous quantities ; correspondingly , describing the gravitational wave detector response in terms of the detector s interaction with spacetime curvature may be expected to be , if nothing else , conceptually more satisfying and physically more revealing . here we derive and describe the response of a wide class of gravitational wave antennas including pulsar timing arrays , spacecraft doppler tracking , and both ground- and space - based laser interferomteric detectors in a way that relies solely upon physical measurements and the physical properties of spacetime as described by the riemann curvature tensor . the resulting expression of the detector response , and each term that comprises it , is separately gauge invariant and has a clear physical interpretation . we show that when the gravitational waves can be described as a gauge independent curvature perturbation of a background spacetime the wave contribution to the response is wholly embodied in an integration of a projection of riemann curvature tensor perturbation along specific null geodesics of the unperturbed spacetime . our principle result is directly applicable to gravitational wave detection via pulsar timing or spacecraft doppler tracking ; is the building - block upon which a physically and pedagogically satisfying description of the response of interferometeric antennas may be based ; can form the basis for a simplified and general derivation of a fully relativistic pulsar timing formula ; and may be used to simplify the use of numerial relativity simulations to aid in the analysis or interpretation of gravitational wave detector observations . @xcite described the operation of the first practical gravitational wave detector in terms of riemann curvature induced excitations of the oscillation modes of a high quality - factor metal bar . throughout the 1960 s a variety of mechanical gravitational wave antenna configurations were proposed ( a qualitative summary of many of these configurations is given in ( * ? ? ? 1013 ) ) and their responses were generally described in an approximate way through a coupling to the riemann curvature as well . by the early 1970s , however , a description of gravitational waves in terms of metric perturbations had taken hold . this was the case from the start for interferometric gravitational wave antennas @xcite and their kin : spacecraft doppler tracking @xcite and pulsar timing @xcite . this trend of describing gravitational wave antenna response to metric perturbations has generally persisted throughout the literature to this day @xcite . as is well known , however , metric perturbations depend on coordinate gauge choices . while the freedom to choose a gauge may be exploited to simplify some computations , it is perilous to give physical meaning to partial or intermediate results of these calculations . nevertheless , throughout the literature research and pedagogical one finds numerous ( and sometimes conflicting ) descriptions of how a gravitational wave physically produces a signal in a detector based on an interpretation of the terms in calculations that depend on a choice of coordinate gague . for example , it is routinely claimed ( though we give only single , recent examples from the literature ) that gravitational waves physically move freely - falling test masses in an ideal interferometric detector ( * ? ? ? 21 ) ; gravitational waves `` push '' the mirrors of an ideal interferometric detector apart and together ( * ? ? ? * lecture 1 page 1 ) ; spacetime curvature influences light differently than it does mirror separations in an interferometer ( * ? ? ? * lecture 1 page 2 ) ; gravitational waves alternately redshift and blueshift the light in an interferometer detector @xcite ; it is the direct affect of gravitational waves on the earth and a distant pulsar that we measure when we detect when we observe gravitational waves via pulsar timing @xcite ; and it is the affect of gravitational waves on the trajectory of the light passing between a pulsar and earth that leads to their detection @xcite . in fact , as is apparent by appropriate application of the equivalence principle , each of these statements is incorrect ; that they are made at all is the result of ascribing physical significance to gauge - dependent quantities . other authors have explored alternative derivations or expressions of the gravitational wave detector response in terms of the riemann curvature @xcite ; however , these discussions either start from the metric perturbation in a preferred gauge or apply , in an approximate way , the geodesic deviation equation to describe the deviation vector between non - geodesics . additionally , these descriptions ( as well as most formulations coupling to the metric perturbation ) generally make the crucial assumption that the background spacetime is minkowski or that the different detector components ( beamsplitter and end mirrors for interferometers or the earth and a pulsar for pulsar timing ) are at some coordinate rest in a preferred gauge . the description of the response we present here makes no assumptions regarding the geometry of the background spacetime and involves , from beginning to end , only physical measurements and gauge - independent quantities : i.e. , it is valid in an arbitrary background spacetime and never requires or invokes any special gauge or gauge - dependent quantities . in section [ sec : doppler ] we provide a general , geometrically motivated derivation for the observed phase evolution of a remote clock . in section [ sec : gwaves ] we review how and when spacetime curvature may be physically and unambiguously separated into background and gravitational wave perturbation contributions , each of which is separately gauge independent , and show that , when such a distrinction is possible , the gravitational wave contribution to the observed clock phase is entirely due to the gravitational wave contribution to the curvature . we discuss the meaning and application of our results in section [ sec : discussion ] . we end in section [ sec : conclusions ] with some brief conclusions and directions for future study .
|
gravitational wave detectors are typically described as responding to gravitational wave metric perturbations , which are gauge - dependent and correspondingly unphysical quantities . the description of gravitational waves , and a gravitational wave detector s response , to the unphysical metric perturbation has lead to a proliferation of false analogies and descriptions regarding how these detectors function , and true misunderstandings of the physical character of gravitational waves . here we provide a fully physical and gauge invariant description of the response of a wide class of gravitational wave detectors in terms of the riemann curvature , the physical quantity that describes gravitational phenomena in general relativity . in the limit of high frequency gravitational waves ,
|
gravitational wave detectors are typically described as responding to gravitational wave metric perturbations , which are gauge - dependent and correspondingly unphysical quantities . this is particularly true for ground - based interferometric detectors , like ligo , space - based detectors , like lisa and its derivatives , spacecraft doppler tracking detectors , and pulsar timing arrays detectors . the description of gravitational waves , and a gravitational wave detector s response , to the unphysical metric perturbation has lead to a proliferation of false analogies and descriptions regarding how these detectors function , and true misunderstandings of the physical character of gravitational waves . here we provide a fully physical and gauge invariant description of the response of a wide class of gravitational wave detectors in terms of the riemann curvature , the physical quantity that describes gravitational phenomena in general relativity . in the limit of high frequency gravitational waves , the riemann curvature separates into two independent gauge invariant quantities : a `` background '' curvature contribution and a `` wave '' curvature contribution . in this limit the gravitational wave contribution to the detector response reduces to an integral of the gravitational wave contribution of the curvature along the unperturbed photon path between components of the detector . the description presented here provides an unambiguous physical description of what a gravitational wave detector measures and how it operates , a simple means of computing corrections to a detectors response owing to general detector motion , a straightforward way of connecting the results of numerical relativity simulations to gravitational wave detection , and a basis for a general and fully relativistic pulsar timing formula .
|
1310.2871
|
c
|
most modern gravitational wave detectors including laser interferometers , pulsar timing arrays , spacecraft doppler tracking rely upon the waves effect on light travel time to detect the wave s presence . the response of these detectors is conventionally expressed in terms of a metric perturbation description of the gravitational waves . metric perturbations , however , are coordinate gauge dependent quantities . their use obscures the physical nature of the gravitational wave phenomena and the principles that governs the operation of these detectors @xcite and lead to `` explanations '' of detector function that are at odds with the equivalence principle , which is the most basic assertion of general relativity . the response of modern gravitational wave detectors may also be expressed in terms of their interaction with spacetime curvature , which is the physically unambiguous property of spacetime that , in general relativity , underlies all gravitational phenomena . such an expression of the detector response , and each term that comprises it , is separately gauge invariant and has a clear physical interpretation . the expression for the response is manifestly consistent with the equivalence principle : i.e. , the role of all gravitational phenonmena in the response wave or otherwise clearly involves measurements made over finite spacetime intervals . the gravitational wave contribution to the response is separately identifiable and the way in which it leads to light - time fluctuations in the detector avoids all the suspect analogies ( moving mirrors , redshifting and blueshifting of light , etc . ) that plague descriptions of detector functioning based upon metric perturbations . contributions from other physical phenomena also appear naturally : e.g. , in the case of pulsar timing , all geometrical and kinematical effects ( rmer delay , parallax , aberration , shklovskii effect @xcite ) , special relativistic and general relativistic effects ( special and general relativistic einstein delay , shapiro time delay and gravitational wave contributions ) all appear naturally . the `` curvature - based '' response function describes all light - time based detectors in all regimes : i.e. , terrestrial laser interferometric detectors , whose physical dimensions and light storage times are shorter than the typical radiation wavelength @xcite ; proposed space - based detectors based upon the lisa heritage , where the detector size and light storage time is on - order the radiation wavelength @xcite ; and spacecraft doppler and pulsar timing , where the radiation wavelength is very much shorter than the detector size or light storage time @xcite . it meshes well with the use of numerical relativity calculation to determine detector response : i.e. , it does not require extracting from these calculations gauge dependent quantities that are not directly calculated in order to compute the director response . in companions papers we shall explore in greater depth how expressing the gravitational wave response in terms of the riemann curvature can be applied to delay - line and fabrey - perot interferometric detectors ; the role of non - uniform motion ( e.g. , rotation ) in lisa - heritage detectors ; the effects of micro - lensing in pulsar timing gravitational wave detection @xcite ; and the use of the riemann curvature response function to provide an alternative derivation of a general - relativistic pulsar timing formula . sometime in the next decade gravitational wave observations will begin to constrain our understanding of astronomical phenomena : when that moment arrives , gravitational wave astronomy will be born . the acceptance of gravitational wave observations as a tool of astronomical discovery can only be hastened by an intuitive , simple and physically correct understanding of gravitational waves and their detection : what they are , how they are generated , how they propagate , and how they interact with a detector . expressing the gravitational wave detector response in terms of the riemann curvature is but one step in that direction . lsf thanks the aspen center for physics for their hospitality and thanks david garfinkle and john baker for discussions . this work was supported by national science foundation grant numbers 09 - 40924 and 09 - 69857 awarded to the pennsylvania state university and supported by the natural science and research council of canada .
|
this is particularly true for ground - based interferometric detectors , like ligo , space - based detectors , like lisa and its derivatives , spacecraft doppler tracking detectors , and pulsar timing arrays detectors . the description presented here provides an unambiguous physical description of what a gravitational wave detector measures and how it operates , a simple means of computing corrections to a detectors response owing to general detector motion , a straightforward way of connecting the results of numerical relativity simulations to gravitational wave detection , and a basis for a general and fully relativistic pulsar timing formula .
|
gravitational wave detectors are typically described as responding to gravitational wave metric perturbations , which are gauge - dependent and correspondingly unphysical quantities . this is particularly true for ground - based interferometric detectors , like ligo , space - based detectors , like lisa and its derivatives , spacecraft doppler tracking detectors , and pulsar timing arrays detectors . the description of gravitational waves , and a gravitational wave detector s response , to the unphysical metric perturbation has lead to a proliferation of false analogies and descriptions regarding how these detectors function , and true misunderstandings of the physical character of gravitational waves . here we provide a fully physical and gauge invariant description of the response of a wide class of gravitational wave detectors in terms of the riemann curvature , the physical quantity that describes gravitational phenomena in general relativity . in the limit of high frequency gravitational waves , the riemann curvature separates into two independent gauge invariant quantities : a `` background '' curvature contribution and a `` wave '' curvature contribution . in this limit the gravitational wave contribution to the detector response reduces to an integral of the gravitational wave contribution of the curvature along the unperturbed photon path between components of the detector . the description presented here provides an unambiguous physical description of what a gravitational wave detector measures and how it operates , a simple means of computing corrections to a detectors response owing to general detector motion , a straightforward way of connecting the results of numerical relativity simulations to gravitational wave detection , and a basis for a general and fully relativistic pulsar timing formula .
|
1205.1486
|
c
|
the simulations of sec . [ sec_results ] demonstrate that two dominant factors control the characteristic stress scale of the stress - strain curve . these are the choice of @xmath11 and the mobile dislocation density . figs . [ different_tau_fig ] and [ different_rho_fig ] demonstrate that @xmath11 sets an upper stress limit for extended plastic flow to occur . this upper limit is approached when the mobile dislocation density is sufficiently low that the sinusoidal stress field is the dominant contribution to the stress field each dislocation feels . by increasing the mobile dislocation content , the characteristic shear - stress scale reduces due to the increasing role of the internal stress fields arising from the elastic interaction between dislocations . at the same time a broadening of the micro - plastic regime is observed , which very much is a general property of micro - yielding , where a greater number of mobile edge dislocations correlates with larger measurable plastic strain @xcite . figs . [ different_lambda_fig ] to [ first_burst_fig ] demonstrate the effect of @xmath12 ( and @xmath14 ) on the stress - strain curve is somewhat subtler . the choice of @xmath12 influences the initial configuration of the mobile dislocation population , thereby affecting the statistics of the stress at which the first strain - burst occurs and also the way in which the extended plastic flow regime is reached . [ strain_burst_dist1_fig ] shows , however , that the statistics of the strain burst magnitude is insensitive to all three discussed parameters , reflecting the universality of soc with respect to micro structural details . how shall one quantitatively interpret the imposed sinusoidal stress field ? in a real material that is nominally free of dislocation structures , some type of length scale will naturally emerge as a function of macroscopic plastic strain due to the evolution of a growing and interacting dislocation structure that eventually leads to the phenomenon of patterning @xcite . although patterning is a term predominantly referring to the effects of latter stage ii and iii hardening regimes , the emergence of micro - structure length scales is expected to occur at all stages of plasticity ranging from slip , dipole and eventually cellular patterns @xcite . in fact , a micro - structural length - scale can equally well be defined for an undeformed as - grown material , where the mean dislocation spacing can be used to describe the initially present internal stress fluctuations a view point that is central to the early work of tinder and co - workers @xcite . from this perspective the imposed sinusoidal stress field , can be viewed as the simplest realisation of the internal stress field arising from such a structure . in the present model this stress field is time independent , implying it is constructed by that part of the dislocation population that is immobile , with the explicit dislocations and their dynamics , arising from the ( much smaller ) mobile component of the dislocation population . thus a typical loading simulation can be seen as the deformation of a model material that has a particular sample preparation or deformation history characterised by @xmath11 and @xmath12 , and a mobile dislocation density that is only a small part of the total dislocation density . much past work exists concerning the emergence of internal stress and length scales as a function of deformation history . in early work on the theory of cell formation , two relationships have emerged in which the total evolving dislocation density , @xmath109 , plays a central role . they are : @xmath110 and @xmath111 where @xmath20 is a representative ( not necessarily pure ) shear modulus and @xmath112 the burgers vector magnitude . in the above , @xmath113 is the evolving flow stress of the material and @xmath114 is an evolving internal length scale that can be referred to as a cell size . the first expression has its earliest origins in a taylor hardening picture in which the total dislocation density is seen as an immobile forest dislocation population . the second equation has its theoretical origins in the early work of holt @xcite who derived it for a dipolar population of screw - dislocations , showing that a uniform arrangement was unstable to fluctuations with one such length scale dominating , characterised by eqn . [ eqcell2 ] . this length scale , which could be related to a fixed self - screening distance of the dislocation network , was postulated to reflect an emerging cell size . the approach was based on an energy minimisation principle , however due to dislocation reactions , the more modern viewpoint is that the dynamics of cell formation lies in a statistical process involving dislocation reactions and that the screening length , and therefore cell - size , is an evolving variable @xcite . thus , the model parameter @xmath12 has a direct counterpart in cell formation theory , @xmath114 , which can represent quite generally , a mean - free path for a mobile dislocation , a dipolar screening length or a well evolved cell length scale . moreover , since an unloading / loading cycle will generally return a system to the flow stress before unloading , and that the present simulations have shown that the flow stress is partly controlled by @xmath11 , @xmath11 should be in some way related to @xmath113 . from this perspective , @xmath11 and @xmath12 are parameters that are not entirely independent from each other . in fact , eqns . [ eqcell1 ] and [ eqcell2 ] express that the cell size decreases inversely as a function of flow stress , a well known experimental observation that is refered to as `` similitude '' @xcite . although similitudity is generally confirmed by experiment , some experimental work does present a more complicated picture . early tensile / tem work on tapered cu single crystals finds an initially broad distribution of cell sizes that narrows and shifts to small lengths with increasing flow stress @xcite . this result suggests that a single structural length - scale might not always be a good statistical description of the evolving micro - structure . indeed , more modern viewpoints , in which dislocation structure evolution is a non - equilibrium process @xcite , tend to suggest a distribution of emerging length - scales leading to a scale - free fractal - like structure . although such micro - structures have been quantitatively established by tem investigations of latter - stage hardened single crystals of cu @xcite , their existence is not universal , depending strongly on material type and deformation history . the current work does not address this aspect . more general forms of an inhomogeneous internal stress field that capture such scale - free micro - structural features can be envisioned ; a direction which will be investigated in future work . whilst the dipolar mat geometry in an external field offers a platform with which to study the depinning transition and more generally the transition to extended plastic flow , when comparing to experiment , careful consideration has to be given to its regime of applicability . to do this , a typical simulation of secs . [ sec_results ] and [ sec_statistics ] is now broadly summarised . upon choosing numerical values for all model parameters , the @xmath82 dislocations are introduced to the system via a distribution of random positions . this unstable configuration is then relaxed to a local minimum energy in which the forces on each dislocation are below a small threshold value . the deformation simulation is then begun using one of the three loading modes of sec . [ sec_loading ] . as the stress increases , intermittent plasticity increasingly occurs until a stress is reached at which extended and overlapping strain events occur , which in the previous sections has loosely been referred to as the plastic flow regime . it is important to emphasise that no attempt has been made to obtain the global energy minimum of the starting configuration . such an initial state turns out to play a crucial role in the observed properties of the model , since many high energy configurations will exist , and it is these that dominate the early stages of plasticity . as a deformation simulation proceeds , such high energy configurations structurally transform eventually leading to a plastic flow regime and often to the homogenisation of the dislocation configuration . in other words , the extended plastic flow regime should be considered to be outside the applicability regime of the present model when a comparison to experiment is made , or equivalently , the present model is only suitable for the study of the micro plastic regime of the stress - strain curve . the rational behind the use of an initial high - energy dislocation configuration originates from the assumption that the explicit dislocation population of the model represents only the mobile dislocation network , which constitutes only a small part of the true population . thus , in the same way as @xmath11 and @xmath12 characterise the sample preparation or deformation history of the model material , so does the initial high energy ( explicit ) mobile dislocation content . this is quite compatible from the perspective of soc in which the dislocation structure reaches a critical configuration that is far from equilibrium , and that structural rearrangements correspond to the system transforming from one soc state to another . by construction , that part mediating the structural transformation will be the current mobile dislocation content . the central simplification of the present model , is that it separates the mobile and immobile populations , associating the former to an explicit mobile dislocation content that represents the non - equilibrium component of the network , and relegating the latter to an effective static internal stress field . that this internal stress field is unchanging and that the same explicit mobile dislocation population exists as a function of strain for the entire deformation simulation , is of course different from a real material , where the structure evolves with strain , and at any particular non - negligible strain interval , quite different dislocations might constitute the mobile dislocation population . this again emphasises that the present model should only be applied to the micro plastic regime , where significant structural evolution is minimal . experimental evidence for a lack of structural evolution in the mico - plastic regime is best seen in low amplitude cyclic deformation experiments of fcc metals , in which the plastic strain per cycle can be as low as @xmath115 leading to significant changes in load stress and internal length scale only after the occurrence of several tens - to - hundreds of thousands of cycles @xcite . it is further noted , that documented experimental studies on the micro - plasticity at room temperature primarily report on movements of edge or non - screw type dislocations , whereas a clear increase in dislocation density or the formation of dislocation structures as a result of multiplication remains absent @xcite . in the bulk case there are exceptions to this trend where in the case of a work - hardened al - mg alloy which exhibits dynamic strain ageing , emerging structural length - scales were already detected in the micro - plastic regime @xcite using high - resolution extensometry methods @xcite . the results of sec . [ sec_statistics_dist ] demonstrate that the developed model exhibits power - law behaviour in the distribution of strain burst magnitudes , and thus the scale - free avalanche phenomenon seen in experiments , either via the stress - strain curve of micro - compression tests @xcite or via in situ acoustic emission experiments @xcite , and in simulation , via two or three dimensional dislocation dynamics simulations in which the entire network is represented by an explicit dislocation population and individual dislocation reaction mechanisms are taken into account . with an exponent of approximately -1.96 , the model gives a value that is somewhat higher than that seen in simple metals @xcite and ice @xcite , but more comparable to that seen in lif crystals @xcite . that such a simple model can admit scale - free behaviour , is connected to the dependence of the intermittent plasticity on the extremal configurations of the explicit dislocation population . this was directly seen in the statistics of the first - burst shear - stress and also the distribution of strain burst magnitudes , where with a large enough increase of @xmath12 ( say from 2 @xmath59 m to 10 @xmath59 m ) the first burst statistics changes from being dominated by extreme value statistics to that being dominated by the statistics of the most probable ( fig . [ first_burst_fig]b ) corresponding to an increased presence of cut - off effects in the statistics of strain burst magnitudes ( fig . [ strain_burst_dist1_fig]a ) . this is a natural result of the observation that quantities that depend on extreme value statistics can exhibit power - law behaviour in their distributions , emphasising a connection with soc that is related to only the mobile dislocation population being in a non - equilibrium state and not to the characteristics of the present simplified immobile dislocation network a manifestation of a scenario referred to as `` nearly critical '' or `` robust critical '' @xcite .
|
here we present a model to study the micro - plastic regime of a stress - strain curve . in this model these model parameters , along with the mobile dislocation density , are found to admit a diversity of micro - plastic behaviour involving intermittent plasticity in the form of a scale - free avalanche phenomenon , with an exponent and scaling - collapse for the strain burst magnitude distribution that is in agreement with mean - field theory and similar to that seen in experiment and more complex dislocation dynamics simulations .
|
here we present a model to study the micro - plastic regime of a stress - strain curve . in this model an explicit dislocation population represents the mobile dislocation content and an internal shear - stress field represents a mean - field description of the immobile dislocation content . the mobile dislocations are constrained to a simple dipolar mat geometry and modelled via a dislocation dynamics algorithm , whilst the shear - stress field is chosen to be a sinusoidal function of distance along the mat direction . the sinusoidal function , defined by a periodic length and a shear - stress amplitude , is interpreted to represent a pre - existing micro - structure . these model parameters , along with the mobile dislocation density , are found to admit a diversity of micro - plastic behaviour involving intermittent plasticity in the form of a scale - free avalanche phenomenon , with an exponent and scaling - collapse for the strain burst magnitude distribution that is in agreement with mean - field theory and similar to that seen in experiment and more complex dislocation dynamics simulations .
|
1205.1486
|
c
|
a simplified two dimensional dislocation modelling framework has been introduced in which the explicit interacting dislocation population , constrained to a simple dipolar mat geometry , represents only the mobile dislocation density component of the total dislocation density , and the much larger immobile dislocation population is described by a static internal sinusoidal shear - stress field defined by an internal shear - stress amplitude and wavelength . these model parameters , along with the initial non - equilibrium explicit mobile dislocation content characterise either the deformation or sample preparation history of the model material . because of the static nature of the internal field and the lack of dislocation - dislocation reactions , upon loading , the present model is restricted to the micro - plastic region of the stress - strain curve , and therefore to a deformation regime for a given material that involves negligible structural evolution . despite the simplicity of the model and the restriction to the micro - plastic regime , the deformation behaviour exhibits a rich variety of properties as a function of the model parameters . in particular , intermittent plasticity is observed whose strain burst magnitude distribution exhibits scale - free avalanche behaviour .
|
an explicit dislocation population represents the mobile dislocation content and an internal shear - stress field represents a mean - field description of the immobile dislocation content . the mobile dislocations are constrained to a simple dipolar mat geometry and modelled via a dislocation dynamics algorithm , whilst the shear - stress field is chosen to be a sinusoidal function of distance along the mat direction .
|
here we present a model to study the micro - plastic regime of a stress - strain curve . in this model an explicit dislocation population represents the mobile dislocation content and an internal shear - stress field represents a mean - field description of the immobile dislocation content . the mobile dislocations are constrained to a simple dipolar mat geometry and modelled via a dislocation dynamics algorithm , whilst the shear - stress field is chosen to be a sinusoidal function of distance along the mat direction . the sinusoidal function , defined by a periodic length and a shear - stress amplitude , is interpreted to represent a pre - existing micro - structure . these model parameters , along with the mobile dislocation density , are found to admit a diversity of micro - plastic behaviour involving intermittent plasticity in the form of a scale - free avalanche phenomenon , with an exponent and scaling - collapse for the strain burst magnitude distribution that is in agreement with mean - field theory and similar to that seen in experiment and more complex dislocation dynamics simulations .
|
1510.03697
|
c
|
in this paper , we have recognized a supersymmetry in the structure of hamiltonians that energetically impose local constraints on their ground states . this allowed us to formally connect recent development in the theory of phonons to a wider range of systems including frustrated and unfrustrated magnets . the connection ultimately arose from the similar role of the vanishing extension of springs @xmath15 of phonon ground states to the vanishing total spin on a bond in unfrustrated magnets and vanishing total spin on a triangle , square , tetrahedron or other local simplex in frustrated magnetic systems . it also highlights the role of dirac s constraint matrix as a central pillar of these recent developments as well as proves that a topological index exists in all these systems . lets end our discussion of these results with some final comments . remarkably , this supersymmetry identifies maxwell s counting of degrees of freedom @xmath338 , where @xmath339 is the total number of degrees of freedom and @xmath340 is the total number of local constraints imposed by the hamiltonian ( i.e. possibly a redundant set ) , to the witten index @xmath110 associated with the supersymmetry . this maxwell counting , though long used for balls and springs and other mechanical structures , was proposed by moessner and chalker as a means of characterizing highly frustrated antiferromagnets@xcite . here we have connected their work with the actual number of zero modes through recognizing it as the topological index . lattice where dark blue are the mn atoms with magnetic moments and yellow atoms are the ir atoms without magnetic moments . one view of the lattice is through the 111 planes for the blue magnetic atoms . they form a kagome lattice . another view is achieved by considering how each triangle shares a side with another triangle in the octahedron . this view suggests irmn@xmath2 is composed of side sharing triangles in 3 dimensions and is therefore more similar to the triangular lattice antiferromagnet than the kagome antiferromagnet . ] the use of analogs of maxwell constraints in magnetism and the associated topological index provides a simple picture of the strength of magnetism in a system . to see this in practice , consider the question of the origin of the industrial strength of antiferromagnetism in irmn@xmath2 . this material magnetically orders in a 120@xmath341 coplanar state at 960 k@xcite but is composed of kagome planes with a dominant isotropic heisenberg spin exchange@xcite that would appear to frustrated the magnetism@xcite . how then can the maxwell - like constraints advanced by this paper explain such industrial strength magnetism ? the answer lies in the many different kagome planes . they all interconnect as shown in fig . [ fig : irmn3 ] . indeed , the lattice is better thought of as side - sharing triangles in three dimensions instead of layers of kagome planes . with eight triangles in the unit cell , the dominant heisenberg term in its hamiltonian can be written @xmath342 where @xmath343 is the total spin on each triangle . we therefore have a topological index @xmath344 ( since there are 8/3 triangles per spin ) . we can contrast this to say an unfrustrated antiferromagnet on the cubic lattice with three bonds in the unit cell and @xmath345 . but upon closer inspection , we should be careful with such a comparison . in irmn@xmath2 , nearest neighbor heisenberg exchange still has a sub - extensive number of zero modes because the side sharing triangles form corner sharing octehedra . so its @xmath346 is likely lower than the unfrustrated antiferromagnet with the same @xmath347 . nevertheless , this effect is subextensive so it is unlikely to be a big effect . therefore , the energetic constraints imposed by the dominant heisenberg exchange in irmn@xmath2 appear to be nearly as redundant as an unfrustrated antiferromagnet on a cubic lattice suggesting great industrial strength even though it is composed of kagome planes . but not only firm magnetism may prove useful for practical applications . fragile isostatic magnetism may prove to be easily manipulated for special design purposes . it may play a role for magnetic systems similar to the role isostatic phonons have played in the design of metameterials . due to the absence of redundant constraints , isostatic lattices can have particular weaknesses that can enable their manipulation . perhaps even metamaterials designs that exploit this could carry over to magnetic materials . for example , one lattice was designed to have selective mechanical failure in ref . where just a local set of bonds associated with a state of self stress broke down under a load . the magnetic analog of this could correspond to a material design with a selective spin flip of a local subset of spins achieved with a global magnetic field . in this way , isostatic magnetism could prove to have an important role in the design of magnetic materials for applications . and given packing fraction @xmath348 . here the isostatic point highlighted by the green dot at the top right occurs for frictionless grains . b ) the phase diagram for quantum disordering a frustrated antiferromagnet obtained in a semi - classical field theory@xcite . the x - axis is a dimensionless temperature and the y - axis a coupling measuring the strength of quantum fluctuations . c ) classical phase diagram of the loss of magnetic order based on the results of this paper . here again the green dot represents the isostatic point . notice in all three cases , a special point exists in the phase diagram that sheds light on the rest of the phase diagram . it would be interesting to understand the relationship between these three points.,title="fig : " ] and given packing fraction @xmath348 . here the isostatic point highlighted by the green dot at the top right occurs for frictionless grains . b ) the phase diagram for quantum disordering a frustrated antiferromagnet obtained in a semi - classical field theory@xcite . the x - axis is a dimensionless temperature and the y - axis a coupling measuring the strength of quantum fluctuations . c ) classical phase diagram of the loss of magnetic order based on the results of this paper . here again the green dot represents the isostatic point . notice in all three cases , a special point exists in the phase diagram that sheds light on the rest of the phase diagram . it would be interesting to understand the relationship between these three points.,title="fig : " ] and given packing fraction @xmath348 . here the isostatic point highlighted by the green dot at the top right occurs for frictionless grains . b ) the phase diagram for quantum disordering a frustrated antiferromagnet obtained in a semi - classical field theory@xcite . the x - axis is a dimensionless temperature and the y - axis a coupling measuring the strength of quantum fluctuations . c ) classical phase diagram of the loss of magnetic order based on the results of this paper . here again the green dot represents the isostatic point . notice in all three cases , a special point exists in the phase diagram that sheds light on the rest of the phase diagram . it would be interesting to understand the relationship between these three points.,title="fig : " ] but isostatic magnetism may also extend our theoretical understanding of magnetic materials . there is likely a separate topological property not discussed in this paper . since the magninos are free fermions , and the ideal isostatic magnetism state is a gapped state , it must fit into our categorization of topological band theory@xcite much like the phoninos where shown to have a topological band theory by kane and lubensky ( though they did nt call them phoninos ) . it is the supersymmetry established by this paper that enables physics that applies to free fermions ( including majorana fermions ) to also apply to phonons and magnons . this is one of the benefits of generalizing kane and lubensky s theory to a broader class of systems . the results of this paper may also shed light on a long standing problem in the field of magnetism beyond band theory : how to melt magnetic order . the connection established by our formal developments shows explicitly that the loss of rigidity in a solid@xcite , the jamming transition@xcite and the loss of magnetic order due to frustration[ramirez1994,moessner1998a , moessner1998b ] may all be different aspects of the same problem ( see fig . [ fig : jamming ] ) . each of these systems are built around constraints , have a topological index @xmath110 , and have isostatic order with @xmath47 that we curiously identify here with spontaneous susy breaking . so from this perspective , isostatic magnetism is at a critical point between magnetic order and a classical spin liquid . it would be interesting to understand how this critical point is related to the quantum melting of antiferromagnetic order through a quantum critical point@xcite particularly given theories of this quantum phase transition that are semi - classical in nature . * practical use in the design of magnetic materials via the topological index , zero modes and self field modes * a novel application of topological band theory of fermions to magnons * insight into the mechanism of quantum spin liquids and other exotic phases of magnetism . 47ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty http://dx.doi.org/10.1038/nphys2835 10.1038/nphys2835 http://www.nature.com/nphys/journal/v10/n1/abs/nphys2835.html{#}supplementary-information [ * * , ( ) ] link:\doibase 10.1016/0020 - 7683(78)90052 - 5 [ * * , ( ) ] link:\doibase 10.1088/1367 - 2630/15/4/043043 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.103.248101 [ * * , ( ) ] link:\doibase 10.1103/physrevx.5.031011 [ * * , ( ) ] link:\doibase 10.1038/nphoton.2014.248 [ * * , ( ) ] , link:\doibase 10.1038/nphys3228 [ * * , ( ) ] link:\doibase 10.1103/physrevb.87.144101 [ * * , ( ) ] link:\doibase 10.1103/physrevb.89.134409 [ * * , ( ) ] link:\doibase 10.1038/nphys3232 [ * * , ( ) ] http://arxiv.org/abs/1503.01324 [ , ( ) ] , link:\doibase 10.1073/pnas.1405969111 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.116.135501 [ * * , ( ) ] link:\doibase 10.1038/nature08917 [ * * , ( ) ] http://article.pubs.nrc-cnrc.gc.ca/ppv/rpviewdoc?issn=1208-6045{&}volume=79{&}issue=11-12{&}startpage=1283{&}ab=y file:///users / mlawler / dropbox / documents / papers/2001/moessner / canadian journal of physics 2001 moessner.pdf [ * * , ( ) ] http://projecteuclid.org/dpubs?service=ui{&}version=1.0{&}verb=display{&}handle=euclid.jdg/1214437492$\backslash$nhttp://www.math.toronto.edu/mgualt/morse theory / witten morse theory and supersymmetry.pdf [ * * , ( ) ] http://arxiv.org/abs/1407.2890 [ , ( ) ] , link:\doibase 10.1126/science.1248253 [ * * , ( ) ] @noop link:\doibase 10.1098/rspa.1958.0141 [ * * , ( ) ] http://link.aps.org/doi/10.1103/physrevlett.80.2929 papers://ab7c6739 - 3c9b-4bbe-8ce8 - 067dfd022c15/paper / p6318 file:///users / mlawler / dropbox / documents / papers/1998/moessner / physical review letters 1998 moessner.pdf [ * * , ( ) ] link:\doibase 10.1007/bf01325811 [ * * , ( ) ] link:\doibase 10.1007/bf02729860 [ * * , ( ) ] link:\doibase 10.1016/0003 - 4916(77)90335 - 9 [ * * , ( ) ] link:\doibase 10.1143/ptp.63.599 [ * * , ( ) ] link:\doibase 10.1016/0370 - 1573(94)00080-m [ * * , ( ) ] , @noop _ _ ( , ) link:\doibase 10.1103/physrevlett.116.135503 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.115131 [ * * , ( ) ] link:\doibase 10.1103/physrevb.86.125119 [ * * , ( ) ] link:\doibase 10.1103/physrevb.90.115141 [ * * , ( ) ] link:\doibase 10.1007/bf01197577 [ * * , ] @noop _ _ ( , ) link:\doibase 10.1103/physrevb.58.12049 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.99.137207 [ * * , ( ) ] link:\doibase 10.1098/rspa.1957.0133 [ * * , ( ) ] link:\doibase 10.1038/nphys3185 [ * * , ( ) ] @noop * * ( ) link:\doibase 10.1103/physrevb.79.020403 [ * * , ( ) ] link:\doibase 10.1038/nature06981 [ * * , ( ) ] link:\doibase 10.1103/physrevb.39.2344 [ * * , ( ) ] @noop * * ( ) \doibase http://dx.doi.org/10.1016/s0022-5096(02)00107-2 [ * * , ( ) ] \doibase http://dx.doi.org/10.1016/j.jmps.2005.10.008 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.81.1841 [ * * , ( ) ] link:\doibase 10.1038/23819 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.61.1029 [ * * , ( ) ]
|
it is natural to wonder if this striking effect is more general and they conclude their study with `` finally , it will be interesting to explore connections with theories of frustrated magnetism . '' this paper s goal is to make this connection and use it to generalize kl theory beyond phonons .
|
i generalize the theory of phonon topological band structures of isostatic lattices to frustrated antiferromagnets . i achieve this with a discovery of a many - body supersymmetry ( susy ) in the phonon problem of balls and springs and its connection to local constraints satisfied by ground states . the witten index of the susy model demands the maxwell - calladine index of mechanical structures . `` spontaneous supersymmetry breaking '' is identified as the need to gap all modes in the bulk to create the topological isostatic lattice state . since ground states of magnetic systems also satisfy local constraint conditions ( such as the vanishing of the total spin on a triangle ) i identify a similar susy structure for many common models of antiferromagnets including the square , triangluar , kagome , pyrochlore nearest neighbor antiferromagnets , and the square lattice antiferromagnet . remarkably , the kagome family of antiferromagnets is the analog of topological isostatic lattices among this collections of models . thus , a solid state realization of the theory of phonon topological band structure may be found in frustrated magnetic materials . recently , kane and lubensky ( kl ) identified topological properties of isostatic lattice phonons . they achieved this by discovering a topological index governing mechanical structures by building on calladine s work and further utilizing a dirac - like square - root of the phonon equations of motion , a problem to which they could apply the theory of topological insulators . remarkably , they showed the existence of lattices with gapped phonons for periodic boundary conditions that must have gapless phonons with open boundary conditions . it is natural to wonder if this striking effect is more general and they conclude their study with `` finally , it will be interesting to explore connections with theories of frustrated magnetism . '' where reference 48 is my study identifying topological gauge dynamics of the zero modes of classical kagome antiferromagnets . this paper s goal is to make this connection and use it to generalize kl theory beyond phonons . the kl theory of isostatic lattices is a different branch of topological phases from the theory of topological band insulators . following the original discovery of topological insulators , topological properties of boson band structures have been studied for a wide variety of systems including phonons , photons , acoustic phononic crystals and magnons . these systems achieve their topological properties in the presence of time reversal symmetry breaking and are built directly from the physics of the integer quantum hall effect . in contrast , kl s theory of isostatic phonons is time reversal symmetric and `` purely geometric in nature'' . a connection to the integer quantum hall effect is made only after a dirac - like square - rooting procedure of the equations of motion . it therefore presents a new direction in the theory of topological phases . remarkably , though its may not apply directly to solid state phonons because they are mechanically stable , the kl theory has already seen a variety of applications due to its insight into the general phenomena of mechanical collapse . these include some topological aspects of the jamming and rigidity percolation transitions , metamaterials made from beams and pins , and , remarkably , origami . ref . has also taken it beyond the linearized level and discovered solitons that can propagate freely in the bulk of the isostatic state . so , given the fundamental insight it provides , any extension of kl theory to a new class of systems , including extensions that go beyond the linearized limit , is likely to shed new light on those systems . in this light , frustrated magnets and/or highly frustrated magnets are a prime target for an extension of kl theory . they are magnets not only `` on the verge of collapse '' but also those that have already `` collapsed '' . here collapsing is the analog of destabilizing the magnetic ordered state into a paramagnetic state such as a quantum spin liquid or valence bond solid . a variety of materials including the organics , kagome family and pyrochlores are heavily studied for this reason . in addition , highly geometrically frustrated magnets have a form of accidental degeneracy that results from a special feature of the spin hamiltoinan . this frustration is toy - like ( fine tuned ) perhaps in a similar way that balls and springs are toy - like versions of a general theory of phonons . so if kl theory were applicable to frustrated magnets , this generalization might apply to many already realized solid state materials . in this paper , we show that the key to generalizing kl theory to other systems lies in connecting it to an abstract theoretical framework that takes the form of a many - body supersymmetric ( susy ) structure that extends the description of balls and springs . the new fermionic degrees of freedom , that i dub `` phoninos '' , are superpartners to phonons and are governed by the kl theory s square - rooted equations of motion . for linearized phonons , the two sets of degrees of freedom are decoupled . the phoninos therefore need not be real degrees of freedom but just reflect the specialness of balls and springs compared to a more general theory of phonons . i then show that the topological index identified in kl theory is demanded by the witten index of susy . a similar conclusion was reached by ref . in the continuum limit . remarkably , spontaneous susy breaking here is just the need to gap all modes in the bulk to create the topological state ( as could be the case for topological superconductors ) . finally , we discuss what protects this topology including the role of quantum effects and non - linearities . i then apply the same many - body susy construction to the case of magnons . it turns out susy is found for both unfrustrated and frustrated magnons for either case has ground states that satisfy local constraints and the susy construction is built on these constraints . remarkably , kagome magninos are governed by a hamiltonian which is the dirac s constraint matrix studied in ref . and is related through susy to the ordinary kagome magnon problem . the witten index in this case is then shown to vanish for periodic boundary conditions but not for open boundary conditions demonstrating that kagome magnons are the analog of the isostatic lattice of kl theory if we can gap all their modes without changing topology such as in a distorted kagome crystal . such a phase we call isostatic magnetism . we conclude by identifying some of the new physical phenomena predicted by these formal development : the design of magnets by exploiting weaknesses in isostatic magnetism such as through a rich set of magnetic field induced spin flip transitions that we call `` self field modes '' ; and an application of the ideas here to the origin of the industrial strength of mnir ( why it is a robust antiferromagnet ) ; the potential application of topological band theory of electrons to magninos in the isostatic magnetic phase and via susy to magnons ; the study of the loss of magnetic order via isostatic magnetism which can be viewed as a classical critical point between magnetic order and a classical spin liquid / cooperative paramagnet .
|
astro-ph0007328
|
i
|
i investigated the use of pearson s chi - square statistic [ eq . ( [ eq : x2p ] ) ] , the maximum likelihood ratio statistic for poisson distributions [ eq . ( [ eq : x2l ] ) ] , and the chi - square - gamma statistic [ eq . ( [ eq : x2 g ] ) ] for the determination of the goodness - of - fit between theoretical models and low - count poisson - distributed data . i concluded that none of these statistics should be used to determine the goodness - of - fit with data values of 10 or less . i modified pearson s chi - square statistic for the purpose of improving its goodness - of - fit performance . i demonstrated that modified pearson s @xmath0 statistic [ eq . ( [ eq : x2pm ] ) ] works well in the perfect case where one has _ a priori _ knowledge of the correct ( true ) model . in a real experiment , however , the true mean of the parent poisson distribution is rarely ( if ever ) known and model parameters must be estimated from the observations . i demonstrated that the modified pearson s @xmath0 statistic has a variance that is significantly smaller than that of the @xmath0 distribution when _ realistic _ models , defined as having parameters estimated from the observational data , are compared with poisson - distributed data . any statistic that fails with models based on reasonable parameter estimates is not a very practical statistic for the analysis of astrophysical observations . i concluded that the modified pearson s @xmath0 statistic should not be used to determine the goodness - of - fit with low - count data values of 10 or less . i modified the chi - square - gamma statistic for the purpose of improving its goodness - of - fit performance . i demonstrated that the modified chi - square - gamma statistic [ eq . ( [ eq : x2gm ] ) ] performs ( nearly ) like an ideal @xmath0 statistic for the determination of goodness - of - fit with low - count data . on average , for correct ( true ) models , the mean value of the modified chi - square - gamma statistic is equal to the number of degrees of freedom @xmath1 and its variance is @xmath2 like the @xmath0 distribution for @xmath3 degrees of freedom . an ideal @xmath0 statistic for the determination of goodness - of - fit with low - count data should _ fail _ in a predictable manner . hypothesis testing of low - count poisson - distributed data with the modified pearson s @xmath0 statistic will produce the peculiar and undesirable result that correct models are more likely to be rejected than realistic models [ cf . [ fig : x2pmt_5mu ] with fig . [ fig : x2pms_5mu ] ] . the modified chi - square - gamma statistic is a _ practical _ statistic to use for hypothesis testing of astrophysical data from counting experiments because it performs ( nearly ) like an ideal @xmath0 statistic for realistic _ and _ correct models in the low - count _ and _ the high - count data regimes ; accurate and believable probabilities for @xmath100 goodness - of - fit values can be calculated with the incomplete gamma function [ figs . [ fig:10000_cumulative_fraction ] and [ fig:10000_sorted_probability ] ] . a lot of nothing can tell you something as long as there are _ some _ observations with signal in them . vincent eke sent me an e - mail asking if i had an expression for the variance of the @xmath31 statistic which described the mysterious second hump of the solid curve of fig . 3 of . after a rapid exchange of email with me over the period of a week , he was the first to derive an analytical formula for @xmath114 [ eq . ( [ eq : x2gr_var ] ) ] . the knowledge that the variance of @xmath31 could in fact be expressed explicitly as an analytical expression turned out to be the breakthrough that i had needed in order to complete the development of the modified @xmath31 statistic . it is a pleasure to acknowledge his contribution to this research . i was supported by a grant from the national aeronautics and space administration ( nasa ) , order no . s-67046-f , which was awarded by the long - term space astrophysics program ( nra 95-oss-16 ) . this research has made use of nasa s astrophysics data system abstract service which is operated by the jet propulsion laboratory at the california institute of technology , under contract with nasa . abramowitz , m. , & stegun , i. 1964 , `` handbook of mathematical functions with formulas , graphs , and mathematical tables '' , applied mathematics series 55 , ed . m. abramowitz & i. stegun ( washington , d.c . : nbs ) 01cap [ fig : x2p_x2l_x2g_mu100 ] a simulated data set of 1000 samples ( `` observations '' ) of @xmath37 poisson deviates ( `` measurements '' ) per sample was created assuming a mean value @xmath12@xmath115@xmath116 for each poisson deviate . each sample in this data set was then analyzed using pearson s @xmath0 statistic [ _ top ; _ definition : eq . ( [ eq : x2p ] ) ] , the maximum likelihood ratio statistic for poisson distributions [ _ middle ; _ definition : eq . ( [ eq : x2l ] ) ] , and the chi - square - gamma statistic [ _ bottom ; _ definition : eq . ( [ eq : x2 g ] ) ] . the model of the @xmath16th deviate in each sample was set to the true mean value of parent poisson distribution ( i.e. , @xmath117 ) and the number of independent degrees - of - freedom was therefore equal to the number of deviates per sample ( i.e. @xmath118 ) . compare the cumulative distribution for each statistic with the cumulative distribution function of a gaussian distribution with a mean of @xmath37 and a variance of @xmath38 [ _ thick curve _ in each panel ] . the number and error shown on the right side of each panel is the mean and rms value of the 1000 samples shown in that panel ; ideally these values should be about @xmath119 . 02cap [ fig : x2l_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using the maximum likelihood ratio statistic for poisson distributions [ definition : eq . ( [ eq : x2l ] ) ] . in all cases , @xmath118 and @xmath8 was set to the true mean value of the data set . other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 03cap [ fig : x2lr_var ] reduced chi - square as a function of the true poisson mean ( @xmath121 with 10 mean values per decade ) for the maximum likelihood ratio statistic for poisson distributions with the model of the @xmath16th deviate set to the mean value of the parent poisson distribution . open squares _ show the results of the analysis of one sample composed of @xmath122 poisson deviates ( @xmath123 ) at each given poisson mean value . the _ filled squares _ show the results of the analysis of 1000 subsamples of the @xmath122 poisson deviates ( @xmath124 ) previously analyzed as one large sample . the scatter of the filled squares with respect to the open squares is real and is due to random fluctuations of the parent poisson distributions . the _ dashed line _ shows the ideal value of one . _ bottom panel : _ the variance of the reduced chi - square values shown in the top panel . the _ dashed line _ shows the ideal value of two . 04cap [ fig : x2p_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using pearson s @xmath0 statistic [ definition : eq . ( [ eq : x2p ] ) ] ( same input data set as for fig . [ fig : x2l_5mu ] ) . in all cases , @xmath118 and @xmath8 was set to the true mean value of the data set . other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 05cap [ fig : x2pr_var ] reduced chi - square as a function of the true poisson mean for pearson s @xmath0 statistic with the model of the @xmath16th deviate set to the true mean value of the parent poisson distribution ( same input data set as for fig . [ fig : x2lr_var ] ) . the _ solid line _ connecting the open squares in the _ bottom panel _ is the formula @xmath125 [ see eq . ( [ eq : x2pr_var ] ) ] . other details as in fig . [ fig : x2lr_var ] . 06cap [ fig : x2pmt_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using the modified pearson s @xmath0 statistic [ definition : eq . ( [ eq : x2pm ] ) ] ( same input data set as for fig . [ fig : x2l_5mu ] ) . in all cases , @xmath118 and @xmath8 was set to the true mean value of the data set . other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 07cap [ fig : x2pmtr_var ] reduced chi - square as a function of the true poisson mean for the modified pearson s @xmath0 statistic with the model of the @xmath16th deviate set to the true mean value of the parent poisson distribution ( same input data set as for fig . [ fig : x2lr_var ] ) . other details as in fig . [ fig : x2lr_var ] . 08cap [ fig : x2pms_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using the modified pearson s @xmath0 statistic ( same input data set as for fig . [ fig : x2l_5mu ] ) . in all cases , @xmath118 and @xmath8 was set to the _ sample mean_. other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 09cap [ fig : x2pmsr_var ] reduced chi - square as a function of the true poisson mean for the modified pearson s @xmath0 statistic with the model of the @xmath16th deviate set to the _ sample mean _ ( same input data set as for fig . [ fig : x2lr_var ] ) . other details as in fig . [ fig : x2lr_var ] . 10cap [ fig : x2gt_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using the @xmath126 statistic [ definition : eq . ( [ eq : x2 g ] ) ] ( same input data set as for fig . [ fig : x2l_5mu ] ) . in all cases , @xmath118 and @xmath8 was set to the true mean value of the data set . other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 11cap [ fig : x2gt_var ] reduced chi - square as a function of the true poisson mean for the @xmath126 statistic statistic with the model of the @xmath16th deviate set to the true mean value of the parent poisson distribution ( same input data set as for fig . [ fig : x2lr_var ] ) . the _ solid line _ connecting the open squares in the _ top panel _ is the formula @xmath127 [ eq . 29 of ] . the _ solid line _ connecting the open squares in the _ bottom panel _ is equation ( [ eq : x2gr_var ] ) . other details as in fig . [ fig : x2lr_var ] . 12cap [ fig : x2gmt_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using the modified @xmath126 statistic [ definition : eq . ( [ eq : x2gm ] ) ] ( same input data set as for fig . [ fig : x2l_5mu ] ) . in all cases , @xmath118 and @xmath8 was set to the true mean value of the data set . other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 13cap [ fig : x2gmt_var ] reduced chi - square as a function of the true poisson mean for the modified @xmath126 statistic with the model of the @xmath16th deviate set to the true mean value of the parent poisson distribution ( same input data set as for fig . [ fig : x2lr_var ] ) . other details as in fig . [ fig : x2lr_var ] . 14cap [ fig : x2gms_5mu ] the cumulative distribution functions for 1000 samples of @xmath37 poisson deviates ( _ top to bottom _ : @xmath120 , and 0.01 ) analyzed using the modified @xmath126 statistic ( same input data set as for fig . [ fig : x2l_5mu ] ) . in all cases , @xmath118 and @xmath8 was set to the _ sample mean_. other details as in fig . [ fig : x2p_x2l_x2g_mu100 ] . 15cap [ fig : x2gms_var ] reduced chi - square as a function of the true poisson mean for the modified @xmath126 statistic with the model of the @xmath16th deviate set to the _ sample mean _ ( same input data set as for fig . [ fig : x2lr_var ] ) . other details as in fig . [ fig : x2lr_var ] . 16cap [ fig : observation ] the simulated x - ray observation . a 40 photon x - ray point source is located at the @xmath72 position of @xmath73 on a background of 0.06 photons per pixel . the point spread function is @xmath128 $ ] where @xmath70 . this is the same psf used by cash ( @xcite ) but with a resolution of 100 pixels per unit area . 17cap [ fig : first_sky ] indicate pixels with a total number of photons within a radius of 10 pixels that are consistent ( within the 99.9% upper and lower single - sided confidence limits of the observed photon total ) with the estimated background flux level of 0.0747 photons per pixel . the _ x _ marks the center of the x - ray point source and the _ dotted circle _ has a radius of 10 pixels which is the maximum size of the psf . other details as in fig . [ fig : observation ] . 18cap [ fig : final_sky ] indicate the pixels with a total number of photons within a radius of 10 pixels that are consistent ( within the 99.9% upper and lower single - sided confidence limits of the observed photon total ) with the true background flux level of photons per pixel . _ dark - gray circles _ indicate the pixels with a total number of photons within a radius of 10 pixels that are consistent ( within the upper and lower single - sided confidence limits of the observed photon total ) with the model of a 40 photon point source _ at that pixel location _ on a background of 0.06 photons per pixel . other details as in fig . [ fig : first_sky ] . 19cap [ fig : green_radius8 ] indicate the pixels with a total number of photons within a radius of that are consistent ( within the 95% upper and lower single - sided confidence limits of the observed photon total ) with the model of a 40 photon point source at that pixel location on a background of 0.06 photons per pixel . all pixels within the _ solid black contour _ are within the as determined by the @xmath100 analysis of the cumulative radial distribution of the data within 10 pixels is compared with the cumulative radial distribution of a model of a 40 photon point source at that pixel location on a background of 0.06 photons per pixel . note how well the 95% confidence interval of the @xmath100 analysis of the cumulative radial distribution matches the region described by the circled pixels . other details as in fig . [ fig : final_sky ] . 20cap [ fig : crd_probability ] the photon distribution of the 317 pixels within a radius of 10 pixels of the location @xmath73 of fig . [ fig : observation ] was transformed to a cumulative radial distribution ( 10 1-pixel - wide bins @xmath99 10 degrees - of - freedom ) and then compared , using the modified chi - square - gamma statistic , with 80 models of the observation : a 1 to 80 photon ( in steps of 1 photon ) x - ray point source at @xmath73 on a background of 0.06 photons per pixel ( i.e. , the true background ) . the 95th percentage point for the chi - square distribution with 10 degrees of freedom is 18.31 [ i.e. @xmath129 . assuming that @xmath100 is distributed like @xmath0 , we see that the upper and lower single - sided 95% confidence limits for the intensity of an x - ray point source at @xmath73 in fig . [ fig : observation ] is 54.5 and 28.0 , respectively . the true intensity of the x - ray source is 40 photons . 21cap [ fig:10000_cumulative_fraction ] a data set of @xmath37 realizations of the the same model used to make fig . [ fig : observation ] was created . each sample in this data set was then analyzed using the modified chi - square - gamma statistic at the location @xmath73 the true location of the simulated x - ray point source of 40 photons on a background of 0.06 photons per pixel . all 317 pixels within a radius of 10 pixels ( the size of the psf ) were compared to the true model value at that location and the number of independent degrees - of - freedom was therefore equal to the number of pixels analyzed ( i.e. @xmath130 ) . compare the cumulative distribution with the cumulative distribution function of a gaussian distribution with a mean of @xmath131 and a variance of @xmath132 [ _ thick curve _ ] . the numbers with errors shown on the right side give the mean and rms value for the @xmath100 ( _ top _ ) and ideal @xmath0 ( _ bottom _ ) statistics . 22cap [ fig:10000_sorted_probability ] the @xmath37 simulations of fig . [ fig:10000_cumulative_fraction ] were sorted by the value of the modified chi - square - gamma statistic . the _ dark _ plot shows the probablity @xmath133 as a function of the sorted @xmath100 values . the _ gray _ plot on the bottom shows the residuals from the ideal one - to - one correspondance . the _ predicted _ probabilities for the 9000th , 9500th , and 9900th sorted simulations were 90.2069% , 94.8666% , and 98.9497% , which agrees very well with the _ expected _ probabilities of 90% , 95% , and 99% , respectively . for the entire simulation , the mean and rms value of the resiuals is @xmath134 percentage points note that the residuals never exceeds 1 percent . figures [ fig:10000_cumulative_fraction ] and [ fig:10000_sorted_probability ] indicate that the assumption that the @xmath100 statistic is distributed like the @xmath0 distribution was valid .
|
i investigate the use of pearson s chi - square statistic , the maximum likelihood ratio statistic for poisson distributions , and the chi - square - gamma statistic ( mighell 1999 , apj , 518 , 380 ) for the determination of the goodness - of - fit between theoretical models and low - count poisson - distributed data . i demonstrate that these statistics should not be used to determine the goodness - of - fit with data values of 10 or less . i modify the chi - square - gamma statistic for the purpose of improving its goodness - of - fit performance . i demonstrate that the modified chi - square - gamma statistic performs ( nearly ) like an ideal statistic for the determination of goodness - of - fit with low - count data . on average , for correct ( true ) models , the mean value of modified chi - square - gamma statistic is equal to the number of degrees of freedom and its variance is like the distribution for degrees of freedom . probabilities for modified chi - square - gamma goodness - of - fit values can be calculated with the incomplete gamma function . i give a practical demonstration showing how the modified chi - square - gamma statistic can be used in experimental astrophysics by analyzing simulated x - ray observations of a weak point source ( s / n ; 40 photons spread over 317 pixels ) on a noisy background ( 0.06 photons per pixel ) .
|
i investigate the use of pearson s chi - square statistic , the maximum likelihood ratio statistic for poisson distributions , and the chi - square - gamma statistic ( mighell 1999 , apj , 518 , 380 ) for the determination of the goodness - of - fit between theoretical models and low - count poisson - distributed data . i demonstrate that these statistics should not be used to determine the goodness - of - fit with data values of 10 or less . i modify the chi - square - gamma statistic for the purpose of improving its goodness - of - fit performance . i demonstrate that the modified chi - square - gamma statistic performs ( nearly ) like an ideal statistic for the determination of goodness - of - fit with low - count data . on average , for correct ( true ) models , the mean value of modified chi - square - gamma statistic is equal to the number of degrees of freedom and its variance is like the distribution for degrees of freedom . probabilities for modified chi - square - gamma goodness - of - fit values can be calculated with the incomplete gamma function . i give a practical demonstration showing how the modified chi - square - gamma statistic can be used in experimental astrophysics by analyzing simulated x - ray observations of a weak point source ( s / n ; 40 photons spread over 317 pixels ) on a noisy background ( 0.06 photons per pixel ) . accurate estimates ( 95% confidence intervals / limits ) of the location and intensity of the x - ray point source are determined . # 1#1 # 1 # 1
|
0802.1113
|
i
|
in 1998 , blaze , bleumer and strauss @xcite proposed a cryptographic primitive where a semi - trusted proxy is given some information that allows turning s signature on a message into bob s signature on the same message . these _ proxy re - signatures _ ( prs ) not to be confused with proxy signatures @xcite require that the proxy be unable to sign on behalf of alice or bob on its own . the last few years saw a renewed interest in proxy re - cryptography @xcite . + this paper presents the first constructions of _ multi - use _ _ unidirectional _ proxy re - signature wherein the proxy can only translate signatures in one direction and messages can be re - signed a polynomial number of times . our constructions are efficient and demand new ( but falsifiable ) diffie - hellman - related intractability assumptions in bilinear map groups . one of our contributions is a secure scheme in the standard model ( _ i.e. _ without resorting to the random oracle model ) . related work . the delegator can easily designate a proxy translating signatures computed using bob s secret key the delegatee into one that are valid w.r.t . her public key by storing her secret key at the proxy . upon receiving bob s signatures , the proxy can check them and re - sign the message using alice s private key . the problem with this approach is that the proxy can sign arbitrary messages on behalf of alice . proxy re - signatures aim at securely enabling the delegation of signatures without fully trusting the proxy . they are related to proxy signatures , introduced in @xcite and revisted in @xcite , in that any prs can be used to implement a proxy signature mechanism but the converse is not necessarily true . + in 1998 , blaze _ @xcite gave the first example of prs where signing keys remain hidden from the proxy . the primitive was formalized in 2005 by ateniese and hohenberger @xcite who pinned down useful properties that can be expected from proxy re - signature schemes . + 1 . re - signature keys can only be used for delegation in one direction ; 2 . a message can be re - signed a polynomial number of times ; 3 . re - signature keys can be kept secret by an honest proxy ; 4 . a user may not even know that a proxy exists ; 5 . a re - signature can not be linked to the one from which it was generated ; 6 . a user is only required to store a constant amount of secret data ; 7 . the delegatee does not act in the delegation process ; 8 . the proxy can not re - delegate signing rights ; [ fig : properties ] blaze _ et al . _ s construction is _ bidirectional _ ( _ i.e. _ the proxy information allows `` translating '' signatures in either direction ) and _ multi - use _ ( _ i.e. _ the translation of signatures can be performed in sequence and multiple times by distinct proxies without requiring the intervention of signing entities ) . unfortunately , ateniese and hohenberger @xcite pinpointed a flaw in the latter scheme : given a signature / re - signature pair , anyone can deduce the re - signature key that has been used in the delegation ( _ i.e. _ the _ private proxy _ property is not satisfied ) . another issue in @xcite is that the proxy and the delegatee can collude to expose the delegator s secret . + to overcome these limitations , ateniese and hohenberger proposed two constructions based on bilinear maps . the first one is a quite simple multi - use , bidirectional protocol built on boneh - lynn - shacham ( bls ) signatures @xcite . their second scheme is unidirectional ( the design of such a scheme was an open problem raised in @xcite ) but single - use . it involves two different signature algorithms : _ first - level _ signatures can be translated by the proxy whilst _ second - level _ signatures can not . a slightly less efficient variant was also suggested to ensure the privacy of re - signature keys kept at the proxy . the security of all schemes was analyzed in the random oracle model @xcite . + our contributions . ateniese and hohenberger left as open challenges the design of multi - use unidirectional systems and that of secure schemes in the standard security model . the present paper solves both problems : * we present a simple and efficient system ( built on the short signature put forth by boneh _ @xcite ) which is secure in the random oracle model under a reasonable extension of the diffie - hellman assumption ; * using an elegant technique due to waters @xcite , the scheme is easily modified so as to achieve security in the standard model . to the best of our knowledge , this actually provides the first unidirectional prs that dispenses with random oracles and thereby improves a recent bidirectional construction @xcite . both proposals additionally preserve the privacy of proxy keys ( with an improved efficiency w.r.t . @xcite in the case of the first one ) . they combine almost all of the above properties . as in prior unidirectional schemes , proxies are not completely transparent since signatures have different shapes and lengths across successive levels . the size of our signatures actually grows linearly with the number of past translations : signatures at level @xmath1 ( i.e. that have been translated @xmath2 times if the original version was signed at level @xmath3 ) consist of about @xmath4 group elements . in spite of this blow - up , we retain important benefits : * signers may want to tolerate a limited number ( say @xmath5 ) of signature translations for specific messages . then , if at most @xmath6 translations are permitted in the global system , users can directly generate a signature at level @xmath7 . * the conversion of a @xmath8 level signature is indistinguishable from one generated at level @xmath9 by the second signer . the original signer s identity is moreover perfectly hidden and the verifier only needs the new signer s public key . the simplicity of our schemes makes them attractive for applications that motivated the search for multi - use unidirectional systems in @xcite . one of them was to provide a proof that a certain path was taken in a directed graph : for instance , u.s . customs only need one public key ( the one of the immigration agent who previously validated a signature on an e - passport ) to make sure that a foreign visitor legally entered the country and went through the required checkpoints . another application was the conversion of certificates where valid signatures for untrusted public keys can be turned into signatures that verify under trusted keys . as exemplified in @xcite , unidirectional schemes are quite appealing for converting certificates between _ ad - hoc _ networks : using the public key of network b s certification authority ( ca ) , the ca of network a can non - interactively compute a translation key and set up a proxy converting certificates from network b within its own domain without having to rely on untrusted nodes of b. + roadmap . in the forthcoming sections , we recall the syntax of unidirectional prs schemes and the security model in section [ sec : model ] . section [ sec : bilinear ] explains which algorithmic assumptions we need . section [ bls - version ] describes our random - oracle - using scheme . section [ waters - version ] details how to get rid of the random oracle idealization .
|
in 1998 , blaze , bleumer , and strauss suggested a cryptographic primitive named _ proxy re - signatures _ where a proxy turns a signature computed under alice s secret key into one from bob on the same message . the semi - trusted proxy does not learn either party s signing key and can not sign arbitrary messages on behalf of alice or bob . at ccs 2005 , ateniese and hohenberger nonetheless , they left open the problem of designing a _ multi - use unidirectional _ scheme where the proxy is able to translate in only one direction and signatures can be re - translated several times . + this paper solves this problem , suggested for the first time years ago , and shows the first _ multi - hop _ _ unidirectional _ proxy re - signature schemes . we describe a random - oracle - using system that is secure in the ateniese - hohenberger model . the same technique also yields a similar construction in the _ standard model _ ( i.e. without relying on random oracles ) . both schemes are efficient and require newly defined but falsifiable diffie - hellman - like assumptions in bilinear groups . * keywords . * multi - use proxy re - signatures , unidirectionality , pairings .
|
in 1998 , blaze , bleumer , and strauss suggested a cryptographic primitive named _ proxy re - signatures _ where a proxy turns a signature computed under alice s secret key into one from bob on the same message . the semi - trusted proxy does not learn either party s signing key and can not sign arbitrary messages on behalf of alice or bob . at ccs 2005 , ateniese and hohenberger revisited the primitive by providing appropriate security definitions and efficient constructions in the random oracle model . nonetheless , they left open the problem of designing a _ multi - use unidirectional _ scheme where the proxy is able to translate in only one direction and signatures can be re - translated several times . + this paper solves this problem , suggested for the first time years ago , and shows the first _ multi - hop _ _ unidirectional _ proxy re - signature schemes . we describe a random - oracle - using system that is secure in the ateniese - hohenberger model . the same technique also yields a similar construction in the _ standard model _ ( i.e. without relying on random oracles ) . both schemes are efficient and require newly defined but falsifiable diffie - hellman - like assumptions in bilinear groups . * keywords . * multi - use proxy re - signatures , unidirectionality , pairings .
|
cond-mat0302503
|
i
|
the presence in a solid of randomly - distributed imperfections causes its electrons to propagate diffusively rather than ballistically . as a result , when two electrons interact , they tarry in each other s vicinity , and become more strongly correlated @xcite . as shown by finkelstein @xcite and castellani , _ et al . _ @xcite , this effect leads to an instability of the diffusive fermi liquid fixed point ( see also refs . ) which manifests itself through divergent corrections to a variety of physical quantities . the instability exists for arbitrarily weak coupling and for arbitrarily weak disorder . its simplest incarnation is in an electron system in a weak magnetic field @xmath1 which suppresses quantum interference effects and relegates them to subleading order in the dimensionless resistance@xcite @xmath2 ( measured in units of @xmath3 ) while interactions make a contribution at leading order . since the relevant interaction is in the spin - triplet particle - hole channel , the @xmath4 factor must be tuned to zero if @xmath5 so that there is no zeeman splitting , which would cut off the divergences . though this instability was identitified nearly 20 years ago , there is , at present , no consensus on its outcome . the recent discovery of evidence for a metal - insulator transition in two - dimensional electron gases ( 2degs ) in si - mosfets @xcite , gaas @xcite heterostructures , and si - ge @xcite has led to a re - examination of the problem of interacting electrons in a random potential . the interesting phenomena seen in these experiments occur in a regime in which interactions are strong ( large @xmath6 ) and quantum interference effects are likely to be important because the conductivity is on the order of the quantum limit @xmath7 and @xmath8 . though a number of interesting theories @xcite have been proposed , none have provided a complete explanation of the experimental data , which is nt surprising , given that the presumably simpler problem of weakly - interacting electrons in a weak random potential with @xmath5 , @xmath9 remains unsolved . it is this simpler problem which is the subject of this paper though it is unresolved , the instability caused by the triplet interaction is , in fact , a crime with only one plausible suspect : ferromagnetism . since the diffusive fermi liquid is unstable , the system must settle into some other phase at low temperature ; since the instability is in the spin - triplet channel , it is presumably one in which the spins are ordered . the basic interaction in this problem is exchange , which is ferromagnetic . there is no available mechanism which generates antiferromagnetic interactions . to see this , note that the instability occurs when the conductance is large , while superexchange occurs in or near an insulating state ( since it arises when the kinetic energy is treated as a perturbation ) there would be rkky interactions between localized spins , such as magnetic impurities , if we had any , but we do nt . in the absence of any antiferromagnetic interactions , we rule out spin density - waves ( sdws)@xcite , spin glass order , and random singlet phases@xcite . the latter two furthermore require large effective disorder while the instability of interest here is manifest for small effective disorder@xcite . it has been suggested that the instability of the triplet interaction signals the formation of ` local moments'@xcite . in the weak - interaction , weak - disorder limit , however , the instability of the diffusive fermi liquid only develops at large length scales , so the resulting magnetic moments are anything but ` local ' . local moments might be expected when the energy cost for flipping a spin at short length scales is very small . this might occur near the stoner transition of the clean system @xcite , where the critical susceptibility , @xmath10 , is of curie form for a mean - field exponent @xmath11 . there is also a substantial local moment regime with curie susceptibility in the large-@xmath12 hubbard model ( because of the weakness of superexchange for large @xmath12 ) and doped semiconductors ( because of competing ferromagnetic and antiferromagnetic interactions ) . however , these scenarios are not relevant to the presumably simpler , though still non - trivial , problem of the weakly - interacting , weakly - disordered limit . this leaves us with one remaining possibility and , indeed , a number of authors have proposed that the singularity of the triplet interaction channel leads to ferromagnetism . the basic underlying physics is simply that diffusing electrons interact more strongly because they move more slowly and , as a result , the interaction energy gained by ( at least partially ) polarizing the system _ always _ outweighs the single - particle energy cost , in contrast to a clean system , where the interaction energy gain is a constant @xmath12 which must exceed the single - particle energy cost @xmath13 , i.e. the stoner criterion ( @xmath14 is the density of states at the fermi energy ) . andreev and kamenev @xcite proposed such a mechanism for finite - size systems by computing the disorder enhancement of the interaction matrix element for a state with two electrons in single - particle states separated by small energies . heuristically , one can say that the two wavefunctions are concentrated in the same minima of the potential . chamon and mucciolo @xcite found a ferromagnetic saddle - point solution of the one - loop effective action of finkelstein s theory . their calculation generalizes stoner s by including quantum corrections resulting from interactions between diffusion modes . belitz and kirkpatrick @xcite advanced the same hypothesis when they noted that the phase transition which they expected to be driven by the growth of the triplet interaction is of the same form as a transition to ferromagnetism in a disordered system . in this paper , we calculate the effective action for the magetization in a weakly - interacting electron system in a slightly dirty metal . the effective action is calculated to lowest order in the dimensionless resistance @xmath2 . it exhibits the physics described above and is clearly unstable to the formation of a ferromagnetic moment . in principle , the emergence of some other type of order could intervene ( this effect would have to be non - perturbative in @xmath2 ) before ferromagnetism develops , but the arguments given above rule out this possibility . thus , we conclude that in the limit of small @xmath2 , the singularity of the spin - triplet interaction channel leads to ferromagnetism . our results substantiate the conclusions of previous authors @xcite . the basic philosophy is simple : since the rg equations tell us that the interaction in the triplet channel grows in importance as the energy scale is lowered , we try to satisfy this term in the action first by allowing the magnetization , @xmath15 , to develop a non - zero expectation value . in order to do this , we compute an effective action @xmath16 $ ] for the magnetization @xmath15 taking finkelstein s ` non - linear @xmath17-model ' for interacting electrons in a disordered system as the point of departure . since our goal is to calculate an effective action @xmath16 $ ] to lowest order in the resistance @xmath2 , finkelstein s @xmath17-model is merely a convenient way of organizing the pertinent diagrams . we believe that it is intuitively appealing to follow finkelstein and work directly with a theory which retains only diffusion modes and drops the superfluous high - energy electronic degrees of freedom , but this is a matter of convenience , and it is a option which we can take regardless of whether new terms are generated at the two - loop or higher level ( i.e. regardless of whether the non - linear terms in the theory are really related by symmetry , as they would be in a @xmath17-model ) . our derivation emphasizes the similarity between , on the one hand , the rg equation for the triplet - channel interaction in a dirty metal and the consequent development of ferromagnetism with , on the other hand , the rg equation for the cooper - channel interaction in a clean fermi liquid and the consequent development of superconductivity . we believe that this illuminates the inevitability of our conclusion . once ferromagnetic order has developed , the system becomes essentially one of spinless electrons whose diffusion is decoupled from the spin waves of the system . in this regime , the suppression of the resistivity which accompanies the growth of the triplet interaction parameter thus , our results indicate that a metallic state , if it exists in two dimensions , does not owe its existence to the growth of the triplet interaction parameter . in two dimensions , our results should be interpreted as follows . as we lower the temperature , the system develops local ferromagnetic order as we cross the mean - field transition temperature @xmath18 . the spin density is controlled by classical thermal fluctuations which prevent order and lead to a correlation length @xmath19 at low - temperatures . in the regime @xmath20 , we have a system in which charge diffuses while the spins are gapped . the localization physics of this system is essentially the same as that of spinless electrons , so the system leaves the diffusive regime and eventually becomes localized . thus , our small @xmath2 analysis can not be used all the way to zero temperature . the trend is towards a ferromagnetic insulator , but the onset of insulating behavior can thwart the development of ferromagnetism . recent experiments @xcite performed near the metal - insulator transition indicate an enhancement of the spin susceptibility which is cut off by the onset of insulating behavior . the metal - insulator transition is outside of the regime of validity of our calculation , but this is , nevertheless , qualitatively consistent with our picture . in @xmath21 dimensions , when the interaction strength is larger than a threshhold value of @xmath22 , there is a ferromagnetic transition at a finite temperature @xmath23 . if the resistance is smaller than a critical value @xmath24 , then the resistance will remain finite down to zero temperature , where the system is in a metallic ferromagnetic state .
|
in a dirty metal , electron - electron interactions in the spin - triplet channel lead to singular corrections to a variety of physical quantities . we show that these singularities herald the emergence of ferromagnetism . we calculate the effective action for the magnetic moment of weakly - interacting electrons in a dirty metal and show that a state with finite ferromagnetic moment minimizes this effective action .
|
in a dirty metal , electron - electron interactions in the spin - triplet channel lead to singular corrections to a variety of physical quantities . we show that these singularities herald the emergence of ferromagnetism . we calculate the effective action for the magnetic moment of weakly - interacting electrons in a dirty metal and show that a state with finite ferromagnetic moment minimizes this effective action . the saddle - point approximation is exact in an appropriate large- limit . we discuss the physics of the ferromagnetic state with particular regard to thermal fluctuations and localization effects .
|
0803.0720
|
i
|
throughout @xmath0 is a field . in @xcite iyama and yoshino consider the following two settings . [ ref-1.1 - 0 ] let @xmath1 $ ] and let @xmath2 be the cyclic group of three elements . consider the action of @xmath3 on @xmath4 via @xmath5 where @xmath6 , @xmath7 . put @xmath8 . [ ref-1.2 - 1 ] let @xmath9 $ ] and let @xmath10 be the cyclic group of two elements . consider the action of @xmath11 on @xmath4 via @xmath12 . put @xmath13 . in both examples iyama and yoshino reduce the classification of maximal cohen - macaulay modules over @xmath14 to the representation theory of certain generalized kronecker quivers . they use this to classify the rigid cohen - macaulay modules over @xmath14 . as predicted by deformation theory , the latter are described by discrete data . the explicit description of the stable category of maximal cohen - macaulay modules over a commutative gorenstein ring ( also known as the singularity category @xcite ) is a problem that has received much attention over the years . this appears to be in general a difficult problem and perhaps the best one can hope for is a reduction to linear algebra , or in other words : the representation theory of quivers . this is precisely what iyama and yoshino have accomplished . the proofs of iyama and yoshino are based on the machinery of mutation in triangulated categories , a general theory developed by them . in the current paper we present two alternative approaches to the examples . hopefully the additional insight obtained in this way may be useful elsewhere . our first approach applies to example [ ref-1.2 - 1 ] and is inspired by the treatment in @xcite of example [ ref-1.1 - 0 ] where the authors used the fact that in this case the stable category @xmath15 of maximal cohen - macaulay @xmath14-modules is a @xmath16-calabi - yau category which has a cluster tilting object whose endomorphism ring is the path algebra @xmath17 of the kronecker quiver with @xmath18 arrows . then they invoke their acyclicity result ( slightly specialized ) : ( * ? ? ? * , thm ) assume that @xmath19 is @xmath0-linear algebraic krull - schmidt @xmath16-calabi - yau category with a cluster tilting object @xmath20 such that @xmath21 is hereditary . then there is an exact equivalence betweem @xmath19 and the orbit category @xmath22)$ ] . from this result they obtain immediately that @xmath15 is the orbit category @xmath23)$ ] . this gives a very satisfactory description of @xmath15 and implies in particular the results by iyama and yoshino . in the first part of this paper we show that example [ ref-1.2 - 1 ] is amenable to a similar approach . iyama and yoshino prove that @xmath15 is a @xmath18-calabi - yau category with a @xmath18-cluster tilting object @xmath20 such that @xmath24 ( * ? ? ? * theorem 9.3 ) . we show that under these circumstances there is an analogue of the acyclicity result of the first author and reiten . [ ref-1.3 - 2 ] ( see [ ref-3.4 - 18 ] ) assume that @xmath19 is @xmath0-linear algebraic krull - schmidt @xmath18-calabi - yau category with a @xmath18-cluster tilting object @xmath20 such that @xmath24 . then there is an exact equivalence of @xmath19 with the orbit category @xmath25)$ ] , @xmath26 , where @xmath27 is the generalized kronecker quiver with @xmath28 arrows and @xmath29 is a natural square root of the auslander - reiten translate of @xmath30 , which on the pre - projective / pre - injective component corresponds to `` moving one place to the left '' . in the second part of this paper , which is logically independent of the first we give yet another approach to the examples [ ref-1.1 - 0],[ref-1.2 - 1 ] based on the following observation which might have independent interest . [ ref-1.4 - 3 ] ( see prop . [ gradedcase ] ) let @xmath31 be a finitely generated commutative graded gorenstein @xmath0-algebra with an isolated singularity . let @xmath32 be the completion of @xmath33 at @xmath34 . let @xmath35 be the stable category of graded maximal cohen - macaulay @xmath33-modules . then the obvious functor @xmath36 induces an equivalence @xmath37 where @xmath38 is the shift functor for the grading . in this proposition the quotient @xmath39 has to be understood as the triangulated / karoubian hull ( as explained in @xcite ) of the naive quotient of @xmath35 by the shift functor @xmath40 . this result is similar in spirit to @xcite which treats the finite representation type case . note however that one of the main results in loc . is that in case of finite representation type case _ every _ indecomposable maximal cohen - macaulay @xmath41-module is gradable . this does not seem to be a formal consequence of proposition [ ref-1.4 - 3 ] . it would be interesting to investigate this further . in [ ref-8 - 57 ] we show that at least rigid cohen - macaulay modules are always gradable so they are automatically in the image of @xmath42 . we expect this to be well known in some form but we have been unable to locate a reference . hence in order to understand @xmath43 it is sufficient to understand @xmath42 . the latter is the graded singularity category @xcite of @xmath33 and by ( * ? ? ? * thm 2.5 ) it is related to @xmath44 where @xmath45 . in examples [ ref-1.1 - 0],[ref-1.2 - 1 ] @xmath14 is the completion of a graded ring @xmath33 which is the veronese of a polynomial ring . hence @xmath46 is simply a projective space . using orlov s results and the existence of exceptional collections on projective space we get very quickly in example [ ref-1.1 - 0 ] @xmath47 and in example [ ref-1.2 - 1 ] @xmath48 ( where here and below @xmath49 actually stands for a quasi - equivalence between the underlying dg - categories ) . finally it suffices to observe that in example [ ref-1.1 - 0 ] we have @xmath50 $ ] and in example [ ref-1.2 - 1 ] we have @xmath51 $ ] ( see [ ref-7 - 51 ] below ) . finally we mention the following interesting side result [ ref-1.5 - 5 ] [ ref-1.5 - 6 ] let @xmath52 be a gorenstein local `` g - ring '' for example @xmath14 may be essentially of finite type over a field with an isolated singularity . then the natural functor @xmath53 is an equivalence up to direct summands . in partular every maximal cohen - macaulay module over @xmath54 is a direct summand of the completion of a maximal cohen - macaulay module over @xmath14 . the original proof ( by the first and the third author ) of this result was unnecessarily complicated . after the paper was put on the arxiv daniel murfet ( who has become the second author ) informed us about the existence of a much nicer proof in the context of singularity categories ( see proposition [ theorem : first_theorem ] ) . the same argument also applies to proposition [ ref-1.4 - 3 ] . so we dropped our original proofs and put the new argument in an appendix to which we refer . meanwhile orlov @xcite has proved ( independently and using different methods ) a very general result which implies in particular proposition [ ref-1.5 - 6 ] .
|
in a recent paper iyama and yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal cohen - macaulay modules in terms of linear algebra data . in this paper we present two new approaches to these examples . in the first approach we give a relation with cluster categories . in the second approach we use orlov s result on the graded singularity category .
|
in a recent paper iyama and yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal cohen - macaulay modules in terms of linear algebra data . in this paper we present two new approaches to these examples . in the first approach we give a relation with cluster categories . in the second approach we use orlov s result on the graded singularity category .
|
1209.3994
|
i
|
in the yang - mills theory @xcite and quantum chromodynamics ( qcd ) for strong interactions , both the renormalizability and the physical unitarity are satisfied , as demonstrated first in @xcite . moreover , it is also known that the massive yang - mills theory satisfies both the renormalizability and the physical unitarity @xcite , if the local gauge invariance is spontaneously broken by the higgs field @xcite and the gauge field acquires the mass through the higgs mechanism by absorbing the nambu - goldstone particle associated with the spontaneous symmetry breakdown . in other words , both the renormalizability and the physical unitarity survive the spontaneous breaking of the gauge symmetry . it is a long - standing problem @xcite to clarify whether it is possible or not to construct a massive yang - mills model blessed with both the physical unitarity and the renormalizability without the higgs fields , in which the local gauge symmetry is not spontaneously broken . here the lagrangian is assumed to be written in polynomials of the fields ( we exclude the nonpolynomial type @xcite from our discussions ) . we are anxious to find such a model for understanding the mass gap and confinement caused by the strong interactions @xcite , since the higgs field does not exist and the color gauge symmetry is kept intact in qcd . indeed , there are continued attempts to look for an alternative way to describe massive non - abelian gauge fields without the higgs field @xcite . however , all these efforts were unsuccessful in coping with both renormalizability and unitarity very well : in all the models proposed so far for the massive yang - mills theory without the higgs fields , it seems that the renormalizability and the physical unitarity are not compatible with each other , although there are some models which satisfy either the renormalizability or the physical unitarity . see @xcite for reviews and @xcite for later developments . for this purpose , we start once again from the curci - ferrari ( cf ) model @xcite , which is a massive extension of the massless yang - mills theory in the most general renormalizable gauge having both the becchi - rouet - stora - tyutin ( brst ) and anti - brst symmetries @xcite . in preceding studies for the cf model @xcite , the cf model is proved to be renormalizable , whereas the cf model has been concluded to violate physical unitary @xcite . however , the preceding studies are restricted to considerations in the perturbation theory . we need a nonperturbative framework to draw a definite conclusion to this issue . in a previous paper @xcite , therefore , we have presented a nonperturbative construction of a massive yang - mills field @xmath0 which describes a non - abelian massive spin - one vector boson with the correct physical degrees of freedom without the higgs field @xcite . this is achieved by finding a nonlinear but local transformation from the original fields in the cf model to the physical massive vector field @xmath0 which is invariant under the modified brst and anti - brst transformation . as an application , we have written down a local mass term for the yang - mills field and a dimension - two condensate , which are exactly invariant under the modified brst transformation , lorentz transformation and color rotation . the resulting massive yang - mills model is regarded as a low - energy effective theory of qcd , which enables us to understand the decoupling solution @xcite characterizing the deep infrared regime responsible for color confinement @xcite . in a series of papers , we give the perturbative and nonperturbative studies on the physical unitarity @xcite in the massive yang - mills theory constructed in the previous paper @xcite . in the ordinary massless yang - mills theory , the physical unitarity is a first step of understanding color confinement @xcite : in the intermediate state , the contributions from the unphysical gauge modes , i.e. , the longitudinal and scalar modes are exactly canceled by those of the ghost and antighost , which is a special case of the quartet mechanism @xcite . we clarify how the situation changes in the massive case . moreover , we clarify the reason for failures of the preceding attempts from our viewpoint . this paper is the first of the planned papers for discussing the perturbative and nonperturbative physical unitarity in the massive yang - mills theory without the higgs field . in this paper we present a new perturbative treatment for the physical unitarity after reviewing the general properties of the massive yang - mills theory . then we reproduce the violation of physical unitarity in a transparent way . in subsequent papers , we present a nonperturbative framework to discuss a possible scenario of restoring the physical unitarity in the massive yang - mills theory . finally , we mention the difference between the unitarity and physical unitarity of the scattering matrix from our point of view . for the tree - level scattering amplitude between two longitudinally polarized vector bosons , it is known @xcite that the scattering probability as a function of the energy @xmath1 becomes greater than 1 above a critical value @xmath2 , since the amplitude grows with the energy @xmath1 like @xmath3 where @xmath4 is the mass of the vector boson and @xmath5 is the coupling constant for the self - interactions among vector bosons . this implies that the perturbative unitarity breaks down in high - energy @xmath6 . therefore , for the perturbative unitarity to be satisfied , the energy must be restricted to low - energy @xmath7 , which is called the unitarity bound . ( the violation of the unitarity condition for the scattering amplitude in high energy is understood from the nambu - goldstone equivalence theorem @xcite and the low - energy theorem . ) in the higgs sector of weak interactions in the standard model , the higgs particle exists and the exchange of the higgs particle affects the amplitude so that the amplitude approaches a constant at energies far above the higgs pole . consequently , the higgs mass must be less than an upper bound . if such new physical degrees of freedom do not exist , this behavior is not modified and the unitarity violation in high energy can not be avoided in the massive yang - mills theory , since the nakanishi - lautrup ( nl ) auxiliary field can be integrated out and the ghosts can play no role in the tree - level amplitude . in our works , we regard the cf model as a low - energy effective theory of the yang - mills theory to be valid in the region @xmath7 for discussing color confinement . we restrict our examination on the physical unitarity to a sufficiently low - energy region below a few gev to evade the unitarity violation and consider only the physical unitarity , i.e. , unphysical mode cancellation in our papers . therefore , the well - known fact about the unitarity violation in the above does not contradict our research on the physical unitarity . this paper is organized as follows . in section ii , we introduce a massive yang - mills theory without the higgs field and define the cf model as a special case . the cf model is not invariant under the usual brst and anti - brst transformations . however , the cf model can be made invariant by modifying the brst and anti - brst transformations . the cost of introducing the mass term is the violation of nilpotency of the modified brst and anti - brst transformations . we point out an important fact that even the modified brst ( anti - brst)-invariant quantity depends on a parameter @xmath8 in the @xmath9 case . this should be compared with the @xmath10 case , in which @xmath8 is a gauge - fixing parameter and the brst - invariant quantity does not depend on @xmath8 , which means that the physics does not depend on @xmath8 in the @xmath10 case . this is not the case for @xmath9 . in section iii , we summarize the result obtained in the previous paper @xcite on a nonperturbative construction of a non - abelian massive yang - mills field @xmath0 under the requirements which guarantee ( i ) the modified brst ( and anti - brst ) invariance , ( ii ) correct degrees of freedom for describing a massive spin - one particle , and ( iii ) the expected transformation rule under color rotation . we write down the massive vector field explicitly in terms of the original yang - mills field , the faddeev - popov ( fp ) ghost field , antighost field and the nl field in the cf model . in section iv , we give a perturbative framework of the cf model in terms of the new field variable @xmath0 . we give the feynman rules up to the order @xmath5 . in section v , we check the physical unitarity in the massless yang - mills theory . using a simple example , it is demonstrated in the lowest order of perturbation theory that the physical unitarity follows from the cancellation among unphysical modes : the longitudinal and scalar modes of the yang - mills field together with the fp ghost and antighost . in section vi , we review a conventional argument for the violation of physical unitarity in the massive yang - mills theory without the higgs field . using a simple example corresponding to the previous section , we show that the violation of physical unitarity follows from the incomplete cancellation among unphysical modes : the scalar mode with the fp ghost and antighost . in section vii , we begin with a new analysis on the physical unitarity of the cf model based on a novel framework using the field @xmath0 given in section iii . in this section , we give a new perturbative analysis using the result of section v. we confirm that the physical unitarity is indeed violated in the cf model in the framework of the perturbation theory in the coupling constant . it is easily seen that the violation of physical unitarity follows from the incomplete cancellation among unphysical modes : the nl field ( corresponding to the scalar mode ) with the fp ghost and antighost . we discuss how to avoid the violation of physical unitarity within the perturbative framework . in the final section , we summarize the results and mention the perspective on the next work . in appendix a , we calculate the jacobian associated with the change of variables from the original cf model to the new theory written in terms of new variables . in appendix b , the feynman rules are given up to the next order @xmath11 , with which we supplement the results of section v.
|
in a series of papers , we examine the physical unitarity in a massive yang - mills theory without the higgs field in which the color gauge symmetry is not spontaneously broken and kept intact . for this purpose , we use a new framework proposed in the previous paper kondo [ arxiv:1208.3521 ] based on a nonperturbative construction of a non - abelian field describing a massive spin - one vector boson field , which enables us to perform the perturbative and nonperturbative studies on the physical unitarity . in this paper , we present a new perturbative treatment for the physical unitarity after giving the general properties of the massive yang - mills theory . then we reproduce the violation of physical unitarity in a transparent way . this paper is a preliminary work to the subsequent papers in which we present a nonperturbative framework to propose a possible scenario of restoring the physical unitarity in the curci - ferrari model .
|
in a series of papers , we examine the physical unitarity in a massive yang - mills theory without the higgs field in which the color gauge symmetry is not spontaneously broken and kept intact . for this purpose , we use a new framework proposed in the previous paper kondo [ arxiv:1208.3521 ] based on a nonperturbative construction of a non - abelian field describing a massive spin - one vector boson field , which enables us to perform the perturbative and nonperturbative studies on the physical unitarity . in this paper , we present a new perturbative treatment for the physical unitarity after giving the general properties of the massive yang - mills theory . then we reproduce the violation of physical unitarity in a transparent way . this paper is a preliminary work to the subsequent papers in which we present a nonperturbative framework to propose a possible scenario of restoring the physical unitarity in the curci - ferrari model .
|
astro-ph0409605
|
c
|
a significant hindrance to a wider acceptance of the primordial scenarios for gc origin is an apparent absence of dm in galactic gcs . many observational facts have been suggested to be evidence for gcs having no dm , including the presence of such features in the outer parts of the gc density profiles as apparent tidal cutoffs or breaks , relatively low values of the apparent central mass - to - light ratio @xmath191 which are consistent with purely baryonic clusters , flat radial distribution of the line - of - sight velocity dispersion in the outskirts of gcs believed to be a sign of tidal heating , and non - spherical shape of the clusters in their outer parts . here we present the results of simulations of stellar clusters relaxing inside live dm minihalos in the early universe ( at @xmath12 ) . we study three distinctly different cases which can correspond to very different gas - dynamic processes forming a gc : a mild warm collapse , a violent cold collapse resulting in a much denser cluster with a significant fraction of stars escaping the gc in the absence of dm , and a hot collapse resulting in a lower density cluster with many would - be stars - escapers . we show that gcs forming in dm minihalos exhibit the same properties as one would expect from the action of the tidal field of the host galaxy on a purely stellar cluster : king - like radial density cutoffs ( for the case of a warm collapse ) , and breaks in the outer parts of the density profile accompanied by a plateau in the velocity dispersion profile ( for a cold collapse ) . also , the apparent mass - to - light ratio for our clusters with dm is generally close to the case of a purely stellar cluster . ( the special case of a hot collapse inside a dm halo which produces inflated values of @xmath191 can be mistaken for a cluster being at its last stage of disruption by the tidal forces . ) we argue that increasingly eccentric isodensity contours observed in the outskirts of some gcs could be created by a stellar cluster relaxing inside a triaxial dm minihalo , and not by external tidal fields as it is usually interpreted . indeed , cosmological dm halos are known to have noticeably non - spherical shapes ; a stellar cluster relaxing inside such a halo would have close to spherical distribution in its denser part where the stars dominate dm , and would exhibit isodensity contours of increasingly larger eccentricity in its outskirts where dm becomes the dominant mass component . it is also important to remember that few galactic gcs show clear signs of a `` tidal '' cutoff in the outer density profiles @xcite . the `` tidal '' features of an opposite nature `` breaks '' in the outer parts of the radial surface brightness profiles in some gcs are often observed at or below the inferred level of contamination by foreground / background objects , and could be in many cases an artifact of the background subtraction procedure which relies heavily on the assumption that the background objects are smoothly distributed across the field of view . a good example is that of draco dwarf spheroidal galaxy . @xcite used simple non - filtered stellar counts from photographic plates followed by a background subtraction procedure to show that this galaxy appears to have a relatively small value of its radial density `` tidal cutoff '' @xmath220 arcmin and a substantial population of `` extratidal '' stars . @xcite used a more advanced approach of multi - color filtering of draco stars from the sloan digital sky survey images and achieved much higher signal - to - noise ratio than in @xcite . new , higher quality results were supposed to make the `` extratidal '' features of draco much more visible . instead , @xcite demonstrated that the draco s radial surface brightness profile is very regular down to a very low level ( 0.003 of the central surface brightness ) , and suggested a larger value for the king tidal radius of @xmath221 arcmin . we argue that the qualitative results presented in this paper are very general , and do not depend much on the the fact that we used ms04 model to set up the initial non - equilibrium stellar core configurations , and on the particular values of the model parameters ( such as @xmath13 , @xmath14 , and @xmath22 ) . as we discussed in section [ results ] , the appearance of `` tidal '' or `` extratidal '' features in our warm and cold collapse models is caused by two reasons : ( 1 ) at radii @xmath181 the potential is dominated by dm , whereas in the stellar core the potential is dominated by stars from the beginning till the end of simulations , and ( 2 ) the collapse is violent enough to eject a fraction of stars beyond the initial stellar cluster radius . as we showed in [ physical ] , for the initial stellar density @xmath222 @xmath9 pc@xmath16 ( from ms04 ) , the whole physically plausible range of stars - to - dm mass ratios @xmath22 and the gc formation redshifts @xmath31 satisfy the above first condition . for warm and cold collapse , this condition can be reexpressed as @xmath223>1 $ ] . one can easily estimate @xmath79 for other values of @xmath13 . the second condition is more difficult to quantify . in ms04 we showed that collapsing homogeneous isothermal spheres produce extended halos for any values of the initial virial ratio @xmath57 ( except for @xmath224 systems which are too hot to form a bound cluster in the absence of dm ) . @xcite simulated cold collapse for a wider spectrum of initial cluster configurations , including power - law density profile , clumpy and rotating clusters . in all their simulations an extended halo is formed after the initial violent relaxation . it appears that in many ( probably most ) stellar cluster configurations , which are not in detailed equilibrium initially , our second condition can be met . of course , not every stellar cluster configuration will result in a gc - like object , with a relatively large core and an extended halo , after the initial violent relaxation phase . @xcite demonstrated that a warm ( @xmath225 ) collapse of a homogeneous isothermal sphere produces clusters with large cores ( their models d and g ) . adiabatic collapse of homogeneous isothermal spheres produces clusters which surface density profiles are very close to those of dynamically young galactic gcs ( ms04 ) . @xcite concluded that any cold stellar system which does not contain significant inhomogeneities relaxes to a large - core configuration . the results of @xcite can also be used to estimate the importance of adiabaticity for core formation . indeed , their initially homogeneous models h and g span a large range of @xmath57 , and include both adiabatic cases ( @xmath226 for their number of particles @xmath227 , from eq.[[eqbeta ] ] ) and non - adiabatic ones ( @xmath228 ) . it appears that all their models ( adiabatic and non - adiabatic ) form a relatively large core . the issue is still open , but it appears that the adiabaticity requirement ( our eq.[[eqbeta ] ] ) is not a very important one for our problem . ( though this requirement is met automatically for real gcs for the values of @xmath13 and @xmath14 derived in ms04 . ) an important point to make is that the simulations presented in this paper describe the collisionless phase of gc formation and evolution , and can not be directly applied to gcs which have experienced significant secular evolution due to encounters between individual stars . in ms04 we showed that such collisionless simulations of purely stellar clusters describe very well the surface brightness profiles of dynamically young galactic gcs ( such as ngc 2419 , ngc 5139 , ic 4499 , arp 2 , and palomar 3 see fig . 1 in ms04 ) . in paper ii we will address ( among other things ) the issue of long - term dynamic evolution of hybrid gcs . we will demonstrate that , at least for the warm - collapse case , secular evolution does not change our qualitative results presented in this paper . in the light of the results presented in this paper and the above arguments , we argue that additional observational evidence is required to determine with any degree of confidence if gcs have any dm presently attached to them , or if they are purely stellar systems truncated by the tidal field of the host galaxy . a decisive evidence would be the presence of obvious tidal tails . a beautiful example is given by palomar 5 @xcite , where tidal tails were observed to extend over @xmath229 in the sky . even if a gc cluster is proven not to have a significant amount of dm , it does not preclude it having been formed originally inside a dm minihalo . in `` semi - consistent '' simulations of @xcite of a dwarf galaxy formation , proto - gcs were observed to form inside dm minihalos , with the dm being lost during the violent relaxation accompanying the formation of the dwarf galaxy . unfortunately , their simulations did not have enough resolution to clarify the fate of dm in proto - gcs . in paper ii we will address the issue of the fate of dm in hybrid proto - gcs experiencing severe tidal stripping in the potential of the host dwarf galaxy .
|
we study the initial relaxation of a stellar core inside a live dark matter minihalo in the early universe . our dark - matter dominated globular clusters show features which are usually attributed to the action of the tidal field of the host galaxy . among them are the presence of an apparent cutoff ( `` tidal radius '' ) or of a `` break '' in the outer parts of the radial surface brightness profile , and a flat line - of - sight velocity dispersion profile in the outskirts of the cluster . the apparent mass - to - light ratios of our hybrid ( stars dark matter ) globular clusters are very close to those of purely stellar clusters . we suggest that additional observational evidence such as the presence of obvious tidal tails is required to rule out the presence of significant amounts of dark matter in present day globular clusters .
|
in a series of two papers , we test the primordial scenario of globular cluster formation using results of high - resolutions-body simulations . in this first paper we study the initial relaxation of a stellar core inside a live dark matter minihalo in the early universe . our dark - matter dominated globular clusters show features which are usually attributed to the action of the tidal field of the host galaxy . among them are the presence of an apparent cutoff ( `` tidal radius '' ) or of a `` break '' in the outer parts of the radial surface brightness profile , and a flat line - of - sight velocity dispersion profile in the outskirts of the cluster . the apparent mass - to - light ratios of our hybrid ( stars dark matter ) globular clusters are very close to those of purely stellar clusters . we suggest that additional observational evidence such as the presence of obvious tidal tails is required to rule out the presence of significant amounts of dark matter in present day globular clusters .
|
1504.02799
|
i
|
at the conclusion of an all - pay auction , all bidders must pay the bids they submitted , with only the highest bidder receiving the item . with this idea in mind , one can play a variant of a two - player game using an all - pay auction to decide who moves next instead of simply alternating between players . for example , one could play all - pay tic - tac - toe with @xmath0 chips . each round both players privately record their bids and then simultaneously reveal them . if player a bids @xmath1 and his opponent bids @xmath2 , player a would get to choose a square to mark and the next round of bidding would begin wih player @xmath3 having @xmath4 chips , player @xmath5 having @xmath6 chips . note that the chips have no value outside the game and only serve to determine who moves - the ultimate goal is still just to get three - in - a - row . another variant of the game could have only the player who wins the move pay his / her bid , i.e. deciding who moves next using a first - price auction . these games were first studied formally in the 1980s by richman , whose work has since then been greatly expanded upon . intuitively , there is less risk in these `` richman games '' for the player losing the bid . if your opponent bids @xmath0 for a certain move , it makes no difference whether your bid was @xmath7 or @xmath8 . all that matters is that your opponent s bid was higher . a surprising consequence of this single - pay structure is that for every state of a game , there exists a `` richman value '' @xmath9 for each player that represents the proportion of the total chips that player would need to hold to have a deterministic winning strategy . in this situation , the player with the winning strategy can tell her opponent what bid she will be making next without affecting her ability to ultimately win . for zero - sum games , this means that unless a player s chip ratio is exactly @xmath9 , then one of the players must have such a winning strategy @xcite , @xcite . our objective is to begin the formal study of all - pay bidding games . returning to the above example where your opponent bids @xmath0 chips and you are indifferent between bidding @xmath7 and @xmath8 , it is clear this is no longer true for an all - pay bidding mechanism . you would be very disappointed had you bid @xmath7 , as your opponent would be paying just @xmath10 chip on net to make a move . had you bid @xmath8 , though , you might feel pretty good about not moving this current turn , as the @xmath0 extra chips may make a bigger difference for the rest of the game . thus , there are at least two bidding scenarios which intuitively seem like very good positions to be in : winning the bid by a relatively small number of chips or losing the bid by a relatively large number of chips . this behavior suggests that , unlike in richman games , in all - pay bidding games one of the players will not necessarily have a deterministic winning strategy . instead , players must randomize their bidding in some way . thus , we must appeal to the concept of mixed bidding strategies in nash equilibria . before presenting formal definitions and results , we provide a sample all - pay bidding game to illustrate some of the main features of playing these games . alice and bob , each with @xmath0 chips , are playing all - pay bidding tic - tac - toe . each turn alice and bob secretly write down a bid , a whole number less than or equal to their total number of chips . they then reveal their bids and whoever bid more gets to decide who makes the next move . we say a player has * advantage * if , when players bid the same amount , that player gets to decide who makes the next move . the question of deciding how to assign advantage is one we encountered early on . for our games , we give advantage to the player with more chips , then arbitrarily let alice have advantage when alice and bob have the same number of chips . a number of other mechanisms would also suffice , such as alternating advantage or having a special `` tie - breaking '' chip that grants advantage and is passed each time it is used . our choice was made in the interest of computational simplicity and to eventually allow extension to real - valued bidding . _ first move . _ both players have @xmath0 chips . alice bids @xmath2 , bob bids @xmath1 . bob wins the right to move and plays in the center of the board . _ second move . _ alice has @xmath6 chips , bob has @xmath4 chips . alice wants to win this move to keep pace with bob , but also does not see why it should be worth more than the first , so she only slightly increases her bid to @xmath11 . bob , thinking that alice may want to win this move more , is content to let alice win and collect chips by bidding @xmath8 . alice wins the right to move and plays in the top - left corner of the board . _ third move . _ alice has @xmath4 chips , bob has @xmath6 chips . alice bids @xmath12 , bob bids @xmath1 , so alice wins the right to move and plays in the top - center of the board . _ fourth move . _ alice has @xmath13 chips , bob has @xmath14 chips . alice is one move away from winning and decides to risk it and bid all of her @xmath13 chips . unfortunately for her , bob has guessed her move and has himself bid @xmath13 as well . because bob has more chips overall , he uses his advantage to win the tie and plays in the top - right corner of the board , blocking alice s victory and setting himself up for one . _ fifth move . _ alice has @xmath13 chips , bob has @xmath14 chips . bob has more chips and is just a move away from winning , so he can bid everything , play in the bottom - left corner and win the game . in normal tic - tac - toe , both players can guarantee a draw by playing well , but as we see from this example , the result of a game of all - pay tic - tac - toe involves far more chance . for example , at the fourth move in the above game , alice could have guessed bob might bid @xmath13 and chosen to `` duck '' by bidding @xmath8 . in this case bob would win the move and play as before , but now the chip counts would be @xmath15 to @xmath1 in alice s favor , and alice can bid @xmath1 and then @xmath13 to win the next two moves and win in the left column . it is easy to see that if a player knows what his opponent will bid at each move , he can win the game easily . thus , in the vast majority of all - pay bidding games , optimal play can not be deterministic . though we do not return to tic - tac - toe in this paper , it served as a test game for much of our research . using our results , we built a computer program to play all - pay bidding tic - tac - toe optimally . the program can be played against at http://biddingttt.herokuapp.com . the theory behind this program , which is not specific to tic - tac - toe , will be the focus of the rest of the paper . our ultimate goal is to characterize the optimal strategies for a general class of all - pay bidding games . the game consists of iterations of both players bidding for the right to move followed by one of the players making a move . in turn , an optimal strategy will also have two parts : the bid strategy and the move strategy . for a given position in the game ( e.g. a configuration of the tic - tac - toe board ) and chip counts for each of the players ( e.g. alice has @xmath6 chips , bob has @xmath4 chips ) , the bid strategy must tell players how to best randomize their bets ( e.g. alice bids @xmath8 chips half the time , @xmath13 chips half the time ) while the move strategy must tell whoever wins the bid the best move to make ( e.g. where to play on the tic - tac - toe board ) . the problem of determining move strategy is largely combinatorial in nature and remains similar to its analog in richman games . we can still represent the space of game states as a directed graph , and there is a not always a single best move that each player can make upon winning the bid . that is , the best move could also depend on each player s chip counts moving forward . the focus of our work , then , will be on determining the optimal bidding strategy for any game position and chip counts . naturally , this should depend on a player s chances of winning in any of the possible subsequent game situations ( i.e. after a single move and updated chip counts ) . for purposes of initial analysis , we will assume that these future winning probabilities are known , and see how the bidding strategy can be determined from this information . then , by using the recursive nature of the directed graph , we will be able to start from the `` win '' and `` loss '' nodes ( where the probability is just @xmath10 or @xmath8 ) to find the optimal bidding strategies and winning probabilities for any game situation . for the rest of this paper , we will often refer to a bidding strategy as just a `` strategy '' when it is clear that the focus is just on the bidding side of the game . here , a strategy will be a probability vector where the @xmath16th coordinate corresponds to the probability a player will bid @xmath16 chips . further , a nash equilibrium for a game situation will just be a pair of strategies so that neither player has an incentive to deviate . this means that each player s strategy will maximize his / her minimum probability of ultimately winning from the next turn of the game . it quickly becomes apparent that a naive recursive algorithm using linear programming is feasible only for games with very few moves . thus , in the interest of being able to practically calculate the optimal bidding strategies for general games , we prove some structural results on the nash equilibria . in particular , useful structure arises when we study a particular class of games that we dubbed `` precise '' , which roughly speaking are games where having one more chip is strictly better than not . the key result is a surprising relationship between opposing optimal strategies that allows one to immediately write a nash equilibrium strategy for the player without advantage if given a nash equilibrium strategy for the player with advantage . this relationship , ( [ reverse_thm ] ) , which we call the reverse theorem , is a critical step toward the calculation of optimal strategies for precise games . further , by assigning an arbitrarily small value in the game to each chip , we get a precise game that is very similar to the original game . we show that the optimal strategies we can calculate for these new precise games will indeed converge to optimal strategies for our possibly imprecise games . our theoretical results ultimately culminate in a fast algorithm for computing optimal probabilistic bidding strategies . together with a move strategy for the combinatorial side of the game , this gives a complete characterization of optimal play for all - pay bidding games .
|
in an all - pay auction , only one bidder wins but all bidders must pay the auctioneer . all - pay bidding games arise from attaching a similar bidding structure to traditional combinatorial games to determine which player moves next . in contrast to the established theory of single - pay bidding games , optimal play involves choosing bids from some probability distribution that will guarantee a minimum probability of winning . in this manner , all - pay bidding games we d the underlying concepts of economic and combinatorial games . we then give a fast algorithm for computing such strategies for a large class of all - pay bidding games . the methods presented provide a framework for further development of the theory of all - pay bidding games .
|
in an all - pay auction , only one bidder wins but all bidders must pay the auctioneer . all - pay bidding games arise from attaching a similar bidding structure to traditional combinatorial games to determine which player moves next . in contrast to the established theory of single - pay bidding games , optimal play involves choosing bids from some probability distribution that will guarantee a minimum probability of winning . in this manner , all - pay bidding games we d the underlying concepts of economic and combinatorial games . we present several results on the structures of optimal strategies in these games . we then give a fast algorithm for computing such strategies for a large class of all - pay bidding games . the methods presented provide a framework for further development of the theory of all - pay bidding games . + michael menz^1^ , justin wang^2^ , jiyang xie^3^ + * 3>p.3 ^1^yale university & ^2^yale university & ^3^yale university michael.menz@yale.edu & justin.wang@yale.edu & jiyang.xie@yale.edu
|
1212.6318
|
c
|
the c and o abundances of main - sequence stars in the hyades cluster are not yet well established despite their astrophysical importance , for which a number of previous studies reported different results . also , the abundances of li ( key element for investigating the physical process in the envelope ) and fe ( representative of metallicity ) are worth reinvestigation by taking this opportunity . motivated by this situation , we decided to carry out a systematic abundance study of these elements for hyades main - sequence stars in the @xmath0 range of @xmath1 50007000 k. practically , we derived these abundances by applying a spectrum - synthesis analysis to four spectral regions at 60806089 @xmath33 , 67076709 @xmath33 , 71107121 @xmath33 , and 61576159 @xmath33 ( comprising lines of fe - group elements , li i 6708 line , c i 71117119 lines , and o i 61568 lines , respectively ) based on the high - dispersion spectra of 68 selected hyades f g type stars obtained with the 188 cm reflector and the hides spectrograph at okayama astrophysical observatory . it turned out that these c , o , and fe abundances similarly exhibit a marginal @xmath0-dependent gradient ( i.e. , slightly increasing with a decrease in @xmath0 ; typically on the order of @xmath8 dex k@xmath9 ) although this might be nothing but an apparent effect due to an improper choice of atmospheric parameters , we found it hard to give a quantitatively reasonable explanation . apart from this small systematic gradient , the abundances of c , o , and fe in these hyades stars were found to be fairly uniform and marginally supersolar with only a small scatter of @xmath107 dex : @xmath2[c / h]@xmath3 @xmath4 , @xmath2[o / h]@xmath5 @xmath6 , and @xmath2[fe / h]@xmath7 @xmath4 , suggesting that the primordial abundances are almost retained . regarding li , we confirmed the well - known @xmath0-dependent trend in the li abundances of hyades f g stars reported so far ( i.e. , a conspicuous li - trough at 6700 k @xmath132 k and a progressive decline with a decrease in @xmath0 below @xmath158 k ) . since @xmath14(li ) at 7000 k @xmath150 k ( a zone encompassed by the deficiency of li in a / am stars and the li chasm of f stars ) is almost the solar - system abundance , the li depletion mechanism seen in a - type stars is considered to be different from the physical process responsible for the li gap of hyades f - type stars . we concluded that the the surface li of hyades stars is essentially controlled only by @xmath0 and other parameters such as the rotational velocity are almost irrelevant . a positive correlation between @xmath14(li ) and stellar rotation , which is observed in field solar - analog stars , is not seen in these younger early g - type stars of the hyades cluster . this may impose an important constraint on the time scale in the build - up of such a rotation - dependent li anomaly . anders , e. , & grevesse , n. 1989 , geochim . acta , 53 , 197 boesgaard , a. m. , & budge , k. g. 1988 , apj , 332 , 410 boesgaard , a. m. , & tripicco , m. j. 1986 , apj , 302 , l49 burkhart , c. , & coupry , m. f. 1989 , a&a , 220 , 197 burkhart , c. , & coupry , m. f. 2000 , a&a , 354 , 216 castelli , f. , gratton , r. g. , & kurucz , r. l. 1997 , a&a , 318 , 841 cayrel , r. , cayrel de strobel , g. , campbell , b. , & dppen , w. 1984 , apj , 283 , 205 de bruijne , j. h. j. , hoogerwerf , r. , & de zeeuw , p. t. 2001 , a&a , 367 , 111 esa 1997 , the hipparcos and tycho catalogues , esa sp-1200 , available from nasa - adc or cds in a machine - readable form ( file name : hip_main.dat ) friel , e. d. , & boesgaard , a. m. 1990 , apj , 351 , 480 garca - lpez , r. , rebolo , r. , herrero , a. , & beckman , j. e. 1993 , apj , 412 , 173 gebran , m. , vick , m. , monier , r. , & fossati , l. 2010 , a&a , 523 , a71 gustafsson , b. , karlsson , t. , olsson , e. , edvardsson , b. , & ryde , n. 1999 , a&a , 342 , 426 herbig , g. h. 1965 , apj , 141 , 588 izumiura , h. 1999 , in proc . 4th east asian meeting on astronomy , observational astrophysics in asia and its future ed . p. s. chen ( kunming : yunnan observatory ) , 77 king , j. r. 1993 , ph . d. dissertation , university of hawaii king , j. r. , & hiltgen , d. d. 1996 , aj , 112 , 2650 kurucz , r. l. 1993 , kurucz cd - rom , no . 13 ( harvard - smithsonian center for astrophysics ) kurucz , r. l. , & bell , b. 1995 , kurucz cd - rom , no . 23 ( harvard - smithsonian center for astrophysics ) paulson , d. b. , sneden , c. , & cochran , w. d. 2003 , aj , 125 , 3185 perryman , m. a. c. , et al . 1998 , a&a , 331 , 81 petigura , e. a. , & marcy , g. w. 2011 , apj , 735 , 41 richer , j. , michaud , g. , & turcotte , s. 2000 , apj , 529 , 338 schuler , s. c. , hatzes , a. p. , king , j. r. , krster , m. , & the , l .- s . 2006a , aj , 131 , 1057 schuler , s. c. , king , j. r. , terndrup , d. m. , pinsonneault , m. h. , murray , n. , & hobbs , l. m. 2006b , apj , 636 , 432 soderblom , d. r. , oey , m. s. , johnson , d. r. h. , & stone , r. p. s. 1990 , aj , 99 , 595 takeda , y. 1994 , pasj , 46 , 53 takeda , y. 1995 , pasj , 47 , 287 takeda , y. 2008 , in the metal - rich universe , eds . g. israelian & g. meynet , ( cambridge : cambridge university press ) , 308 takeda , y. , & honda , s. 2005 , pasj , 57 , 65 takeda , y. , kambe , e. , sadakane , k. , & masuda , s. 2010 , pasj , 62 , 1239 takeda , y. , kang , d .- , han , i. , lee , b .- , & kim , k .- m . 2009 , pasj , 61 , 1165 takeda , y. , kang , d .- , han , i. , lee , b .- c . , kim , k .- m . , kawanomoto , s. , & ohishi , n. 2012 , pasj , 64 , 38 takeda , y. , & kawanomoto , s. 2005 , pasj , 57 , 45 takeda , y. , kawanomoto , s. , honda , s. , ando , h. , & sakurai , t. 2007 , a&a , 468 , 663 takeda , y. , kawanomoto , s. , honda , s. , ando , h. , & sakurai , t. 2010 , a&a , 515 , a93 takeda , y. , kawanomoto , s. , takada - hidai , m. , & sadakane , k. 1998 , pasj , 50 , 509 takeda , y. , ohkubo , m. , sato , b. , kambe , e. , & sadakane , k. 2005 , pasj , 57 , 27 takeda , y. , & sadakane , k. 1997 , pasj , 49 , 367 takeda , y. , sato , b. , & murata , d. 2008 , pasj , 60 , 781 takeda , y. , takada - hidai , m. , jugaku , j. , sakaue , a. , & sadakane , k. 1999 , pasj , 51 , 961 thorburn , j. a. , hobbs , l. m. , deliyannis , c. p. , & pinsonneault , m. h. 1993 , apj , 415 , 150 tomkin , j. , & lambert , d. l. 1978 , apj , 223 , 937 varenne , o. , & monier , r. 1999 , a&a , 351 , 247 wallerstein , g. , herbig , g. h. , & conti , p. s. 1965 , apj , 141 , 610 ccrccspecies & @xmath159 & @xmath160 & @xmath73 & remark + + v i & 6081.441 & 1.05 & @xmath1250.58 & + co i&6082.422 & 3.51 & @xmath1250.52 & + fe i&6082.708 & 2.22 & @xmath1253.57 & + fe ii&6084.111 & 3.20 & @xmath1253.81 & + ti i&6085.228 & 1.05 & @xmath1251.35 & + fe i&6085.260 & 2.76 & @xmath1253.21 & + ni i&6086.276 & 4.27 & @xmath1250.53 & + co i&6086.658 & 3.41 & @xmath1251.04 & + si i&6087.805 & 5.87 & @xmath1251.60 & + + fe i & 6703.568 & 2.76 & @xmath1253.02 & ( adjusted ) + fe i & 6705.101 & 4.61 & @xmath1251.02 & ( adjusted ) + fe i & 6707.441 & 4.61 & @xmath1252.35 & + li i & 6707.756 & 0.00 & @xmath1250.43 & li 6708 + li i & 6707.768 & 0.00 & @xmath1250.21 & li 6708 + li i & 6707.907 & 0.00 & @xmath1250.93 & li 6708 + li i & 6707.908 & 0.00 & @xmath1251.16 & li 6708 + li i & 6707.919 & 0.00 & @xmath1250.71 & li 6708 + li i & 6707.920 & 0.00 & @xmath1250.93 & li 6708 + + ni i & 7110.892 & 1.94 & @xmath1252.88 & ( adjusted ) + c i & 7111.472 & 8.64 & @xmath1251.24 & ( adjusted ) + fe i & 7112.168 & 2.99 & @xmath1252.89 & ( adjusted ) + c i & 7113.178 & 8.65 & @xmath1250.80 & ( adjusted),c 7113 + c i & 7115.172 & 8.64 & @xmath1250.96 & ( adjusted ) + c i & 7116.991 & 8.65 & @xmath1250.91 & + fe i & 7118.119 & 5.01 & @xmath1251.39 & ( adjusted ) + c i & 7119.656 & 8.64 & @xmath1251.13 & ( adjusted ) + fe i & 7120.022 & 4.56 & @xmath1251.91 & ( adjusted ) + + o i & 6156.737 & 10.74 & @xmath1251.52 & + o i & 6156.755 & 10.74 & @xmath1250.93 & + o i & 6156.778 & 10.74 & @xmath1250.73 & + fe i&6157.725 & 4.08 & @xmath1251.26 & + o i & 6158.149 & 10.74 & @xmath1251.89 & o 6158 + o i & 6158.172 & 10.74 & @xmath1251.03 & o 6158 + o i & 6158.187 & 10.73 & @xmath1250.44 & o 6158 + note . @xmath159 is the air wavelength ( in @xmath33 ) , @xmath160 is the lower excitation potential ( in ev ) , and @xmath73 is the logarithm of @xmath55 ( statistical weight of the lower level ) times @xmath161 ( absorption oscillator strength ) . these data were taken primarily from the compilation of kurucz and bell ( 1995 ) , though empirically adjusted `` solar @xmath74 values '' were applied in several cases ( remarked as `` adjusted '' in column 5 ) . regarding lithium , we considered only the component lines of @xmath162li , neglecting those of @xmath163li .
|
in an attempt to carry out a systematic study on the behavior of the photospheric abundances of li , c , and o ( along with fe ) for hyades main - sequence stars in the range of 50007000 k , we conducted an extensive spectrum - synthesis analysis applied to four spectral regions ( comprising lines of fe - group elements , li i 6708 line , c i 71117119 lines , and o i 61568 lines ) based on the high - dispersion spectra of 68 selected f g type stars belonging to this cluster . the abundances of c and o turned out to be fairly uniform in a marginally supersolar level such like the case of fe :[c / h] ,[o / h] , and[fe / h] , suggesting that the primordial abundances are almost retained for these elements . strictly , however , they show a slightly increasing trend with a decrease in ( typically on the order of dex k ) ; while this might be due to an improper choice of atmospheric parameters , we found it hard to give a quantitatively reasonable explanation . regarding li , we confirmed the well - known-dependent trend in the li abundance reported so far ( a conspicuous li - trough at 6300 k k and a progressive decrease toward a lower at k ) , which means that the surface li of hyades stars is essentially controlled only by and other parameters such as the rotational velocity are almost irrelevant .
|
in an attempt to carry out a systematic study on the behavior of the photospheric abundances of li , c , and o ( along with fe ) for hyades main - sequence stars in the range of 50007000 k , we conducted an extensive spectrum - synthesis analysis applied to four spectral regions ( comprising lines of fe - group elements , li i 6708 line , c i 71117119 lines , and o i 61568 lines ) based on the high - dispersion spectra of 68 selected f g type stars belonging to this cluster . the abundances of c and o turned out to be fairly uniform in a marginally supersolar level such like the case of fe :[c / h] ,[o / h] , and[fe / h] , suggesting that the primordial abundances are almost retained for these elements . strictly , however , they show a slightly increasing trend with a decrease in ( typically on the order of dex k ) ; while this might be due to an improper choice of atmospheric parameters , we found it hard to give a quantitatively reasonable explanation . regarding li , we confirmed the well - known-dependent trend in the li abundance reported so far ( a conspicuous li - trough at 6300 k k and a progressive decrease toward a lower at k ) , which means that the surface li of hyades stars is essentially controlled only by and other parameters such as the rotational velocity are almost irrelevant .
|
cond-mat0204264
|
i
|
recently , many studies have been devoted to fractal structures induced by the chaotic dynamics in the phase - space of non - equilibrium statistical systems and , in particular , to the fractal character of the hydrodynamic modes governing the exponential relaxation of the system to the thermodynamic equilibrium @xcite . this fractal character has been related to the entropy production in the approach of equilibrium , for a diffusive multibaker model @xcite . previous works had shown similar results for the entropy production in non - equilibrium steady states @xcite . on the other hand , the fractal dimension characterizing the hydrodynamic modes of diffusion has been related to the diffusion coefficient and to the positive lyapunov exponent by a relation , first obtained for diffusive multibaker maps @xcite , and proved for general chaotic systems with two degrees of freedom @xcite . the construction of the hydrodynamic modes in refs . @xcite gives us a new approach to relate macroscopic transport coefficients and microscopic chaotic properties . it establishes a link between the irreversible transport processes and the reversible microscopic dynamics . two other approaches are the so - called escape rate formalism and the thermostated system approach . the first approach can be applied to open systems with absorbing boundaries @xcite . the escape of trajectories leads to the formation of a fractal repeller , consisting in the set of orbits forever trapped within the absorbing boundaries . under such conditions , the transport properties can be related to the positive lyapunov exponents and the kolmogorov - sinai entropy per unit time as well as to fractal dimensions @xcite . in the second approach , an external field is applied to the system , and in order to keep the kinetic energy constant , a special force acts on the particles as a heat pump @xcite . the system is still time - reversible but does not preserve phase - space volumes anymore . the trajectories converge to a fractal attractor . in thermostated systems , the transport properties have been related to the sum of lyapunov exponents @xcite . in the new approach of refs . @xcite , the advantage is that the fractal curve considered is directly related to the hydrodynamic modes of relaxation towards thermodynamic equilibrium . it does not require absorbing boundaries neither a thermostat . in the present paper , we extend the new approach of refs . @xcite to reaction - diffusion processes . reaction - diffusion processes are of fundamental importance to understand self - organization in physico - chemical systems @xcite . in chemical systems , the microscopic mechanisms of the relaxation toward the thermodynamic equilibrium are still poorly understood already for simple reactions . for the purpose of contributing to this important question , we consider here a simple reactive process of isomerization . the particle is supposed to perform a deterministic motion among static scatterers and to carry a color @xmath0 or @xmath1 . when colliding on some special scatterers , playing the role of catalysts , its color changes instantaneously with a given probability @xmath2 , @xmath3 for this class of systems , we obtain an expression of the hausdorff dimension of the reactive modes of relaxation in terms of the reactive dispersion relation and of a function @xmath4 generalizing the ruelle topological pressure @xmath5 . in the long wavelength limit , we then infer a relation between the hausdorff dimension of the modes , the reactive diffusion coefficient , the reaction rate , and two derivatives of the function @xmath4 . in the limit @xmath6 , we recover a relation previously derived for the diffusive case @xcite . this new expression relates the macroscopic transport and reaction processes to the microscopic underlying dynamics . the paper is organized as follows . in section [ theory ] , we generalize the work of refs . @xcite to the reactive case considered here . in section [ triadic ] , we test our relation on a reactive multibaker model . we next study the case of a two - dimensional periodic reactive lorentz gas in section [ lorentz ] . conclusions are drawn in section [ conclusions ] .
|
in chaotic reaction - diffusion systems with two degrees of freedom , the modes governing the exponential relaxation to the thermodynamic equilibrium present a fractal structure which can be characterized by a hausdorff dimension . for long wavelength modes , this dimension is related to the lyapunov exponent and to a reactive diffusion coefficient . = msbm10
|
in chaotic reaction - diffusion systems with two degrees of freedom , the modes governing the exponential relaxation to the thermodynamic equilibrium present a fractal structure which can be characterized by a hausdorff dimension . for long wavelength modes , this dimension is related to the lyapunov exponent and to a reactive diffusion coefficient . this relationship is tested numerically on a reactive multibaker model and on a two - dimensional periodic reactive lorentz gas . the agreement with the theory is excellent . * key words:*reaction - diffusion systems ; hydrodynamic modes ; fractals ; microscopic chaos . = msbm10
|
1008.3567
|
i
|
there is growing interest in systems composed of individual monomers that interact via resonant dipole - dipole interaction . upon electronic excitation this transition dipole - dipole interaction between the monomers is responsible for a collective behaviour of these systems . besides the classical examples like van - der - waals crystals @xcite , aggregates of organic dyes @xcite , and light harvesting units of plants , algae , and bacteria @xcite many new systems have emerged . examples are ultra - cold rydberg atoms @xcite , quantum dots @xcite , assemblies of nano - particles @xcite , and recently also hybrid systems @xcite . the common approach to describe these aggregates is to treat the monomers as electronic two - level systems . besides the electronic degrees of freedom , however , one often has to take into account nuclear degrees of freedom explicitly . for instance in the case of molecular aggregates @xcite , which will serve as the primary example in this work , the electronic excitation of a monomer couples strongly to internal vibrational modes of the monomer and to modes of the surroundings . often , the monomer spectrum is dominated by one vibrational progression which is considerably broadened . a commonly applied approximation is then to only consider one effective mode corresponding to this progression @xcite and to take the broadening into account by convoluting with some lineshape function which is usually assumed to be gaussian . it has been shown that using this approach already important features of experimental spectra can be reproduced @xcite . although the resulting spectra reveal many characteristics of the aggregates , important aspects like the detailed shapes of the j - band @xcite and the h - band @xcite can not adequately be described by considering only one vibrational mode . on the other hand , the exact inclusion of only _ one _ vibrational mode already complicates the treatment of molecular aggregates considerably , so that this approach is restricted to small aggregates . this problem becomes even more serious when one attempts to include more modes in this manner . when the interaction of the chromophore monomers with the environment is negligible ( e.g. in high resolution spectroscopy in helium nanodroplets @xcite ) , then the _ explicit _ inclusion of vibrational modes is of great importance . however , for typical spectra in solution or in a solid state matrix ( where a strong coupling between the chromophores and the environment is present ) it seems better to use a continuum of vibrations that couple to the electronic excitation to account for the large number of environmental degrees of freedom . this interaction between the electronic excitation and the vibrations is conveniently encoded in the so - called spectral density . it describes the frequency - dependent coupling between the system ( the electronic degrees of freedom ) and the ( continuum of ) harmonic oscillators . in the markov case the spectral density is assumed to be flat in the relevant frequency regions . clearly , for the considered monomers this assumption does not hold . due to strong interaction with some internal vibrational modes , the spectral density will be highly structured ( i.e. frequency - dependent ) , indicating that a non - markovian theoretical framework is required . an approach to tackle this complicated problem was recently presented in ref @xcite . the method is based on the non - markovian quantum state diffusion ( nmqsd ) description of open quantum systems @xcite . here , the system part is chosen to contain only the electronic degrees of freedom which interact with a non - markovian environment ( the bath ) comprising all vibrations ( internal modes of the monomers as well as external modes ) . one then can derive a stochastic evolution equation for states in the ( small ) space of the system part . however , solving the exact evolution equation turns out to be very difficult due to the appearance of a functional derivative w.r.t . functionals containing the bath degrees of freedom . to overcome these difficulties , in an approximation only the zeroth order of a functional expansion ( we will refer to it as zofe approximation ) of the problematic term is taken into account @xcite . for several ( simple ) problems this procedure has been shown to give the exact result @xcite . however , for more complex problems like the molecular aggregates studied in this work , the range of validity of the approximation is not clear . it should be noted that the nmqsd approach in combination with the zofe approximation provides a very efficient calculation scheme : in order to obtain the absorption spectrum of the aggregate and the energy transfer between the monomers , the equations one has to solve are in the small hilbert space of the electronic degrees of freedom alone @xcite . one aim of the present paper is to examine the validity of the zofe approximation leading to the calculation scheme presented in ref . @xcite . to this end , we compare with an approach where so - called pseudomodes @xcite are included into the system part together with the electronic degrees of freedom . the electronic degrees of freedom now couple only to the pseudomodes , the pseudomodes in turn are then coupled to a markovian bath . for a spectral density consisting of a sum of lorentzians the pseudomode method is exact ( taking one pseudomode for each lorentzian ) . this allows to directly compare the approximative nmqsd - zofe treatment with exact calculations . however , due to the inclusion of the pseudomodes into the system part , the numerical solution of the corresponding evolution equation is limited to a rather small number of monomers in the aggregate with only a few pseudomodes , i.e. only a few lorentzians in the spectral density . besides the possibility of comparing the nmqsd - zofe approach with exact calculations , the pseudomode method has also some physical significance : one can think of the pseudomodes as internal vibrational modes of a chromophore that strongly couple to the electronic excitation and which are damped by the coupling to the remaining vibrations . the comparison between zero temperature absorption spectra of small aggregates calculated using the nmqsd - zofe approach and spectra obtained from the exact pseudomode approach shows that in the cases considered there is mostly quite good agreement between the two approaches . we will discuss in which situations the approximative result of the nmqsd - zofe approach is expected to deviate from the exact solution . the structure of this paper is as follows : in section [ mod_of_agg ] , we introduce the hamiltonian of the aggregate . the hamiltonian is written as the sum of a system part ( containing only electronic degrees of freedom ) , an environmental part ( containing all vibrational modes ) , and the part of the interaction between electronic degrees of freedom and vibrations . in the following section [ sec : absofagg ] , the basic formulas that are used to calculate the absorption spectrum are given by specifying the initial state and introducing the dipole correlation function . in section [ sec : qsd_approach ] , the general non - markovian schrdinger equation ( nmqsd ) approach is applied to the case of an aggregate . it is shown how the absorption spectrum can be obtained in this approach . next , in section [ sec_zofe ] , the zofe approximation is introduced . then , in section [ sec : exact_solv_model ] , the pseudomode ( pm ) approach is presented . in section [ sec : compar_nmqsd_pm ] , the nmqsd - zofe absorption spectra are compared with exact pm spectra . we conclude in section [ conclusion ] by summarizing our findings . details of the calculations and minor results have been placed in the appendices . in appendix [ ap_exact_solvable ] , two exactly solvable cases ( namely that of non - interacting monomers and the case where the coupling to the vibrations can be considered to be markovian ) are discussed . in appendix [ sec : ap_absorppm ] , it is shown how to obtain the absorption spectrum using the pm approach . the numerical implementation is discussed . finally , in appendix [ sec : ap_equiv_models_with_and_without_pm ] , it is shown that for a lorentzian spectral density the absorption spectrum obtained from the exact nmqsd approach is equal to the spectrum obtained from the pm method .
|
in many molecular systems one encounters the situation where electronic excitations couple to a quasi - continuum of phonon modes . that continuum may be highly structured e.g. due to some weakly damped high frequency modes . to handle such a situation , an approach combining the non - markovian quantum state diffusion ( nmqsd ) description of open quantum systems with an efficient but abstract approximation was recently applied to calculate energy transfer and absorption spectra of molecular aggregates [ roden , eisfeld , wolff , strunz , prl 103 ( 2009 ) 058301 ] . to explore the validity of the used approximation for such complicated systems , in the present work we compare the calculated ( approximative ) absorption spectra with exact results . these are obtained from the method of pseudomodes , which we show to be capable of determining the exact spectra for small aggregates and a few pseudomodes . it turns out that in the cases considered , the results of the two approaches mostly agree quite well .
|
in many molecular systems one encounters the situation where electronic excitations couple to a quasi - continuum of phonon modes . that continuum may be highly structured e.g. due to some weakly damped high frequency modes . to handle such a situation , an approach combining the non - markovian quantum state diffusion ( nmqsd ) description of open quantum systems with an efficient but abstract approximation was recently applied to calculate energy transfer and absorption spectra of molecular aggregates [ roden , eisfeld , wolff , strunz , prl 103 ( 2009 ) 058301 ] . to explore the validity of the used approximation for such complicated systems , in the present work we compare the calculated ( approximative ) absorption spectra with exact results . these are obtained from the method of pseudomodes , which we show to be capable of determining the exact spectra for small aggregates and a few pseudomodes . it turns out that in the cases considered , the results of the two approaches mostly agree quite well . the advantages and disadvantages of the two approaches are discussed .
|
1505.04256
|
i
|
radio relics are diffuse radio sources found in the outskirts of galaxy clusters and they are thought to trace synchrotron - emitting cosmic - ray ( cr ) electrons accelerated via diffusive shock acceleration ( dsa ) at cluster shocks ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? so far several dozens of clusters have been observed to have radio relics with a variety of morphologies and most of them are considered to be associated with cluster merger activities ( see for reviews , e.g. , * ? ? ? * ; * ? ? ? for instance , double radio relics , such as the ones in zwcl0008.8 + 5215 , are thought to reveal the bow shocks induced by a binary major merger @xcite . on the other hand , recently it was shown that shocks induced by the infall of the warm - hot intergalactic medium ( whim ) along adjacent filaments into the hot intracluster medium ( icm ) can efficiently accelerate cr electrons , and so they could be responsible for some radio relics in the cluster outskirts ( see , e.g. , * ? ? ? * ) . the radio relic 1253 + 275 in coma cluster observed in both radio @xcite and x - ray @xcite provides an example of such infall shocks . the so - called sausage relic in ciza j2242.8 + 5301 ( @xmath7 ) contains a thin arc - like structure of @xmath8 kpc width and @xmath9 mpc length , which could be represented by a portion of spherical shell with radius @xmath5 mpc @xcite . unique features of this giant radio relic include the nearly uniform surface brightness along the length of the relic and the strong polarization of up to @xmath10 with magnetic field vectors aligned with the relic @xcite . a temperature jump across the relic that corresponds to a @xmath11 shock has been detected in x - ray observations @xcite . this was smaller than @xmath12 estimated from the above radio observation . several examples of mpc - scale radio relics include the toothbrush relic in 1rxs j0603.3 with a peculiar linear morphology @xcite and the relics in a3667 @xcite and a3376 @xcite . the shock mach numbers of radio relics estimated based on x - ray observation are often lower than those inferred from the radio spectral index using the dsa model , for instance , in the toothbrush relic @xcite and in the radio relic in a2256 @xcite . although such giant radio relics are quite rare , the fraction of x - ray luminous clusters hosting some radio relics is estimated to be @xmath13 % or so @xcite . through a number of studies using cosmological hydrodynamical simulations , it has been demonstrated that during the process of hierarchical structure formation , abundant shocks are produced in the large - scale structure of the universe , especially in clusters ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? considering that the characteristic time - scale of cluster dynamics including mergers is @xmath14 gyr , typical cluster shocks are expected to last for about the same period . yet , the number of observed radio relics , which is thought to trace such shocks , is still limited . so it is plausible to conjecture that cluster shocks may ` turn on ' to emit synchrotron radiation only for a fraction of their lifetime . one feasible scenario is that a cluster shock lights up in radio when it sweeps up a fossil cloud , a magnetized icm gas with fossil relativistic electrons left over from either a radio jet from agn or a previous episode of shock / turbulence acceleration ( see the discussions in section 2.5 and figure 1 ) . pre - exiting seed electrons and/or enhanced magnetic fields are the requisite conditions for possible lighting - up of relic shocks . in particular , the elongated shape with _ uniform _ surface brightness and high polarization fraction of radio emission in the sausage relic , may be explained , if a mpc - scale thermal gas cloud , containing fossil relativistic electrons and permeated with regular magnetic field of a few to several @xmath15 g , is adopted . a more detailed description will be given later in section 2.5 . in this picture , fossil electrons are expected to be re - accelerated for less than cloud - crossing time ( @xmath16 ) , which is much shorter than the cluster dynamical time - scale . in addition , only occasional encounters with fossil clouds combined with the short acceleration duration could alleviate the strong constraints on the dsa theory based on non - detection of @xmath17-ray emission from clusters by fermi - lat @xcite . a similar idea has been brought up by @xcite , who reported the discovery of a mpc - scale , elongated relic in the bullet cluster 1e 0657 - 55.8 . they also proposed that the arc - like shape of uniform surface brightness in some radio relics may trace the underlying regions of pre - existing , seed electrons remaining from old radio lobes . on the other hand , @xcite suggested that radio relics could be explained by revival of fossil radio plasma by compression due to a passage of a shock , rather than dsa . in a follow - up study , @xcite showed using mhd simulations that a cocoon of hot radio plasma swept by a shock turns into a filamentary or toroidal structure . although this scenario remains to be a viable explanation for some radio relics , it may not account for the uniform arc - like morphology of the sausage relic . it is now well established , through observations of radio halos / relics and faraday rotation measures of background radio sources , that the icm is permeated with @xmath15g - level magnetic fields ( e.g. * ? ? ? * ; * ? ? ? the observed radial profile of magnetic field strength tends to peak at the center with a few @xmath15 g and decrease outward to @xmath18 g in the cluster outskirts @xcite . a variety of physical processes that could generate and amplify magnetic fields in the icm have been suggested : primordial processes , plasma processes at the recombination epoch , and biermann battery mechanism , combined with turbulence dynamo , in addition to galactic winds and agn jets ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? given the fact that @xmath19 g fields are required to explain the amplitude and width of the observed radio flux profile of the sausage relic ( e.g. * ? ? ? * ; * ? ? ? * ) , the presence of a cloud with enhanced magnetic fields of several @xmath15 g might be preferred to the background fields of @xmath20 g in the cluster periphery . alternatively , @xcite showed that the postshock magnetic fields can be amplified to @xmath21 g level leading to high degrees of polarization , if there exists dynamically significant turbulence in the upstream region of a curved shock . although it is well accepted that magnetic fields can be amplified via various plasma instabilities at collisionless shocks , the dependence on the shock parameters such as the shock sonic and alfvnic mach numbers , and the obliquity of background magnetic fields remains to be further investigated ( see * ? ? ? for example , the acceleration of protons and ensuing magnetic field amplification via resonant and non - resonant streaming instabilities are found to be ineffective at perpendicular shocks @xcite . in several studies using cosmological hydrodynamical simulations , synthetic radio maps of simulated clusters were constructed by identifying shocks and adopting models for dsa of electrons and magnetic field amplification @xcite . in particular , @xcite demonstrated , by generating mock radio maps of simulated cluster samples , that radio emission tends to increase toward the cluster periphery and peak around @xmath22 ( where @xmath23 is the virial radius ) , mainly because the kinetic energy dissipated at shocks peaks around @xmath24 . as a result , radio relics are rarely found in the cluster central regions . re - acceleration of fossil relativistic electrons by cosmological shocks during the large scale structure formation has been explored by @xcite . the radio emitting shocks in these studies look like segments of spherical shocks , moving from the cluster core region into the periphery . we presume that they are generated mostly as a consequence of major mergers or energetic infalls of the whim along adjacent filaments . so it seems necessary to study spherical shocks propagating through the cluster periphery , rather than interpreting the radio spectra by dsa at _ steady planar _ shocks , in order to better understand the nature of radio relics @xcite . according to the dsa theory , in the case of a _ steady planar _ shock with _ constant _ postshock magnetic field , the electron distribution function at the shock location becomes a power - law of @xmath25 , and so the synchrotron emissivity from those electrons becomes a power - law of @xmath26 . the power - low slopes depend only on the shock sonic mach number , @xmath27 , and are given as @xmath28 and @xmath29 for the gasdynamic shock with the adiabatic index @xmath30 @xcite . here we refer @xmath31 as the injection spectral index for a steady planar shock with constant postshock magnetic field . then , the volume - integrated synchrotron spectrum downstream of the shock also becomes a simple power - law of @xmath32 with the spectral index @xmath33 above the break frequency , @xmath34 , since electrons cool via synchrotron and inverse - compton ( ic ) losses behind the shock ( e.g. , * ? ? ? * ; * ? ? ? ' as the spectral index of the flux density , @xmath35 for unresolved sources , so in that case @xmath36 is the same as @xmath37 . here @xmath38 is defined as the spectral index of the _ local _ emissivity , @xmath39 . see equations ( [ alpha])-([anu ] ) . ] such predictions of the dsa theory have been applied to explain the observed properties of radio relics , e.g. , the relation between the injection spectral index and the volume - integrated spectral index , and the gradual steepening of spatially resolved spectrum downstream of the shock . @xcite performed time - dependent , dsa simulations of cr electrons for _ steady planar _ shocks with @xmath40 and constant postshock magnetic fields . several models with thermal leakage injection or pre - existing electrons were considered in order to reproduce the surface brightness and spectral aging profiles of radio relics in ciza j2242.8 + 5301 and zwcl0008.8 + 5215 . adopting the same geometrical structure of radio - emitting volume as described in section 2.5 , they showed that the synchrotron emission from shock accelerated electrons could explain the observed profiles of the radio flux , @xmath41 , of the sausage relic , and the observed profiles of both @xmath41 and @xmath42 of the relic in zwcl0008.8 + 5215 . here @xmath43 is the distance behind the projected shock edge in the plane of the sky . in the case of spherically expanding shocks with varying speeds and/or nonuniform magnetic field profiles , on the other hand , the electron spectrum and the ensuing radio spectrum could deviate from those simple power - law forms , as shown in @xcite . then even the injection slope should vary with the frequency , @xmath44 . here we follow the evolution of a spherical shock expanding outward in the cluster outskirts with a decreasing density profile , which may lead to a curvature in both the injected spectrum and the volume - integrated spectrum . moreover , if the shock is relatively young or the electron acceleration duration is short ( @xmath45 myr ) , then the break frequency falls in @xmath46 ghz and the volume - integrated spectrum of a radio relic would steepen gradually with the spectral index from @xmath31 to @xmath47 over @xmath2 ( e.g. * ? ? ? * ) . in the case of the sausage relic , @xcite and @xcite originally reported observations of @xmath48 and @xmath49 , which imply a shock of @xmath12 . @xcite , however , found a spectral steepening of the volume - integrated spectrum at 16 ghz , which would be inconsistent with the dsa model for a steady planar shock . moreover , @xcite , by performing a spatially - resolved spectral fitting , revised the injection index to a steeper value , @xmath50 . then , the corresponding shock mach number is reduced to @xmath51 . they also suggested that the spectral age , calculated under the assumption of freely - aging electrons downstream of a steady planar shock , might not be compatible with the shock speed estimated from x - ray and radio observations . also @xcite reported that for the relic in a2256 , the volume - integrated index steepens from @xmath52 for @xmath53 mhz to @xmath54 for @xmath55 ghz , which was interpreted as a broken power - law . discoveries of radio relic shocks with @xmath56 in recent years have brought up the need for more accurate understanding of injections of protons and electrons at weak collisionless shocks , especially at high plasma beta ( @xmath57 ) icm plasmas ( e.g. * ? ? ? here @xmath58 is the ratio of the gas to magnetic field pressure . injection of electrons into the fermi 1st - order process has been one of long - standing problems in the dsa theory for astrophysical shocks , because it involves complex plasma kinetic processes that can be studied only through full particle - in - cell ( pic ) simulations ( e.g. * ? ? ? * ; * ? ? ? it is thought that electrons must be pre - accelerated from their thermal momentum to several times the postshock thermal proton momentum to take part in the dsa process , and electron injection is much less efficient than proton injection due to smaller rigidity of electrons . several recent studies using pic simulations have shown that some of incoming protons and electrons gain energies via shock drift acceleration ( sda ) while drifting along the shock surface , and then the particles are reflected toward the upstream region . those reflected particles can be scattered back to the shock by plasma waves excited in the foreshock region , and then undergo multiple cycles of sda , resulting in power - law suprathermal populations ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? . such ` self pre - acceleration ' of thermal electrons in the foreshock region could be sufficient enough even at weak shocks in high beta icm plasmas to explain the observed flux level of radio relics . in these pic simulations , however , subsequent acceleration of suprathermal electrons into full dsa regime has not been explored yet , because extreme computational resources are required to follow the simulations for a large dynamic range of particle energy . the main reasons that we implement the fossil electron distribution , instead of the shock injection only case , are ( 1 ) the relative scarcity of radio relics compared to the abundance of shocks expected to form in the icm , ( 2 ) the peculiar uniformity of the surface brightness of the sausage relic , and ( 3 ) curved integrated spectra often found in some radio relics , implying the acceleration duration @xmath45 myr , much shorter than the cluster dynamical time . in this paper , we consider a dsa model for radio relics ; a spherical shock moves into a magnetized gas cloud containing fossil relativistic electrons , while propagating through a density gradient in the cluster outskirts . specifically , we perform time - dependent dsa simulations for several spherical shock models with the parameters relevant for the sausage relic . we then calculate the surface brightness profile , @xmath59 , and the volume - integrated radio spectrum , @xmath60 , by adopting a specific geometrical structure of shock surface , and compare them with the observational data of the sausage relic . in section 2 , the dsa simulations and the model parameters are described . the comparison of our results with observations is discussed in section 3 . a brief summary is given in section 4 .
|
in order to understand certain observed features of arc - like giant radio relics such as the rareness , uniform surface brightness , and curved integrated spectra , we explore a diffusive shock acceleration ( dsa ) model for radio relics in which a spherical shock impinges on a magnetized cloud containing fossil relativistic electrons . toward this end , the surface brightness profile of radio - emitting postshock region and the volume - integrated radio spectrum are calculated and compared with observations . we find that the observed width of the sausage relic can be explained reasonably well by shocks with speed and sonic mach number .
|
in order to understand certain observed features of arc - like giant radio relics such as the rareness , uniform surface brightness , and curved integrated spectra , we explore a diffusive shock acceleration ( dsa ) model for radio relics in which a spherical shock impinges on a magnetized cloud containing fossil relativistic electrons . toward this end , we perform dsa simulations of spherical shocks with the parameters relevant for the sausage radio relic in cluster ciza j2242.8 + 5301 , and calculate the ensuing radio synchrotron emission from re - accelerated electrons . three types of fossil electron populations are considered : a delta - function like population with the shock injection momentum , a power - law distribution , and a power - law with an exponential cutoff . the surface brightness profile of radio - emitting postshock region and the volume - integrated radio spectrum are calculated and compared with observations . we find that the observed width of the sausage relic can be explained reasonably well by shocks with speed and sonic mach number . these shocks produce curved radio spectra that steepen gradually over with break frequency ghz , if the duration of electron acceleration is myr . however , the abrupt increase of spectral index above ghz observed in the sausage relic seems to indicate that additional physical processes , other than radiative losses , operate for electrons with .
|
1505.04256
|
i
|
we propose a model that may explain some characteristics of giant radio relics : the relative rareness , uniform surface brightness along the length of thin arc - like radio structure , and spectral curvature in the integrated radio spectrum over @xmath336 ghz . in the model , a spherical shock encounters an elongated cloud of the icm thermal gas that is permeated by enhanced magnetic fields and an additional population of fossil relativistic electrons . as a result of the shock passage , the fossil electrons are re - accelerated to radio - emitting energies ( @xmath337 ) , resulting in a birth of a giant radio relic . in order to explore this scenario , we have performed time - dependent , dsa simulations of spherical shocks with the parameters relevant for the sausage radio relic in cluster ciza j2242.8 + 5301 . in the fiducial model , the shock decelerates from @xmath338 ( @xmath339 ) to @xmath340 ( @xmath164 ) at the acceleration age of 60 myr . the seed , fossil electrons with @xmath341 are assumed to be injected into the cr population , which is subsequently re - accelerated to higher energies . such shocks are expected to produce the electron energy spectrum , @xmath342 , resulting in the synchrotron radiation spectrum with the injection index , @xmath343 , and the integrated index , @xmath330 , at high frequencies ( @xmath344 ghz ) . we consider various models with a range of shock parameters , different upstream gas density profiles , different downstream magnetic field profiles , and three types of fossil electron populations , as summarized in table 1 . adopting a ribbon - like curved shock surface and the associated downstream volume , which are constrained by the extension angle ( or viewing depth ) of @xmath203 as detailed in section 2.5 ( e.g * ? ? ? * ; * ? ? ? * ) , the radio surface brightness profile , @xmath242 , and the volume - integrated spectrum , @xmath60 , are calculated . \1 ) two observables , the break frequency in the integrated synchrotron spectrum , @xmath34 , and the width of the synchrotron emission region behind the shock , @xmath231 , can have identical values for two values of postshock magnetic field strength ( see equations [ [ fbr ] ] and [ [ lwidth ] ] ) . \2 ) the observed width of the surface brightness projected onto the sky plane , @xmath234 , strongly depends on the assumed value of @xmath237 ( see figure 4 ) . so @xmath234 may not be used to estimate the postshock magnetic field strength , unless the projection effects can be modeled properly . \3 ) the integrated synchrotron spectrum is expected to have a spectral curvature that runs over a broad range of frequency , typically for @xmath345 . for a shock of @xmath346 with the postshock magnetic field strength , @xmath347 or @xmath348 , the integrated spectral index increases gradually from @xmath343 to @xmath349 over @xmath285 ghz , if the duration of the shock acceleration is @xmath286 myr \4 ) assuming that the upstream sound speed is @xmath350 ( @xmath351 kev ) as inferred from x - ray observation , a shock of @xmath346 and @xmath352 ( e.g. , * sa1 * model ) can reasonably explain the observed width , @xmath353 kpc @xcite , and the curved integrated spectrum of the sausage relic @xcite . * sb1 * model with a shock of @xmath354 , however , produces the integrated spectrum that seems too flat to explain the observed spectrum above @xmath355 ghz . \5 ) we also consider two toy models with power - law electron populations with exponential cutoffs at @xmath356 , @xmath357 $ ] ( * sc1pex1 * and * sc1pex2 * models ) . they may represent the electron populations that were produced earlier and then have cooled down to @xmath356 . * sc1pex1 * model with a weaker shock ( @xmath189 ) reproduces better the characteristics of the observed integrated spectrum . but the steepening of the integrated spectrum due to radiative cooling alone may not explain the strong spectral curvature above 1.5 ghz toward 16 ghz . \6 ) this strong curvature at @xmath5 ghz may imply that the downstream electron energy spectrum is influenced by some additional physical processes other than radiative losses , because the integrated spectrum of radiatively cooled electrons steepens with the frequency only gradually . this conclusion is likely to remain unchanged even in the case where the observed spectrum consists of the synchrotron emission from multiple shocks with different mach numbers , as long as the postshock electrons experience only simple radiative cooling . the authors thank the anonymous referee for his / her thorough review and constructive suggestions that lead to a significant improvement of the paper . hk was supported by basic science research program through the national research foundation of korea ( nrf ) funded by the ministry of education ( 2014r1a1a2057940 ) . dr was supported by the national research foundation of korea through grant nrf-2014m1a7a1a03029872 and nrf-2012k1a3a7a03049606 . lccccccc & @xmath358 & 2.5 & @xmath359 & @xmath360 & @xmath361 & @xmath362 & @xmath363 + * sa1b * & @xmath358 & 0.25 & @xmath364 & @xmath360 & @xmath361 & @xmath362 & @xmath363 + * sa1p * & @xmath358 & 2.5 & @xmath359 & @xmath360 & @xmath361 & @xmath362 & @xmath365 + * sa2*@xmath366 & @xmath358 & 2.0 & @xmath367 & @xmath360 & @xmath361 & @xmath362 & @xmath363 + * sa3 * & @xmath368 & 2.5 & @xmath359 & @xmath360 & @xmath369 & @xmath370 & @xmath363 + * sa4*@xmath366 & @xmath78 & 2.0 & @xmath367 & @xmath360 & @xmath371 & @xmath372 & @xmath363 + * sb1 * & @xmath358 & 2.5 & @xmath373 & @xmath374 & @xmath361 & @xmath375 & @xmath363 + * sc1pex1*@xmath376 & @xmath358 & 2.5 & @xmath377 & @xmath360 & @xmath378 & @xmath379 & @xmath380 + * sc1pex2*@xmath381 & @xmath358 & 2.5 & @xmath377 & @xmath360 & @xmath378 & @xmath379 & @xmath382 + -0.8 cm , and the magnetic field , @xmath177 , of the spherical shock in * sa1 * ( top two panels ) and * sa4 * ( bottom two panels ) models at the acceleration age , @xmath383 , 60 , 110 myr ( black solid , red dotted , and blue dashed lines , respectively).,title="fig : " ] 1.5 cm kev ( red solid line ) , fossil electrons with @xmath384 in * sa1 * model ( black solid ) , a power - law of @xmath385 in * sa1p * model ( blue solid ) , and a power - law with an exponential cutoff in * sc1pex1 * model ( green solid ) . see table 1 for the different model parameters . the vertical dashed line demarcates the injection momentum , @xmath139 . an unspecified suprathermal distribution is shown in the red dotted line , but the electron distribution below @xmath139 is not relevant for dsa . , title="fig : " ] , @xmath241 , and @xmath242 for a steady _ planar _ shock with @xmath238 and @xmath239 at the acceleration age of 80 myr . here , @xmath386 is the downstream distance away from the shock , while @xmath43 is the distant behind the shock projected in the sky plane . top : @xmath387 is plotted for the electron lorentz factor , @xmath388 ( black solid line ) , @xmath389 ( red dotted ) , @xmath390 ( blue dashed ) , @xmath391 ( green long dashed ) , @xmath392 ( magenta dot - dashed ) , and @xmath393 ( cyan dot - long dashed ) . middle : @xmath394 is plotted for the observation frequency , @xmath395 ghz ( black solid line ) , 0.594 ghz ( red dotted ) , 1.33 ghz ( blue dashed ) , 2.36 ghz ( green long dashed ) , 16.7 ghz ( magenta dot - dashed ) , and 29.8 ghz ( cyan dot - long dashed ) . bottom : @xmath396 is plotted for the extension angle @xmath397 , and @xmath398 ( solid , dotted , and dashed lines , respectively ) . the red ( blue ) curves are for @xmath399 ( 1.33 ghz ) . the redshift of the host cluster is assumed to be @xmath7 here . ] -0.2 cm ( upper four panels ) and * sa1b * model with @xmath167 ( lower four panels ) at @xmath383 , 60 , 110 myr ( solid , dotted , and dashed lines , respectively ) . left : electron distribution function at the shock position , @xmath400 ( black lines ) , volume - integrated electron distribution function , @xmath401 ( red lines ) , and slopes of electron distribution functions , @xmath402 ( black lines ) and @xmath403 ( red lines ) . right : synchrotron spectrum at the shock position , @xmath404 ( black lines ) , volume - integrated synchrotron spectrum , @xmath405 ( red lines ) , and synchrotron spectral indices , @xmath406 ( black lines ) and @xmath407 ( red lines).,title="fig : " ] -0.3 cm , 600 , and 1400 mhz ( black solid , red dotted , blue dashed lines ) in * sa1 * , * sa1b * , * sa1p * , and * sc1pex * models ( from top to bottom panels ) . see table 1 for the model parameters . the results are shown at the acceleration age , @xmath136 , specified in each panel . for the extension angle , @xmath408 is adopted . the quantity @xmath409 is plotted with a scale factor @xmath298 to present it in the same ordinate scale for all the models.,title="fig : " ] -0.3 cm , 600 , and 1400 mhz ( black solid , red dotted , blue dashed lines ) in * sa2 * , * sa3 * , * sa4 * , and * sb1 * models ( from top to bottom panels ) . see table 1 for the model parameters . the results are shown at @xmath410 30 , 60 , and 110 myr ( from left to right panels ) . for the extension angle , @xmath408 is adopted . the quantity @xmath409 is plotted with a scale factor @xmath298 to present it in the same ordinate scale for all the models.,title="fig : " ] -0.3 cm , and its spectral index , @xmath210 , at @xmath411 , 60 , and 110 myr ( black solid , red dotted , and blue dashed lines , respectively ) for * sa1 * , * sa1b * , and * sa1p * models ( top three panels ) , and at @xmath411 , 80 , and 126 myr ( black solid , red dotted , and blue dashed lines , respectively ) for * sc1pex1 * model ( bottom panel ) . filled circles show the data from @xcite , scaled to fit by eye the red dotted line ( @xmath412 ) for * sa1 * model . observational errors are small , about @xmath413 , except for the data at 16 ghz with @xmath414 ( shown in a vertical bar).,title="fig : " ] -0.3 cm , and its spectral index , @xmath210 , at @xmath411 , 60 , and 110 myr ( black solid , red dotted , and blue dashed lines , respectively ) for * sa2 * , * sa3 * , * sa4 * , and * sb1 * models . filled circles show the data from @xcite , scaled as in figure 8 . , title="fig : " ] -1.2 cm , for all the models in table 1 . in the left panel , * sa1 * ( black solid line ) , * sa1b * ( red dotted ) , * sa1p * ( blue dashed ) , * sc1pex1 * ( green long dashed ) , and * sc1pex2 * ( magenta dot - dashed ) models are shown . in the right panel , * sa2 * ( black solid line ) , * sa3 * ( red dotted ) , * sa4 * ( blue dashed ) , and * sb1 * ( green long dashed ) models are shown . the results are shown at 60 myr , except for * sc1pex1 * and * sc1pex2 * models for which the results are shown at 80 myr . filled circles show the data from @xcite , scaled as in figure 8 . observational errors ( vertical bars ) are small , about @xmath413 , except for the data at 16 ghz with @xmath414.,title="fig : " ] -0.3 cm , averaged over @xmath415$]behind the shock , where @xmath334 , @xmath416 1 , 2 , 3 , 4 , 5 , 6 ( black solid , red dotted , blue dashed , green long dashed , magenta dot - dashed , cyan dot - long dashed lines , respectively ) for * sa1 * , * sa1b * and * sc1pex1 * models . the results are shown at the acceleration age , @xmath136 , specified in each panel.,title="fig : " ]
|
we perform dsa simulations of spherical shocks with the parameters relevant for the sausage radio relic in cluster ciza j2242.8 + 5301 , and calculate the ensuing radio synchrotron emission from re - accelerated electrons . three types of fossil electron populations are considered : a delta - function like population with the shock injection momentum , a power - law distribution , and a power - law with an exponential cutoff . these shocks produce curved radio spectra that steepen gradually over with break frequency ghz , if the duration of electron acceleration is myr . however , the abrupt increase of spectral index above ghz observed in the sausage relic seems to indicate that additional physical processes , other than radiative losses , operate for electrons with .
|
in order to understand certain observed features of arc - like giant radio relics such as the rareness , uniform surface brightness , and curved integrated spectra , we explore a diffusive shock acceleration ( dsa ) model for radio relics in which a spherical shock impinges on a magnetized cloud containing fossil relativistic electrons . toward this end , we perform dsa simulations of spherical shocks with the parameters relevant for the sausage radio relic in cluster ciza j2242.8 + 5301 , and calculate the ensuing radio synchrotron emission from re - accelerated electrons . three types of fossil electron populations are considered : a delta - function like population with the shock injection momentum , a power - law distribution , and a power - law with an exponential cutoff . the surface brightness profile of radio - emitting postshock region and the volume - integrated radio spectrum are calculated and compared with observations . we find that the observed width of the sausage relic can be explained reasonably well by shocks with speed and sonic mach number . these shocks produce curved radio spectra that steepen gradually over with break frequency ghz , if the duration of electron acceleration is myr . however , the abrupt increase of spectral index above ghz observed in the sausage relic seems to indicate that additional physical processes , other than radiative losses , operate for electrons with .
|
cond-mat0207571
|
i
|
since the discovery of the high - temperature superconductors in 1986 , there has been intense study of a number of two - dimensional models that are believed to model the electronic properties of the cuo@xmath2 plane of the cuprate superconductors , for example , the hubbard model , the @xmath3 model , and the heisenberg model.@xcite two - dimensional quantum models with short - range kinetic and interaction terms are difficult to study . in one dimension , there are exact solutions using the bethe ansatz and a host of related analytical techniques,@xcite and there is a very accurate numerical method , the density - matrix renormalization group ( dmrg),@xcite that can be applied to large systems relatively easily . in two dimensions , on the other hand , there are few exact solutions ( one famous nontrivial case is the hubbard model with one hole in a half - filled background , the nagaoka state@xcite ) , and current numerical methods are not satisfactory ( quantum monte carlo is plagued by the negative sign problem@xcite at low temperatures and at many fillings of interest and the dmrg in two dimensions@xcite is still in early development stage ) . the most reliable method for studying complicated quantum systems is exact diagonalization , which means enumerating all basis states and diagonalizing the resulting hamiltonian matrix . of course , this method is computationally limited by the growth of the hilbert space which is in general exponential in the number of particles and the lattice size . the @xmath4 hubbard model with 16 electrons , 8 spin - up and 8 spin - down , after reduction by particle conservation , translation , and the symmetries of the square , has 1,310,242 states in the largest matrix block,@xcite and can be diagonalized using the well - known lanczos method . the hubbard model has been diagonalized for the @xmath4 lattice ( see e.g. , ref . ) , and at low filling ( four electrons ) for @xmath5@xcite with extensive employment of symmetries . we have asked the question : _ is there a model that contains the basic ingredient of short - range hopping and interaction but is simpler , in the exact diagonalization sense , than the hubbard model ? _ the answer is yes : we can neglect the spin . we obtain the following hamiltonian for _ spinless _ fermions , @xmath6 where @xmath7 and @xmath8 are spinless fermion creation and annihilation operators at site @xmath9 , @xmath10 the number operator , @xmath11 the nearest - neighbor hopping amplitude , and @xmath12 the nearest - neighbor interaction . note that with spinless fermions , there can be at the most one particle per site ; no on - site interaction ( as that in the hubbard model ) is possible , and we have included in our hamiltonian nearest - neighbor repulsion . the spinless fermion model , eq . ( [ eq - ham ] ) , is a two - state model , and the number of basis states for a @xmath13-site system is @xmath14 , which is a significant reduction from the @xmath15 of the hubbard model . we can further reduce the number of basis states by taking the nearest - neighbor interaction @xmath16 , i.e. , infinite repulsion , which excludes nearest neighbors , giving roughly @xmath17 states . the spinless fermion model with infinite repulsion eq . ( [ eq - ham ] ) contains a significant reduction of the hilbert space . after using particle conservation and translation symmetry ( but not point group symmetry ) , the largest matrix for the @xmath0 system has @xmath18 states ( for 11 particles ) , and we can therefore compute for all fillings the @xmath0 system whereas for the hubbard model @xmath4 is basically the limit . this of course means that for certain limits we can also go much further than the hubbard model , for example , we can handle four particles on a @xmath1 lattice where the number of basis states is @xmath19 . this extended capability with our model has enabled us to obtain a number of results that are difficult to obtain with the hubbard model . an added feature of our model is that the basis set for the spinless fermion problem is identical to that for the hardcore boson problem , because with hardcore repulsion , there can be at the most one boson at one site also . therefore , without computational difficulty , we can study numerically both the spinless fermion and hardcore boson problem . spinless fermions can also be realized in experiments , for example , the spin polarized @xmath20he due to a strong magnetic field , or ferro or ferri - magnetic electronic systems where one spin - band is filled . the one - dimensional spinless fermion model with finite repulsion is solved exactly using bethe ansatz.@xcite the infinite - dimensional problem is studied in ref . . a very different approach using the renormalization group for fermions is done in ref . . a monte carlo study of the two - dimensional model at half - filling only and low temperatures is in ref . , which , dating back to 1985 , is one of the earliest quantum monte carlo simulations for fermions . ( it is no coincidence that they chose the model with the smallest hilbert space . ) considering the tremendous effort that has been devoted to the hubbard model and the close resemblance of our model , eq . ( [ eq - ham ] ) , to the hubbard model , it is surprising that works on this spinless model have been rather sparse , though it has been commented that the spinless model offers considerable simplifications.@xcite this paper is one of the two that we are publishing to study systematically the two - dimensional spinless fermion and hardcore boson model with infinite nearest - neighbor repulsion . the present paper focuses on the dilute limit , treating the problem of a few particles , and the other paper@xcite will focus on the dense limit , near half - filled,@xcite where stripes ( that are holes lining up across the lattice ) are natural objects ( see ref . for a condensed study of stripes in this model ) . we will use lanczos exact diagonalization , exploiting the much - reduced hilbert space of our model , and a number of analytical techniques , for example , in this paper , lattice green functions and the t - matrix . one of the goals of these two papers is to advertise this model of spinless fermions to the strongly - correlated electron community , as we believe that it is the simplest model of correlated fermions and deserves more research effort and better understanding . the prior work most comparable to ours may be the studies of four spinless electrons in a @xmath5 lattice , with coulomb repulsion , by pichard _ et al _ ; @xcite their motivation was the wigner crystal melting and the competition of coulomb interactions with anderson localization when a disorder potential is turned on . at the dilute limit of our model , the scattering t - matrix is of fundamental importance . for two particles , we expect that , at least when the potential @xmath12 is small , we can write a perturbative equation for energy , @xmath21 which is to say that the exact interacting energy of two particles is the noninteracting energy @xmath22 , for a pair of momenta @xmath23 and @xmath24 , plus a correction term @xmath25 due to the interaction @xmath12 . and with more than two particles , at least when the particle density is low , we expect to have @xmath26 eq . ( [ eq - perturbmany ] ) is central in fermi liquid theory , where it is justified by the so - called `` adiabatic continuation '' idea , which says that interacting fermion states correspond one - to - one to noninteracting ones as we slowly switch on a potential . in the boson case , because many bosons can occupy one quantum mechanical state and form a condensate , eq . ( [ eq - perturbmany ] ) should be modified , but with only two bosons , we expect eq . ( [ eq - perturb ] ) should be valid ( in that the correction vanishes in the dilute limit ) . ( [ eq - perturb ] ) and ( [ eq - perturbmany ] ) are used when we look at a list of noninteracting energies and draw correspondences with the interacting energies , the energy shift being packaged in the term @xmath25 . one possible objection to the above formulas ( eqs . ( [ eq - perturb ] ) and ( [ eq - perturbmany ] ) ) is that they appear to be perturbative , yet the interaction potential in our problem is infinitely strong , so the first - order ( first born approximation ) scattering amplitude , being proportional to the potential , is infinite too . however , this singular potential scattering problem ( e.g. , hard - sphere interaction in 3d ) has been solved ( see ref . ) by replacing the potential with the so - called scattering length , which is finite even when the potential is infinite . as we review in appendix [ sec - physical ] , a perturbation series ( born series ) can be written down ( that corresponds to a series of the so - called ladder diagrams ) and even though each term is proportional to the potential , the sum of all terms ( the t - matrix , @xmath25 in eqs . ( [ eq - perturb ] ) and ( [ eq - perturbmany ] ) ) is finite . because the t - matrix captures two - body interaction effects , it is the centerpiece of dilute fermion and boson calculations with strong interactions . field - theoretical calculations in both three and two dimensions are based on the ladder diagrams and the t - matrix . see fetter and walecka@xcite for the 3d problem , schick@xcite for the 2d boson problem and bloom@xcite the 2d fermion problem . for lattice fermion problems , kanamori@xcite derived the t - matrix for a tight - binding model that is essentially a hubbard model ( this work is also described in yosida@xcite ) . and in ref . , the t - matrix is worked out explicitly for the hubbard model , and kanamori s result is obtained . ref . also evaluated the t - matrix for the dilute limit in three dimensions and obtained a functional dependence on particle density . rudin and mattis@xcite used the t - matrix expression derived in refs . and and found upper and lower bounds of the fermion t - matrix in two dimensions in terms of particle density . rudin and mattis s result for the low - density limit of the two - dimensional hubbard model is of the same functional form as bloom s diagrammatical calculation for the two - dimensional fermion hard disks.@xcite since the discovery of high - temperature superconductors , bloom s calculation has received a lot of attention because of its relevance to the validity of the fermi liquid description of dilute fermions in two dimensions . there have been a number of works on the 2d dilute fermi gas@xcite and on the dilute limit of 2d hubbard model,@xcite all using the t - matrix , but these results have not been checked by numerical calculations . in fact , we are not aware of a systematic study of the t - matrix for a lattice model . in this paper , we present the first such study for the two - particle problem in sec . [ sec - tmat ] ( for bosons and fermions ) and the few - fermion problem in sec . [ sec - tmat2 ] . we check the t - matrix results with exact diagonalization data and show that our t - matrix on a lattice is the sum of the two - body scattering terms to infinite order . in this paper , we will study systematically the dilute limit of our model eq . ( [ eq - ham ] ) , focusing on the problem of a few particles . our paper is divided into four parts . in sec . [ sec - green ] , the two - particle ( boson and fermion ) problem is studied . we formulated the two - particle schrodinger equation using lattice green functions , employ some of its recursion relations to simplify the problem , and obtain the two - boson ground state energy in the large - lattice limit . using the two - particle result , we then study the problem of a few particles and obtain an expression for ground state energy on a large lattice . in sec . [ sec - tmat ] , the two - particle problem is then cast into a different form , emphasizing the scatterings between the two particles . the result is the t - matrix , that is exact for the two - particle problem and contains all two - body scattering terms . we will study the two - particle t - matrix in great detail , showing the differences between the boson and fermion cases , and demonstrating that the first t - matrix iteration is often a good approximation for fermion energy . in appendix [ sec - physical ] , we show explicitly that the t - matrix we obtain is the sum total of all two - body scattering terms . the problem of a few fermions is taken up in sec . [ sec - tmat2 ] . first , the fermion shell effect is discussed and demonstrated from diagonalization , and we show the difference for bosons and fermions . we show the modifications to the two - fermion t - matrix that enable us to calculate energies for three , four , and five particles . using this t - matrix , we can compute the interaction corrections to the noninteracting energy and trace the change in the energy spectrum from the nointeracting one to the interacting one . finally , in sec . [ sec - dilute ] , we discuss the energy per particle curve for dilute bosons and fermions . we have studied the two - dimensional results derived by schick@xcite for bosons and bloom@xcite for fermions by fitting the data from diagonalization for a number of lattices . schick s result for dilute bosons is checked nicely , and we explain that for spinless fermions in our model we will need the p - wave scattering term , which is not included in bloom s calculation . in appendix [ sec - diag ] , we discuss briefly our exact diagonalization computer program , which can handle arbitary periodic boundaries specified by two vectors on the square lattice and uses translation symmetry to reduce the matrix size .
|
this is the simplest model of correlated electrons and is more tractable for exact diagonalization than the hubbard model . we study systematically the dilute limit of this model by a combination of analytical and several numerical approaches : the two - particle problem using lattice green functions and the t - matrix , the few - fermion problem using a modified t - matrix ( demonstrating that the interaction energy is well captured by pairwise terms ) , and for bosons the fitting of the energy as a function of density to schick s analytical result for dilute hard disks . we present the first systematic study for a strongly - interacting lattice model of the t - matrix , which appears as the central object in older theories of the existence of a two - dimensional fermi liquid for dilute fermions with strong interactions . for our model , we can ( lanczos ) diagonalize the system at all fillings and the system with four particles , thus going far beyond previous diagonalization works on the hubbard model .
|
in our model , spinless fermions ( or hardcore bosons ) on a square lattice hop to nearest neighbor sites , and also experience a hard - core repulsion at the nearest neighbor separation . this is the simplest model of correlated electrons and is more tractable for exact diagonalization than the hubbard model . we study systematically the dilute limit of this model by a combination of analytical and several numerical approaches : the two - particle problem using lattice green functions and the t - matrix , the few - fermion problem using a modified t - matrix ( demonstrating that the interaction energy is well captured by pairwise terms ) , and for bosons the fitting of the energy as a function of density to schick s analytical result for dilute hard disks . we present the first systematic study for a strongly - interacting lattice model of the t - matrix , which appears as the central object in older theories of the existence of a two - dimensional fermi liquid for dilute fermions with strong interactions . for our model , we can ( lanczos ) diagonalize the system at all fillings and the system with four particles , thus going far beyond previous diagonalization works on the hubbard model .
|
1407.7480
|
i
|
there are many problems in computational physics that involve solving partial differential equations ( pdes ) in complex geometries . examples include fluid flows in complicated systems , vein networks in plant leaves , and tumours in human bodies . standard solution methods for pdes in complex domains typically involve triangulation and unstructured grids . this rules out coarse - scale discretizations and thus efficient geometric multi - level solutions . also , mesh generation for three - dimensional complex geometries remains a challenge , in particular if we allow the geometry to evolve with time . in the past several years , there has been much effort put into the development of numerical methods for solving partial differential equations in complex domains . however , most of these methods typically require tools not frequently available in standard finite element and finite difference software packages . examples of such approaches include the extended and composite finite element methods ( e.g. , @xcite ) , immersed interface methods ( e.g. , @xcite ) , virtual node methods with embedded boundary conditions ( e.g. , @xcite ) , matched interface and boundary methods ( e.g. , @xcite ) , modified finite volume / embedded boundary / cut - cell methods / ghost - fluid methods ( e.g. , @xcite ) . in another approach , known as the fictitious domain method ( e.g. , @xcite ) , the original system is either augmented with equations for lagrange multipliers to enforce the boundary conditions , or the penalty method is used to enforce the boundary conditions weakly . see also @xcite for a review of numerical methods for solving the poisson equation , the diffusion equation and the stefan problem on irregular domains . an alternate approach for simulating pdes in complex domains , which does not require any modification of standard finite element or finite difference software , is the diffuse - domain method . in this method , the domain is represented implicitly by a phase - field function , which is an approximation of the characteristic function of the domain . the domain boundary is replaced by a narrow diffuse interface layer such that the phase - field function rapidly transitions from one inside the domain to zero in the exterior of the domain . the boundary of the domain can thus be represented as an isosurface of the phase - field function . the pde is then reformulated on a larger , regular domain with additional source terms that approximate the boundary conditions . although uniform grids can be used , local grid refinement near domain boundaries improves efficiency and enables the use of smaller interface thicknesses than are achievable using uniform grids . a related approach involves the level - set method @xcite to describe the implicitly embedded surface and to obtain the appropriate surface operators ( e.g. , @xcite ) . the diffuse - domain method ( ddm ) was introduced by kockelkoren et al.@xcite to study diffusion inside a cell with zero neumann boundary conditions at the cell boundary ( a similar approach was also used in @xcite using spectral methods ) . the ddm was later used to simulate electrical waves in the heart @xcite and membrane - bound turing patterns @xcite . more recently , diffuse - interface methods have been developed for solving pdes on stationary @xcite and evolving @xcite surfaces . diffuse - domain methods for solving pdes in complex evolving domains with dirichlet , neumann and robin boundary conditions were developed by li et al.@xcite and by teigen et al . @xcite who modelled bulk - surface coupling . the ddm was also used by aland et al . @xcite to simulate incompressible two - phase flows in complex domains in 2d and 3d , and by teigen et al . @xcite to study two - phase flows with soluble surfactants . li et al . @xcite showed that in the ddm there exist several approximations to the physical boundary conditions that converge asymptotically to the correct sharp - interface problem . li et al . presented some numerical convergence results for a few selected problems and observed that the choice of boundary condition can significantly affect the accuracy of the ddm . however , li et al . did not perform a quantitative comparison between the different boundary - condition approximations , nor did they estimate convergence rates . further , li et al . did not address the source of the different levels of accuracy they observed for the different boundary - condition approximations . in the context of dirichlet boundary conditions , franz et al.@xcite recently presented a rigorous error analysis of the ddm for a reaction - diffusion equation and found that the method converges only with first - order accuracy in the interface thickness parameter @xmath0 , which they confirmed numerically . similar results were obtained numerically by reuter et al . @xcite who reformulated the ddm using an integral equation solver . reuter et al . demonstrated that their generalized ddm , with appropriate choices of approximate surface delta functions , converges with first - order accuracy to solutions of the poisson equation with dirichlet boundary conditions . here , we focus on neumann and robin boundary conditions and we present a matched asymptotic analysis of general diffuse - domain methods in a fixed complex geometry , focusing on the poisson equation for robin boundary conditions and a steady reaction - diffusion equation for neumann boundary conditions . however , our approach applies to transient problems and more general equations in the same way as shown in @xcite . our analysis shows that for certain choices of the boundary condition approximations , the ddm is second - order accurate in @xmath0 , which in practice is proportional to the smallest mesh size . however , for other choices the ddm is only first - order accurate . this helps to explain why the choice of boundary condition approximation is important for rapid global convergence and high accuracy . further , inspired by the work of karma and rappel @xcite and almgren @xcite , who incorporated second - order corrections in their phase field models of crystal growth and by the work of folch et al.@xcite who added second - order corrections in phase - field models of advection , we also suggest correction terms that may be added to yield a more accurate version of the diffuse - domain method . simple modifications of first - order boundary condition approximations are proposed to achieve asymptotically second - order accurate schemes . our analytic results are confirmed numerically for selected test problems . the outline of the paper is as follows . in [ sec : dda ] we introduce and present an analysis of general diffuse - domain methods . in [ sec : discretization ] the numerical methods are described , and in [ sec : results ] the test cases are introduced and numerical results are presented and discussed . we finally give some concluding remarks in [ sec : conclusion ] .
|
in recent work , li et al . ( comm . math . sci . , 7:81 - 107 , 2009 ) developed a diffuse - domain method ( ddm ) for solving partial differential equations in complex , dynamic geometries with dirichlet , neumann , and robin boundary conditions . the diffuse - domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness that is typically proportional to the minimum grid size . the original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms . the resulting equations can be solved with standard finite difference and finite element software packages . here , we present a matched asymptotic analysis of general diffuse - domain methods for neumann and robin boundary conditions . our analysis shows that for certain choices of the boundary condition approximations , the ddm is second - order accurate in . however , for other choices the ddm is only first - order accurate . this helps to explain why the choice of boundary - condition approximation is important for rapid global convergence and high accuracy . simple modifications of first - order boundary condition approximations are proposed to achieve asymptotically second - order accurate schemes . our analytic results are confirmed numerically in the and norms for selected test problems .
|
in recent work , li et al . ( comm . math . sci . , 7:81 - 107 , 2009 ) developed a diffuse - domain method ( ddm ) for solving partial differential equations in complex , dynamic geometries with dirichlet , neumann , and robin boundary conditions . the diffuse - domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness that is typically proportional to the minimum grid size . the original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms . the resulting equations can be solved with standard finite difference and finite element software packages . here , we present a matched asymptotic analysis of general diffuse - domain methods for neumann and robin boundary conditions . our analysis shows that for certain choices of the boundary condition approximations , the ddm is second - order accurate in . however , for other choices the ddm is only first - order accurate . this helps to explain why the choice of boundary - condition approximation is important for rapid global convergence and high accuracy . our analysis also suggests correction terms that may be added to yield more accurate diffuse - domain methods . simple modifications of first - order boundary condition approximations are proposed to achieve asymptotically second - order accurate schemes . our analytic results are confirmed numerically in the and norms for selected test problems . karl yngve lervg and john lowengrub numerical solution of partial differential equations , phase - field approximation , implicit geometry representation , matched asymptotic analysis .
|
astro-ph0405146
|
i
|
the error boxes of unidentified gamma - ray sources are usually large , and thus the task of finding appropriate candidate counterparts at other wavelengths has not been easy . about 20 bright point - like gamma - ray sources were found near the galactic plane using cos - b ( swanenburg et al . 1981 ) some of which may be concentrations of molecular hydrogen ( mayer - hasselwander & simpson 1990 ) . another few , such as the crab and vela pulsars , were identified based on their periodic emission ( thompson et al . the nature of the other objects remained unknown . the much more sensitive egret telescope on the _ compton gamma - ray observatory _ ( _ cgro _ ) was expected to contribute decisively to the identification of the cos - b sources . and indeed , the higher count rates and tighter source locations provided by egret confirmed the existence of most of the cos - b sources and led to the identification of several other sources . most prominent were the gamma - ray pulsars geminga and psr b170644 which could be identified on the basis of detections at x - ray and radio wavelengths ( see e.g. kanbach 2002 and becker & pavlov 2001 for a review and references ) . at high galactic latitudes about 90 new high - energy sources could be correlated with blazars . the final egret catalog of gamma - ray sources lists 271 objects ( hartman et al . 1999 ) of which about 170 are unidentified . their distribution suggests that most of them are galactic . the origin and nature of this population of extremely energetic objects is clearly of interest . seven of the galactic gamma - ray sources are rotation - powered pulsars , identified through the periodic modulation of their gamma - ray fluxes . these seven are also persistent , point - like sources at gamma - ray energies . in the 100 mev 1 gev energy range , these sources have hard , power - law - like spectra with high - energy cut - offs at a few gev . although rotation - powered pulsars are best known as radio sources , this is not true for all geminga , for example , shows at best marginal evidence of pulsed radio emission ( kuzmin & losovkii 1997 ) . geminga is thus taken as the prototype of a ` radio - quiet ' gamma - ray pulsar of which many more should exist in the galaxy . although geminga s gamma - ray luminosity is rather low ( its small distance of about 160 pc makes it a bright source ) the property of radio faintness could be indicative of pulsar emission where the beamed radiation at different wavelengths is emitted into widely different directions . such a model may be applicable to young , high luminosity pulsars as well ( yadigaroglu & romani 1995 ) . there are other models to explain geminga s radio faintness though ( e.g. gil et al . 1998 ) . a review of the spectra of unidentified low galactic latitude egret sources ( bertsch et al . 2000 ; merck et al . 1996 ) shows that about 10 objects exhibit the very hard power - law type spectra with a cut - off at several gev as seen also in the identified pulsars . these objects would be prime targets for identification efforts at other wavelengths . relatively deep radio searches ( at 770 mhz ) at the positions of several of these sources have not found radio counterparts ( nice & sayer 1997 ) . population studies of the unidentified gamma - ray sources close to the galactic plane indicate that their luminosities are also quite compatible with the luminosities of the younger identified pulsars ( kanbach et al . suggestions , other than pulsars , for the nature of these gamma - ray sources have also been widely discussed . energetic objects , like massive young stars or ob associations and snrs have been correlated with the 3eg catalog ( e.g. romero et al . 2000 ) and certainly indicate a close relationship with the gamma - ray sources . multi - wavelength observations focusing on promising candidate sources have been quite successful in recent years . observations in x - rays have been useful : e.g. in the cases of 3eg j2006@xmath112321 = pmn2005@xmath112310 ( wallace et al . 2002 ) and 3eg j2016 + 3657 = b2013 + 370 ( halpern et al . new pulsar / isolated neutron star identifications were reported , e.g. 3eg j2227 + 6122 ( halpern et al . 2001b ) by discovery of the characteristic pulsar period of rx / ax j2229.0 + 6114 . x - ray observations were used to relate the high galactic latitude source 3eg j1835 + 5918 to an isolated neutron star , rx j1836.2 + 5925 ( reimer et al . 2001 ; mirabal & halpern 2001 ; halpern et al . 2002 ) . with the wealth of incoming discoveries from the parkes multi - beam pulsar survey , promising associations between newly discovered radio pulsars and egret sources have also been discussed . these associations include the two young pulsars psr j14206048 and psr j18370604 ( damico et al . 2001 ) in the vicinity of 3eg j14206038 and 3eg j18370606 , respectively ; psr j10165857 near the snr g284.31.8 is a plausible counterpart for 3eg j10135915 ( camilo et al . 2001 ) . in a recent survey of 56 unidentified egret sources roberts et al . ( 2004 ) found a radio pulsar located inside the 95% likelihood map in six of the investigated gamma - ray sources . the discovery of psr j2021 + 3651 in the error box of gev 2020 + 3658 using the 305 m arecibo radio telescope is another positive example ( roberts et al . 2002 ; 2004 ; hessels et al . however , torres , butt & camilo ( 2001 ) , and more recently kramer et al . ( 2003 ) , who have summarized the observational status of the radio pulsars and egret - detected gamma - ray sources concluded that , in many cases , further multi - frequency investigations are required in order to conclusively translate a proposed association into a final source identification . 3eg j2020 + 4017 is among the brightest persistent sources in the egret sky . originally listed as a cos - b source ( 2cg@xmath12 ) it is still unidentified . its gamma - ray flux is consistent with constant flux ( hartman et al . 1999 ) , and the spectrum is hard and best described by a power - law with photon - index of @xmath13 . merck et al . ( 1996 ) found evidence for a spectral break at @xmath14 gev which has been confirmed in recent studies by bertsch et al . ( 2000 ) and reimer & bertsch ( 2001 ) . examining all archival egret data and using photons @xmath15 gev , brazier et al . ( 1996 ) found a best position at @xmath16 , @xmath17 with a @xmath18 95%-confidence error box . this position was consistent with the 2eg catalog position and placed the egret source within the @xmath0-cygni supernova remnant g78.2 + 2.1 . the remnant g78.2 + 2.1 consists of a @xmath19-diameter , circular radio shell with two bright , broad opposing arcs on its rim ( higgs , landecker & roger 1977 ; wendker , higgs & landecker 1991 ) . g78.2 + 2.1 has a kinematic distance of @xmath20 kpc @xmath21 ( landecker , roger & higgs 1980 ; green 1989 ) and is estimated to have an age of 5400 yr ( sturner & dermer 1995 ) . a very bright star , @xmath0-cygni ( @xmath22 , spectral type f8iab ) lies on the eastern edge and lends its name to the remnant . a small hii region , located close to the star , forms the so - called @xmath0-cygni nebula . brazier et al . ( 1996 ) analyzed rosat pspc data viewing the @xmath0-cygni region . six pspc observations were targeted at celestial positions within 40 arcmin of the egret source . these rosat observations are combined and shown in figure [ f : rosat ] . the point source rx j2020.2 + 4026 is located within the 95% confidence contour of the 2eg position of 2cg078 + 2 ( brazier et al . 1996 ) , and was suggested by these authors to be the x - ray counterpart to the gamma - ray source . brazier et al . ( 1996 ) and carraminana et al . ( 2000 ) provided a possible optical counterpart for rx j2020.2 + 4026 . optical follow - up observations revealed a 14.5 magnitude k0v star nearby and within the @xmath23 rosat error circle . the x - ray to optical flux ratio of this star was found to be marginally consistent with that found for late - type stars ( stocke et al . 1991 ; fleming et al . 1995 ) , so that an association of rx j2020.2 + 4026 with the gamma - ray source could not be excluded ( brazier et al . 1996 ) . with the 3eg catalog ( hartman et al . 1999 ) , an improved position of 2cg078 + 2 became available : @xmath24 , @xmath25 , i.e. shifted in right ascension and declination by a few arc - minutes with respect to the 2eg position used by brazier et al . ( 1996 ) . with this improved position the proposed counterpart rx j2020.2 + 4026 is no longer located within the 95% contour of 3eg j2020 + 4017 . the 99% likelihood contour , however , still includes rx j2020.2 + 4026 ( figure [ f : rosat ] ) . in this paper we report on follow - up studies of rx j2020.2 + 4026 with _ chandra _ and the green bank radio telescope . the _ chandra _ observations were taken with the aim to determine the position and spectrum of rx j2020.2 + 4026 with high precision and to explore the possible connection with 3eg j2020 + 4017 . gbt observations at 820 mhz were made in order to search the egret error box of 3eg j2020 + 4017 for a young radio pulsar .
|
we also report on reanalysis of archival rosat data . with _ chandra _ it became possible for the first time to measure the position of the putative gamma - ray counterpart rx j2020.2 + 4026 with sub - arcsec accuracy and to deduce its x - ray spectral characteristics . these observations demonstrate that rx j2020.2 + 4026 is associated with a k field star and therefore is unlikely to be the counterpart of the bright gamma - ray source 2cg078 + 2 in the snr g78.2 + 2.1 as had been previously suggested . the _ chandra _
|
in search of the counterpart to the brightest unidentified gamma - ray source 3eg j2020 + 4017 ( 2cg078 + 2 ) we report on new x - ray and radio observations of the-cygni field with the _ chandra _ x - ray observatory and with the green bank telescope ( gbt ) . we also report on reanalysis of archival rosat data . with _ chandra _ it became possible for the first time to measure the position of the putative gamma - ray counterpart rx j2020.2 + 4026 with sub - arcsec accuracy and to deduce its x - ray spectral characteristics . these observations demonstrate that rx j2020.2 + 4026 is associated with a k field star and therefore is unlikely to be the counterpart of the bright gamma - ray source 2cg078 + 2 in the snr g78.2 + 2.1 as had been previously suggested . the _ chandra _ observation detected 37 additional x - ray sources which were correlated with catalogs of optical and infrared data . subsequent gbt radio observations covered the complete 99% egret likelihood contour of 3eg j2020 + 4017 with a sensitivity limit of which is lower than most of the recent deep radio search limits . if there is a pulsar operating in 3eg j2020 + 4017 , this sensitivity limit suggests that the pulsar either does not produce significant amounts of radio emission or that its geometry is such that the radio beam does not intersect with the line of sight . finally , reanalysis of archival rosat data leads to a flux upper limit of for a putative point - like x - ray source located within the 68% confidence contour of 3eg j2020 + 4017 . adopting the snr age of 5400 yrs and assuming a spin - down to x - ray energy conversion factor of this upper limit constrains the parameters of a putative neutron star as a counterpart for 3eg j2020 + 4017 to be , and g.
|
astro-ph0405146
|
c
|
we have searched a portion of the @xmath0-cygni field for possible x - ray counterparts to the intriguing gamma - ray source 3eg j2020 + 4017 ( 2cg078 + 2 ) using _ chandra _ and rosat . we have shown that a previous candidate , rx j2020.2 + 4026 , is almost certainly not the gamma - ray source but identified with a normal star . this conclusion is based on the refined position of the x - ray source , its spectrum and coincidence with both optical and infrared sources and the inferred x - ray luminosity . further , we have found a total of 38 x - ray sources in the _ chandra _ s2- , s3- and s4-fields which covers only part of the much larger error box containing the location of the egret source . a re - analysis of archival rosat hri data revealed three more x - ray sources within the egret error box which are not detected in the _ chandra _ observations . two of these sources are surely variable whereas the third source was found in a region not covered by the _ chandra _ observation . we found that some of the _ chandra _ sources have counterparts that may be main - sequence stars based on their identification with optical objects and 2mass sources of normal colors . of course the x - ray emission may not be due to the main sequence star , but can arise from an accreting compact companion . none of the x - ray sources appear to be radio pulsars , down to a limiting sensitivity of @xmath86 for an assumed pulse duty cycle of 4% . this limit also applies to the entire region associated with the 99%-confidence position contours of 3eg j2020 + 4017 . determining an upper limit for a putative x - ray point source located within the 68% confidence contour of 3eg j2020 + 4017 using archival rosat pspc data we found a @xmath64 luminosity upper limit of @xmath87 which is four times smaller than the rosat pspc - deduced luminosity observed from the vela pulsar ( @xmath88 , e.g. table 3 in becker & aschenbach 2002 ) but about four times higher than the total rosat observed x - ray luminosity from geminga ( @xmath89 ) . we therefore consider it as a valid option that the counterpart of 3eg j2020 + 4017 is a neutron star with an x - ray luminosity similar to that observed from vela - like to middle - aged pulsars . an object with such luminosity would not have been detected in the x - ray data from rosat which cover that region of the sky . adopting the snr age of 5400 yrs and assuming a spin - down to x - ray energy conversion factor of @xmath3 ( becker & trmper 1997 ) we are able to constrain the spin - parameters of such a putative neutron star to be @xmath90 , @xmath91 and @xmath92 g , which are consistent with the properties of known vela- to middle - aged pulsars ( e.g. gonzalez & safi - harb 2003 , becker & pavlov 2001 ) , given the uncertainty of this approach . the ratio of the @xmath0-ray to soft x - ray flux deduced from our upper limit , @xmath93 , is consistent with this conclusion . in order to obtain a full census of the x - ray population in the error box of 3eg j2020 + 4017 further observations with _ chandra _ are required . as the discovery of geminga has taught us , deep follow - up optical observations of new x - ray sources can also lead to the identification of the nature of a high - energy source . finally , the measurements we expect from the _ glast _ mission ( launch 2007 ) should provide a much improved signal - to - noise ratio and a source location better than @xmath94 for this gamma - ray source . this will open the possibility to directly search for pulsar periodicities in the gamma - ray data . in case no pulsar is found , the restricted number of _ chandra _ sources in the _ glast _ error box will then be prime candidates for even deeper searches for counterparts . those of us at the marshall space flight center acknowledge support from the _ chandra _ project . mcw acknowledges with gratitude conversations with marshall joy and roc cutri that clarified some of the mysteries of the infrared portion of the spectrum . za was supported by nasa grant nra-99 - 01-ltsa-070 . drl is a university research fellow funded by the royal society . fc is supported in part by nsf grant ast-02 - 05853 . this publication makes use of data products from the two micron all sky survey , which is a joint project of the university of massachusetts and the infrared processing and analysis center / california institute of technology , funded by the national aeronautics and space administration and the national science foundation . in addition , this research has made use of data obtained from the high energy astrophysics science archive research center ( heasarc ) , provided by nasa s goddard space flight center . backer , d.c . , dexter , m.r . , zepka , a. , ng , d. , werthimer , d.j . , ray , p.s . , foster , r.s . , 1997 , pasp , 109 , 61 becker , w. , aschenbach , b. , 2002 , in proceedings of the we - heraeus seminar on neutron stars , pulsars and supernova remnants , eds . w.becker , h.lesch & j / trmper , mpe - report 278 , 64 , ( available from astro - ph/0208466 ) camilo , f. , stairs , i.h . , lorimer , d.r . , backer , d.c . , ransom , s.m . , klein , b. , wielebinski , r. , kramer , m. , mclaughlin , m.a . , arzoumanian , z. , mller , p. , 2002 , apj , 571 , l41 camilo , f. , 2003 , in _ radio pulsars _ , eds m.bailes , d.j.nice , and s.e.thorsett , astronomical society of the pacific , san francisco carraminana , a. , chavushyan , v. , zharikov , s. , et al . , 2000 , in _ proc . 5th compton symposium _ , aip - cp 510 , p49 gil , j.a . , khechinashvili , d.g . , melikidze , g.i . , 1998 , mnras , 298 , 1207 gonzalez , m. , safi - harb , s. , 2003 , apj , 591 , 143 halpern , j.p . , gotthelf , e.v . , mirabal , n. , camilo , f. , 2002 , apj , 573 , l41 halpern , j.p . , eracleous , m. , mukherjee , r. , et al . 2001a , apj 551 , 1016 hessels , j.w.t . , roberts , m.s.e . , ransom , s.m . , kaspi , v.m . , romani , r.w . , ng , c.y . , freier , p.c.c . , gaensler , b.m . , 2004 , astro - ph/0403632 higgs , l.a . , landecker , t.l . , and roger , r.s . , 1977 , aj , 82 , 718 lorimer , d.r . , kramer , m. , mller , p. , wex , n. , jessner , a. , lange , c. , wielebinski , r. , 2000 , a&a , 358 , 169 lorimer , d.r . , yates , j.a . , lyne , a.g . , gould , d.m . , 1995 , mnras , 273 , 411l maeda , y. , koyama , k. , yokogawa , j. , skinner , s. , 1999 , apj 510 , 967 mayer - hasselwander , h.a . , simpson , g. , 1990 , in _ the egret science symposium _ , eds c.fichtel , s.huntre , p.sreekumar , f.stecker , nasa , vp-3071 , p153 & & & & & & & & & + & & & ( @xmath95 ) & & & ( @xmath95 ) & ( @xmath95 ) & ( @xmath95 ) & ( @xmath95 ) + s401 & 304.86758 & 40.436279 & 13.6 & 49 & 5.7 & 1.25 & 0.61&1.55 & & + s402 & 304.88129 & 40.373795 & 13.8 & 25 & 3.9 & 1.72 & & 1.65 & + s403 & 304.92621 & 40.362309 & 10.1 & 69 & 6.7 & 0.87 & 1.05 & 1.13 & + s301 & 304.97339 & 40.455490 & 4.54 & 13 & 3.0 & 0.88 & 0.70 & 0.63 & + s302 & 304.97769 & 40.468494 & 4.61 & 36 & 5.2 & 0.65 & & & + s303 & 304.99615 & 40.398434 & 3.75 & 14 & 3.1 & 0.76 & 0.85 & 1.29 & + s304 & 305.00381 & 40.365112 & 5.03 & 14 & 3.0 & 0.94 & 1.15 & 1.41 & + s305 & 305.00769 & 40.434021 & 2.63 & 176 & 11.2 & 0.47 & & 0.56 & + s306 & 305.01837 & 40.457535 & 2.47 & 23 & 4.3 & 0.55 & & & + s307 & 305.02713 & 40.412449 & 2.19 & 25 & 4.5 & 0.52 & 0.37 & 0.39 & + s308 & 305.02737 & 40.374905 & 3.60 & 28 & 4.8 & 0.61 & 0.25 & 0.36 & + s309 & 305.05789 & 40.394272 & 2.04 & 15 & 3.3 & 0.55 & & 0.17 & + s310 & 305.06354 & 40.486912 & 2.46 & 12 & 3.1 & 0.63 & 0.95 & 1.05 & + s311 & 305.07104 & 40.446281 & 1.21 & 44 & 5.8 & 0.47 & 0.30 & 0.25 & + s312 & 305.07147 & 40.437424 & 1.14 & 253 & 13.7 & 0.46 & 0.27 & 0.30 & 5.9@xmath96 + s313 & 305.08884 & 40.448517 & 1.21 & 20 & 3.7 & 0.48 & & & + s314 & 305.09262 & 40.490631 & 2.57 & 22 & 4.0 & 0.56 & & 0.13 & + s315 & 305.10596 & 40.484375 & 2.37 & 12 & 3.0 & 0.61 & 0.65 & 0.41 & + s316 & 305.11560 & 40.440331 & 1.40 & 11 & 3.1 & 0.52 & & & + s201 & 305.14136 & 40.473434 & 2.68 & 10 & 2.9 & 0.68 & 0.34 & 0.62 & + s202 & 305.15140 & 40.494373 & 3.97 & 17 & 3.5 & 0.73 & 0.80 & 0.64 & + s203 & 305.15570 & 40.495060 & 4.18 & 11 & 2.8 & 0.88 & & & + s204 & 305.17081 & 40.451115 & 3.26 & 85 & 7.9 & 0.50 & 0.22 & 0.35 & 1.2@xmath97 + s205 & 305.17349 & 40.380219 & 4.65 & 29 & 4.7 & 0.69 & 0.73 & & + s206 & 305.18216 & 40.450073 & 3.82 & 213 & 12.4 & 0.48 & & 0.31 & + s207 & 305.18600 & 40.430450 & 3.93 & 10 & 2.9 & 0.87 & 0.78 & 1.03 & + s208 & 305.21335 & 40.403984 & 6.05 & 16 & 3.4 & 1.02 & & 0.84 & + s209 & 305.21414 & 40.509041 & 8.20 & 28 & 4.2 & 1.04 & 1.73 & 1.66 & + s210 & 305.21881 & 40.473904 & 6.70 & 24 & 3.8 & 0.95 & 0.55 & 0.55 & + s211 & 305.21906 & 40.408849 & 6.35 & 66 & 7.0 & 0.65 & 0.43 & 0.62 & + s212 & 305.22256 & 40.507294 & 8.70 & 47 & 5.9 & 0.89 & 1.54 & 1.61 & + s213 & 305.23041 & 40.474827 & 7.60 & 33 & 4.2 & 0.95 & 0.91 & 0.63 & + s214 & 305.24078 & 40.474091 & 8.44 & 101 & 8.2 & 0.68 & 0.60 & 0.40 & 2.9@xmath98 + s215 & 305.24927 & 40.378075 & 9.95 & 17 & 3.1 & 1.52 & 0.91 & 0.95 & + s216 & 305.25110 & 40.485874 & 9.80 & 33 & 4.3 & 1.12 & 1.63 & 1.10 & + s217 & 305.27808 & 40.430069 & 11.2 & 24 & 3.8 & 1.47 & 0.97 & 0.97 & + s218 & 305.29422 & 40.513210 & 15.8 & 35 & 4.0 & 1.66 & & 0.57 & + s219 & 305.29767 & 40.468163 & 13.9 & 201 & 11.5 & 0.74 & 0.38 & 0.70 & 2.4@xmath99 + & & & & & & & + & & & & & & & ( @xmath95 ) + & & & & & & & + s401 & 304.867803 & 40.436275 & 0.041 & & & & + s401 & 304.868070 & 40.436495 & 0.041 & & & & + s402 & & & & 304.881105 & 40.373360 & 0.117 & + s403 & 304.926575 & 40.362225 & 0.020 & 304.926589 & 40.362183 & 0.030 & 0.16 + s301 & 304.973164 & 40.455400 & 0.021 & 304.973295 & 40.455330 & 0.031 & 0.44 + s303 & 304.996045 & 40.398212 & 0.015 & 304.995913 & 40.398125 & 0.023 & 0.48 + s304 & 305.003723 & 40.364800 & 0.023 & 305.003740 & 40.364723 & 0.035 & 0.28 + s305 & & & & 305.007841 & 40.434124 & 0.009 & + s307 & 305.027000 & 40.412425 & 0.007 & 305.027043 & 40.412365 & 0.011 & 0.25 + s308 & 305.027320 & 40.374964 & 0.010 & 305.027281 & 40.374832 & 0.015 & 0.49 + s309 & & & & 305.057841 & 40.394245 & 0.012 & + s310 & 305.063206 & 40.486845 & 0.010 & 305.063218 & 40.486755 & 0.015 & 0.33 + s311 & 305.070998 & 40.446359 & 0.006 & 305.071003 & 40.446217 & 0.009 & 0.51 + s312 & 305.071387 & 40.437464 & 0.005 & 305.071389 & 40.437366 & 0.008 & 0.35 + s314 & & & & 305.092591 & 40.490601 & 0.012 & + s315 & 305.105898 & 40.484550 & 0.010 & 305.105912 & 40.484482 & 0.015 & 0.25 + s201 & 305.141242 & 40.473467 & 0.012 & 305.141188 & 40.473324 & 0.018 & 0.54 + s202 & 305.151131 & 40.494459 & 0.014 & 305.151193 & 40.494289 & 0.021 & 0.64 + s204 & 305.170775 & 40.451170 & 0.007 & 305.170747 & 40.451031 & 0.010 & 0.51 + s205 & 305.173353 & 40.380392 & 0.013 & & & & + s206 & & & & 305.182074 & 40.450016 & 0.009 & + s207 & 305.186253 & 40.430348 & 0.020 & 305.186255 & 40.430241 & 0.030 & 0.39 + s208 & & & & 305.213189 & 40.403786 & 0.041 & + s209 & 305.214453 & 40.508625 & 0.028 & 305.213771 & 40.508675 & 0.042 & 1.88 + s210 & 305.218609 & 40.473920 & 0.024 & 305.218624 & 40.473850 & 0.035 & 0.26 + s211 & 305.219192 & 40.408425 & 0.011 & 305.219198 & 40.408352 & 0.017 & 0.26 + s212 & 305.223092 & 40.507159 & 0.021 & 305.223083 & 40.507088 & 0.031 & 0.26 + s213 & 305.230259 & 40.475053 & 0.024 & 305.230230 & 40.474934 & 0.036 & 0.44 + s214 & 305.240995 & 40.474125 & 0.012 & 305.240870 & 40.474003 & 0.018 & 0.56 + s215 & 305.249534 & 40.378228 & 0.061 & 305.249506 & 40.378269 & 0.090 & 0.17 + s216 & 305.250842 & 40.486821 & 0.033 & 305.250918 & 40.486145 & 0.049 & 0.53 + s217 & 305.278420 & 40.429992 & 0.057 & 305.278386 & 40.429932 & 0.085 & 0.24 + s218 & & & & 305.294173 & 40.513363 & 0.109 & + s219 & 305.297639 & 40.468059 & 0.015 & 305.297601 & 40.467976 & 0.022 & 0.32 + ccccccccccccccc source & j & @xmath30 & h & @xmath30 & k@xmath100 & @xmath30 & j@xmath11h & @xmath30 & h@xmath11k@xmath100 & @xmath30 & j@xmath11k@xmath100 & @xmath30 + + s201 & 14.821 & 0.038 & 14.136 & 0.053 & 13.706 & 0.061 & 0.685 & 0.065 & 0.430 & 0.081 & 1.115 & 0.072 + s202 & 14.458 & 0.038 & 13.702 & 0.037 & 13.485 & 0.05 & 0.756 & 0.053 & 0.217 & 0.062 & 0.973 & 0.063 + s204 & 13.243 & 0.029 & 12.566 & 0.032 & 12.359 & 0.036 & 0.677 & 0.043 & 0.207 & 0.048 & 0.884 & 0.046 + s206 & 14.516 & 0.051 & 13.921 & 0.051 & 13.796 & 0.069 & 0.595 & 0.072 & 0.125 & 0.086 & 0.72 & 0.086 + s207 & 15.235 & 0.052 & 14.678 & 0.070 & 14.58 & 0.12 & 0.557 & 0.087 & 0.098 & 0.139 & 0.655 & 0.131 + s208 & 17.597@xmath101 & & 15.246 & 0.133 & 14.614 & 0.12 & @xmath102 & & 0.632 & 0.179 & @xmath103 & + s209 & 14.281 & 0.052 & 13.232 & 0.068 & 12.725 & 0.063 & 1.049 & 0.086 & 0.507 & 0.093 & 1.556 & 0.082 + s210 & 14.648 & 0.038 & 13.685 & 0.039 & 13.249 & 0.043 & 0.963 & 0.054 & 0.436 & 0.058 & 1.399 & 0.057 + s211 & 13.503 & 0.026 & 12.695 & 0.021 & 12.338 & 0.03 & 0.808 & 0.033 & 0.357 & 0.037 & 1.165 & 0.040 + s212 & 11.468 & 0.022 & 11.075 & 0.020 & 10.9 & 0.023 & 0.393 & 0.030 & 0.175 & 0.030 & 0.568 & 0.032 + s213 & 14.99 & 0.045 & 14.151 & 0.052 & 13.802 & 0.063 & 0.839 & 0.069 & 0.349 & 0.082 & 1.188 & 0.077 + s214 & 12.674@xmath101 & & 11.851 & 0.035 & 11.606 & 0.039 & @xmath104 & & 0.245 & 0.052 & @xmath105 & + s215 & 14.361 & 0.036 & 13.29 & 0.036 & 11.96@xmath101 & & 1.071 & 0.051 & @xmath106 & & @xmath107 & + s216 & 13.736 & 0.028 & 12.929 & 0.031 & 12.615 & 0.034 & 0.807 & 0.042 & 0.314 & 0.046 & 1.121 & 0.044 + s217 & 11.733 & 0.021 & 11.497 & 0.018 & 11.463 & 0.018 & 0.236 & 0.028 & 0.034 & 0.025 & 0.27 & 0.028 + s218 & 14.9@xmath101 & & 15.591 & 0.136 & 13.743@xmath101 & & @xmath108 & & @xmath109 & & n / a & + s219 & 11.927 & 0.021 & 11.252 & 0.018 & 11.155 & 0.017 & 0.675 & 0.028 & 0.097 & 0.025 & 0.772 & 0.027 + s301 & 14.915 & 0.039 & 13.742 & 0.037 & 13.337 & 0.042 & 1.173 & 0.054 & 0.405 & 0.056 & 1.578 & 0.057 + s303 & 15.776 & 0.069 & 14.678 & 0.064 & 14.283 & 0.091 & 1.098 & 0.094 & 0.395 & 0.111 & 1.493 & 0.114 + s304 & 15.764 & 0.068 & 14.914 & 0.076 & 14.479 & 0.103 & 1.750 & 0.102 & -0.465 & 0.128 & 1.285 & 0.123 + s305 & 11.185 & 0.023 & 10.783 & 0.018 & 10.673 & 0.016 & 0.402 & 0.029 & 0.11 & 0.024 & 0.512 & 0.028 + s307 & 14.559 & 0.033 & 13.225 & 0.026 & 12.636 & 0.026 & 1.334 & 0.042 & 0.589 & 0.037 & 1.923 & 0.042 + s308 & 14.29 & 0.039 & 13.375 & 0.063 & 13.007 & 0.036 & 0.915 & 0.074 & 0.368 & 0.073 & 1.283 & 0.053 + s309 & 17.863@xmath101 & & 15.472 & 0.118 & 14.683 & 0.125 & @xmath102 & 0.118 & 0.789 & 0.172 & @xmath110 & + s310 & 15.37 & 0.052 & 14.331 & 0.049 & 14.127 & 0.076 & 1.039 & 0.071 & 0.204 & 0.090 & 1.243 & 0.092 + s311 & 15.181 & 0.054 & 14.246 & 0.054 & 13.957 & 0.067 & 0.935 & 0.076 & 0.289 & 0.086 & 1.224 & 0.086 + s312 & 12.373 & 0.022 & 11.83 & 0.021 & 11.686 & 0.017 & 0.543 & 0.030 & 0.144 & 0.027 & 0.687 & 0.028 + s314 & 16.492 & 0.135 & 15.4 & 0.107 & 14.906 & 0.158 & 1.092 & 0.172 & 0.494 & 0.191 & 1.586 & 0.208 + s315 & 15.365 & 0.052 & 14.738 & 0.062 & 14.38 & 0.101 & 0.627 & 0.081 & 0.358 & 0.119 & 0.985 & 0.114 + s402 & 14.456 & 0.038 & 13.702 & 0.035 & 13.477 & 0.045 & 0.754 & 0.052 & 0.225 & 0.057 & 0.979 & 0.059 + s403 & 13.262 & 0.023 & 12.749 & 0.024 & 12.634 & 0.028 & 0.513 & 0.033 & 0.115 & 0.037 & 0.628 & 0.036 + cccccc source & model@xmath101 & @xmath37 & @xmath111 & @xmath112 & @xmath113 or kt ( kev ) + & & & & @xmath114 & + + s312 & pl & 16.6 & 15 & 1.07 ( 0.80@xmath1151.45 ) & 8.2 ( 6.8@xmath11510.0 ) + s312 & mekal & 22.3 & 15 & 0.00 ( 0.00@xmath1150.05 ) & 0.77 ( 0.68@xmath1150.83 ) + s312 & bb & 14.6 & 15 & 0.37 ( 0.22@xmath1150.61 ) & 0.14 ( 0.12@xmath1150.15 ) + s206 & pl & 18.7 & 16 & 0.01 ( 0.00@xmath1150.15 ) & 1.92 ( 1.67@xmath1152.36 ) + s206 & mekal & 43.4 & 16 & 1.2 & 1.0 + s206 & bb & 32.8 & 16 & 0.0 & 0.5 + s219 & pl & 29.8 & 16 & 0.5 & 5.4 + s219 & mekal & 49.2 & 16 & 0.0 & 1.0 + s219 & bb & 33.9 & 16 & 0.0 & 0.2 + s305 & pl & 18.2 & 12 & 0.48 ( 0.16@xmath1150.82 ) & 2.98 ( 2.51@xmath116 3.89 ) + s305 & mekal & 17.8 & 12 & 1.28 ( 1.07@xmath1151.51 ) & 0.70 ( 0.59@xmath117 0.82 ) + s305 & bb & 21.7 & 12 & 0.0 ( 0.0@xmath1180.20 ) & 0.39 ( 0.32@xmath117 0.44 ) + s214 & pl & 12.3 & 6 & 0.2 & 3.2 + s214 & mekal & 9.7 & 6 & 0.9 & 0.9 + s214 & bb & 14.6 & 6 & 0.0 & 0.3 + s204 & pl & 7.75 & 4 & 0.0 & 2.6 + s204 & mekal & 11.6 & 4 & 1.0 & 1.0 + s204 & bb & 12.0 & 4 & 0.0 & 0.4 + . ccccc seq . & start date & exposure & r.a . ( j2000 ) & dec . ( j2000 ) + number & ( ymd ) & ( sec ) & ( hms ) & ( dms ) + + 202033h & 1994 11 15 & 10536 & 20 20 28.08 & 41 21 36 + 202534h & 1997 06 06 & 10370 & 20 22 14.04 & 40 15 36 + 400899h & 1996 11 02 & 36552 & 20 21 04.08 & 40 26 24 + 500339h & 1994 11 14 & 18761 & 20 19 48.00 & 40 03 00 + cccccc data & source & r.a . ( j2000 ) & dec . ( j2000 ) & hri rate & _ chandra _ + & & & & cts / s @xmath119 & + + 400899h & rx j202137.6 + 402959 & 305.407035 & 40.499979 & @xmath120 & + 400899h & rx j202057.8 + 402829 & 305.240996 & 40.474830 & @xmath121 & s214 + 400899h & rx j202111.3 + 402806 & 305.297479 & 40.468571 & @xmath122 & s219 + 400899h & rx j202040.8 + 402704 & 305.170322 & 40.451285 & @xmath123 & s204 + 400899h & rx j202130.5 + 402649 & 305.377281 & 40.447124 & @xmath124 & + 400899h & rx j202016.8 + 402614 & 305.070074 & 40.437480 & @xmath125 & s312 + 202534h & rx j202240.0 + 401900 & 305.666926 & 40.316753 & @xmath126 & + 202534h & rx j202150.5 + 401837 & 305.460420 & 40.310447 & @xmath127 & + 500339h & rx j201950.8 + 395752 & 304.961886 & 39.964551 & @xmath128 & +
|
in search of the counterpart to the brightest unidentified gamma - ray source 3eg j2020 + 4017 ( 2cg078 + 2 ) we report on new x - ray and radio observations of the-cygni field with the _ chandra _ x - ray observatory and with the green bank telescope ( gbt ) . observation detected 37 additional x - ray sources which were correlated with catalogs of optical and infrared data . finally , reanalysis of archival rosat data leads to a flux upper limit of for a putative point - like x - ray source located within the 68% confidence contour of 3eg j2020 + 4017 . adopting the snr age of 5400 yrs and assuming a spin - down to x - ray energy conversion factor of this upper limit constrains the parameters of a putative neutron star as a counterpart for 3eg j2020 + 4017 to be , and g.
|
in search of the counterpart to the brightest unidentified gamma - ray source 3eg j2020 + 4017 ( 2cg078 + 2 ) we report on new x - ray and radio observations of the-cygni field with the _ chandra _ x - ray observatory and with the green bank telescope ( gbt ) . we also report on reanalysis of archival rosat data . with _ chandra _ it became possible for the first time to measure the position of the putative gamma - ray counterpart rx j2020.2 + 4026 with sub - arcsec accuracy and to deduce its x - ray spectral characteristics . these observations demonstrate that rx j2020.2 + 4026 is associated with a k field star and therefore is unlikely to be the counterpart of the bright gamma - ray source 2cg078 + 2 in the snr g78.2 + 2.1 as had been previously suggested . the _ chandra _ observation detected 37 additional x - ray sources which were correlated with catalogs of optical and infrared data . subsequent gbt radio observations covered the complete 99% egret likelihood contour of 3eg j2020 + 4017 with a sensitivity limit of which is lower than most of the recent deep radio search limits . if there is a pulsar operating in 3eg j2020 + 4017 , this sensitivity limit suggests that the pulsar either does not produce significant amounts of radio emission or that its geometry is such that the radio beam does not intersect with the line of sight . finally , reanalysis of archival rosat data leads to a flux upper limit of for a putative point - like x - ray source located within the 68% confidence contour of 3eg j2020 + 4017 . adopting the snr age of 5400 yrs and assuming a spin - down to x - ray energy conversion factor of this upper limit constrains the parameters of a putative neutron star as a counterpart for 3eg j2020 + 4017 to be , and g.
|
1205.2059
|
c
|
correlations between different parts of the genome are usually referred to as linkage disequilibrium , suggesting that due to genetic linkage , i.e. , a high probability of coinheritance , the allele frequencies at different loci are not independent . here , we have used a different measure to characterize correlations in the population . instead of looking at correlations between individual loci , we have characterized the distribution of clone sizes , or equivalently haplotype frequencies , in adapting populations . our analysis is not restricted to additive fitness functions , but we have also analyzed a simple model where fitness is partly heritable and partly random . while macroscopic condensation , @xmath210 , sets in only for @xmath211 in the absence of epistasis , we observe condensation of the population at recombination rates of order @xmath81 in presence of epistasis . here , @xmath81 is the standard deviation in fitness and sets the strength of selection . the reason for this behavior is the fact that the velocity at which the population adapts is smaller than the fitness variance of the recombinant offspring . the additional epistatic variance allows the seeding of new clones far ahead of the population mean , which is only slowly catching up . hence fit clones can out - grow any traveling gaussian . at the same time , condensed clones cause the average epistatic fitness to be significantly greater than 0 . since this epistatic fitness is lost upon outcrossing , the population has a tendency to partition into a few fit clones with high epistatic fitness and a large number of recombinant genotypes with random epistatic fitness . this co - existence between `` condensed '' and `` dust '' phases is seen in fig . [ fig : clones ] in the panels on the right . as long as the heritability , i.e. , the fraction of additive variance , is larger than zero , the population seeds new clones even at low recombination rates and the rate of coalescence will be given by @xmath205 times the characteristic turn - over rate of clones . in the absence of any additive variance , the observed behavior is quite different . in this case , the fitness function is completely random ( a.k.a . house - of - cards model @xcite , or random epistasis / energy model @xcite ) . the only way the population adapts is by sampling fitter and fitter individuals from the same distribution . in other words , the population dynamics amounts to a record process where the total number of samples taken increases as @xmath212 . records establish and grow with the rate @xmath213 . one additional complication that is of particular importance in the case @xmath94 is the fact that whenever a clone recombines with itself , it does not generate a novel genotype . this has the tendency to shut off recombination and stabilize clones as soon as they grow large , resulting in a rapid loss of genetic diversity . in a previous publication @xcite , some of us have studied the onset of condensation in a more descriptive manner . here , we have extended this work by explicitly calculating @xmath0 , both during an initial transient as well as in a steady state where variance is maintained . the model we have used is extremely simplistic , and one might wonder about its relation to real world populations . nevertheless , it accounts for a number of features of real populations such as heritabilities between @xmath60 and @xmath214 , outcrossing , and mimics a large number of loci in the sense of quantitative genetics . these features give rise to qualitatively different dynamical regimes , which will also be observable in more realistic models . some facultatively sexual populations are in fact remarkably close to this simple `` @xmath55 and @xmath54 '' model . many plants and microbial populations are facultative outcrossers . in the event of outcrossing , a large number of crossovers on many chromosomes produces many independently inherited genomic segments . hiv , for example , recombines via template switching following coinfection at an outcrossing rate of a few percent @xcite . in each of these outcrossing events , roughly 10 crossovers are observed @xcite . if populations are polymorphic at many loci , the resulting off - spring distribution is approximately described by eq . ( [ eq : rec_kernel ] ) . we have made a further simplification by assuming that the fitness of a recombinant offspring is independent of its parents and simply drawn from a comoving distribution . this assumption is expected to approximate recombination processes where the offspring and parent fitness decorrelate rapidly over a few out - crossing events , as for example in eq . ( [ eq : rec_kernel ] ) @xcite . note , however , that loci close together on the chromosome decorrelate only slowly . the other dramatic simplification made above was the partitioning of fitness into additive and random components corresponding to high order epistatic interactions . more generally we expect to find epistatic interactions of various orders , which are heritable to different extents . however , the `` @xmath55 and @xmath54 '' model should not be thought of as a parameterization of the genotype - fitness map , but rather as a partitioning into variance that can be explained by the best fitting additive model , and the remaining variance @xcite . the best - fitting additive model is in general time dependent , and the heritability can change as the allele frequencies change @xcite . we know rather little about the genetic architecture of fitness , which justifies the the use of such simple models . in specific scenarios where the genotype - phenotype map is known , more detailed modeling should be guided by the general conclusions drawn from the `` @xmath55 and @xmath54 '' model . the variation that we assume is always present among the recombinant off - spring , it is ultimately fueled by mutations the balance between the influx of beneficial mutations and selection in facultatively sexual population has been investigated in @xcite . even in sexual populations , the dynamics of mutation can be strongly affected by the selection through the clonal structure of the population . the participation ratio , @xmath0 , is exactly the probability for two individuals to be genetically identical . therefore , it immediately gives a measure of the clonal structure of the population . if the two genotypes are identical , they have had a common ancestor in the recent past . hence , @xmath205 is proportional to the rate of coalescence , and as soon as @xmath205 is no longer proportional to @xmath160 , coalescence is greatly accelerated relative to a neutral model . it is well known that selection accelerates coalescence since fit individuals have more offspring and dominate future generations . recombination tends to reduce the effects of linked selection since it decouples different regions of the genome . we have calculated the rate of coalescence in our model , which is set by a balance between selection and recombination . we have shown that there is a critical recombination rate , where recombination is overwhelmed by selection and the population structure changes qualitatively . in the case of additive fitness functions , we have found that @xmath215 $ ] , which is in agreement with earlier work @xcite . in this case , the population consists of clones apparent as little `` bubbles '' in the representation of fig . [ fig : clones ] . any such bubble originates from a common ancestor @xmath216 generations in the past , and @xmath217 is the probability that two individuals belong to the same `` bubble '' . this bubble - coalescent is similar to ideas developed for structured populations @xcite or the fitness class coalescent @xcite . note , however , that the clone - size distribution is very long tailed and the bubble coalescent is not in the universality class of the kingman coalescent @xcite , but possibly of bolthausen - sznitman type @xcite . genetic identity between some of the individuals reduces the effect of outcrossing , since identical parents produce identical offspring . since the probability of such an occurrence is equal to @xmath0 , the effective rate of recombination in the partly clonal population is @xmath218 . hence , strictly speaking our calculations of @xmath219 deep in the clonal condensation regime must be taken through a self - consistency step , which replaces @xmath5 that was hereto an independent variable by a dependent variable @xmath220 . this however would not change our estimates for @xmath221 and @xmath222 , which are defined by the point of first emergence of clones ( when @xmath0 rises above @xmath223 ) . going beyond the mft description of recombination , one may define a participation ratio _ density _ @xmath224 in terms of which @xmath225 which picks up the fitness dependence of @xmath0 : individuals with relatively high fitness are much more likely to be clonally related . the significance of @xmath0 is not limited to the case of exact genetic identity within clones . in particular , mutations would introduce additional polymorphic loci into the clones " that were the focus of our study . yet the genetic structure of the population introduced by clonal condensation still survives : one only needs to distinguish between high - frequency polymorphisms , which are being reshuffled by recombination as approximated by our model and the low frequency new polymorphisms due to recent ( on the time scale of clonal growth ) mutations . the latter would appear on the background that is clonal with respect to the high - frequency alleles . thus the clones " emerging at low @xmath5 should be thought of as haplotypes @xcite and the clonal condensation " is the process that suppresses the number of haplotypes on small length scales . the participation ratio can be readily generalized to allow for a degree of genetic distance within a pair of individuals . as currently defined @xmath226 ( where @xmath227 stands for the genetic distance between @xmath228 and @xmath229 ) . this is immediately recognizable as a special case of the parisi order parameter @xmath230 @xcite . the latter therefore provides an interesting representation of the haplotypic structure of populations . it would be interesting to understand whether more realistic models of fitness landscapes ( with low order , rather than random epistasis ) would generate more complex hierarchical structure of @xmath231 than the simple dust / clone " dichotomy found in our simply model . the relation between the rem and sherrington - kirkpatrick models of spin glasses @xcite could provide useful guidance and ultimately yield better understanding of haplotype distributions and recombinant coalescents @xcite . the analysis of clonal condensation " presented above can and should be extended in several ways . within the confines of the model considered , one may want to obtain a better understanding of the mixed phase " lying between the two transition lines identified in fig . [ fig : v_and_y ] . this phase is characterized by large fluctuations in clone size distribution , and hence in @xmath0 , even in the approximation imposing a fixed traveling wave velocity @xmath52 . in reality , the population sets its own instantaneous rate of change of average fitness , which depends sensitively on the fitness of the leading clones and changes with time as new fitness records " are established by fresh recombinants . we have already described in fig . [ fig : stick_slip ] the stick - slip " dynamics , which is characteristic of the mixed phase regime . ( needless to say , the existence of the mixed phase region corresponds to the 1st order nature of the clonal condensation transition for @xmath232 . ) a fully quantitative description of this behavior would require going beyond mft . so far , attempts to describe fluctuations in the dynamics of adaptation have been few and far apart @xcite : a quantitative description of the stick - slip " progress of adaptation would represent a major step forward . another necessary extension of the model involves mutational influx . a non - zero mutation rate would provide an influx of genetic variation and define true statistically stationary states corresponding to adaptive traveling waves @xcite or , in the presence of both deleterious and beneficial mutations , to a dynamic mutation selection balance @xcite . in that case , emergent clones become fuzzy " as they accumulate mutations , and the participation ratio should be replaced by a more general parisi order parameter as explained above . the result should provide interesting quantitative insight into the expected genetic structure of facultatively sexual populations . perhaps the most interesting and important extension of the present work would involve a generalization of the model and its analysis to linear genomes and obligatory sexual populations . in contrast to the current model where recombination freely re - assorts the loci , a more realistic linear chromosome model would describe recombination in terms of relatively infrequent crossover . this would naturally tie the frequency of recombination to the length of the segment considered . we expect that on sufficiently short scales , where recombination is infrequent , a tendency for haplotype condensation would be manifest if the population is diverse enough . whether epistasis plays a significant role in the condensation process will depend on the distribution of epistatic interactions along the genome @xcite . if there is a strong tendency of mutations to interact with mutations nearby @xcite , the heritability decreases as smaller and smaller segments are considered . this could result in condensation of mutations into `` super - alleles '' . however , the embedding of the haplotype in question into a larger genome gives rise to complications related to hill - robertson interference @xcite . transient associations with other genomic regions will either boost ( the hitch - hiking process ) or suppress ( background selection ) the spread of a haplotype in the population . this reduces the efficacy of selection and gives rise to a stochastic dynamics with rather different properties than genetic drift @xcite . bridging the different scales and resulting dynamical regimes represent an important challenge for the future . in conclusion , we stress the distinction between the qle and clonal condensation " regimes . in the qle regime , recombination is sufficiently rapid to overwhelm any clonal amplification due to selection , and correlations between alleles at different loci are relatively weak . in this regime , the correlations between loci are well described by linkage disequilibrium , which measures population averaged pair correlation of loci . by contrast , clonal condensation is a non - perturbative , large deviation from linkage equilibrium ( under which loci are completely uncorrelated ) , which in particular results in a stratification of the population depending on its fitness : clones appear predominantly in the upper reaches of the fitness distribution . strong heterogeneity among individuals along the fitness axis is not captured by measuring linkage disequilibrium and other traditional approaches . understanding strong interactions in multi - locus systems requires new ideas and tools . we have found simple models such as the rem to be a very useful source of insight into these non - trivial aspects of population genetics . we are grateful for stimulating discussions with h. teotonio , a. dayarian , i. rouzine , d. fisher , m. desai and m. lynch and would like to thank t. kessinger for careful reading of the manuscript . ran is supported by an erc - starting grant 260686 and bis acknowledges support of the hfsp rfg0045/2010 .
|
in sexual population , recombination reshuffles genetic variation and produces novel combinations of existing alleles , while selection amplifies the fittest genotypes in the population . if recombination is more rapid than selection , populations consist of a diverse mixture of many genotypes , as is observed in many populations . in the opposite regime , which is realized for example in the facultatively sexual populations that outcross in only a fraction of reproductive cycles , selection can amplify individual genotypes into large clones . clonal condensation leads to a strong genetic heterogeneity of the population which is not adequately described by traditional population genetics measures , such as linkage disequilibrium . here of course the number of polymorphic loci , , itself changes as new mutations arise forming new polymorphisms while older " polymorphisms disappear from the population . two parents , if chosen from different clones , produce offspring that are distinct from either parent . in obligate sexually reproducing species , the formation of clones is prevented since reproduction is coupled to recombination and no parent can produce genetically identical offspring . many species , in particular microbial species and plants , can reproduce both by clonal reproduction ( e.g. , budding in yeast , selfing or vegetative reproduction in plants ) or by sexual propagation . such facultatively sexual species display a great variety in their mode of propagation , the frequency of outcrossing , and the heritability of fitness . whenever two individuals are part of the same clone , they share a recent common ancestor , such that is proportional to the rate of pair coalescence . in the canonical theory of neutral coalescence , much of our analysis is presented in the context of a facultatively sexual population , where the evolving entities are individuals and their genotypes . we shall return to this important question in the discussion . in an earlier publication , we have shown that clones are absent in the so called quasi linkage equilibrium ( qle ) phase " corresponding to frequent outcrossing limit but appear in a regime of small out - crossing frequency , with depending on the complexity of the fitness landscape ( i.e. the extent of fitness additivity ) and weakly dependent on . our results therefore provide insight into how recombination and epistasis affect the dynamics and structure of adapting population waves and define conditions under which genetic diversity is maintained or lost .
|
in sexual population , recombination reshuffles genetic variation and produces novel combinations of existing alleles , while selection amplifies the fittest genotypes in the population . if recombination is more rapid than selection , populations consist of a diverse mixture of many genotypes , as is observed in many populations . in the opposite regime , which is realized for example in the facultatively sexual populations that outcross in only a fraction of reproductive cycles , selection can amplify individual genotypes into large clones . such clones emerge when the fitness advantage of some of the genotypes is large enough that they grow to a significant fraction of the population despite being broken down by recombination . the occurrence of this clonal condensation " depends , in addition to the outcrossing rate , on the heritability of fitness . clonal condensation leads to a strong genetic heterogeneity of the population which is not adequately described by traditional population genetics measures , such as linkage disequilibrium . here we point out the similarity between clonal condensation and the freezing transition in the random energy model of spin glasses . guided by this analogy we explicitly calculate the probability , , that two individuals are genetically identical as a function of the key parameters of the model . while is the analog of the spin - glass order parameter , it is also closely related to rate of coalescence in population genetics : two individuals that are part of the same clone have a recent common ancestor . genetic diversity is the fodder for natural selection and the fuel of evolution . it is generated by mutations and by recombination , which reshuffles genomes and thereby accelerates the exploration of the space of genotypes . the latter consists of all of the possible combinations of the genetic variants , a.k.a . alleles , present at ( biallelic ) polymorphic loci . of course the number of polymorphic loci , , itself changes as new mutations arise forming new polymorphisms while older " polymorphisms disappear from the population . the population itself consists of individuals which sample only a small fraction of the possible genotypes , i.e. . the dynamics in genotype space is therefore highly stochastic . of particular importance are those genetic polymorphisms that affect the fitness of individuals , the fitness being defined as the expected number of offspring in the next generation . selection on its own would amplify the number of high fitness individuals and condense the population into a few clones " comprising a large fraction of the population . in populations of sexually reproducing organisms , the growth of such clones and the subsequent decline of genetic diversity are checked by recombination . two parents , if chosen from different clones , produce offspring that are distinct from either parent . in obligate sexually reproducing species , the formation of clones is prevented since reproduction is coupled to recombination and no parent can produce genetically identical offspring . many species , in particular microbial species and plants , can reproduce both by clonal reproduction ( e.g. , budding in yeast , selfing or vegetative reproduction in plants ) or by sexual propagation . such facultatively sexual species display a great variety in their mode of propagation , the frequency of outcrossing , and the heritability of fitness . the latter is very important in sexual populations , as it determines to what extend recombinant offspring benefit from the same fitness advantages that made their parents successful . the aim of this article is to describe quantitatively the competing tendencies of natural selection and recombination with regard to genetic diversity , focusing on facultatively sexual organisms . the competition of natural selection with recombination is the dominant mechanism of evolution on relatively short time scales , on which mutational input is negligible compared to diversification by recombination . this situation is particularly relevant to adaptation following a major outcrossing event , or within a so called hybrid zone , where diverged genotypes have come together to generate a hybrid population . as this hybrid population continues to breed within itself , it can give rise to a bout of rapid adaptation , as beneficial alleles from both original populations are combined to form novel fit genotypes that spread within the hybrid population . we will focus on the probability , , that two random individuals sampled from the population have the same genotype , i.e. , are clones of each other . this quantity is important for population genetics , since it characterizes genetic diversity ( its inverse is a measure of the number of dominant clones ) as well as the dynamics of coalescence . whenever two individuals are part of the same clone , they share a recent common ancestor , such that is proportional to the rate of pair coalescence . in the canonical theory of neutral coalescence , this rate is equal to the inverse population size . we will find here that , and with it the rate of coalescence , is determined by the clonal structure rather than the population size at low outcrossing rates . much of our analysis is presented in the context of a facultatively sexual population , where the evolving entities are individuals and their genotypes . note , however , that some of our considerations also hold for contiguous segments of chromosomal dna that are short enough to undergo only infrequent recombination even in obligatory sexual reproduction . in that case , we would be interested in the probability of a given chromosomal segment to be identical for a random pair of individuals drawn from the population . in this context , is the homozygosity of the population at this extended locus at which many different alleles segregate ( this is of course a much weaker condition than clonal relation of whole genomes ) . a complementary view of this probability relates it to the haplotype diversity " , i.e. , the number and population distribution of distinct genomic sequences for the chromosomal segment in question . we shall return to this important question in the discussion . in an earlier publication , we have shown that clones are absent in the so called quasi linkage equilibrium ( qle ) phase " corresponding to frequent outcrossing limit but appear in a regime of small out - crossing frequency , with depending on the complexity of the fitness landscape ( i.e. the extent of fitness additivity ) and weakly dependent on . we will review this finding and present a more detailed analysis of the time dependence of this condensation phenomenon , as well as its quantitative dependence on fitness additivity and hence heritability . furthermore , we also study the extend of clonal condensation in a steady state where the fitness distribution is moving towards higher fitness at a constant velocity . we will put these results into the context of the random energy model ( rem ) of statistical physics , introduced and solved by bernard derrida . in fact , is closely related to the parisi order parameter and the onset of clonality is closely related to the spin - glass transition observed in simple models of disordered media such as the rem . below , we will first draw the analogy between the dynamics of selection in finite populations and the rem . this analogy is particularly simple for . whereas the condensation transition in the rem occurs below a certain critical temperature , the transition to clonal population structure occurs beyond a certain critical time , . hence the population genetic analog of temperature will be the inverse time . we shall then generalize the model in order to include ( facultative ) recombination and fitness landscapes with varying degrees of epistasis , i.e. , genetic interactions . the results for mixed epistatic and additive models enables us to set the analysis into the context of the traveling wave " approximation , which has recently emerged as a powerful representation of adaptive population dynamics in genetically diverse populations . our results therefore provide insight into how recombination and epistasis affect the dynamics and structure of adapting population waves and define conditions under which genetic diversity is maintained or lost .
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.