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Extended hybrid model

# An extended hybrid magnetohydrodynamics gyrokinetic model for numerical simulation of shear Alfvén waves in burning plasmas

## Abstract

Adopting the theoretical framework for the generalized fishbonelike dispersion relation, an extended hybrid magnetohydrodynamics gyrokinetic simulation model has been derived analytically by taking into account both thermal ion compressibility and diamagnetic effects in addition to energetic particle kinetic behaviors. The extended model has been used for implementing an eXtended version of Hybrid Magnetohydrodynamics Gyrokinetic Code (XHMGC) to study thermal ion kinetic effects on Alfvénic modes driven by energetic particles, such as kinetic beta induced Alfvén eigenmodes in tokamak fusion plasmas.

## 1 Introduction and motivation

Nonlinear numerical simulations of magnetohydrodynamics (MHD) and Alfvén modes driven by energetic particles (EPs) mostly rely on hybrid MHD gyrokinetic codes, such as HMGC [1], M3D [2], and MEGA [3]. In the hybrid MHD gyrokinetic model, the thermal plasma component is described by MHD, while EP dynamics, in the so-called pressure coupling equation [2], is accounted for via the divergence of the EP pressure tensor, which is computed by solving the gyrokinetic equation with particle in cell (PIC) techniques. Kinetic treatments of the thermal plasma component are well known and generally implemented in (linear) spectral codes, such as NOVA-K [5, 6] and MARS-K [7]. More recently, significant developments in gyrokinetic simulation codes, such as GTC [8] and GYRO [9], have also allowed investigating the kinetic effects of thermal plasma and EP dynamics on long wavelength electro-magnetic fluctuations, which were previously investigated only with codes based on the hybrid MHD-gyrokinetic approach [1, 2, 3, 4]. In HMGC [1], the thermal plasma description is originally limited to the reduced MHD model [10]. In the present work, our goal is to extend the hybrid MHD-gyrokinetic model implemented in HMGC [1] to the low-frequency domain of the beta induced Alfvén eigenmode (BAE) - shear Alfvén wave (SAW) continuous spectrum [11], where the mode frequency can be generally comparable with thermal ion diamagnetic and/or transit frequencies, i.e. . In this frequency range, where kinetic thermal ion (KTI) gap generally exists and influences plasma dynamics [11], there is a continuous transition between various MHD and SAW fluctuation branches, as predicted theoretically [12, 13, 14, 15, 16, 17, 18, 19, 20] and confirmed experimentally [21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Another notable feature of these low frequency fluctuations is that they may be resonantly excited by wave-particle interactions with EPs as well as thermal plasma particles, depending on the perpendicular wavelength [20, 31]. With the extended hybrid MHD gyrokinetic model discussed here, it will be possible to investigate various problems related with resonant excitation of Alfvénic and MHD fluctuations by EPs in the BAE-SAW continuous spectrum, consistent with gyrokinetic codes, e.g., GTC [32], in a common validity domain. Therefore, both the eXtended HMGC (XHMGC) and GTC codes can be verified using different models; yielding more detailed understanding of the underlying physics. In fact, theoretical and numerical work, presented in this article and partly developed within the framework of the SciDAC project on “Gyrokinetic Simulation of Energetic Particle Turbulence and Transport” (GSEP), was the prerequisite for successful verification of XHMGC predictions against analytic theories [33] as well as GTC numerical simulation results [34, 35], reported recently.

In this work, we extend the hybrid MHD-gyrokinetic model, derived originally in [2] for applications to numerical simulations of EP driven Alfvén modes. The main differences with respect to the usual pressure coupling equation [2] are due to renormalization of the inertia term, to properly account for finite thermal ion diamagnetic effects, as well as to the gyrokinetic treatment of the thermal ion pressure tensor, which allows us to properly handle wave-particle resonant interactions in the low frequency regime, where they can be of crucial importance for the analysis of linear and nonlinear behaviors of collisionless burning plasmas. The extended model has been developed assuming ideal Ohm’s law as well as ignoring finite Larmor radius (FLR) effects in order to simplify the technical complications while still maintaining all essential physics ingredients [36]. In practice, maintaining the ideal MHD Ohm’s law as limiting case implies assuming and neglecting ion FLR effects, although finite magnetic drift orbit widths (FOW) are fully retained [37]. A more general approach without these simplifying assumptions will be developed in a separate work. For demonstrating the validity of the modified equations, we show that they are equivalent to the quasi-neutrality and vorticity equations derived in [36] for the frequency range from the kinetic ballooning mode (KBM) and BAE to the toroidal Alfvén eigenmode (TAE). The XHMGC model equations in the linear limit are equivalent to the extended kinetic MHD used in spectral codes, such as NOVA-K [5, 6] and MARS-K [7], but with EP dynamics treated non-perturbatively and on the same footing as the thermal plasma response (see Section 2 for more details). The possibility of investigating nonlinear dynamics, however, makes XHMGC more suitable to direct comparisons with M3D [2] or gyrokinetic codes [32, 38] in a common validity domain.

The paper is organized as follows. In Section 2, the extended hybrid model equations are presented and discussed within the theoretical framework of [36]. In Section 3, we describe the numerical implementation of the extended model into HMGC, by adding both thermal ion compressibility and diamagnetic effects (of thermal ions as well as EPs) into MHD equations and a thermal ion population in the PIC module. In Section 4, possible applications and validity limits of XHMGC are discussed. A synthetic summary of current BAE numerical simulation results [33] are also provided. Finally, conclusions and discussions are given in Section 5.

## 2 Derivation of the extended hybrid model

Reference [36] presents a general theoretical framework for stability analyses of various modes and the respective governing equations. It shows that all modes of the shear Alfvén branch having frequencies in the range between the thermal ion transit and Alfvén frequency can be consistently described by one single general fishbone-like dispersion relation (GFLDR) [12, 16, 36, 15, 14]. Reference [36] discusses various reduced equations governing the evolution of SAW fluctuations in burning plasmas, using the general approach of reference [39]. In this sense, reference [36] could seem not the optimal eliminate reference framework for further generalizing the HMGC hybrid model equations  [1, 40], which are to be used for nonlinear studies as well. However, the detailed analyses of reduced model equations, reported in [36], on the basis of specializations of ordering of dimensionless parameters in the case of burning plasmas of fusion interest, starting from the somewhat different orderings of interest to space plasmas given in [39], allow to fully grasp the physics implications of the underlying approximations. Moreover, on the basis of our discussions, it is straightforward to motivate the extension of the derived model equations to the nonlinear case, as shown at the end of this section.

Considering that the characteristic frequency, , is much lower than the ion cyclotron frequency, , we may adopt the gyrokinetic theoretical approach and closely follow reference [39]. The low-frequency plasma oscillations can, thus, be described in terms of three fluctuating scalar fields: the scalar potential perturbation , the parallel (to , with the equilibrium magnetic field) magnetic field perturbation and the perturbed field , which is related to the parallel vector potential fluctuation by

 δA∥≡−i(cω)b⋅∇δψ. (1)

The governing equations for describing the excitation of the shear Alfvén frequency spectrum by energetic ions precession, precession-bounce and transit resonances in the range , covering the entire frequency range from KBM/BAE [14, 13, 21, 41] to TAE [42, 43, 44], are generalized kinetic vorticity equation and quasi-neutrality condition, which can be written as followings, in the limit of vanishing FLR (see equation (16) and equation (17) in reference [36]):

 B0⋅∇(k2⊥k2θB20B0⋅∇δψ)+ω(ω−ω∗pi−nEmEnimiω∗pE)v2Ak2⊥k2θδϕ −⟨∑s≠e4πesk2θc2ω^ωdsδKs⟩+∑s4πk2θB20k×b⋅∇(Ps⊥+Ps∥)Ωκδψ=0, (2)
 ⟨∑s≠Ee2sms∂F0s∂ε⟩(δϕ−δψ)+∑s=ies⟨δKs⟩=0, (3)

where the non-adiabatic particle response, , is obtained via the drift-kinetic equation

 [ωtr∂θ−i(ω−ωd)]sδKs=i(em)sQF0s[(δϕ−δψ)+(^ωdω)sδψ]. (4)

Here, angular brackets stand for velocity space integration, denotes all particle species ( bulk electrons, bulk ions, energetic particles), and are the species electric charge and mass, is the equilibrium distribution function (generally anisotropic), the energy per unit mass, , , is the wave vector, is the cyclotron frequency, is the perpendicular wave vector, is the diamagnetic frequency, and are, respectively, the total perpendicular and parallel plasma pressures, is the transit frequency and , with and . Note that the difference between and , with , has been discussed in [36, 39] and, generally, must be handled properly; although, for many applications in low pressure () plasmas, one can consider after solving for from perpendicular pressure balance [39, 36], as implicitly assumed in equations 23 and 4. Note, also, that we have maintained the EP contribution to the divergence of the polarization current, which is represented by its leading term in equation 2. This term is readily derived from the last term on the left hand side (LHS) of equation (13) in reference [36] (see also A for further details) and was neglected in there due to the ordering , valid in a burning plasma dominated by fusion alpha particle self-heating. Here, and denote the beta values of EP and bulk plasma components (electrons and thermal ions), respectively, while and are the energy confinement time and EP slowing down time. More generally [12, 13, 14, 15], the ordering better represents nowadays magnetized plasmas of fusion interest and, thus, , as assumed in equation 2.

Equations 2 and 3, together with the drift-kinetic equation, equation 4, are the simplest yet relevant equations for analyzing the resonant excitations of SAW by EPs. Equation 2 demonstrates that both resonant as well as non-resonant responses due to the term enter via the magnetic curvature drift coupling. In the high frequency case, , the thermal ion kinetic compression response can be neglected. Thus, the quasi-neutrality condition, equation 3, reduces to the ideal MHD approximation, ; i.e.  [39]. Meanwhile, neglecting the terms, equation 2 becomes equivalent to equation (3) in [2], i.e. the following pressure coupling equation in the hybrid MHD-gyrokinetic approach

 ρbdvbdt=−∇Pb−(∇⋅PE)⊥+J×Bc; (5)

where the subscript denotes the bulk plasma (electrons and thermal ions), while and are, respectively, bulk plasma mass density and fluid velocity. Here, the EP contribution to the perpendicular momentum change of the plasma has been neglected, due to  [36, 2], and thermal ion diamagnetic effects are consistently dropped since .

In order to extend the hybrid model to the low-frequency regime where , we need to include the effects of the thermal ion compressibility within the hybrid simulation scheme. That is, we need to include effects associated with the terms in equations 2 and 3. First, in order to simplify the discussions, we formally assume in the present work; the general case with finite will be considered elsewhere. Thus, according to equation 3, we have and the ideal MHD condition remains valid. Next, we proceed to establish correspondences between the pressure coupling equation, equation 5, and the generalized kinetic vorticity equation, equation 2.

Applying the operator to the linearized equation 5 and noting the quasi-neutrality condition , we readily derive

 1c∂∂tB0⋅∇δJ∥B0i+1c∂∂tδB⋅∇(J∥0B0)ii+∂∂t∇⋅⎛⎜⎝B0×ρb0dδvbdtB20⎞⎟⎠iii +∂∂t∇δPb⋅(∇×bB0)iv+∂∂t∇⋅(b×(∇⋅δPE)⊥B0)v=0. (6)

Noting also the parallel Ampère’s law along with ,

 4πδJ∥=−c∇2δA∥, (7)

and equation 1, term (i) can be seen to correspond to the field line bending term; i.e. the first term in equation 2. Term (ii), on the contrary, does not have any direct correspondence in equation 2. This term is the usual kink drive and it was dropped in the analysis of [36], focusing on drift Alfvén fluctuations with high mode numbers, for it is formally of , with the toroidal mode number. However, as noted in equation (A1) of [36], term (ii) is readily recovered in a form that can be straightforwardly reduced to that reported here. Meanwhile, from the linearized Ohm’s law

 δE⊥+1cδvb×B0=0, (8)

and , term (iii) corresponds to the second term in equation 2 with the terms neglected. To establish correspondences between the pressure responses in equations 2 and 2, we first denote . It can then be shown (see appendix B) that term (iv) corresponds to the thermal ion and electron contributions to the last term on the LHS of equation 2, when kinetic compression effects of the background thermal plasma are neglected.

Finally, let us discuss term (v), due to EP pressure perturbation, which can be expressed as (see C).

 ∂∂t∇⋅(b×(∇⋅δPE)⊥B0)=ωB0Ωκ(δPE∥+δPE⊥). (9)

Meanwhile, noting the definition of  [39, 36], the term in equation 2 can be shown to be related with the pressure perturbations as (see D)

 ⟨4πesk2θc2ω^ωdsδKs⟩ = 4πωk2θcB0Ωκ(δPs⊥+δPs∥) (10) +4πk2θB20(k×b)⋅(∇P0s⊥+∇P0s∥)Ωκδψ.

Note that the 2nd term in the right hand side (RHS) disappears in the ideal MHD limit. Equation 10, thus, clearly demonstrate that the contribution in equation 2, combined with the 3rd term on the RHS of equation 10 (or the last term on the LHS in equation 2), has the same form of equation 9 and recovers the total pressure response of term (iv) in equation 2 for . In other words, the term in equation 2 corresponds to the kinetic compressibility component of the pressure perturbations.

Summarizing the above discussions, it is clear that, in order to include effects due to finite thermal ion compressibility and diamagnetic drift as well as the finite EP contribution to the divergence of the polarization current, the pressure coupling equation in the MHD-gyrokinetic approach, equation 5, has to be modified such that its perpendicular components are given by equation 26 of A, which we rewrite here for the reader’s convenience

 [ρb(∂∂t+vb⋅∇)+b×∇P0E⊥ωcE⋅∇]δvb= −∇⊥Pe−(∇⋅Pi)⊥−(∇⋅PE)⊥+(J×Bc)⊥. (11)

Here, , and the “unshifted” pressure tensors and need to be calculated from solutions of the gyrokinetic equations as specified in A, while is consistently neglected in the present approach, assuming . Reminding the concluding remark of A, this equation readily reduces to the well-known pressure coupling equation 2, in the limit where thermal ion diamagnetic effects and EP contribution to the divergence of the polarization current are neglected.

As anticipated above, in the present work, we followed reference [36], since that has a detailed discussion of validity limits of different reduced models of the whole vorticity and quasi-neutrality equations, derived for fusion applications and following the trace of reference [39]. Equation 2 includes equilibrium parallel current effects, as discussed earlier in this section and in [36] (Appendix). This simple remark readily follows from the discussion presented in [34] as well as the modified momentum balance equation implemented in XHMGC, i.e. equation 2 itself. The present model is valid in the nonlinear case too, as shown by the simple derivation provided in A and by the following discussion. This is deduced easily from direct inspection of equation (5) in reference [45]. That equation clearly shows that, for the small FLR limit considered in HMGC [1, 40], the nonlinear terms, treated explicitly, are those that are coming from convective nonlinearity and from the Maxwell stress nonlinearity, when the thermal ion response is taken in the fluid limit, both of which are readily obtained from equation 2 upon application of the operator , as it was done for equation 5 earlier in the section. Other nonlinear dynamics, which are implicitly included in and terms, are fully retained via equation 2. Thus, the back reaction of zonal structures onto SAW fluctuations is fully accounted for, i.e. that of zonal flows (ZFs) and fields as well as radial modulation of equilibrium profiles [9, 11, 46] which also enter via the diamagnetic terms in equation 2, computed on the whole (slowly evolving) thermal ion and EP pressure profile, obtained from the respective toroidally and poloidally averaged distribution functions. This choice is consistent with known approaches to nonlinear MHD equations, accounting for finite diamagnetic drift corrections [47, 48, 49, 50, 51].

Thus, the approximations involved with the extended implementation within XHMGC on the basis of equation 2 consist of neglecting FLR, assuming electron as a massless fluid, considering (such that parallel Ohm’s law is recovered in the ideal MHD limit) and accounting for Reynolds stress in the thermal ion fluid limit. The possible further extension of the present model to include finite and generalizing the parallel Ohm’s law, while maintaining other simplifying assumptions, is straightforward on the basis of the present discussion and will be reported in a separate work [52]. Here we note that the present extended hybrid model, based on equation 2, with clearly formulated assumptions that limit its applicability, includes very rich physics; e.g. it is capable to correctly evaluate the renormalized inertia for ZFs, for which the trapped thermal ion dynamics is of crucial importance, and to account for geodesic acoustic mode (GAM) kinetic response, including Landau damping.

So far, XHMGC has been used for moderate EP drive [33, 34, 35, 37], where the EP diamagnetic correction to the divergence of the polarization current can be neglected, as argued in [36]. Actually, in the studies reported in [33], thermal ion diamagnetic contribution to the polarization current is also neglected, since the case of uniform thermal ion pressure profiles is investigated in there for facilitating comparisons of numerical simulation results with analytic theory predictions (see also section 4).

## 3 Numerical implementation

HMGC [1] is used for investigating linear and nonlinear properties of moderate toroidal number (n) shear Alfvén modes in tokamaks. It solves the coupled set of O() reduced-MHD equations [10] for the electromagnetic fields and the gyro-center Vlasov equation for a population of energetic ions, where large aspect ratio is assumed, i.e. , with and the tokamak minor and major radius, respectively . Energetic particles contribute to the dynamic evolution of the wave fields via the pressure tensor term in the MHD equations, as described by the pressure coupling equation [2]. This code allows us to describe both self-consistent mode structures in toroidal equilibria and EP dynamics, as well as to get a deeper insight into how the Alfvénic modes affect the confinement of such particles.

The extended model, described in section 2, has been implemented into the eXtended version of HMGC (XHMGC). Following the general procedure, described in references [1, 40], for the formal manipulation of equation 2, the relevant equations for the MHD solver are in terms of the poloidal magnetic field stream function and , which is proportional to the scalar potential and defined as , can be written in the following form in the cylindrical coordinate system :

 ∂Ψ∂t=R2R0∇Ψ×∇φ⋅∇U+B0R0∂U∂φ+ηc24π△∗Ψ+O(ϵ4vABφ), (12)
 ^ρ(DDt+2R0∂U∂Z)∇2⊥U+∇^ρ⋅(DDt+1R0∂U∂Z)∇⊥U (13) −(R2R30ωci0∂P0i⊥∂Z+R2R30ωcE0∂P0E⊥∂Z)∇2⊥U −∇(R2R30ωci0∂P0i⊥∂Z+R2R30ωcE0∂P0E⊥∂Z)⋅∇⊥U +∇⋅[R4R30(∇φ×∇P0E⊥ωcE0+∇φ×∇P0i⊥ωci0)⋅∇∇⊥U] = 14πB⋅∇△∗Ψ+1R0∇⋅[R2(∇Pe+∇⋅Πi+∇⋅ΠE)×∇φ] +O(ϵ4ρv4Aa2),

where we have maintained the same notation of reference [1] and explicitly show the additional terms that have been added to implement the extended XHMGC model. Thus,

 ^ρ=R2R20ρ,  DDt=∂∂t+R2R0∇U×∇φ⋅∇, ∇2⊥≡1R∂∂RR∂∂R+∂2∂Z2,

the Grad-Shafranov operator is defined by

 △∗≡R∂∂R1R∂∂R+∂2∂Z2, (14)

is the vacuum magnetic field on the magnetic axis at  1, , and the subscript denotes components perpendicular to . In the above equations, is the fluid velocity, is the bulk plasma mass density, is the scalar pressure of bulk electrons, and are, respectively, the pressure-tensor of the EP and thermal ions, computed with the definitions of equations 20 and 24, given in A, and, is the resistivity and is the speed of light. These equations have been first derived in reference [10], limited to the MHD description of the thermal plasma, while the inclusion of energetic particle dynamics has been discussed in references. [1, 40]. Here, in the proposed further extension of the numerical model, thermal ion dynamics as well as diamagnetic effects are also taken into account, according to equation 2, derived in A and section 2. At the leading order in , , the reduced-MHD equations describe the thermal plasma in the cylindrical approximation. Toroidal geometry enters the equations as corrections at the next order in the inverse aspect ratio.

In order to close equations 12 and 13, the EP and thermal ion pressure tensor components can be obtained by directly calculating the appropriate velocity moments of the distribution function for the particle population interacting with the perturbed electromagnetic field. As discussed in section 2, we initially assume the limit for the sake of simplicity, i.e. . Meanwhile, with cold electron assumption and ignoring thermal ion finite Larmor radius (FLR), ideal MHD parallel Ohm’s law can be readily recovered.

As to numerical formulation, the equations of motion in gyro-center coordinates for thermal ions are in the same form, mutatis mutandis, as those reported in [1] for EPs. In the gyrocenter-coordinate system , where is the gyrocenter position, is the conserved magnetic moment, is the parallel speed and is the gyrophase, the equations of motion take the form

 d¯Rdt = ¯Vb+esmsΩsb×∇ϕ−¯VmsΩsb×∇a∥ +[¯Mms+¯VΩs(¯V+a∥ms)]b×∇lnB, d¯Mdt = 0, d¯Vdt = 1msb⋅{[esΩs(¯V+a∥ms)∇ϕ+¯Mms∇a∥]×∇lnB (15) +esmsΩs∇a∥×∇ϕ}−ΩE¯Mmsb⋅∇lnB.

Here, the subscript denotes either EP or thermal ion species and, using the same notations as in [1], is the corresponding cyclotron frequency. The fluctuating potential is related to the poloidal magnetic field stream function through the relationship . The parallel electric field term in the equation for has been suppressed, neglecting, thus, small resistive corrections to the ideal-MHD parallel Ohm’s law. Meanwhile, the pressure tensor can be written, in terms of the gyrocenter coordinates, as

 Πs(t,x)=1m2s∫d¯ZDZc→¯Z¯Fs(t,¯R,¯M,¯V) [Ωs¯MmsI+bb(¯V2−Ωs¯Mms)]δ(x−¯R), (16)

where is the unit tensor, , is the gyrocenter distribution function and is the Jacobian of the transformation from canonical to gyrocenter coordinates. The distribution function satisfies the Vlasov equation

 (∂∂t+d¯Rdt⋅∇+d¯Vdt∂∂¯V)¯Fs=0, (17)

where and are given by equation 3. In the numerical implimentation of XHMGC, equations 3 and 17 can be readily solved as a full-F simulation. On the other hand, a algorithm [40, 53, 54, 55] is also implemented in order to minimize the discrete particle noise. The latter is recommended as far as , the former when .

## 4 Applications

In general, the extended version of HMGC can have two species of kinetic particles. On one hand, one can use XHMGC for investigating thermal ion kinetic effects on Alfvénic modes driven by EP. On the other hand, it may be interesting to use XHMGC as a tool to simulate two coexisting EP species, generated e.g. by both ion cyclotron resonance heating (ICRH) and neutral beam injection (NBI) heating, in order to study linear excitation of Alfvénic fluctuations and Energetic Particle Modes (EPM) [15], as well as the interplay between the respective nonlinear physics controlled by the different heating sources [56].

HMGC has been extensively used in [1] and [57] to investigate the linear physics (damping and EP drive mechnisms), and in [40] and [58] to analyze the nonlinear dynamics of EPM. XHMGC has been verified against those previous findings and can recover numerical simulation results in the above studies. Furthermore, by accounting for the kinetic thermal ion effects, XHMGC shows the existence of Kinetic BAE (KBAE) which can be seen as radially trapped eigenstates due to discretization of BAE-SAW continuum by FLR/FOW effects, as well as KBAE resonantly excited by wave-particle interactions with EPs [33, 37].

As an example to demonstrate the capability of XHMGC, we briefly report simulation results of KBAE, which is discussed in detail in [33]. The results show that a fully kinetic treatment of thermal ions is necessary for a proper description of the low frequency Alfvénic fluctuation spectrum. By including thermal ion compressibility, our numerical simulations do show the existence of a finite-frequency BAE accumulation point in the SAW continuum, which was demonstrated analytically and numerically using MHD codes [59]. Meanwhile, when effects due to finite ion drift orbit width (FOW) are included, our simulations clearly demonstrate that the BAE-SAW continuum becomes discretized; yielding a series of discrete kinetic eigenmodes with small frequency separation [33, 60]. In figure 1, we have plotted the BAE accumulation frequencies in the fluid limit which are defined as with in the current case, as well as eigenmode frequencies obtained from simulation results. The analytically predicted KBAE frequencies [33] are in good agreement with observations from numerical simulations. The results also indicate that FOW kinetic effects increase with the toroidal mode number, as expected [33, 60].

On the other hand, our simulations also show that KBAE can be driven by EPs. In figure 2, we can see that the frequencies scale properly with the KBAE frequencies; and the growth rates decrease with the thermal ion temperature due to the stronger ion Landau damping and/or the weaker EP drive due to the increased frequency mismatch between mode and characteristic EP frequencies. In the absence of thermal ion kinetic effects, the excited modes may be identified as energetic particle mode (EPM); which requires sufficiently strong drive to overcome the SAW continuum damping. Including the thermal ion kinetic effects not only introduce a finite kinetic thermal ion frequency gap at the BAE accumulation frequency but also discretize the BAE-SAW continuum. In that case, the continuum damping is greatly reduced or nullified, and the discrete KBAE’s are more readily excited by the EP drive.

## 5 Conclusions and discussions

In the present work, we have employed the theoretical framework (generalized kinetic vorticity and quasi-neutrality equations) of the generalized linear fishbone dispersion relation and derived an extended hybrid MHD-gyrokinetic simulation model applicable to the low-frequency regime, where effects of thermal ion compressibility and diamagnetic drifts play significant roles in the dynamics of Alfvén waves and energetic particles in tokamak plasmas. The extended simulation model has been implemented into an eXtended version of HMGC (XHMGC). Initial simulations of XHMGC have discovered the existence of KBAE discretized by the thermal ion FOW effects, which are absent in conventional MHD codes. Simulations also demonstrate that KBAE can be readily excited by EPs. In the current model, we have taken and neglected finite Larmor radius effects in order to simplify the presentation and focus on the most important qualitative new physics connected with implementation of the thermal ion compressibility. In addition, XHMGC is limited to circular shifted magnetic surfaces equilibria, with relatively large aspect ratio; XHMGC includes kinetic effects related to both bulk and fast ions; however, it is typically used for retaining only the perturbed pressure for two EP species; XHMGC doesn’t include rotation (see A), while it retains the perturbed electrostatic potential. These additional effects will be considered in future works.

More recently, the electromagnetic formulation [8] of global gyrokinetic particle simulation in toroidal geometry has been implemented in GTC [32]. In such a code, ions are treated by the gyrokinetic equation, while electrons are simulated using an improved fluid-kinetic electron model [8]. In [34], the connection between the extended hybrid MHD-gyrokinetic model and gyrokinetic simulation model has been verified in the drift kinetic limit as well as ignoring the terms on the order of . Instead of directly calculating the pressure tensor, lower moments of the kinetic equation have been calculated, i.e. the perturbed density and parallel current. Using charge neutrality condition, it can be demonstrated that the combination of the perturbed density and parallel current contribution is totally equivalent to the pressure tensor in equation 2. Therefore, both GTC and XHMGC can be verified using different models in a common validity regime; yielding more detailed understanding of the underlying physics.

On the other hand, kinetic compressibility is also included in (linear) spectral codes, such as NOVA-K [5, 6] and MARS-K [7]. While in NOVA-K only perturbative treatment of the kinetic effects is considered and continuum damping is not taken into account, as in MARS-K, since both are eigenvalue codes, the spectral approach allows the study of the linear stability of all eigenmodes in a general equilibria; meanwhile, kinetic effects are generally related to bulk plasmas only, although the inclusion of fast ions is quite straightforward. As to other hybrid MHD gyrokinetic codes, M3D [2] is based on the pressure coupling equation; MEGA [3, 4] uses a hybrid model for MHD and energetic particles, where the effect of the energetic ions on the MHD fluid is taken into account in the MHD momentum equation through the energetic ion current. The diamagnetic drift effect is evaluated in the MHD equations by adding the diamagnetic advection term to the equation of motion [47, 48, 49, 50, 61]. At present, XHMGC can handle two species kinetic particles self-consistently, but is limited to circular shifted magnetic flux surfaces equilibria with vanishing bulk plasma equilibrium pressure. Meanwhile, a new version of the code with general equilibria is being developed, with the capability of solving fully compressible gyrokinetic particle response.

## Acknowledgments

This work is supported by the ITER-CN under Grant No.2009GB105005, the NSF of China under Grant No. 11075140, Euratom Communities under the contract of Association between EURATOM/ENEA, USDOE GRANTS, SciDAC, and GSEP.

## Appendix A Simple derivation of model equations

Adopting a multi-fluid moment description of plasma dynamics, the force balance equation can be written as

 ρb(∂∂t+vb⋅∇)vb+ρE(∂∂t+vE⋅∇)vE =−∇Pe−∇⋅Pi−∇⋅PE+J×Bc. (18)

Here, and are bulk plasma and EP mass densities, , and from equation (8), having omitted terms that are or higher with respect to the RHS. Furthermore, thermal ion and EP pressure tensors on the RHS have to be interpreted as usual, i.e. with the conventional fluid velocity shift in the definition

 Psij=ms∫dv(vi−usi)(vj−usj)fs, (19)

with the particle distribution function and . When the pressure tensor is computed form the particle distribution function within the gyrokinetic description, some subtleties are connected with the ordering in the plane orthogonal to , with the Larmor radius of the -species, its thermal speed and the characteristic equilibrium radial scale-length. Thus, in the drift-kinetic limit used in this work, , with the unit diagonal tensor and

 Ps⊥ = ms∫dvv2⊥2fs, (20) ^Ps∥ = ms∫dv(v∥−us∥0)2fs. (21)

Note the difference between , used here, and , used in equation 19, being the (slowly evolving) equilibrium particle distribution function. In equation 18, we assumed that only EPs can carry significant parallel fluid velocity.

The perpendicular component of equation 18 can be further simplified, by noting that , with and and the tokamak minor and major radii, and that we are using the optimal ordering . This allows us to rewrite

 [ρE(∂∂t+vE⋅∇)vE]⊥=b×∇P0E⊥ωcE⋅∇vE⊥+ρEu2E∥κ, (22)

where we have dropped the terms, for they are and, similarly, the term, since it is – or, equivalently, – for ; at shorter wavelength or higher frequency, this term would be negligible anyway with respect to the thermal ion inertia response, as negligible would be diamagnetic responses of both EPs and thermal ions. So, equation 22 well describes the physics we want to incorporate in the present analysis. Recalling the definition of , we also have

 −(∇⋅Ps)⊥=−∇⊥Ps⊥−κ(^Ps∥−Ps⊥) (23)

Thus, the second term on the RHS of equation 22 can be combined with the term on the RHS of equation 23, computed for EPs, and actually be reabsorbed into that (up to the relevant order), provided that the pressure tensor is reinterpreted as , with the “unshifted” expression

 Ps∥=ms∫dvv2∥fs. (24)

replacing the usual definition given in equation 21. With this convention on the pressure tensor, the perpendicular components of equation 18 can be rewritten as

 ρb(∂∂t+vb⋅∇)vb+b×∇P0E⊥ωcE⋅∇vE⊥= −∇⊥Pe−(∇⋅Pi)⊥−(∇⋅PE)⊥+(J×Bc)⊥. (25)

Actually, equation 25 can be reduced further when residual terms that are or higher with respect to the RHS are omitted, as noted below equation 18. In fact, one readily obtains

 [ρb(∂∂t+vb⋅∇)+b×∇P0E⊥ωcE⋅∇]δvb= −∇⊥Pe−(∇⋅Pi)⊥−(∇⋅PE)⊥+(J×Bc)⊥. (26)

This equation readily reduces to the well-known pressure coupling equation [2], in the limit where thermal ion diamagnetic effects and EP contribution to the divergence of the polarization current are neglected.

## Appendix B Study of term (iv) in equation 2

In the low- approximation (),

 ∇×(bB0)≅2b×κB0. (27)

Meanwhile, in the incompressible limit,

 ∂∂tδPb+δvb⋅∇P0b=0, (28)

where

 δvb⊥=cB0×∇⊥δϕB20. (29)

Then, with equations 2728 and 29

 ∂t(∇×bB0)⋅∇δPb = 2b×κB0⋅∇∂δPb∂t (30) = c2b×κB0⋅∇(−B0×∇⊥δϕB20⋅∇P0b) = c2b×κB0⋅∇(B0×∇P0bB20⋅∇⊥δϕ) = −2cB20k×b⋅∇P0bΩκδϕ,

where .

## Appendix C Study of term (v) in equation 2

Assuming

 δPE=bbδPE∥+(I−bb)δPE⊥, (31)

we can show

 ∇⋅δPE = (δPE∥−δPE⊥)(b∇⋅b+κ) (32) + b∇∥(δPE∥−δPE⊥)+∇δPE⊥,

where . Thus,

 b×∇⋅PE=b×∇δPE⊥+(δPE∥−δPE⊥)b×κ. (33)

Now

 ∇⋅(bB0×∇δPE⊥)≅2b×κB0⋅∇δPE⊥, (34)

and

 ∇⋅[(δPE∥−δPE⊥)bB0×κ] ≅ b×κB0⋅∇(δPE∥−δPE⊥). (35)

In equation 35, we have used the large aspect ratio assumption, , consistent with the reduced MHD description used in HMGC [1, 40]. Combining equations 33 to 35, we obtain

 ∂∂t∇⋅(b×(∇⋅δPE)⊥B0) = b×κB0⋅∇∂∂t(δPE∥+δPE⊥) (36) = ωB0Ωκ(δPE∥+δPE⊥)

## Appendix D Study of the ∝δKs term in equation 2

Here, we assume the definition of  [36]

 eiLksδKs=δfs−(em)s[∂F0∂εδϕ−QF0ωeiLkJ0(k⊥ρs)δψ]s, (37)

where is the fluctuating particle distribution function, , on the RHS, we have dropped all terms , for they generate contributions of higher order in what follows [36]. Thus, in our treatment, is generally anisotropic, although terms do not appear explicitly. One then finds

 ⟨4πesk2θc2J0(k⊥ρs)ω^ωdsδKs⟩ = ⟨4πesk2θc2ω^ωdsδfs⟩I−⟨4πe2sk2θmsc2ω^ωds∂F0s∂ε⟩δϕII (38) +⟨4πe2sk2θmsc2ω^ωdsQF0sωJ20(k⊥ρs)⟩δψIII,

with denoting velocity integration and is the zero order Bessel function.

For term (I), we obtain

 (I) = 4πωk2θcΩκ⟨ms(μ+v2∥/B)δfs⟩=4πωk2θcBΩκ(δPs⊥+δPs∥), (39)

where

 δPs⊥=⟨m2v2⊥δfs⟩ (40)

and

 δPs∥=⟨mv2∥δfs⟩ (41)

are, respectively, perturbed perpendicular and parallel pressure.

Meanwhile, for , term (III) can be written as

 (III) = ⟨4πesk2θc(μ+v2∥B)Ωκ(ω∂εF0s+mscesB(k×b)⋅


text:
# \newcommand bug

I am trying to define the following new command:

\newcommand{\kernel}[2]{k\left( #1, #2 \right)}

which when implemented via

$\kernel{\cdot}{Y}$

yields k(.,Y) (not sure how to reproduce the \cdot here)

which works, however I would like to set a default for the first argument to be \cdot, and I tried to do it via the following:

\newcommand{\kernel}[2][\cdot]{k\left( #1, #2 \right)}

however if I implement it via

$\kernel{}{Y}$

it gives k(.,)Y and not k(.,Y)!!!

This seems like a weird bug and I was wondering if anyone knew how to fix it, I tried using xparse instead to define the command but I got the same bug. Thanks in advance!

• Since it is an optional argument, you would say $\kernel[]{Y}$ with square brackets around the optional (null) argument. – Steven B. Segletes Mar 28 '17 at 10:16
• This is no bug but wrong usage ;-) – user31729 Mar 28 '17 at 10:20
• Isn't there already a \ker command that indicates the kernel of a vector space? – user31729 Mar 28 '17 at 10:27

The given definition of the \kernel command is ok, but the calling is wrong. [2][\cdot] means that the first argument is optional and has be to called with \kernel[foo]{Y},i.e. [...] indicates the optional argument.

In the code below I show an xparse version, also replacing k with a operator (expanding to k) and removing the \left(...\right) pair.

\documentclass{article}

\usepackage{xparse}
\usepackage{mathtools}
\DeclareMathOperator{\kernelop}{k}

\NewDocumentCommand{\kernel}{O{\cdot}m}{\kernelop(#1,#2)}

\begin{document}

$\kernel{Y}$

$\kernel[\oplus]{X}$

\end{document}

• I usually give some room to the dot, with \,\cdot or even \,\cdot\,. – Manuel Mar 28 '17 at 10:45
• @Manuel: That depends on personal preferences. The issue was the wrong usage of the command, I only improved it somewhat :-) – user31729 Mar 28 '17 at 12:15

format: plain


text:
## Seminar:

Spring 2013, Thursdays, Milner 216, 4:00-4:50 PM

Date: February 7

"  Mini-Talks

* Unitary Yang-Baxter Operators

* Can you hear the density of a taut string "

* Prof. Eric Rowell

* Prof. Willaim Rundell

Abstract

* The Yang-Baxter equation (YBE) goes back at least to the late 1960s appearing in the papers of physicists working in statistical mechanics. In the 1980s the YBE fell into the hands of algebraists, who built machines for producing solutions to the YBE, namely quantum groups. Now the YBE is of interest to physicists again, with potential applications to quantum information. I will briefly describe this recent connection and a very simple stated conjecture that would have important information-theoretic ramifications.

* The title could also have been "can you determine the interior density of the sun from vibrational models". I admit these are a bit cryptic from a mathematical perspective, so here is a more concrete problem: If $A$ is an $n \times n$ matrix with unknown entries can I determine these if I know the eigenvalues $\{\mu_k\}_1^n$ of $A$? Clearly, the answer is no, certainly no in a unique way: we have $n^2$ unknowns but only $n$ pieces of information. What if now $A$ is completely known but I add an unknown diagonal matrix $D$ to it and give you the eigenvalues $\{\lambda_k\}_1^n$ of $A+D$? Can I determine $D$? Now the known/unknown count is correct. Good, but certainly not enough to give a firm answer. I actually want to pose this problem but in a pde setting and the answers will give insight into the two original questions.


text:
### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

### A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

# Trapezium Arch

##### Stage: 3 and 4 Short Challenge Level:

Consider the ten trapezia. Add an eleventh trapezium to one edge of the arch.
To move from the left of the arch to the right, each trapezium has to turn the same number of degrees.

Consider the blue edgeof the bottom left hand stone. To complete the arch it has to turn a total of $180$ degrees in 10 steps. That is $18$ degrees for each step, or $9$ degrees for each of the base angles of each trapezium.

Therefore the largest angles of each trapezium are $(90+9)$ degrees and the smallest $(90-9)$ degrees = 81 degrees .

This problem is taken from the UKMT Mathematical Challenges.
View the archive of all weekly problems grouped by curriculum topic

View the previous week's solution
View the current weekly problem


text:
# MathGloss/D/Derivative

< MathGloss | D
Template:Derivative

This glossary is far from complete. We are constantly adding math terms.
For instructions on adding new terms, please refer to Math Glossary Main Page

Definition
 Derivative If $y=f(x)$ is is a function of x then rate of change of $y$ with respect to $x$ is called the derivative of $y$ with respect to $x$, denoted by $dy/dx$

## Examples

$y=2x+5, dy/dx=2$
$y=x^2+4x+5, dy/dx=2x+4$


text:
Skip to main content

No project description provided

## Project description

# C Speech Features

This library is a complete port of [python_speech_features](https://github.com/jameslyons/python_speech_features) to C.

## Usage

Clone this repository in your C/C++ project:

 git clone https://github.com/cwiiis/c_speech_features

and add these files to your build:

 c_speech_features.c kiss_fft130/kiss_fft.c kiss_fft130/tools/kiss_fftr.c

Alternatively, you can build shared and static libraries to use with your project instead of importing the files directly. Using cmake, from within a check-out:

 cmake -DCMAKE_C_FLAGS="-O2" -DCMAKE_INSTALL_PREFIX="/usr/local" -DENABLE_DOUBLE=OFF make

This will result in libc_speech_features.so and libc_speech_features_static.a being built (or platform equivalents), which you can link in your project. An install target is also provided, allowing system-wide use of the library, but this is not recommended.

Refer to [c_speech_features.h](c_speech_features.h) for documentation.

## Bindings

### Python

Building python bindings requires [SWIG](http://www.swig.org/) and [numpy](http://www.numpy.org/). After building the library, as documented above, run:

 python setup.py install

This will build and install the statically linked c_speech_features module, which can then be used in any python script. Refer to [python/test.py](python/test.py) to see examples of its use. For the most part, it is a drop-in replacement for python_speech_features.

### Differences to python_speech_features

• c_speech_features only supports one datatype, which is defined at compile-time. Currently, this can be either 32-bit (float), or 64-bit (double). python_speech_features operates in 64-bit by default.

• python_speech_features.sigproc functions are located directly on the c_speech_features module.

• The winfunc parameter is an array instead of a function, and is expected to be as wide as necessary depending on context.

### Uses

It is not recommended to replace python_speech_features with c_speech_features in a Python context. python_speech_features is safer and will output better error messages when fed with incorrect parameters. c_speech_features is most likely to crash in such situations. On the other hand, this module can be useful to debug issues when using c_speech_features in more appropriate contexts, or if you have very specific needs. c_speech_features is likely to perform faster when compiled with optimisation than python_speech_features. In 64-bit, the gains are minimal, except for particular functions, where as in 32-bit the gains can be substantial (at the cost of precision).

## Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.

### Source Distribution

c_speech_features-0.4.6.zip (140.5 kB view hashes)

Uploaded source


text:
Worst-Case Analysis for Randomly Collected Data
We introduce a framework for statistical estimation that leverages knowledge of how samples are collected but makes no distributional assumptions on the data values. Specifically, we consider a population of elements [n]={1,...,n} with corresponding data values x1,...,xn. We observe the values for a "sample" set A \subset [n] and wish to estimate some statistic of the values for a "target" set B \subset [n] where B could be the entire set. Crucially, we assume that the sets A and B are drawn according to some known distribution P over pairs of subsets of [n]. A given estimation algorithm is evaluated based on its "worst-case, expected error" where the expectation is with respect to the distribution P from which the sample A and target sets B are drawn, and the worst-case is with respect to the data values x1,...,xn. Within this framework, we give an efficient algorithm for estimating the target mean that returns a weighted combination of the sample values–-where the weights are functions of the distribution P and the sample and target sets A, B--and show that the worst-case expected error achieved by this algorithm is at most a multiplicative pi/2 factor worse than the optimal of such algorithms. more »
Authors:
; ;
Award ID(s):
Publication Date:
NSF-PAR ID:
10275102
Journal Name:
NeurIPS 2021
4. Abstract Estimating the mean of a probability distribution using i.i.d. samples is a classical problem in statistics, wherein finite-sample optimal estimators are sought under various distributional assumptions. In this paper, we consider the problem of mean estimation when independent samples are drawn from $d$-dimensional non-identical distributions possessing a common mean. When the distributions are radially symmetric and unimodal, we propose a novel estimator, which is a hybrid of the modal interval, shorth and median estimators and whose performance adapts to the level of heterogeneity in the data. We show that our estimator is near optimal when data are i.i.d. and when the fraction of ‘low-noise’ distributions is as small as $\varOmega \left (\frac{d \log n}{n}\right )$, where $n$ is the number of samples. We also derive minimax lower bounds on the expected error of any estimator that is agnostic to the scales of individual data points. Finally, we extend our theory to linear regression. In both the mean estimation and regression settings, we present computationally feasible versions of our estimators that run in time polynomial in the number of data points.


text:
Tag Info

37

Hint: The probability that an equal number of tails and heads appear is $\large{{2k \choose k} \frac{1}{2^{2k}}}$ The two remaining outcomes (that there are more heads than tails or more tails than heads) are equally likely.

27

I think this is a slightly more intuitive way of looking at the question: Suppose $f (x)=k$. Once we have chosen $k$, there are $200$ possible values for $g (x)$, one of which is $k$, hence we get an answer of $\frac{1}{200}$.

13

Hint: Fair coin $\implies$ Probability of tails occurring more $=$ probability of heads occurring more $= p$, say. Probability of exactly equal number of heads and tails $=1-2p$. Can you find this one?

13

is it possible that the event has in fact zero probability? Yes. For a situation where this always happens, assume that one observes a random number $x$ drawn from the uniform distribution on $(0,1)$. Then the probability to observe $x$ is zero. (Proof: For every interval $I\subseteq(0,1)$ the probability to observe a number in $I$ is the length of ...

12

Let me get you started: if $(X,Y)$ is i.i.d., $(Y,X)$ is distributed like $(X,Y)$ hence $Y-X$ is distributed like $X-Y$ hence $E(u(Y-X))=E(u(X-Y))$ for every $u$ such that the expectations exist. For $u(t)=t^3$ (or any odd function), this yields $E((X-Y)^3)=E((Y-X)^3)=-E((X-Y)^3)$ hence... Thus, the answer is "Yes, provided $E(|X|^3)$ is finite" (otherwise ...

12

Assuming the selections are independent and uniform, there are $100 \times 200$ possible equiprobable pairs of which $100$ can be a match, so the probability is ${100 \over 200 \cdot 100 }$.

12

Start with $x_1=0$ and then just let $x_{n+1}=1$ whenever $f(n)/n\leq p$ and $x_{n+1}=0$ whenever $f(n)/n> p$. It's easy to prove by induction that we will always have $\left|\frac{f(n)}n-p\right|\leq\frac 1n$ and so $f(n)/n\rightarrow p$.

11

The proof of this result has nothing to do with probability theory in general or with Borel-Cantelli lemmas in particular, it is only plain-old-deterministic real analysis: Let $(x_n)$ denote a real valued sequence such that $0\leqslant x_n\lt1$ for every $n$, then $\prod\limits_n(1-x_n)=0$ if and only if $\sum\limits_nx_n$ diverges. Proof: If ...

10

Looking at the final balls is the way to go, as you can regard drawing all $60$ balls as choosing one of the equally probable permutation. Working from the back, you can also ignore a colour once it has been drawn. The probability that the last ball is blue and that the last green comes after the last red is $\dfrac{30}{10+20+30}\times \dfrac{20}{10+20} ... 10 This is false, when$\lim_{n \to \infty} p_n$doesn't exist. For instance, consider$p_{2n-1} = \dfrac1{2^n}$and$p_{2n} = 1-\dfrac1{2^n}$. This gives us$\sum_{n=1}^{2N} p_n = \sum_{n=1}^{2N} (1-p_n) = N$, whereas $$\sum_{n=1}^{2N} p_n(1-p_n) < \sum_{n=1}^N \dfrac1{2^{n-1}} < 2$$ However, the claim is true when$\lim_{n \to \infty} p_n$exists. ... 9 Assuming that by "number" you mean "integer," and by "random" you mean "uniformly at random," AND ASSUMING THAT THE CHOICES ARE INDEPENDENT, then the desired probability is $$\sum_{k=1}^{200} \Pr[X = k]\Pr[Y = k] = \sum_{k=1}^{100} \frac{1}{100} \cdot \frac{1}{200} + \sum_{k=101}^{200} 0 \cdot \frac{1}{200} = \frac{1}{200}.$$ 8 The problem is the claim that all$x^2$values between$0$and$4$are equally likely. That's just not true, except in the sense that each one of them has probability zero. But in any more reasonable sense, it fails. In particular, if all were equally likely, then the probability that the square was between 0 and 2 would be the same as the probability that ... 8 Wouldn't this be needlessly complicating things? Stirling formula tells you that $$\Gamma(x+1/2)\sim(2\pi)^{1/2}(x+1/2)^{x}e^{-x-1/2}$$ and that $$x^{1/2}\,\Gamma(x)\sim(2\pi)^{1/2}x^{x}e^{-x}$$ hence $$... 8 Write A \setminus B as A \cap B^{c}. 8 Since \mathbb E[X^2] = \mathbb E[X\mathbb E[Y|X]] = \mathbb E[\mathbb E[XY|X]] = \mathbb E[XY]= \mathbb E[\mathbb E[XY|Y]] = \mathbb E[Y\mathbb E[X|Y]] = \mathbb E[Y^2], observe that$$ \mathbb E[(X-Y)^2]=\mathbb E[X^2+Y^2-2XY]=0, $$which implies that X and Y differ on a set of measure zero. For the weaker condition where X and Y are ... 8 Note that the integral can be written as$$ \int_0^\infty \int_x^\infty f(t)\,\mathrm dt\,\mathrm dx=\int_0^\infty \int_0^\infty f(t)\mathbf{1}_{t\geq x}\,\mathrm dt\,\mathrm dx $$and using Fubini we get$$ \int_0^\infty \int_0^\infty f(t)\mathbf{1}_{t\geq x}\,\mathrm dx\,\mathrm dt=\int_0^\infty \int_0^t f(t)\,\mathrm dx\,\mathrm dt. $$Here ... 8 There is a difference between mutually exclusive events and independent events (indeed, a very strict difference in the sense that a couple of non-trivial events cannot be both mutually exclusive and independent). Events A and B are mutually exclusive if it is impossible for both of them to occur simultaneously. To borrow from your example, it is ... 8 I'll try to give you an example that shows why it is very useful. Suppose that the random variable X follows the binomial distribution with parameters n and p. Then the expected value of X is given by$$ \operatorname EX=\sum_{k=0}^nk\binom{n}{k}p^k(1-p)^{n-k}. $$But we know that X=\sum_{j=1}^nY_j, where Y_1,\ldots,Y_n are independent and ... 7 With such a large number of trials and with p = 0.5, a normal approximation to the binomial distribution would also work. if X \sim \mathrm{Binomial}(n = 10^6, p = 0.5), then$$\Pr[X = n/2] = \Pr\left[\frac{X - np}{\sqrt{np(1-p)}} = \frac{n/2 - np}{\sqrt{np(1-p)}}\right] \approx \Pr[-1/\sqrt{n} \le Z \le 1/\sqrt{n}]$$using continuity correction, where ... 7 Then can you help me to show that \alpha is in fact stopping time? Well, not very easily I am afraid, since \alpha is not always a stopping time for the filtration (\mathcal F_t)_{t\geqslant0} defined by \mathcal F_t=\sigma(X_s;0\leqslant s\leqslant t). For a counterexample, consider some Bernoulli random variable U such that P(U=1)=\frac12 ... 7 Your final expression:$$a_k = \frac{1}{5} \cdot\left( \frac{25}{36} \right)^k $$This correctly gives the probability that your first 6 will occur on the 2kth roll. So, a_1 is the probability that your first 6 is on roll 2; a_2 is the probability that your first 6 is on roll 4; etc. In order to find the probability p that your first 6 is on any ... 7 The measure P_X (not P_x) is the image measure of P by X, defined by the identity P_X(B)=P(X^{-1}(B)) for every B in \mathcal B(\mathbb R) (and X being a random variable is exactly the hypothesis one needs to be sure that P(X^{-1}(B)) exists). The fact that, for every measurable function u suitably integrable,$$ \int_\Omega u(X)\,\mathrm ... 7 The probability is$0$, assuming that each point is chosen with uniform distribution on the circle. For a right-angled triangle you need two of the points to be the two endpoints of a diameter. The probability for that is obviously$0$, since there isn't any range of possibilities, and the distribution is continuous. For anyone who is not satisfied so ... 7 Note that $$\bigcup_{n=1}^\infty A_n \supset \bigcup_{n=1}^\infty A_n \cap \bigcup_{n=2}^\infty A_n \cap \ldots \cap \bigcup_{n=k}^\infty A_n \cap \ldots = \bigcap_{k=1}^\infty \bigcup_{n=k}^\infty A_n = \limsup_{n\to\infty} A_n$$ And use$A\subset B \Rightarrow P(A) \le P(B)$7 Let$S_{100} = \sum_{k=1}^{100}X_k$denote the winnings after 100 bets. Then $$E(S_{100})= 100E(X)= 25$$ and $$var(S_{100})= \sum_{k=1}^{100}var(X)= 100(27/16)\approx 168.75.$$ As a sum of$100$independent random variables,$S_{100}$is approximately normally distributed with mean$25$and standard deviation$\sqrt{168.75} \approx 12.99$. The ... 7 Suppose that I have a set$S$of white balls. The calculation $$\binom{n}k\binom{n-k}\ell$$ counts the ways to perform the following operation: pick$k$of the balls in$S$and paint them red, then pick$\ell$of the remaining white balls and color them blue. I could get the same results by first picking any$k+\ell$balls, then choosing$k$of those to ... 7 For concreteness, assume that we have a biased coin that has unknown probability of landing "heads" and we want to give a$95\%$confidence interval for$p$. Rrepeat the experiment of tossing the coin$n$times. Let$\bar{X}$be the sample proportion of heads. We will use$\bar{X}$to estimate$p$. Note that$\bar{X}$has mean$p$and variance ... 6 $$T=ZX+(1-Z)Y\implies E(\mathrm e^{\mathrm itT})=pE(\mathrm e^{\mathrm itX})+(1-p)E(\mathrm e^{\mathrm itY})$$ 6 It turns out the numerical findings about$\mathbb{E}[N_3]$by David E is not a coincidence, it is exact! $$\mathbb{E}[N_3] = \frac{1}{1-\sin(1)}$$ Let$X_1, X_2, \ldots $be a sequence of random variables. For each$n$, we will assume$X_n$take value from the set$\langle n \rangle \stackrel{def}{=}\{ 1, 2, \ldots, n \}\$ with uniform probability. After ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# zbMATH — the first resource for mathematics

Learning, mutation, and long run equilibria in games. (English) Zbl 0776.90095
Summary: We analyze an evolutionary model with a finite number of players and with noise or mutations. The expansion and contraction of strategies is linked — as usual — to their current relative success, but mutations — which perturb the system away from its deterministic evolution — are present as well. Mutations can occur in every period, so the focus is on the implications of ongoing mutations, not a one-shot mutation. The effect of these mutations is to drastically reduce the set of equilibria to what we term “long-run equilibria”. For $$2\times 2$$ symmetric games with two symmetric strict Nash equilibria the equilibrium selected satisfies (for large populations) the criterion of risk-dominance of J. C. Harsanyi and R. Selten [“A general theory of equilibrium selection in games” (1988; Zbl 0693.90098)]. In particular, if both strategies have equal security levels, the Pareto dominant Nash equilibrium is selected, even though there is another strict Nash equilibrium.

##### MSC:
 91A15 Stochastic games, stochastic differential games 92D15 Problems related to evolution 91E40 Memory and learning in psychology
Full Text:


text:
Enter the degree of operating leverage and the degree of financial leverage into the Calculator. The calculator will evaluate the Total Leverage.

## Total Leverage Formula

TL = DOL * DFL

Variables:

• TL is the Total Leverage ()
• DOL is the degree of operating leverage
• FDL is the degree of financial leverage

To calculate Total Leverage, multiply the degree of operating leverage by the degree of financial leverage.

## How to Calculate Total Leverage?

The following steps outline how to calculate the Total Leverage.

1. First, determine the degree of operating leverage.
2. Next, determine the degree of financial leverage.
3. Next, gather the formula from above = TL = DOL * DFL.
4. Finally, calculate the Total Leverage.
5. After inserting the variables and calculating the result, check your answer with the calculator above.

Example Problem :

Use the following variables as an example problem to test your knowledge.

degree of operating leverage = 0.74

degree of financial leverage = 0.89


text:
# Thread: SAT QOTD - Sept 10

1. ## SAT QOTD - Sept 10

I can't seem to solve this question, can someone give me examples on how to solve it and please not solve it for me.

2. ## Re: SAT QOTD - Sept 10

Can you solve this $6r+10=4r+22~?$

3. ## Re: SAT QOTD - Sept 10

Originally Posted by Plato
Can you solve this $6r+10=4r+22~?$
Yea $2r = 12$ Which equals 6


text:
# Categorial grammar

Categorial grammar is a term used for a family of formalisms in natural language syntax motivated by the principle of compositionality and organized according to the view that syntactic constituents should generally combine as functions or according to a function-argument relationship. Most versions of categorial grammar analyze sentence structure in terms of constituencies (as opposed to dependencies) and are therefore phrase structure grammars (as opposed to dependency grammars).

## Basics

A categorial grammar consists of two parts: a lexicon, which assigns a set of types (also called categories) to each basic symbol, and some type inference rules, which determine how the type of a string of symbols follows from the types of the constituent symbols. It has the advantage that the type inference rules can be fixed once and for all, so that the specification of a particular language grammar is entirely determined by the lexicon.

A categorial grammar shares some features with the simply typed lambda calculus. Whereas the lambda calculus has only one function type $A \rightarrow B$, a categorial grammar typically has two function types, one type which is applied on the left, and one on the right. For example, a simple categorial grammar might have two function types $B/A\,\!$ and $A\backslash B$. The first, $B/A\,\!$, is the type of a phrase that results in a phrase of type $B\,\!$ when followed (on the right) by a phrase of type $A\,\!$. The second, $A\backslash B\,\!$, is the type of a phrase that results in a phrase of type $B\,\!$ when preceded (on the left) by a phrase of type $A\,\!$.

The notation is based upon algebra. A fraction when multiplied by (i.e. concatenated with) its denominator yields its numerator. As concatenation is not commutative, it makes a difference whether the denominator occurs to the left or right. The concatenation must be on the same side as the denominator for it to cancel out.

The first and simplest kind of categorial grammar is called a basic categorial grammar, or sometimes an AB-grammar (after Ajdukiewicz and Bar-Hillel). Given a set of primitive types $\text{Prim}\,\!$, let $\text{Tp}(\text{Prim})\,\!$ be the set of types constructed from primitive types. In the basic case, this is the least set such that $\text{Prim}\subseteq \text{Tp}(\text{Prim})$ and if $X, Y\in \text{Tp}(\text{Prim})$ then $(X/Y), (Y\backslash X) \in \text{Tp}(\text{Prim})$. Think of these as purely formal expressions freely generated from the primitive types; any semantics will be added later. Some authors assume a fixed infinite set of primitive types used by all grammars, but by making the primitive types part of the grammar, the whole construction is kept finite.

A basic categorial grammar is a tuple $(\Sigma, \text{Prim}, S, \triangleleft)$ where $\Sigma\,\!$ is a finite set of symbols, $\text{Prim}\,\!$ is a finite set of primitive types, and $S \in \text{Tp}(\text{Prim})$.

The relation $\triangleleft$ is the lexicon, which relates types to symbols $(\triangleleft) \subseteq \text{Tp}(\text{Prim}) \times \Sigma$. Since the lexicon is finite, it can be specified by listing a set of pairs like $TYPE\triangleleft\text{symbol}$.

Such a grammar for English might have three basic types $(N,NP, \text{ and } S)\,\!$, assigning count nouns the type $N\,\!$, complete noun phrases the type $NP\,\!$, and sentences the type $S\,\!$. Then an adjective could have the type $N/N\,\!$, because if it is followed by a noun then the whole phrase is a noun. Similarly, a determiner has the type $NP/N\,\!$, because it forms a complete noun phrase when followed by a noun. Intransitive verbs have the type $NP\backslash S$, and transitive verbs the type $(NP\backslash S)/NP$. Then a string of words is a sentence if it has overall type $S\,\!$.

For example, take the string "the bad boy made that mess". Now "the" and "that" are determiners, "boy" and "mess" are nouns, "bad" is an adjective, and "made" is a transitive verb, so the lexicon is {$NP/N\triangleleft\text{the}$, $NP/N\triangleleft\text{that}$, $N\triangleleft\text{boy}$, $N\triangleleft\text{mess}$, $N/N\triangleleft\text{bad}$, $(NP\backslash S)/NP\triangleleft\text{made}$}.

and the sequence of types in the string is

${\text{the}\atop {NP/N,}} {\text{bad}\atop {N/N,}} {\text{boy}\atop {N,}} {\text{made}\atop {(NP\backslash S)/NP,}} {\text{that}\atop {NP/N,}} {\text{mess}\atop {N}}$

now find functions and appropriate arguments and reduce them according to the two inference rules $X\leftarrow X/Y,\; Y$ and $X\leftarrow Y,\; Y\backslash X$:

$.\qquad NP/N,\; N/N,\; N,\; (NP\backslash S)/NP,\; \underbrace{NP/N,\; N}$
$.\qquad NP/N,\; N/N,\; N,\; \underbrace{(NP\backslash S)/NP, \quad NP}$
$.\qquad NP/N,\; \underbrace{N/N,\; N}, \qquad (NP\backslash S)$
$.\qquad \underbrace{NP/N,\; \quad N},\; \qquad (NP\backslash S)$
$.\qquad \qquad\underbrace{NP,\; \qquad (NP\backslash S)}$
$.\qquad \qquad\qquad\quad\;\;\; S$

The fact that the result is $S\,\!$ means that the string is a sentence, while the sequence of reductions shows that it must be parsed as ((the (bad boy)) (made (that mess))).

Categorial grammars of this form (having only function application rules) are equivalent in generative capacity to context-free grammars and are thus often considered inadequate for theories of natural language syntax. Unlike CFGs, categorial grammars are lexicalized, meaning that only a small number of (mostly language-independent) rules are employed, and all other syntactic phenomena derive from the lexical entries of specific words.

Another appealing aspect of categorial grammars is that it is often easy to assign them a compositional semantics, by first assigning interpretation types to all the basic categories, and then associating all the derived categories with appropriate function types. The interpretation of any constituent is then simply the value of a function at an argument. With some modifications to handle intensionality and quantification, this approach can be used to cover a wide variety of semantic phenomena.

## Lambek calculus

A Lambek grammar is an elaboration of this idea which has a concatenation operator for types, and several other inference rules. Pentus has shown that these still have the generative capacity of context-free grammars.

For the Lambek calculus, there is a type concatenation operator $\star\,\!$, so that $\text{Prim}\subseteq \text{Tp}(\text{Prim})$ and if $X, Y\in \text{Tp}(\text{Prim})$ then $(X/Y), (X\backslash Y), (X\star Y)\in \text{Tp}(\text{Prim})$.

The Lambek calculus consists of several deduction rules which specify how type inclusion assertions can be derived. In the following rules, upper case roman letters stand for types, upper case Greek letters stand for sequences of types. A sequent of the form $X \leftarrow \Gamma$ can be read: a string is of type $X\,\!$ if it consists of the concatenation of strings of each of the types in $\Gamma\,\!$. If a type is interpreted as a set of strings, then the $\leftarrow$ may be interpreted as $\supseteq\,\!$, that is, "includes as a subset". A horizontal line means that the inclusion above the line implies the one below the line.

The process is begun by the Axiom rule, which has no antecedents and just says that any type includes itself.

$(Axiom)\quad {{}\over X \leftarrow X}$

The Cut rule says that inclusions can be composed.

$(Cut) \quad {Z \leftarrow \Delta X \Delta' \qquad X \leftarrow \Gamma \over Z \leftarrow \Delta \Gamma \Delta'}$

The other rules come in pairs, one pair for each type construction operator, each pair consisting of one rule for the operator in the target, one in the source, of the arrow. The name of a rule consists of the operator and an arrow, with the operator on the side of the arrow on which it occurs in the conclusion.

Target Source
$(\backslash \leftarrow) \quad {Y\leftarrow X \Gamma \over X\backslash Y\leftarrow\Gamma}$ $(\leftarrow \backslash) \quad {Z \leftarrow \Delta Y \Delta' \qquad X\leftarrow\Gamma \over Z \leftarrow \Delta \Gamma(X\backslash Y) \Delta'}$
$(/\leftarrow) \quad {Y\leftarrow \Gamma X \over Y/X\leftarrow\Gamma}$ $(\leftarrow/) \quad {Z\leftarrow \Delta Y \Delta' \qquad X\leftarrow\Gamma \over Z\leftarrow \Delta (Y/X)\Gamma \Delta'}$
$(\star\leftarrow) \quad {X\leftarrow \Gamma \qquad Y \leftarrow \Gamma' \over X \star Y \leftarrow \Gamma\Gamma'}$ $(\leftarrow\star) \quad {Z\leftarrow \Delta X Y \Delta' \over Z\leftarrow \Delta (X \star Y) \Delta'}$

For an example, here is a derivation of "type raising", which says that $(B/A)\backslash B \leftarrow A$. The names of rules and the substitutions used are to the right.

$\dfrac {\dfrac{}{B \leftarrow B} \qquad \dfrac{}{A \leftarrow A} } {\dfrac {B \leftarrow (B/A), \;\; A} {(B/A)\backslash B \leftarrow A} } \qquad \begin{matrix} \mbox{(Axioms)}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{ }\\ {(\leftarrow/)\,\,[Z=Y=B,X=A,\Gamma=(A),\Delta=\Delta'=()]}\\ {(\backslash\leftarrow)\,\,[Y=B,X=(B/A),\Gamma=(A)]}\qquad\qquad\qquad{ }\\ \end{matrix}$

### Relation to context-free grammars

Recall that a context-free grammar is a 4-tuple:

$G = (V,\, \Sigma,\, ::=,\, S)$ where

1. $V\,$ is a finite set of non-terminals or variables.

2. $\Sigma\,$ is a finite set of terminal symbols.

3. $::=\,$ is a finite set of production rules, that is, a finite relation $(::=)\subseteq V \times (V \cup \Sigma)^*$.

4. $S\,$ is the start variable.

From the point of view of categorial grammars, a context-free grammar can be seen as a calculus with a set of special purpose axioms for each language, but with no type construction operators and no inference rules except Cut.

Specifically, given a context-free grammar as above, define a categorial grammar $(\text{Prim},\, \Sigma,\, \triangleleft,\, S)$ where $\text{Prim}=V\cup\Sigma$, and $\text{Tp}(\text{Prim})=\text{Prim}\,\!$. Let there be an axiom ${x \leftarrow x}$ for every symbol $x \in V\cup\Sigma$, an axiom ${X \leftarrow \Gamma}$ for every production rule $X ::= \Gamma\,\!$, a lexicon entry ${s \triangleleft s}$ for every terminal symbol $s \in \Sigma$, and Cut for the only rule. This categorial grammar generates the same language as the given CFG.

Of course, this is not a basic categorial grammar, since it has special axioms that depend upon the language; i.e. it is not lexicalized. Also, it makes no use at all of non-primitive types.

To show that any context-free language can be generated by a basic categorial grammar, recall that any context-free language can be generated by a context-free grammar in Greibach normal form.

The grammar is in Greibach normal form if every production rule is of the form $A ::= s A_0 \ldots A_{N-1}$, where capital letters are variables, $s \in \Sigma$, and $N\ge 0$, that is, the right side of the production is a single terminal symbol followed by zero or more (non-terminal) variables.

Now given a CFG in Greibach normal form, define a basic categorial grammar with a primitive type for each non-terminal variable $\text{Prim}=V\,\!$, and with an entry in the lexicon $A/A_{N-1}/ \ldots /A_0 \triangleleft s$, for each production rule $A ::= s A_0 \ldots A_{N-1}$. It is fairly easy to see that this basic categorial grammar generates the same language as the original CFG. Note that the lexicon of this grammar will generally assign multiple types to each symbol.

The same construction works for Lambek grammars, since they are an extension of basic categorial grammars. It is necessary to verify that the extra inference rules do not change the generated language. This can be done and shows that every context-free language is generated by some Lambek grammar.

To show the converse, that every language generated by a Lambek grammar is context-free, is much more difficult. It was an open problem for nearly thirty years, from the early 1960s until about 1991 when it was proven by Pentus.

The basic idea is, given a Lambek grammar, $(\text{Prim},\, \Sigma,\, \triangleleft,\, S)$ construct a context-free grammar $(V,\, \Sigma,\, ::=,\, S)$ with the same set of terminal symbols, the same start symbol, with variables some (not all) types $V\subset \text{Tp}(\text{Prim})\,\!$, and with a production rule $T::=\text{s}\,\!$ for each entry $T\triangleleft\text{s}$ in the lexicon, and production rules $T::=\Gamma\,\!$ for certain sequents $T\leftarrow\Gamma$ which are derivable in the Lambek calculus.

Of course, there are infinitely many types and infinitely many derivable sequents, so in order to make a finite grammar it is necessary put a bound on the size of the types and sequents that are needed. The heart of Pentus's proof is to show that there is such a finite bound.

### Notation

The notation in this field is not standardized. The notations used in formal language theory, logic, category theory, and linguistics, conflict with each other. In logic, arrows point to the more general from the more particular, that is, to the conclusion from the hypotheses. In this article, this convention is followed, i.e. the target of the arrow is the more general (inclusive) type.

In logic, arrows usually point left to right. In this article this convention is reversed for consistency with the notation of context-free grammars, where the single non-terminal symbol is always on the left. We use the symbol $::=$ in a production rule as in Backus-Naur form. Some authors use an arrow, which unfortunately may point in either direction, depending on whether the grammar is thought of as generating or recognizing the language.

Some authors on categorial grammars write $B\backslash A$ instead of $A\backslash B$. The convention used here follows Lambek and algebra.

## Historical notes

The basic ideas of categorial grammar date from work by Kazimierz Ajdukiewicz (in 1935) and Yehoshua Bar-Hillel (in 1953). In 1958, Joachim Lambek introduced a syntactic calculus that formalized the function type constructors along with various rules for the combination of functions. This calculus is a forerunner of linear logic in that it is a substructural logic. Montague grammar uses an ad hoc syntactic system for English that is based on the principles of categorial grammar. Although Montague's work is sometimes regarded as syntactically uninteresting, it helped to bolster interest in categorial grammar by associating it with a highly successful formal treatment of natural language semantics. More recent work in categorial grammar has focused on the improvement of syntactic coverage. One formalism which has received considerable attention in recent years is Steedman and Szabolcsi's combinatory categorial grammar which builds on combinatory logic invented by Moses Schönfinkel and Haskell Curry.

There are a number of related formalisms of this kind in linguistics, such as type logical grammar and abstract categorial grammar.

## Some definitions

Derivation
A derivation is a binary tree that encodes a proof.
Parse tree
A parse tree displays a derivation, showing the syntactic structure of a sentence.
Functor and Argument
In a right (left) function application, the node of the type A\B (B/A) is called the functor, and the node of the type A is called an argument.
Functor-argument structure

## Refinements of categorial grammar

A variety of changes to categorial grammar have been proposed to improve syntactic coverage. Some of the most common ones are listed below.

### Features and subcategories

Most systems of categorial grammar subdivide categories. The most common way to do this is by tagging them with features, such as person, gender, number, and tense. Sometimes only atomic categories are tagged in this way. In Montague grammar, it is traditional to subdivide function categories using a multiple slash convention, so A/B and A//B would be two distinct categories of left-applying functions, that took the same arguments but could be distinguished between by other functions taking them as arguments.

### Function composition

Rules of function composition are included in many categorial grammars. An example of such a rule would be one that allowed the concatenation of a constituent of type A/B with one of type B/C to produce a new constituent of type A/C. The semantics of such a rule would simply involve the composition of the functions involved. Function composition is important in categorial accounts of conjunction and extraction, especially as they relate to phenomena like right node raising. The introduction of function composition into a categorial grammar leads to many kinds of derivational ambiguity that are vacuous in the sense that they do not correspond to semantic ambiguities.

### Conjunction

Many categorial grammars include a typical conjunction rule, of the general form X CONJ X → X, where X is a category. Conjunction can generally be applied to nonstandard constituents resulting from type raising or function composition..

### Discontinuity

The grammar is extended to handle linguistic phenomena such as discontinuous idioms, gapping and extraction.

## References

• Curry, Haskell B.; Feys, Richard (1958), Combinatory Logic 1, North-Holland
• Jacobson, Pauline (1999), "Towards a variable-free semantics.", Linguistics and Philosophy 22: 117–184
• Lambek, Joachim (1958), "The mathematics of sentence structure", Amer. Math. Monthly 65: 154–170
• Pentus, Mati (1997), Lambek Calculus and Formal Grammars, Amer. Math. Soc. Transl.
• Steedman, Mark (1987), "Combinatory grammars and parasitic gaps", Natural Language and Linguistic Theory 5: 403–439
• Steedman, Mark (1996), Surface Structure and Interpretation, The MIT Press
• Steedman, Mark (2000), The Syntactic Process, The MIT Press
• Szabolcsi, Anna (1989), "Bound variables in syntax (are there any?)", in Bartsch; van Benthem; van Emde Boas, Semantics and Contextual Expression, Foris, pp. 294–318
• Szabolcsi, Anna (1992), Sag; Szabolcsi, eds., "Lexical Matters", CSLI Lecture Notes (Stanford: CSLI Publications) 24: 241–269 |chapter= ignored (help)
• Szabolcsi, Anna (2003), "Binding on the fly: Cross-sentential anaphora in variable-free semantics", in Kruijff; Oehrle, Resource Sensitivity in Binding and Anaphora, Kluwer, pp. 215–229
• Morril, Glynn (1995), "Discontinuity in categorial grammar", Linguistics and Philosophy (Springer): 175–219


text:
# pi/12

• March 25th 2011, 01:26 AM
JazKW347
pi/12
how can i use the difference identity to find pi/12, because i keep getting 0 when i try it.
• March 25th 2011, 01:29 AM
Prove It
Are you sure you're not trying to find $\displaystyle \sin{\frac{\pi}{12}}$ or $\displaystyle \cos{\frac{\pi}{12}}$?
• March 25th 2011, 01:32 AM
JazKW347
Ii is Cos(pi/12) i guess i looked at it wrong, i did not see the Cosine
• March 25th 2011, 01:44 AM
Prove It
Either evaluate $\displaystyle \cos{\left(\frac{\pi}{3} - \frac{\pi}{4}\right)}$ using the angle difference identity, or evaluate $\displaystyle \cos{\left(\frac{1}{2}\cdot \frac{\pi}{6}\right)}$ using the half angle identity.


text:
# Work with proxies

##### In this topic

Under some circumstances, Esri Maps for SAP BusinessObjects uses an internal proxy mechanism to broker communication with the ArcGIS platform (in addition to any web services referenced by retrieved ArcGIS items). When required, Esri Maps for SAP BusinessObjects sends HTTP requests to the proxy, and the proxy forwards the requests to the remote web server (such as ArcGIS). The proxy then relays the response back to Esri Maps for SAP BusinessObjects.

The internal proxy is not used if Esri Maps for SAP BusinessObjects and all the data and web services it uses are on the same domain, or if both the client browser and all servers support Cross-Origin Resource Sharing (CORS). In most production environments, however, this configuration is not common, and configuring the internal proxy is recommended.

## Specify CORS-enabled servers (optional)

Most modern browsers use the recommended standard, Cross-Origin Resource Sharing (CORS). The CORS standard adds headers to the HTTP specification that allow servers to describe the set of origins that are permitted to read resources and services using a web browser.

Esri Maps for SAP BusinessObjects automatically detects servers that use Cross-Origin Resource Sharing (CORS) and communicates with these servers without issue. In some cases, however, the initial request (an HTTP POST, for example) is automatically routed through the Esri Maps for SAP BusinessObjects internal proxy, even though the server is CORS-enabled. As an Esri Maps for SAP BusinessObjects administrator, you can list the URLs for known CORS-enabled servers in the Esri Maps for SAP BusinessObjects configuration file (settings.js) to ensure that all calls originating from those locations are automatically considered CORS-enabled, thereby bypassing the proxy.

##### Note:

Servers listed in the corsEnabledServers property must be preconfigured to support CORS. Before adding a server URL to this list, verify that the server is, in fact, CORS-enabled. If a server specified in this setting is not CORS-enabled, communication with the Esri Maps for SAP BusinessObjects server will fail. For more information, see Enable Cross-Origin Resource Sharing

To specify CORS-enabled servers:

1. Use a standard text editor to open the application's configuration file.

The configuration file is in the following location: <Tomcat installation directory>\webapps\em4bobj\em4bobj\dashb\configuration\settings.js

2. In the corsEnabledServers property, specify the URLs of the servers to be automatically considered CORS-enabled.

Each URL must be enclosed within quotation marks, with multiple URLs separated by commas.

For example: corsEnabledServers: ["mapserver.mycompany.com", "otherserver.othercompany.com"]

##### Note:

You cannot include an instance of Portal for ArcGIS that uses web tier authentication in this list. In such an instance, however, automatic detection of CORS will always occur without involving the proxy.

3. Save and close the settings.js file.
4. Restart your web application server. Clearing the browser cache may be required to implement changes.

## Configure the internal proxy

Although the internal proxy is designed to work with minimal custom configuration, you can customize its settings to allow only requests to specific servers; that is, you can specify the resources to which the proxy will forward requests.

To configure the internal proxy settings, modify the proxy configuration file using a standard text editor.

The proxy configuration file (proxy.config) is located at <Tomcat installation directory>\webapps\em4bobj\em4bobj\dashb\proxy.config.

##### Note:

After you save the proxy configuration file, restart your web application server. It may be necessary to clear the browser cache to activate changes to the proxy file.

To define the URLs that Esri Maps for SAP BusinessObjects is allowed to access, specify each individual URL in a separate serverUrl child element. Create a separate serverUrl element for every server that Esri Maps for SAP BusinessObjects must access to retrieve map services.

##### Note:

Setting the serverUrls element's mustMatch attribute to true specifies that the proxy can access only the servers explicitly listed. Setting this attribute to false means that the proxy can access any server, and ignores any server listed as a serverUrl.

For example:

<ProxyConfig>
<serverUrls mustMatch="true">
<serverUrl url="http://www.arcgis.com"/>
<serverUrl url=”http://geoenrich.arcgis.com”/>
<serverUrl url=”http://geocode.arcgis.com”/>
<serverUrl url=”http://gis.mycompany.com/arcgis/rest/services”/>
</serverUrls>

## Test the proxy

To test the internal proxy, route an HTTP request through it. If the proxy returns the expected value, it is working correctly. The following steps provide examples of calls that simulate the HTTP traffic that Esri Maps for SAP BusinessObjects would generate.

1. To verify the location of the Esri Maps for SAP BusinessObjects internal proxy, type its URL in a web browser's address bar.

http://<mySAPBIPlatformWebServer>/em4bobj/em4bobj/dashb/proxy.jsp

When you successfully access the internal proxy, the page displays a time stamp, for example, Thu Oct 02 07:34:52 PDT 2014. If you do not receive the expected response, verify your internal proxy configuration settings.

2. Verify that you can connect to a known resource.

For example, each of the following URLs should return a JSON object that describes the version of the software installed on the server.

• ArcGIS Online: http://www.arcgis.com/sharing?f=pjson
• Portal for ArcGIS instance: http://myPortal/gis/sharing/?f=pjson

3. Route traffic through the internal proxy.

For example, each of the following URLs should return a JSON object that describes the version of the software installed on the server:

• ArcGIS Online: http://<mySAPBIPlatformWebServer>/em4bobj/em4bobj/dashb/proxy.jsp?http://www.arcgis.com/sharing?f=pjson
• Portal for ArcGIS instance: http://<mySAPBIPlatformWebServer>/em4bobj/em4bobj/dashb/proxy.jsp?http://myPortal/gis/sharing/?f=pjson

##### Note:

If routing a call to the internal proxy does not return the expected result, you may need to use an upstream proxy to access the resources. In this case, configure the upstream proxy, and then run the tests in this section again. The internal proxy will route traffic through the upstream proxy and return the expected result to the web browser. For more information, see Configure an optional upstream proxy.

A successful result means that your proxies are properly configured and working as expected.

## Configure an optional upstream proxy

In situations in which Esri Maps for SAP BusinessObjects must use its internal proxy, the internal proxy must be able to forward HTTP requests to the intended target resources (such as map services). In some environments, firewall rules prevent these requests from being sent; for example, the firewall may prevent HTTP traffic from leaving the internal network.

To solve this problem, you can configure an upstream forwarding proxy.

An upstream proxy allows connections outside the network without compromising the security of the internal network. One common approach to using an upstream proxy is to place a proxy server in a perimeter network to protect the internal network.

The upstream proxy itself represents an enterprise-level, third-party (non-Esri) solution. It is not part of Esri Maps for SAP BusinessObjects.

##### Note:

Esri Maps for SAP BusinessObjects does not support upstream proxy servers that require authentication.

The Esri Maps for SAP BusinessObjects proxy configuration file contains elements that allow you to configure an upstream proxy.

To configure upstream proxy settings, modify the proxy configuration file using a standard text editor.

1. Set the useUpstreamProxy element's enabled attribute to true.
2. Specify the URL of the upstream proxy server using the following convention:
http://<address>:<port>

where:

• port—The port number assigned to the proxy server, for example, 8888

For example:

<useUpstreamProxy enabled = true>
http://proxy.mycompany.com:8888
</useUpstreamProxy>

##### Note:

If you use an IP address to specify the location of your proxy server, do not type leading zeroes; for example, use 130.25.0.1 instead of 130.025.000.001.


text:
# zbMATH — the first resource for mathematics

$$L^2$$-invariants: Theory and applications to geometry and $$K$$-theory. (English) Zbl 1009.55001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 44. Berlin: Springer. xv, 595 p. EUR 119.00/net; sFr. 192.50; £83.50; \$ 129.00 (2002).
$$L^2$$-methods were introduced into topology by Atiyah in the 1970’s, from an analytic perspective (elliptic operators on infinite covering spaces of Riemannian manifolds). He defined $$L^2$$-Betti numbers by means of the von Neumann dimension associated to the von Neumann algebra $${\mathcal N}(\pi)$$ of the covering group $$\pi$$. Subsequently, two further types of $$L^2$$-invariants were introduced: the Novikov-Shubin invariants, which measure the asymptotic behaviour of the spectrum of the Laplace operator, and the $$L^2$$-torsion, which is an analogue of the Reidemeister torsion. Although these invariants were all originally defined in terms of elliptic operators on the $$L^2$$-completions of spaces of differential forms on the covering space, the analysis has been stripped away and the method given a more algebraic appearance, notably by Lück and Farber.
The von Neumann algebra $${\mathcal N}(\pi)$$ is the algebra of bounded (left) $$\mathbb{C}[\pi]$$-linear operators on $$\ell^2(\pi)$$, the Hilbert space completion of $$\mathbb{C}[\pi]$$. The von Neumann trace gives rise to a dimension function on finitely generated projective $${\mathcal N}(\pi)$$-modules, with values in $$[0,\infty)$$, and which extends to a $$[0,\infty]$$-valued function on arbitrary $${\mathcal N}(\pi)$$-modules. The $$L^2$$-cohomology modules of a cell-complex $$X$$ with finite skeleta and $$\pi_1(X)=\pi$$ are determined by the complex of cellular cocycles with square summable values on the universal covering of $$X$$, and the $$L^2$$-Betti numbers of $$X$$ are the dimensions of these modules. They are in general non-negative real numbers, rather than integers, but their alternating sum is the Euler characteristic $$\chi(X)$$, if $$X$$ is finitely dominated, and they satisfy Poincaré duality, if $$X$$ is a Poincaré duality complex. In addition, they are multiplicative in finite covers, and in degrees 0 and 1 depend only on the fundamental group. In Farber’s version of the theory the Novikov-Shubin invariants are invariants of the “torsion” of the $$L^2$$-cohomology modules. (The $$L^2$$-Betti numbers are invariants of the reduced $$L^2$$-cohomology, obtained by factoring out the torsion part). The $$L^2$$-torsion is defined when all the $$L^2$$-Betti numbers are 0. (We shall not attempt detailed definitions here).
The above-mentioned properties of the $$L^2$$-Betti numbers all hold for formal reasons, and the proofs are straightforward. The hard work lies in determing the nature of these invariants, in particular whether they are rational, integral or 0. The Atiyah conjecture is that the $$L^2$$-Betti numbers of a finite complex are rational, and are integral if the fundamental group is torsion-free. This is known to imply that the complex group rings of torsion-free groups have no zero-divisors. It is similarly conjectured that the Novikov-Shubin invariants of a finite complex are rational numbers (when finite). The known properties of Riemannian manifolds of constant negative curvature suggest the Singer conjecture, that the $$L^2$$-Betti numbers of an aspherical closed manifold should be nonzero only in the middle dimension. In particular, if the dimension is odd the $$L^2$$-Betti numbers should all be 0. There is then a related conjecture for the sign of the $$L^2$$-torsion of such a manifold. Vanishing criteria for some or all of the $$L^2$$-Betti numbers have surprisingly strong consequences, particularly in combinatorial group theory and low-dimensional topology.
The present book is the first substantial monograph on this topic. There are 17 numbered chapters: §0. Introduction; §1. $$L^2$$-Betti numbers; §2. Novikov-Shubin invariants; §3. $$L^2$$-torsion; §4. $$L^2$$-Invariants of 3-manifolds; §5. $$L^2$$-Invariants of symmetric spaces; §6. $$L^2$$-Invariants for general spaces with group action; §7. Application to groups; §8. The algebra of affiliated operators; §9. Middle $$K$$- and $$L$$-theory of von Neumann algebras; §10. The Atiyah conjecture; §11. The Singer conjecture; §12. The zero-in-the-spectrum conjecture; §13. The approximation and determinant conjectures; §14. $$L^2$$-Invariants and the simplicial volume; §15. Survey on other topics related to $$L^2$$-invariants; and §16. Solutions to the exercises; and a list of over 500 references.
This is an impressive account of much of what is presently known about these invariants, focused on the major applications and a number of key conjectures, with strong implications for other areas of mathematics. It combines features of a text and a reference work; to a considerable degree the chapters can be read independently, and there are numerous nontrivial exercises, with nearly 50 pages of detailed hints at the end. As the author states, it would require another book to treat in detail the topics outlined in §15. In particular, the applications to the classical knot concordance group (due to Cochran, Orr and Teichner) are probably still evolving too rapidly to be incorporated in a reference work. However, this reviewer would have liked to have seen a more substantial discussion of Farber’s approach, here confined to one paragraph in §6.8. (It should be noted also that operator algebra methods have had significant impact on other areas of manifold topology and $$K$$-theory, notably in connection with the Baum-Connes conjecture and the Novikov conjecture, but these applications involve $$C^*$$-algebras rather than von Neumann algebras).

##### MSC:
 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 19K99 $$K$$-theory and operator algebras 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 55N25 Homology with local coefficients, equivariant cohomology 58J99 Partial differential equations on manifolds; differential operators