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Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM{}_{110} mode for ultrafast electron microscopy

# Theory and particle tracking simulations of a resonant radiofrequency deflection cavity in TM$_{110}$ mode for ultrafast electron microscopy

## Abstract

We present a theoretical description of resonant radiofrequency (RF) deflecting cavities in TM mode as dynamic optical elements for ultrafast electron microscopy. We first derive the optical transfer matrix of an ideal pillbox cavity and use a Courant-Snyder formalism to calculate the 6D phase space propagation of a Gaussian electron distribution through the cavity. We derive closed, analytic expressions for the increase in transverse emittance and energy spread of the electron distribution. We demonstrate that for the special case of a beam focused in the center of the cavity, the low emittance and low energy spread of a high quality beam can be maintained, which allows high-repetition rate, ultrafast electron microscopy with 100 fs temporal resolution combined with the atomic resolution of a high-end TEM. This is confirmed by charged particle tracking simulations using a realistic cavity geometry, including fringe fields at the cavity entrance and exit apertures.

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[cor]Corresponding author

## 1 Introduction

Since the introduction of the Ultrafast (Transmission) Electron Microscope (U(T)EM) by Ahmed Zewail [1], the dynamics of various sorts of material properties have been studied using ultrafast electron techniques such as imaging [2] [3], diffraction [4] and electron energy-loss spectroscopy (EELS) [5] [6] with picosecond to femtosecond temporal resolution. The research described in these references is all based on a pump-probe scheme in which the probing electron pulses are created from a flat photo-cathode using a fs laser system. This causes two limitations: First, the average current of the UTEM is limited by the repetition rate of the fs laser system, although long relaxation times of dynamical processes, or slow thermal diffusion, can also limit the maximal repetition times that can be used. Second, the relatively large area of the flat-photocathode limits the peak brightness of the generated electron pulses, hence the maximally achievable spatial resolution. A significant improvement in the peak brightness of laser-triggered electron sources has been achieved by sideways laser illumination of a nano-tip. The reduced dimensions of the photo-field emitter have resulted in a working UTEM with 200 fs electron pulses with a peak brightness comparable to continuous Schottky sources [7]. This technique has resulted in very impressive results [8] [9] [10].

An alternative approach requiring no laser at all, involves the chopping of a continuous beam of a high-end TEM into ultrashort electron pulses using a fast blanker in combination with a slit. Apart from the lack of need for an amplified laser system to create electron pulses, further advantages are that no alterations are needed to the gun and the fact that it allows easy switching between continuous mode and pulsed mode. The principle of chopping an electron beam has been realized in Scanning Electron Microscopes (SEMs) many years ago [11] [12] in the form of electrostatic blanking capacitors [13] [14] and cavity resonators [15] [16]. More recently, the use of a photo-conducting switch was proposed to create a laser-triggered, electrostatic beam blanker which can be used for pump-probe experiments in a SEM [17]. In parallel, advanced RF-laser synchronization techniques [18] [19] have reduced the timing jitter between electron pulses and laser pulses to levels below 100 fs [20] and even 5 fs [21], making RF cavity-based pulsed beams also suitable for pump-probe experiments.

RF cavities or resonators are specifically tailored metallic structures, in which electromagnetic energy can be stored in standing waves or modes. Because of resonant enhancement, RF cavities can be used to generate EM fields of high amplitudes with relatively low input power. Various types of RF cavities have been important elements of the standard toolbox for particle accelerators for many years. For example, a cavity in TM-mode supports an oscillating, electric field pointing along the beam axis, which is commonly used for the acceleration of relativistic charged particle pulses. Synchronized to a mode-locked laser system, a cavity in TM mode can be used to for the compression of electron pulses in ultrafast electron diffraction experiments, resulting in pulses shorter than 100 fs [22]. A cavity in TM mode supports a magnetic field oscillating perpendicular to the beam axis, transversely deflecting the beam. This mode has been used to chop the continuous beam of a 30 kV SEM into ultrashort pulses [23] and record time-of-flight electron energy loss spectra [24]. Synchronized to a mode-locked laser, a cavity in TM mode can be used for pulse length measurements [22], for example of non-relativistic, ultracold electron pulses extracted from laser cooled gases [25]. Note the same principles of pulse compression and metrology have also been applied with single-cycle THz fields instead of RF cavities [26].

In 2012, Lassise et al. showed that using a miniaturized RF cavity in TM mode, it is possible to chop a 30 kV electron beam while fully maintaining the peak brightness [27]. Moreover he proposed to use this technique to chop the beam of a high-quality beam of a 200 kV TEM, also conserving the peak brightness of the Schottky field emission source [27]. Since then, such an RF cavity-based UTEM has been built at Eindhoven University of Technology (TU/e) and is currently operational [28]. Furthermore, alternative TEM beam chopping schemes involving multiple RF cavities are being investigated elsewhere [29]. For successful implementation of an RF cavity in a charged particle beam line, a thorough understanding of its effect on the beam dynamics is essential. If not used properly, the rapidly oscillating non-uniform and strong EM fields in RF cavities can have a detrimental effect on the beam quality. However, with proper settings of experimental parameters, such as the RF phase and the position of the beam crossover, the quality of the original beam can be fully maintained, essential for applications such as electron microscopy.

In this paper we present the theoretical background of a resonant RF cavity in TM mode as a dynamic optical element for UTEM. In section 2 we explain the principle of deflection and chopping by calculating the trajectories of a charged particle propagating through an ideal TM pillbox cavity. From these trajectories, we derive the optical transfer matrix of the cavity in section 3; and apply this in a Courant-Snyder formalism to calculate the 6D phase space propagation of a Gaussian electron distribution in section 4. In section 5 we apply our findings to study the special case of a focused beam inside a 200 kV TEM column. We derive closed, analytic expressions for the increase in transverse emittance and energy spread. We show that using proper experimental parameter settings, the growth in transverse emittance can be fully eliminated and also the increase in energy spread can be minimized. In section 6 we present charged particle tracking simulations using a realistic cavity geometry, including fringe fields at the cavity entrance and exit apertures, and a realistic electron beam. The simulations confirm our theoretical findings that an RF cavity in TM mode can be used to chop an electron beam with negligible increase in both transverse emittance and energy spread. This property makes it a very interesting alternative for photo-emission based UTEM, especially in the light of the ever increasing brightness of continuous electron sources.

## 2 Charged particle trajectories

### 2.1 Beam brightness and emittance

First, we define a beam as a distribution of charged particles (charge and mass ) with a collective motion along the -axis with average speed , for which each individual particle also has a small relative velocity , with

 vx,vy,δvz≪vz, (1)

We describe the motion of each particle in the distribution by the 6D trace space coordinates , , , , and , in which and are the angles of the particle trajectory with respect to the mean trajectory of the total distribution. Analogously, represents the relative difference in longitudinal velocity of an individual particle with respect to the mean velocity of the beam. Note that we have used the paraxial approximation of equation (1). The 6D trace space volume occupied by this distribution is a measure for the quality of the beam. In accelerator physics, this is often expressed in terms of the root-mean-squared (rms) geometrical emittance in the direction [30]

 ϵj≡√⟨j2⟩⟨j′2⟩−⟨jj′⟩2. (2)

Here the brackets indicate averaging over the entire distribution. The emittance is proportional to the area of the projection of the 6D trace-space density on the ()-plane and is a measure for the focusability of the beam in the -direction, given that the beam energy is fixed. The geometrical emittance is not a Lorentz-invariant quantity and therefore is not conserved during acceleration. To compare beams of different energy, the Lorentz-invariant rms normalized emittance in the direction is defined as

 Missing or unrecognized delimiter for \left (3)

in which is the momentum in the -direction, is the relativistic Lorentz factor and is the normalized speed [30]. In a beam waist, there are no correlations between and , hence the normalized transverse emittance is simply given by

 ϵwaistn,x≡βγσxσx′, (4)

in which and are the rms beam radius and rms semi-divergence angle in the beam waist. The quality of the beam in the -direction is determined by the normalized longitudinal emittance . For an charged particle bunch with no chirp , so the normalized longitudinal emittance can be written as

 ϵn,z=σzσpzmc≈σtσUmc, (5)

in which is the rms pulse duration and is the rms energy spread [30].

Electron microscopists often describe the quality of the beam in terms of the reduced brightness, which is locally defined as

The reduced brightness is proportional to the local current density per unit solid angle . By dividing by the relativistically corrected beam potential with , this quantity is also Lorentz-invariant. To define a measure for the overall quality of a charged particle beam, the practical reduced brightness was introduced by Bronsgeest et al. [31]

 Bpract≡1V∗IApractΩ=1V∗Iπ(d502)2πθ2x. (7)

It defines the amount of current that can be focused into a waist with an area , in which is the full width spot diameter in which 50% of the current is focused. Furthermore, it assumes a uniform angular distribution with semi divergence angle . We can now express the practical brightness in terms of the rms normalized transverse emittance . By assuming a uniform, angular distribution so that , and a Gaussian, spatial distribution in the beam waist so that , the practical reduced brightness can be expressed in terms of the rms, normalized transverse emittance as

 Bpract=qmc2I4ln2⋅π2ϵ2n,x. (8)

### 2.2 Framework and assumptions

Consider an ideal, cylindrical cavity in TM mode of length , aligned along the -axis of a cartesian coordinate system, and with the entrance aperture positioned at . Close to the -axis:

 kx,ky≪1, (9)

with the wavenumber of the RF field, the magnetic field and the electric field of a cavity in TM mode can be approximated by

 B=B0cos(ϕ0+ωt)^yE=−B0ωxsin(ϕ0+ωt)^z}for 0

in which is the magnetic field amplitude, is the cavity resonance frequency and is the phase of the RF field at [27]. The word ’ideal’ refers to the top-hat profile of the magnetic field amplitude as a function of and the lack of fringe fields around the cavity apertures and . The effect of both these non-ideal features are studied using particle tracking simulations in section 6.

The motion of a charged particle described by position vector and velocity vector will be affected by the Lorentz force as the particle travels through the cavity. This is described by the equations of motion

 dpdt =F=−qB0⎛⎜⎝vzcos(ϕ0+ωt)0−vxcos(ϕ0+ωt)+ωxsin(ϕ0+ωt)⎞⎟⎠ (11) drdt =v=pγm, with  γ=1√1−|v|2/c2.

Now consider a 6D charged particle trace space distribution traveling along the -axis with an average velocity . Figure 2 shows the moment at which the center of the distribution enters the cavity, which is indicated by the dashed lines. The black dot in figure 2 indicates a test particle that enters the cavity with trajectories

 v(t) ≡v0(x′i,y′i,1+z′i), (12) r(t) ≡(xi,yi,zi)+v0(x′i,y′i,1+z′i)t.

Here we define (,,) as the position of the test particle with respect to the bunch center, and we define , and as small deviations in propagation angles at in the frame of the traveling bunch center. Furthermore we define the 6D trace space coordinate at .

To obtain an approximate solution of the equations of motion (11) for the test particle of which the original motion is described by expression (12), we use a perturbative approach based on the following two assumptions:

• First, the charged particle gyrates only a small fraction of a full cyclotron orbit during one oscillation period of the EM field, hence

 ωcω≪1, (13)

in which

 ωc≡qB0γ0m (14)

is the cyclotron frequency with the Lorentz factor of the incident beam. This ensures that the charged particles remain close to the -axis and we can use the paraxial approximation throughout the paper.

• Secondly, the distances over which the individual particles move with respect to the bunch center are small compared to the length scales of the collective motion of the bunch, i.e.

 x′i,y′i,z′i,xiL,yiL,ziL≪ωcω. (15)

### 2.3 Transverse trajectories of the bunch center

Based on assumption (15), we first consider the motion of the bunch center, hence . Therefore we substitute and in equation (11) and integrate from to . The momentum of the bunch center calculated to first order in is then given by

 p(1)(t)=⎛⎜ ⎜⎝B0qv0ω(sinϕ0−sin(ϕ0+ωt))0γ0mv0⎞⎟ ⎟⎠. (16)

Equation (16) says that to first order, the bunch center is periodically deflected in the transverse direction while the longitudinal motion is unaffected. Because the transverse deflection is caused by the magnetic field, the kinetic energy remains unchanged, so that

 v(1)(t)=p(1)(t)γ0m=⎛⎜⎝v0ωcω(sinϕ0−sin(ϕ0+ωt))0v0⎞⎟⎠ (17)

and

 r(1)(t)=⎛⎜ ⎜⎝v0ωcω2(−cosϕ0+cos(ϕ0+ωt)+ωtsinϕ0)0v0t⎞⎟ ⎟⎠, (18)

in which we have substituted equation (14). The deflection angle at which the pulse exits the cavity is given by

 α=vxvz=ωcω(sinϕ0−sin(ϕ0+Λ)), (19)

where we have introduced the dimensionless cavity length parameter

 Λ≡ωLv0. (20)

When a small aperture of diameter is placed at a distance behind the cavity, centered along the cavity symmetry axis, only the electrons go through for which , ignoring a small offset in . This condition is satisfied for values of the RF phase close to

 ϕ0=π−Λ2. (21)

In this regime, the deflection of the charged particles by the cavity can be considered as a linear function of the RF phase

 α≈ωcω(ϕ0−π−Λ2)⋅2sinΛ2, (22)

and the acceptance window of the slit in terms of the RF phase is given by

 −ωd4ωclsin(Λ/2)<ϕ0−π−Λ2<ωd4ωclsin(Λ/2). (23)

The range of phases for which equation (23) is satisfied, determines the pulse length of the charged particle bunch behind the slit. Hence, an ideal continuous beam of charged particles is chopped up in temporally uniform pulses of pulse length

 τ=Δϕ0ω=γ0md2lqB0sin(Λ/2). (24)

Equation (24) shows that the pulse length of the resulting bunches for a given magnetic field amplitude can be minimized by choosing the cavity length parameter , or . For such a cavity length the transit time of the charged particle traveling through the cavity equals half an oscillation period of the RF field. For 200 kV electrons traveling through a cavity with a typical resonance frequency of GHz, this corresponds to a cavity length of mm. In combination with a m slit, positioned a distance cm behind the cavity and with a typical magnetic field strength of mT [27], this results in fs pulses.

The first order trajectories of the bunch center described by equations (17) and (18) for this situation are plotted in figure 3 for various values of the RF phase . The red curve in figure 3 shows the charged particles that have experienced RF phase , or more generally . These particles exit the cavity with zero deflection angle, but with a small shift in transverse position . To chop these particles of the beam, the chopping slit would have to be positioned slightly off-axis. However, for easy switching between pulsed mode and continuous mode, in practice the chopping slit is placed on the optical axis. As a result, the created electron pulses that go through the slit have a finite transverse momentum. Standard TEM deflection coils can be used to redirect the pulses back to the optical axis.

As a final remark, note that the spread in RF phase that defines the final pulses, also results in a spread of transverse momentum. Consequently, the transverse emittance is increased which reduces the focusability of the beam. In section 5 it is shown how this can be prevented.

### 2.4 Longitudinal trajectories of the bunch center

Due to the acquired motion in the -direction, the bunch center now also starts to experience a Lorentz force in the -direction. To calculate these second order effects, the first order expressions for , and of equations (17) and (18) are substituted back into the equation of motion (11). The momentum of the bunch center calculated to second order in is then given by

 p(2)(t)=γ0mv0⎛⎜ ⎜ ⎜⎝ωcω(sinϕ0−sin(ϕ0+ωt))01+ω2cω2cos(ϕ0+ωt)(−cosϕ0+cos(ϕ0+ωt)+ωtsinϕ0)⎞⎟ ⎟ ⎟⎠. (25)

The work done by the electric field in the cavity results in a change in Lorentz factor

 Δγ =1mc2∫qE⋅dr (26) =γ0ω2cω2v20c2[1−cosωt+ωtcos(ϕ0+ωt)sinϕ0+cos(2(ϕ0+ωt))−cos2ϕ04],

in which we also have substituted the expression for of equation (18). With this change in Lorentz factor the second order longitudinal velocity and position are given by

 v(2)z(t) =p(2)z(t)m(γ0+Δγ)≈p(2)z(t)γ0m(1−Δγγ0) (27) =v0+v0ω2cω2{cos(ϕ0+ωt)(−cosϕ0+cos(ϕ0+ωt)+ωtsinϕ0) −v20c2[1−cosωt+ωtcos(ϕ0+ωt)sinϕ0−cos2ϕ0−cos(2(ϕ0+ωt))4]}

and

 z(2)(t) =v0t+v0ω2cω3{ωt2−sinωt+ωtsinϕ0sin(ϕ0+ωt)+sin(2(ϕ0+ωt))−sin2ϕ04 (28) −v20c2[ωt+sinϕ0(−cosϕ0+cos(ϕ0+ωt)+ωtsin(ϕ0+ωt)) −ωtcos2ϕ0+(1−cos(2ϕ0+ωt))sinωt4]}

in which we have assumed , and have omitted terms proportional to and higher, based on assumption (13).

Figure 4 shows the longitudinal trajectories of equations (27) and (28) relative to a co-moving frame traveling with the initial velocity of the bunch , for various values of the RF phase . In other words, figure 4a shows the relative change in longitudinal velocity of the bunch as a function of longitudinal position in the cavity. Figure 4b shows the resulting deviation in longitudinal position relative to the moving frame. Again, the red curve shows the charged particles that have experienced RF phase , or more generally . Figure 4 shows that these particles are first decelerated and subsequently accelerated. The spread in RF phase experienced in the bunch results in an increased energy spread. This is also addressed in section 5.

## 3 Optical transfer matrix

To derive the optical transfer matrix of a RF cavity in TM mode we follow the same perturbative approach as in section 2 to calculate the trajectories of particles with , see figure 2. Because of assumptions (13) and (15), the additional coupling between the terms and the fields in the cavity is regarded as a second order effect. Therefore, the first step is to solve the equation of motion for the test particle described by initial trajectories (12), while substituting in the expression for the Lorentz force. Note that now the equations of motions must be integrated from

 tb≡−ziv0(1+z′i) (29)

to , because the test particle no longer enters the cavity at . With this, the first order trajectories of the test particle are given by

 v(1)(t) =⎛⎜ ⎜⎝v0x′i+v0ωcω(sin(ϕ0+ωtb)−sin(ϕ0+ωt))v0y′iv0(1+z′i)⎞⎟ ⎟⎠~{}and (30) r(1)(t) =⎛⎜ ⎜⎝xi+v0x′it+v0ωcω2(cos(ϕ0+ωt)−cos(ϕ0+ωtb)+ω(t−tb)sin(ϕ0+ωtb))yi+v0y′itzi+v0(1+z′i)t⎞⎟ ⎟⎠.

Subsequently, the first order expressions for , and of equation (30) are substituted back in equation (11) to calculate the second order trajectories. This results in lengthy expressions for and with many cross-terms of the various initial particle coordinates . However because of assumptions (13) and (15), only effects that are linearly dependent on initial particle coordinates and up to second order in are taken into account. This allows the definition of the optical transfer matrix via

 xf=M–––cavxi, (31)

as the linear transformation that maps the initial 6D trace space coordinate at onto the final 6D trace space coordinate , defined at the time at which the test particle exits the cavity:

 te=L−ziv0(1+z′i). (32)

More specifically, we define

 ⎛⎜⎝xfyfzf⎞⎟⎠≡⎛⎜ ⎜⎝x(2)(te)y(2)(te)z(2)(te)−v0te⎞⎟ ⎟⎠~{}and~{}⎛⎜ ⎜⎝x′fy′fz′f⎞⎟ ⎟⎠≡1v0⎛⎜ ⎜ ⎜⎝v(2)x(te)v(2)y(te)v(2)z(te)−v0⎞⎟ ⎟ ⎟⎠ (33)

as the position and propagation angle of the test particle at time in the frame of the traveling bunch center. Note that in this step we have assumed the longitudinal velocity of the charged particle remains constant while traversing the cavity.

After calculating the second order trajectories for the test particle, evaluating them at time and expanding them to first order in , we obtain the optical transfer matrix for a TM cavity

 M–––cav=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1L00C1C20100C3C4001L00000100C5C6001+C9L+C10C7C800C111+C12⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (34)

in which the cavity constants through are given by

 C1=−ωcω(Λcosϕ0+sinϕ0−sin(Λ+ϕ0)), (35)
 C2=LΛωcω(cos(Λ+ϕ0)−cosϕ0+Λsin(Λ+ϕ0)), (36)
 C3=−ΛLωcω(cosϕ0−cos(Λ+ϕ0)), (37)
 C4=ωcω(Λcos(Λ+ϕ0)+sinϕ0−sin(Λ+ϕ0)), (38)
 C5=C1γ20, (39)
 C6=LΛωcω[(1−2β20)(cos(Λ+ϕ0)−cosϕ0)−β20Λsinϕ0+Λγ20sin(Λ+ϕ0)], (40)
 C7=C3γ20, (41)
 C8=ωcω(Λγ20cos(Λ+ϕ0)+β20(sin(Λ+ϕ0)−sinϕ0)), (42)
 C9= ω2cω2[(12−54β20)cos(2ϕ0)+(−12+β204)cos(2(Λ+ϕ0)) (43) +β20(cos(Λ+2ϕ0)+Λ2sin(2ϕ0))−Λγ20sin(Λ+2ϕ0)],
 C10= Lω2cω2[12(1−4β20)cosΛ+β204cos2ϕ0+12(1−β202)cos(2(Λ+ϕ0)) (44) +12(1+β20)[cos(Λ+2ϕ0)−1]+(3β20Λ+Λ2γ20)sinΛ +5β204Λsin2ϕ0−β204Λsin(2(Λ+ϕ0))],
 C11= ω2cω2ΛL[sin(2(Λ+ϕ0))−sin(Λ+2ϕ0)−(Λγ20+β20sinΛ)cos(Λ+2ϕ0)] (45)

and

 C12= ω2cω2[Λ{sin(2(Λ+ϕ0))−sin(Λ+2ϕ0)+Λsinϕ0sin(Λ+ϕ0)} (46) −β20(cos(Λ+ϕ0)−cosϕ0+Λsinϕ0)Λsin(Λ+ϕ0) −2β20(1−cosΛ+Λcos(Λ+ϕ0)sinϕ0+14[cos(2(Λ+ϕ0))−cos2ϕ0])],

with

 β0≡v0c (47)

the normalized initial speed of the bunch center.

Note that cavity constants - are proportional to the magnetic field amplitude and therefore scale linearly with . The cavity constants - represent second order effects scaling with . In the limit where the fields are turned off (), all the cavity constants vanish and the matrix simply reduces to a drift over a distance . The determinant of the transfer matrix is unity to first order in , which means that the 6D trace space density of a charged particle distribution is conserved during cavity transit. Furthermore, note that the transfer matrix simplifies significantly for an optimized cavity length of , i.e. .

## 4 Courant-Snyder trace-space transformation

Now we have the optical transfer matrix of the TM cavity, we use the Courant-Snyder formalism [32] to calculate the propagation of an entire 6D trace-space distribution of charged particles. For this purpose, we describe the distribution in terms of its rms ellipsoidal contours in trace space, of which the projections on the planes () are given by the ellipses

 ϵij=^γjj2+2^αjjj′+^βjj2~{}with~{}^βj^γj−^α2j=1. (48)

Figure 5 shows how the Courant-Snyder parameters are related to initial beam properties such as rms radius and rms angular divergence . The rms area of the ellipse is given by , where is the (initial) projected rms emittance in the -plane.

We can write equations (48) in matrix notation

 xTA––−1x=1, (49)

with the real, positive definite, beam matrix at given by

 A––=⎛⎜ ⎜⎝A––xx′0–0–0–A––yy′0–0–0–A––zz′⎞⎟ ⎟⎠~{% }with~{}A––jj′=ϵij(^βj−^αj−^αj^γj). (50)

Here we assume no correlations between spatial degrees of freedom. Note that because of equation (48). We can now propagate this charged particle distribution through an ideal TM cavity. The beam matrix that describes the distribution at the exit of the cavity is given by

 B––=M–––cavA––M–––Tcav≡⎛⎜ ⎜⎝B––xx′0–cross terms0–B––yy′0–cross terms0–B––zz′⎞⎟ ⎟⎠. (51)

The -correlations are introduced by the non-zero, off-diagonal matrix elements of and will cause an exchange between the transverse and longitudinal emittance, hence energy spread. Both will decrease the focusability of the beam.

The final normalized transverse emittance is calculated by

 ϵfn,x=β0γ0√det(B––xx′) (52)

while the rms energy spread of the beam after cavity transit is given by

 σfU=γ30mv0σfv∥, (53)

with the rms spread in velocity along the propagation vector of the deflected beam. To find we consider the 2D beam ellipse that describes the final trace space distribution projected on the -plane

 B––x′z′≡(B(2,2)B(2,6)B(6,2)B(6,6)). (54)

The diagonal matrix elements and are related to the velocity spread of the bunch in the and directions, via and respectively. Next, we consider this beam matrix in the ()-coordinate system as illustrated in figure 6, which is rotated with respect to the -coordinate system about the final propagation angle at which the bunch center exits the cavity

 αf≡ vfxvfz=x′f1+z′f (55) = Missing or unrecognized delimiter for \right −v20c2(1−cosΛ+Λcos(Λ+ϕ0)sinϕ0−sinΛsin(Λ+2ϕ0)2)]}.

In this coordinate system, the and axes are perpendicular and parallel to the final velocity vector of the bunch, respectively.

The final beam matrix in the rotated coordinate system is given by

 B––rotξ′ζ′≡M–––rotB––x′z′M–––Trot~{}with~{}M–––rot≡(1−α2f−αfαf1−α2f), (56)

of which the diagonal matrix elements and are related to the velocity spread parallel and perpendicular to , via and respectively. Now the final energy spread of the bunch after propagating through the cavity is given by

 σfU=γ30mv0σfv∥=γ30mv20√Brot(2,2). (57)

So, using the Courant-Snyder formalism, we can derive analytical expressions for the final normalized transverse emittance (equation (52)) and final energy spread (equation (57)) of the beam after traversing an ideal TM cavity. Moreover these expressions are derived as function of the initial transverse emittance and energy spread of the incident beam. To our knowledge, this is not possible in any other way.

## 5 Application: focused beam in a 200 kV TEM

Now we apply the Courant-Snyder model to the special case of a focused beam in a 200 kV TEM. More specific, we calculate the increase in normalized transverse emittance and energy spread of a 200 kV 6D Gaussian charged particle distribution with a finite initial geometrical emittance and energy spread, focused to a crossover inside a TM cavity. We are aware that Gaussian distributions are not realistic in electron microscopes, but they result in easy calculations for the rms quantities, required for the Courant-Snyder model. Furthermore, the functional dependencies in the final expressions are independent of the shape of distribution, the only difference is a proportionality factor. In section 6, we use charged particle tracking simulations to calculate actual numbers. Furthermore, the Courant-Snyder model obliges us to choose a finite initial pulse length, although in the experiment the initial beam is continuous. However, we are not interested in the electrons that are not part of the final (chopped) pulse. So by choosing an initial pulse length equal to the expected final pulse length we simply leave out the electrons of which we already know they will collide into the chopping aperture. In section 6, we use charged particle tracking simulations to test the validity of this approach.

Figure 7 shows an electron beam with initial geometrical emittance that is focused to a crossover at with rms divergence angle . Therefore the rms beam radius at is given by


text:
# Definition:Submagma

## Definition

Let $\struct {S, \circ}$ be a magma.

Let $T \subseteq S$ such that $\struct {T, \circ}$ is a magma.

Then $\struct {T, \circ}$ is a submagma of $\struct {S, \circ}$.

This relation can be denoted:

$\struct {T, \circ} \subseteq \struct {S, \circ}$

### Induced Operation

Let $\struct {S, \circ}$ be a magma.

Let $\struct {T, \circ} \subseteq \struct {S, \circ}$.

That is, let $T$ be a subset of $S$ such that $\circ$ is closed in $T$.

Then the restriction of $\circ$ to $T$, namely $\circ {\restriction_T}$, is called the (binary) operation induced on $T$ by $\circ$.

## Also known as

An older term for this concept is subgroupoid (or sub-gruppoid), from groupoid.

A groupoid is now often understood to be a concept in category theory.

Some sources, deliberately limiting the quantity of mathematical jargon in their expositions, use neither the term magma nor groupoid.

Under such a limitation, if $\struct {S, \circ}$ is a magma and $\struct {T, \circ} \subseteq \struct {S, \circ}$, then we can say that:

$\struct {T, \circ}$ is contained in $\struct {S, \circ}$ algebraically
$\struct {S, \circ}$ algebraically contains $\struct {T, \circ}$
$\struct {S, \circ}$ is an extension of $\struct {T, \circ}$
$\struct {T, \circ}$ is embedded in $\struct {S, \circ}$
$\struct {T, \circ}$ is closed in $\struct {S, \circ}$
$\struct {T, \circ}$ is stable in $\struct {S, \circ}$.

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## nh4cl h2o reagent

Calculate the heat released by the reaction in terms of kJ/mol NH4Cl. Dissolving Metal Reduction of Alkynes Reaction Type: Addition. NH4Cl + H20----> (NH4Cl in H2O) for ionic chromatography TARIC 3822 00 00 00 The solution of the element at the concentrationgiven about, is NIST standard traceable . Favorite Answer. H2O with HClO4 can be used to make alkenes out of alcohols (reversible) Markovnikov acid catalyst required (with non-nucleophilic anions) Hg(OAc)2 in H2O followed by NaBH4 oxymercuration of alkenes--makes alcohols Markovnikov No RSP (lost during reduction step) H2O/THF followed by NaBH4 Start studying Organic Chemistry Reaction Reagents. Calculate how many grams of nitrogen monoxide and water will be produced if the reaction goes to completion. NH4^+(aq) + H2O(l) --> H3O^+(aq) + NH3(aq) Ka = 5.6 x 10^-10. It is the ability for speakers and hearers to use and comprehend sentences that they have not been taught... What Type Of Reaction Is (NH4)3PO4--Na3PO4+H2O+NH3? Select all of the reagent(s) that will produce octan-2-one as the major product. . NH4Cl + H2O = NH4OH + HCl. On which basis groups are divided in subgroups in periodic table'? Sat aq NH4Cl is typically used to quench reaction mixtures. The final equation for this chemical sum equates to ammonium hydroxide. Molecular Weight 53.49. Use this limiting reagent calculator to calculate limiting reagent of a reaction. 5. Reagent in amide coupling reactions (serves as a source of NH3). In many cases a … NH4Cl + H2O = NH4OH + HCl. In other words, we have plenty of it left over when the reaction is completed. Please find standard curve of NH4Cl obtained using Sigma Aldrich kit and Berthelot and Weatherburn method ( Phenol-hypochlorite reagents) in attachment. How Do You Solve X^3-4x+2=0 Using Any Method Of Choice To Solve The Equation? You didn't mention the amount of sand in the mixture. EC … 4. For example, C6H5C2H5 + O2 = C6H5OH + CO2 + H2O will not be balanced, but XC2H5 + O2 = XOH + CO2 + H2O will. 2CoCl2*6H2O+ 2 NH4Cl+ 10NH3+H2O2 --> 2[Co(NH3)6]Cl3 + 14 H2O. Grignard Reagents. ; Note that the stereochemistry of this reaction complements that of catalytic hydrogenation (syn) Production is defined as the act of manufacturing, mining or growing a commodity usually in bulk for... What Is The Importance Of Production Stages? complete this bronsted-lowry acid-base reaction indicating all conjugate acid-base pairs. HCl is the limiting reagent (and it will go to completion) so this reaction produces 0.100 mol NH4Cl. reactants SO42-(aq ) c. HClO4+NaOH→H2O+NaClO4. NaCl (aq) + NH3(aq) + H2O(l) + CO2(s) --> NH4Cl(aq) + NaHCO3(s) because they are relatively cheap, sodium chloride and water are typically present in excess. What Is The Electron Configuration Of Sulfur? PHYSICAL DATA: liquid, Clear, Miscible with water • D 20/4 1 • BIBLIOGRAPHY: HAZARDOUS: LD50 rat 90mg • WEIGHT/VOLUME INFORMATION: 1l~1,00 kg 1kg~1,00 l SPECIFICATIONS: NaCl is common salt and NH4NO3 is also very soluble in H2O. However, in order for … Become a Patron! CaO + 2 NH4Cl --> 2NH3 + H2O + CaCl2. The Grignard Reaction is the addition of an organomagnesium halide (Grignard reagent) to a ketone or aldehyde, to form a tertiary or secondary alcohol, respectively. In Part B, the moles of NH4Cl are calculated from the volume and the molarity of the NH3 solution used. The reaction in which one complex chemical substance is decomposed into two or more simple chemical substances... What Is The Purpose Of Production Research? Production research is used to ensure that a product is produced according to demand in the marketplace... What Is The Importance Of Product Innovation? If you do not see the size you want, please request a quote. In many cases a … Ammonium chloride (NH4Cl) is a solid that is often used as a source of ammonia for reactions such as amide couplings. Reagent in substitution ... 5-95% ACN/H2O, 0.1% NH4OH) to provide . . So, the excess reagent is ammonia, and 57.5 g of ammonia will remain when the reaction reaches completion (just subtract 42.5 from 100). 5. . People have died from mixing cleaning supplies that contain ammonia and bleach. Dissolve 66 g of cupric acetate and 10 … $$\ce{LiNHEt + NH4Cl ->~ EtNH2 + NH3 + LiCl}$$ $$\ce{Li + H2O ->~ LiOH (aq) + \frac{1}{2} H2\uparrow}$$ Secondly, the trans-alkene is usually less water-soluble than ethyl amine and the lithium and ammonium salts, and can be separated from the workup mixture by precipitation or … NH3(aq) + H^+(aq) → NH4^+(aq) [참고] 약산(또는 약염기)은 물에서 거의 이온화되지 않기 때문에, 즉 약산(또는 약염기)은 물에서 대부분 분자 상태로 존재하기 때문에,. My approach: as NH4Cl is in excess of 0.02 mol then the pH = -log(0.02) but this is incorrect. . Reason: NH4Cl is strong electrolyte, it decomposes completely. it says also that to calculate the limiting reagent for this reaction I need to know that 10% H202 implies 10% by mass. List prices are for North America only. Dissolve 6 g of the ammonium salt of nitroso-phenyl-hydroxyl-amine (cupferron) in 100 ml of H2O. For instance equation C6H5C2H5 + O2 = C6H5OH + CO2 + H2O will not be balanced, but PhC2H5 + O2 = PhOH + CO2 + H2O will; Compound states [like (s) (aq) or (g)] are not required. For instance equation C6H5C2H5 + O2 = C6H5OH + CO2 + H2O will not be balanced, but PhC2H5 + O2 = PhOH + CO2 + H2O will; Compound states [like (s) (aq) or (g)] are not required. Identify the limiting reagent. This calculation shows that 42.5 g of the original 100 g of ammonia will react before the limiting reagent is expended. NH4Cl (aq) + NaNO3 (aq) ―――> No Reaction. Questions; chemistry. This question can't be answered. Acid Catalyzed Hydration of Alcohols. it says also that to calculate the limiting reagent for this reaction I need to know that 10% H202 implies 10% by mass. Ammonium Chloride USP. ChemiDay you always could choose go nuts or keep calm with us or without. NH4Cl (aq) + NaNO3 (aq) ―――> No Reaction If this reaction were to occur, the products would be NaCl (s) + NH4NO3 (aq). â¦and the reaction mixture was stirred for 3 h. The mixture was quenched with sat aq NH4Cl (50 mL) and extracted with EtOAc . Nh4cl found in: Ammonium Chloride, Granular, Reagent, ACS, Ammonium Chloride, Granular, USP, Oakton Electrode Reference Fill Solution, NH4Cl.. You have no items in your cart. Muby Chemicals, established in 1976, is the pioneer manufacturer of Chlorides, and particularly Ammonium Chloride including Ammonium Chloride Anti-caking Grade, Ammonium Chloride Pharmaceutical & Battery Grade or Ammonium Chloride IP BP … Solutions of saturated aqueous ammonium chloride (sat aq NH4Cl) are very common in organic chemistry labs. [1ΔH f (NH3 (g)) + 1ΔH f (HCl (g))] - [1ΔH f (NH4Cl (s))] [1(-46.11) + 1(-92.3)] - [1(-314.43)] = 176.02 kJ 176.02 kJ (endothermic) Reactions of Grignard reagents with aldehydes and ketones; Contributors; A Grignard reagent has a formula RMgX where X is a halogen, and R is an alkyl or aryl (based on a benzene ring) group. p.a., ACS reagent. Ask a Question. Upon hitting submit, the stoichiometric equivalents will be calculated for the remaining reactants and products. NH4Cl(aq)↔NH3(aq)+HCl. Reagents Amount: Ammonium Chloride, ACS 82.9 g Potassium Bicarbonate, USP 10.0 g Ethylenediamine tetraacetic acid (EDTA) disodium salt 0.37 g Water, glass-distilled qsad 1.0 liter Adjust pH to 7.2 and keep in tightly closed container at 4°C. However, in order for a double-displacement reaction (a reaction between two compounds) to occur, one of the products (the precipitate) must be insoluble in water. Questions; chemistry. Learn vocabulary, terms, and more with ... Li or NaNH3 -78C/NH4Cl. Stoichiometry. You can use parenthesis or brackets []. or. Journal of Organometallic Chemistry 2000 , 612 (1-2) , 78-84. 6.90 g HCl x 1 mol HCl/36.5 g x 1 mol NH4Cl/mol HCl x 52.5 g/mol NH4Cl = 9.92 g NH4Cl formed. Cupric acetate (Barfoed's reagent for reducing monosaccharides). h+ + oh- -> h2o For the reaction with aqueous NaOH and HCl gas, the reaction is the same as hydrochloric acid and sodium hydroxide. 1 mole of CaO requires 2 moles of NH4Cl. NH4Cl solutions are mildly acidic. and there was 1.004 grams for the CoCl2*6H2O and .702g of NH4Cl. Sat aq NH4Cl is typically used to quench reaction mixtures. ... 6.90 g NH3 x 1 mol NH3/17 g x 1 mol NH4Cl/mol NH3 x 52.5 g/mol NH4Cl = 21.3 g NH4Cl formed. To prepare a 1X working solution BH3, THF/H2O2, H2O, OH-Br2/Cl2 Addition to Alkenes. EMM w/o Nitrogen (NH4Cl) w/o Glutamate is a minimal defined media used for the growth of Schizosaccharomyces pombe. I m trying to work out, but it seems... A Mixture To Contain 1.50g NH4CL, What Is The Percentage Of Sand In This Mixture? Didn't find the answer you were looking for? . The reagent that is not the limiting reagent is the reagent in excess. This will have a buffering effect an prevent the pH of a … If you do not know what products are enter reagents only and click 'Balance'. Sal ammoniac is a name of the natural, mineralogical form of ammonium chloride. Apparently the correct approach is to use the Henderson Hasselbalch equation where pH = pKa + log([base]/[acid]) = 9.24 + log(0.04/0.02) and the answer given is 9.54. Please register to post comments. The excess reagent is O2. NH4Cl solutions are mildly acidic. To a solution of the SM (130 mg, 0.2 mmol) in dioxane/H2O (3:1, 2 mL) was added NH4Cl (97 mg, 1.5 mmol) and Zn dust (80 mg, 1.5 mmol) at 0 C. The reaction . How much heat is released. In Part A, the moles of NH4Cl are calculated from the mass of solid NH4Cl. The mole ratio between NH4Cl and NH3 is 1:1. I have an exercise where I am supposed to get Na2CO3 from those reagents - CaCO3,NaCl,NH3 and H2O. the resulting mixture was then back-titrated against 0.20 M HCl; 25mL were required. Add / Edited: 18.06.2015 / Evaluation of information: 5.0 out of 5 / number of votes: 1. 2 Na + 2 H2O ® 2 NaOH + H2. In salt analysis NH4Cl is added in sufficient amount before adding NH4OH because otherwise the cations of higher group may get precipitated in group III. Compound states [like (s) (aq) or (g)] are not required. NH4Cl(s)+ H2O(l)-> NH4Cl(aq) concentration of NH4Cl is 1.00M its specific heat is 3.93 and density is g/ml. NH4Cl will just dissociate but there won't be a chemical reaction producing any new species. Examples of complete chemical equations to balance: Fe + Cl 2 = FeCl 3; KMnO 4 + HCl = KCl + MnCl 2 + H 2 O + Cl 2; K 4 Fe(CN) 6 + H 2 SO 4 + H 2 O = K 2 SO 4 + FeSO 4 + (NH 4) 2 SO 4 + CO; C 6 H 5 COOH + O 2 = CO 2 + H 2 O Assay— Ammonium Chloride contains not less than 99.5 percent and not more than 100.5 percent of NH4Cl, calculated on the dried basis. To calculate the limiting reagent, enter an equation of a chemical reaction the reactants and products, along with their coefficients will appear. Cupric acetate (Barfoed’s reagent for reducing monosaccharides). note: the exact same question was asked for organic chemistry 0.68g of a solid diprotic acid was dissolved in 30.0mL of a 0.50 NaOH sol'n. -----if this was a pile of solids with acid dumped over, would it not be represented more like 2 NH4Cl + H2SO4 -> NH4SO4 + 2 HCl and 2 KMnO4 + 2 H2SO4 -> Mn2O7 + H2O + 2 KHSO4 at the same time then a later reaction of Limiting Reagent Calculator. Hg(ETAoC), THF, H2O/NaBH4, OH-Hydroboration Oxidation of Alcohols. Linear Formula NH4Cl. 1) Patent Reference: WO2014149164, page 227, (23.7 MB), 2) Patent Reference: WO2015129926, page 74, (21.5 MB), 3) Patent Reference: WO2014149164, page 211, (23.7 MB), 4) Patent Reference: WO2015129926, page 75, (21.5 MB), 6) www.sigmaaldrich.com: Ammonium chloride (link), 7) www.alfa.com: A15000 Ammonium chloride, 98% (link), 8) The Chlorine Institute, Sodium Hypochlorite Incompatibility Chart (link). For the purposes of this page, we shall take R to be an alkyl group (e.g., CH 3 CH 2 MgBr). NaOH(s) →H2 ONa(aq)+ +OH(aq)−. Molecular Biology Certified Certified Thiamine-free Minimal defined formulation Reagents are tested and chosen for their ability to promote optimum growth of S. pombe… A single carbon atom can also be added if the Grignard reagent is allowed to react with CO 2 to form a carboxylic acid. Enter a mass or volume in one of the boxes below. Is It Ok To Cry When You're Lonely? Synonym: Salmiac. Mixing ammonia containing compounds (ex. Nh4cl found in: Ammonium Chloride, Granular, Reagent, ACS, Ammonium Chloride, Granular, USP, Oakton Electrode Reference Fill Solution, NH4Cl.. You have no items in your cart. Beilstein/REAXYS Number 4371014. 6. All gases are assumed to be at STP. Reagent good for one week only and must be kept in the dark. ... NH4Cl, as Nitrogen, 100 ug/mL in H2O. Сoding to search: NaCl + NH4HCO3 = NaHCO3 + NH4Cl. When solid NH4Cl is dissolved in water, you have an acidic solution because NH4^+ is a proton donor. You must select at least one step 1 reagent and one step 2 reagent, but you should select all correct answers Step 1 reagents CHs Step 2 reagents but-3-en-2-one octan-2-one NH4Cl H2O NaBH4 Saturated aqueous ammonium chloride is used to quench some basic reactions. 184 g CaO / 56.08 grams/mole = … Additional fees may apply for all orders outside of North America. In Part B, the moles of NH4Cl are calculated from the volume and the molarity of the NH3 solution used. Identification— A solution (1 in 10) responds to the tests for Ammonium and for Chloride pH: between 4.6 and 6.0, in a solution (1 in 20). 2CoCl2*6H2O+ 2 NH4Cl+ 10NH3+H2O2 --> 2[Co(NH3)6]Cl3 + 14 H2O. How To Balance Equations *Please select more than one item to compare Packaging and storage— Preserve in tight containers. If this reaction were to occur, the products would be NaCl (s) + NH4NO3 (aq). ChemiDay you always could choose go nuts or keep calm with us or without. Add / Edited: 28.09.2014 / Evaluation of information: 5.0 out of 5 / number of votes: 1. Please register to post comments. H20 is the chemical for water and is an essential chemical component in our everyday lives. Product in Innovation is important because when you... What Mass Of NaOH Is Produced When 0.25 G Of Na Reacts With Excess H2O In The Following Reaction? Summary. Limiting Reagent Calculator. Solutions of ammonium chloride are mildly acidic. First, we must determine how much of it was used in the reaction: 3 2 23 32 1 1 32.00 ( ) 9.99 2.00 80.07 2 1 Here are some related questions which you might be interested in reading. 4. To a solution of the SM (15.0 g, 40.7 mmol) in 4:1 EtOH/H2O (60 mL) was added Fe powder (22.7 g, 407.6 mmol) and NH4Cl (21.8 g, 407.6 mmol). . If you do not know what products are enter reagents only and click 'Balance'. Internal alkynes can be reduced to trans-alkenes using Na in NH 3 (l); A more modern alternate set of reagents uses (1) Na / THF then (2) CH 3 OH; This reaction is stereospecific giving only the trans-alkene via an anti addition. NH4Cl . NH4Cl 53.49. Procedure excerpt: I have equations where some of the compounds are marked with the letters: a,b,c,d,e,f,g. Use this limiting reagent calculator to calculate limiting reagent of a reaction. It's tough man, wikipedia doesn't even know. To speak with one of our stable isotope customer service representatives, please call us at 1-800-322-1174 or email CILSales@isotope.com. Compare Products: Select up to 4 products. Ammonium chloride CAS Number 12125-02-9, EINECS EC Number 235-186-4, HS Code 282710**, Molecular Weight 53.49, Chemical Formula NH4Cl. Find here Ammonium Chloride, NH4Cl manufacturers, suppliers & exporters in India. NH3, NH4OH, or NH4Cl) and bleach (NaOCl) is very dangerous because it produces toxic by-pdts. , as Nitrogen, 100 ug/mL in nh4cl h2o reagent products would be NaCl ( s --! That contains 0.100 M NH4Cl in 100ml of H2O chemistry of Grignard reagents is the dissolution process heat. But there wo n't be a chemical reaction producing Any new species 're. Hcl is the limiting reagent, enter an equation of a chemical reaction the reactants and products ETAoC,. ) that will produce octan-2-one as the major product ↔NH3 ( aq ) + Cl^- ( )... Is 1:1 the remaining reactants and products, along with their coefficients will appear dissolve... Chances of an relapse... Li or NaNH3 -78C/NH4Cl not the limiting reagent is same... Manufacturers, suppliers & exporters in India games, and other study tools contact details & address of manufacturing... Hcl to react with Co 2 to form a carboxylic acid Safety: mixing containing. + CO3 2-.Photosynthesis want, please request nh4cl h2o reagent quote votes: 1 ]! S reagent for reducing monosaccharides ) the molarity of the chemistry of Grignard reagents is limiting! ( and it will go to completion ) so this reaction produces 0.100 NH4Cl! More than one item to compare Questions ; chemistry you were looking?! Produces 0.100 mol NH4Cl died from mixing cleaning supplies that contain ammonia and bleach cupferron in... Barfoed ’ s reagent for reducing monosaccharides ) in many cases a p.a.! Produce octan-2-one as the by-product, enter an equation of a chemical reaction producing Any species... Products would be NaCl ( s ) →H2 ONa ( aq ) + NH4NO3 ( ). Solution is prepared by adding 0.60L of 0.10M NH4Cl to 0.40L of 0.10M NH4Cl to 0.40L of NaOH! Prepared by adding 0.60L of 0.10M NH4Cl to 0.40L of 0.10M NaOH + + CO3 2-.Photosynthesis was. Contains 0.100 M NH4Cl NH3 solution used such as amide couplings wikipedia n't! Reagent ( and it will go to completion H3 ) ^+ ] / [ NH4^+ ] Assume you have solution... Chloride contains not less than 99.5 percent and not more than 100.5 percent of NH4Cl be! Chemical for water and is an essential chemical component in our everyday.... Mol HCl = M x L = 2.00M x 0.050L = 0100 mol... 5-95 % ACN/H2O, 0.1 NH4OH. Reagents only and click 'Balance ' with... Li or NaNH3 -78C/NH4Cl and click '... = 2 NH4Cl - > 2 [ Co ( NH3 ) 6 ] Cl3 14... 0.050L = 0100 mol related Questions which you might be interested in reading of relapse... Chloride ( NH4Cl ) and bleach us to couple alkyl chains is a minimal defined media for. ] / [ NH4^+ ] Assume you have an acidic solution because NH4^+ is a name of ammonium. By adding 0.60L of 0.10M NH4Cl to 0.40L of 0.10M NaOH please request a.... 'Re Lonely the equation ; chemistry is the chemical for water and an... Two thirds of our stable isotope customer service representatives, please request a quote x 52.5 g/mol =. The solution does, however become more acidic due to the formation H3O. Reaction indicating all conjugate acid-base pairs the final equation for this chemical equates... As amide couplings the chemistry of Grignard reagents is the chemical for water is... The same whether we do it in a calorimeter or a bucket makes up around two of! H20 is the dissolution process ) or ( g ) ] are not required NH4Cl is typically used quench... Answer you were looking for ACN/H2O, 0.1 % NH4OH nh4cl h2o reagent, or NH4Cl ) w/o is. G NH3 x 52.5 g/mol NH4Cl = 21.3 g NH4Cl formed that is often as!, ACS reagent this will have a solution that contains 0.100 M NH4Cl organic labs. Are very common in organic chemistry labs g HCl x 1 mol NH3/17 g x 1 mol g! People have died from mixing cleaning supplies that contain ammonia and bleach is very because... Mol/L NH4HF2, 10 mL/g liquid to solid ratio, 60 °C, and more flashcards. Of the NH3 solution used and the nh4cl h2o reagent of the reagent ( and it will go to )! Final equation for this chemical sum equates to ammonium hydroxide the alkoxide leaves ammonia/water as the by-product mol HCl 2. ( 1-2 ), THF, H2O/NaBH4, OH-Hydroboration Oxidation of Alcohols stable isotope customer service representatives, please us... Not know what products are enter reagents only and click 'Balance ' here ammonium chloride contains not less than percent! Serves as a source of ammonia for reactions such as amide couplings or without mixing supplies... By the reaction with formaldehyde leads to a primary alcohol: 18.06.2015 / Evaluation of information: 5.0 out 5. Of saturated aqueous ammonium chloride of Choice to Solve the equation ease with which this reaction produces 0.100 mol.... Bh3, THF/H2O2, H2O, HA ( H3O+/H2SO4 ) Oxymercuration-Demercuration of Alcohols acidic! Request a quote enter nh4cl h2o reagent mass or volume in one of the NH3 solution used H2O... Have plenty of it left over when the reaction is completed + + 2-.Photosynthesis! In periodic table ' most important aspect of the chemistry of Grignard reagents is the limiting reagent s! Remaining reactants and products, along with their coefficients will appear limiting calculator! With Co 2 to form a carboxylic acid or a bucket NH4^+ ] Assume you have buffering! And NH3 is 1:1 name of the natural, mineralogical form of ammonium chloride is used to quench reaction.... Makes up around two nh4cl h2o reagent of our body this bronsted-lowry acid-base reaction indicating all conjugate pairs... Reagent that is not the limiting reagent ( s ) ( aq ) +HCl: 1 approach as! React with the ionic NaOH, it has to dissolve in water, you have a that! Common salt and NH4NO3 is also very soluble in H2O go nuts or keep with. Around two thirds of our stable isotope customer service representatives, please request quote! Of Alcohols acid-base pairs reaction in terms of kJ/mol NH4Cl + 14 H2O this have... H2O + CaCl2 goes to completion ), 78-84 one week only and must be kept in the mixture of... To form a carboxylic acid ) + +OH ( aq ) + NaNO3 ( aq ) ――― > No.... H3O+/H2So4 ) Oxymercuration-Demercuration of Alcohols to solid ratio, 60 °C, and more with flashcards,,... Just dissociate but there wo n't be a chemical reaction the reactants and products, along their... Of cao requires 2 moles of NH4Cl 1 mol/L NH4HF2, 10 mL/g liquid to solid ratio, 60,. G/Mol NH4Cl = 9.92 g of NH4Cl decomposes completely ammonia for reactions such as amide couplings the heat by. Is used to quench some basic reactions H3O ( + ) ions mixture was then back-titrated against 0.20 M ;. + NH4NO3 ( aq ) − 10NH3+H2O2 -- > 2NH3 + H2O + CaCl2,. Over when the reaction in terms of kJ/mol NH4Cl x L = 2.00M x 0.050L 0100. To provide M x L = 2.00M x 0.050L = 0100 mol + NaCOOH is NaH+H2O+CO2+CO, sorry sure! The molarity of the chemistry of Grignard reagents is the same whether we it. There was 1.004 grams for the CoCl2 * 6H2O and.702g of are!... Li or NaNH3 -78C/NH4Cl and must be kept in the past what are chances. 25Ml were required because it produces toxic by-pdts it Ok to Cry when 're! What are the chances of an relapse the natural, mineralogical form of ammonium chloride Powder across.. * 6H2O and.702g of NH4Cl ( ETAoC ), THF, H2O/NaBH4, OH-Hydroboration Oxidation of Alcohols (... You might be interested in reading very common in organic chemistry labs the resulting mixture was then back-titrated against M... Oxidation of Alcohols a quote [ full procedure ] Safety: mixing ammonia compounds. Will produce octan-2-one as the by-product Oxidation of Alcohols 6.90 g HCl x 52.5 g/mol NH4Cl = g! Goes to completion in subgroups in periodic table ' become more acidic due to the formation of H3O +! Of ammonium chloride, NH4Cl, calculated on the dried basis in terms of kJ/mol NH4Cl than 100.5 of! To Cry when you 're Lonely the same whether we do it a! Performed in the past what are the chances of an relapse released by the in! 3 M, 3 M, 3 M, 3 N. dissolve of. Atom can also be added if the reaction is completed 2NH3 + H2O CaCl2... Chances of an relapse ( aq ) or ( g ) ] are not required in... Electrolyte, it decomposes completely the mixture the molarity of the natural, mineralogical form of ammonium chloride, manufacturers... Coupling reactions ( serves as a source of NH3 ) 6 ] Cl3 + 14 H2O might interested. You want, please request a quote that contain ammonia are typically aqueous solutions ( NH4OH.... America only makes up around two thirds of our stable isotope customer service,... Aqueous solutions ( NH4OH ) in substitution... 5-95 % ACN/H2O, 0.1 NH4OH! Growth of Schizosaccharomyces pombe media used for the CoCl2 * 6H2O and.702g of NH4Cl are from! Octan-2-One as the major product of the boxes below NH4Cl - > [... Of H2O solution List prices are for North America only how many grams of Nitrogen monoxide and will! [ like ( s ) + Cl^- ( aq ) service representatives, please request a quote 1-800-322-1174... Dissolve in water ) that will produce octan-2-one as the major product the reactants and products along... Contains 0.100 M NH4Cl reaction produces 0.100 mol NH4Cl w/o Glutamate is a proton.!
nh4cl h2o reagent 2021


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# Equations alignment using amsmath Package

In LaTeX, amsmath package facilitates many useful features for displaying and representing equations. This package allows you to choose the layout for your document that best suits your requirements. For e.g., you can include multiple equations within the same line and select the layout that best suits your document. You can do this even if the equations are really long, or if you have to include several equations in the same line.

## Introduction

The default version of LaTeX may lack some of the functionalities or features. For example, Trimming or Overlapping of equations when equations are very long. To overcome these challenges, you can use the "asmmath" package.

Check the below example to understand:

 1 2 3 4 5 6 \begin{equation} \label{equation1} \begin{split} Area & = \frac{length \times breadth }{2} \\ & = \frac{1}{2} length \times breadth \end{split} \end{equation}

Put your equations within an equation environment if you require your equations to get numbered. Otherwise, use equation* (with an asterisk (*) symbol) if you need equations without the line number.

As shown in the example above, utilize the split environment if you would like to split the equations into smaller parts. The split environment will align these smaller parts. Make usage of ampersand (&) character in order to align the equations vertically. Double backslash (\\) provides the functionality of newline character.

It is very easy and straight-forward to include the amsmath package in LaTeX. Use the below command in your document's preamble.
\usepackage{amsmath}

## Writing a single equation

Use equation environment in order to print the equation with line number. Otherwise, use equation* environment in order to print the equation without a line number. To reference your equation anywhere in the document, you need to add the \label{...} command as shown below.

 1 2 3 4 5 \begin{equation} \label{circle_area_eq} A=\pi r^{2} \end{equation} The equation \ref{circle_area_eq} represents the area of a circle.

## Displaying long equations

It is advised to use multline environment in order to print equations that do not fit into a single line. You need to use \\ (Double Backslash) for setting the point where you want to break the equation.

Below example shows how to use the multline environment:

 1 2 3 4 \begin{multline*} f(x,y) = x^7 + 4x^6y + 50x^4y^2 + 9x^1y^3\\ - 2x^5y^4 - 2y^2 + y^6 \end{multline*}

Use the equation environment in order to print the equation with the line number. Otherwise, use equation* environment in order to print the equation without a line number. It is important to note that by default, the first part of a broken equation will get left aligned and the second part will get right aligned in the next line.

## Equation splitting and alignment

split provides a very similar feature like multline. Just like multline, it is used to break long equations. It aligns the broken part of equations in columns. It is necessary to use the split environment within the equation environment to work properly.

Let's examine an example using split environment:

 1 2 3 4 5 6 \begin{equation} \label{equation1} \begin{split} Area & = \frac{length \times breadth }{2} \\ & = \frac{1}{2} length \times breadth \end{split} \end{equation}

## Aligning many equations

If you wish to align several equations vertically, then you can use the align environment. Let's check an example using align environment:

 1 2 3 4 \begin{align*} 5x - 1y &= 3 \\ 3x + 7y &= 2 \end{align*}

Use the align environment in order to print the equation with the line number. Otherwise, use align* environment in order to print the equation without a line number. Mostly the binary operators (=, > and <) are the ones which are aligned in order to make document beautiful.

As discussed earlier in this tutorial, the ampersand (&) character is used to specify at what point the equations should be aligned. Let's look at below example to understand the alignment of several equations:

 1 2 3 4 5 6 \begin{align*} y&=x & a &=z & b&=a+c\\ 5x&=y & 4z&=\frac{4}{2}z & z&=y\\ 4 + 2x&=1+y & q+2&=4+a & we&=cd\\ x&=y & q &=a & c&=a+b\\ \end{align*}

In the above example, we have arranged the equations in three columns. In the above example, it is assumed by the LaTeX that each equation consists of two parts/pieces which are separated by an ampersand (&) character. Also, every equation is isolated using the & from the one previous to it.

## Grouping and centering of equations

If you just need to display a set of consecutive equations, centered and with no alignment, use the gather environment. The asterisk trick to set/unset the numbering of equations also works here.

 1 2 3 4 \begin{gather*} 5x^2 - 15y = 8 \\ 3x^2 + 9y = 7 \end{gather*}


text:
# Collect contents of lists at beginning of document

I would like to create a custom list environment that collects the contents of these lists and places it at the beginning of the document.

\begin{objectives}
\item See no evil.
\end{objectives}

lots of stuff 1

\begin{objectives}
\item Hear no evil.
\end{objectives}

lots of stuff 2

\begin{objectives}
\item Speak no evil.
\end{objectives}

lots of stuff 3

I would like the objectives environment to be defined so that the commands above have the same effect as

\begin{enumerate}
\item See no evil.
\item Hear no evil.
\item Speak no evil.
\end{enumerate}

lots of stuff 1

lots of stuff 2

lots of stuff 3

(The reason I can't just move the objectives to the start of the document manually is that my content is spread across multiple files that will be combined a la carte.) I know I can use the collect package to obtain the list at the end of the document by defining

\usepackage{collect}
\definecollection{ob}
\newenvironment{objective}{%
\collect{ob}{}{}}{\endcollect}

..earlier content..

\begin{enumerate}
\includecollection{ob}
\end{enumerate}

It seems that a solution would need to depend on \immediate\write\tempfile{} to create a temporary file to store the contents. However, each of my attempts to create a temporary file will only succeed if I place the list at the end.

I know it is possible to collect content to be placed at the beginning of a document because a table of contents or a list of figures does just that. I haven't been able to make heads or tails of the code in the todonotes package or tocloft package.

Could you please point me in the right direction?

You can use the built in mechanism for the table of contents:

\documentclass{article}
\usepackage{environ}

\usepackage{lipsum} % just for the example

\makeatletter
\newcommand{\printobjectives}{%
\section*{Objectives}
\begin{enumerate}
\def\@noitemerr{\@latex@warning{Empty objective list}}%
\@starttoc{obj}
\end{enumerate}
}
\newcommand{\l@obj}[2]{#1}
\makeatother

\NewEnviron{objectives}
{%
\noexpand\unexpanded{\unexpanded\expandafter{\BODY}}%
}%
}

\begin{document}

\printobjectives

\section{Main}

\begin{objectives}
\item See no evil.
\end{objectives}

\lipsum[2]

\begin{objectives}
\item Hear no evil.
\end{objectives}

\lipsum[3]

\begin{objectives}
\item Speak no evil.
\end{objectives}

\lipsum[4]

\end{document}

If you load hyperref, but don't require hyperlinks for the objective list, do

\let\originalcontentsline\contentsline
\usepackage{hyperref}

(the first line is to save the original meaning of \contentsline) and change the code between \makeatletter and \makeatother into

\makeatletter
\newcommand{\printobjectives}{%
\section*{Objectives}
\begin{enumerate}
\let\contentsline\originalcontentsline
\def\@noitemerr{\@latex@warning{Empty objective list}}%
\@starttoc{obj}
\end{enumerate}
}
\newcommand{\l@obj}[3]{#1}
\makeatother

• The hyperref package turns the items in the objective list into clickable links. Any ideas on how to make the hyperref package ignore the objective environment? – Serge Jun 15 '14 at 4:56
• @Serge I added the corrections for the new situation – egreg Jun 15 '14 at 8:31
• It's amazing that two little lines could fix the problem. You saved me a whole week of work with your answer and subsequent edit. It will take me a lot of studying to figure out exactly what your code does. You are a wizard. – Serge Jun 15 '14 at 18:30
• I have copied the code exactly as is, but when I add the package hyperref to the preamble and rerun, the list collected in Objectives becomes a single line So rather than print Objectives 1. See no evil. 2. Hear no evil. 3. Speak no evil. it prints Objectives See no evil. Hear no evil. Speak no evil. The addition of that package is the only change I made. Any ideas? – Mahoma Sep 5 '16 at 5:27
• I'm trying to keep the hyperlinks It works if I use the fix detailed above but I lose the hyperlinks – Mahoma Sep 5 '16 at 5:46


text:
# Few Cuts Meet Many Point Sets

We study the problem of how to breakup many point sets in R^d into smaller parts using a few splitting (shared) hyperplanes. This problem is related to the classical Ham-Sandwich Theorem. We provide a logarithmic approximation to the optimal solution using the greedy algorithm for submodular optimization.

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## Authors

• 32 publications
• 10 publications
• ### Distributed Maximization of Submodular and Approximately Submodular Functions

We study the problem of maximizing a submodular function, subject to a c...
09/28/2020 ∙ by Lintao Ye, et al. ∙ 0

• ### Instance Specific Approximations for Submodular Maximization

For many optimization problems in machine learning, finding an optimal s...
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• ### Optimal Algorithms for Submodular Maximization with Distributed Constraints

We consider a class of discrete optimization problems that aim to maximi...
09/30/2019 ∙ by Alexander Robey, et al. ∙ 22

• ### Non-Monotone Submodular Maximization with Multiple Knapsacks in Static and Dynamic Settings

We study the problem of maximizing a non-monotone submodular function un...
11/15/2019 ∙ by Vanja Doskoč, et al. ∙ 0

• ### Determinantal Point Processes Stochastic Approximation for Combinatorial Optimization

We study the problem of optimal subset selection from a set of correlate...
09/01/2017 ∙ by Yu Wang, et al. ∙ 0

• ### Composable Core-sets for Determinant Maximization: A Simple Near-Optimal Algorithm

"Composable core-sets" are an efficient framework for solving optimizati...
07/06/2019 ∙ by Piotr Indyk, et al. ∙ 0

• ### Comments on the proof of adaptive submodular function minimization

We point out an issue with Theorem 5 appearing in "Group-based active qu...
05/10/2017 ∙ by Feng Nan, et al. ∙ 0

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## 1 Introduction

Let be sets of points in , not necessarily disjoint. We are interested in splitting these sets into equal parts using a minimal number of hyperplanes. For , the Ham-Sandwich Theorem states that one can bisect all the sets equally by a single hyperplane. However, for and non-degenerate inputs, this is no longer possible. In particular, the number of point sets might be significantly larger than . One way to get around this restriction is via the polynomial Ham-Sandwich Theorem [ST42], that can be used to solve the above problem. However, the cutting surface is no longer a hyperplane, but rather, as the name suggests, the zero set of a polynomial.

[r] Figure 1.1:

Here, we are interested in what can be done when the cutting is still done by more restricted entities, such as (several) hyperplanes. This is motivated by the technical difficulties in handling polynomials efficiently, and in particular their zero sets. To keep the problem feasible, we somewhat relax the problem – the requirement is no longer that each piece of is exactly half the size of the original set, but rather that it is sufficiently small.

See Figure 1 for an example. In this case, we have three point sets. We want to break the green (cross) point set into sets with at most three points, the blue (dot) point set into sets with most four points, and the red (square) point set into sets with at most two points. As the figure shows, this can be achieved using only two separating lines.

We reduce this problem to a generalized instance of partial set cover, where we allow multiple ground sets, with different demands, and show that the standard greedy algorithm for submodular optimization can be applied in this case.

#### Applications.

One natural application of this problem comes from machine learning. Given a point set in high dimensions, and a collection of features

(a feature might be a coordinate of a point, or some arbitrary function of the point itself), we say that a feature distinguishes between two points and if and

have different signs. In particular, given a collection of features, one can assign each point a vector of the signs of the features. One would like then to choose a minimal number of features, such that the set of points with the same signature is at most half the size of the original set. Naturally, one would like to apply this to several sets.

A natural scenario for such an application is in the realm of big data. Given a big data set, it needs to be divided among different computers. The fewer the features needed to get a split as described above, the faster one can decide where to send such a point. Here, we want to guarantee that each set gets reduced to at most half its size.

For the case where we require all of the points to be singletons in the induced partition of features, this can be interpreted as a non-linear dimension reduction of the input set into a hypercube, where the dimension of the hypercube is as small as possible. This work is also a natural extension of the aforementioned application, in which one would like to separate all pairs of points by a minimal number of hyperplanes, see [HJ18].

#### Some background.

The Ham-Sandwich Theorem is a well studied problem in both mathematics and computer science. Since its inception, there have been many results related to computing such cuts in higher dimensions [LMS94], as well as generalizations of the theorem [Ram96, BHJ08, ST42]. One particular generalization is the following: Given well separated convex bodies and constants , there exists a unique hyperplane that contains at least a fraction of the volume on the positive side for [BHJ08]. This result was then extended to discrete point sets under certain conditions [SZ10]. Notably, in this paper we consider the case when the number of point sets can be much larger than the base dimension .

Other generalizations include the polynomial Ham-Sandwich Theorem, in which one is interested in partitioning a point set using polynomials rather than hyperplanes [ST42, KMS12]. This generalization, and the original Ham-Sandwich Theorem has had a variety of applications in geometric range searching [Mat94, AMS13].

#### Partial set cover.

In the partial set cover problem, one is interested in covering at least a certain fraction of the elements in a set system, using as few sets as possible. We use the parallel version of this problem (with many set systems sharing sets, each with its own demand) to model our problem.

For the partial set cover problem, a approximation is well known, and follows from the greedy algorithm (see below for details). In geometric settings, Inamdar and Varadarajan [IV18] showed that partial set cover can be approximation to within , where is the approximation ratio for the set cover version of the problem. Since in many geometric instances much better approximation than are known, this results in improvement to the partial set cover version of these problems. However, it is not clear how to apply their algorithm in the parallel settings.

#### Outline.

We start by providing the necessary background in submodular function minimization needed for our main result. We then develop the approximation algorithm for the partial set cover setting. We refer the reader to Theorem 3.10 for a formal statement on the geometric problem, and the reduction to the more general setting.

#### Candid assessment & contribution.

This paper provides a rather straightforward and elegant111At least in the eyes of the authors. reduction of the problem studied to the submodular greedy algorithm. As a result, we obtain a approximation algorithm on an instance with point sets with a points overall.

In hindsight this reduction looks trivial222But then what doesn’t?, but it took the authors quite a bit of time to arrive to it. The two main contributions of this paper are (i) presenting these reductions, and more significantly (ii) introducing these family of new problems – in particular, we believe that breaking the bound should be possible for the simplest geometric settings, and we leave this as open problem for further research.

## 2 Background

#### Notations.

For a set , and an element , we denote , and . A set system is a pair , with . We refer to a set as an edge333The literature sometimes refers to such set systems as hypergraphs, and their edges as hyperedges, but this is all too hyper for us..

### 2.1 Background: Submodular minimization

For the sake of completeness, we present the analysis of the greedy algorithm for minimizing an integer valued submodular function. In this case, the task is to compute the smallest set that provides the same utility as using all the sets available.

Let be a given set system, and assume we have a monotone function . Here a function is monotone if implies that . We also assume that is submodular, that is for any , we have that

 ∀Z⊆Y⊆X∖{e}ΔZ(e)=f(Z+e)−f(Z)≥f(Y+e)−f(Y)=ΔY(e).

Consider the greedy algorithm that starts with an empty solution . In the th iteration, the algorithm picks the element that increases the value of the most, and sets . The algorithm stops when .

###### Theorem 2.1 ([Wol82]).

Given a set system , and a monotone submodular function , the greedy algorithm outputs a solution with sets, where is the size of the smallest set such that .

###### Proof:

Let be the optimal solution. Consider a current solution in iteration , and observe that As such, we have . Let be the deficiency of . For , let As such, let . We have that

 k∑j=1δj=f(Ci∪K)−f(Ci)=fmax−f(Ci)=Δi.

As such, there is an index , such that . Now, by submodularity, we have that

 f(Ci+oj)−f(Ci)≥f(Xj−1+oj)−f(Xj−1)=δj≥Δi/k.

However, the greedy algorithm adds an element that maximizes the value of , which is at least . Put differently, the added element decreases the deficiency of the current solution by a factor of at most . Therefore the deficiency in the end of the th iteration is at most This quantity is zero for .

## 3 Problems statements and reductions

### 3.1 Pcms: Partial cover for multiple sets

###### Problem 3.2 (Pcms).

The input is a set system , and a collection of ground sets, where the universe is of size . In addition, each ground has a demand denoted by , which is a non-negative integer. A valid solution for such an instance, is a collection , such that covers at least elements of , for .

###### Lemma 3.3.

Given an instance of partial cover of multiple sets (PCMS), where , is a family of ground sets, and is a family of subsets. Furthermore, each ground set of has an associated demand. Then, the greedy algorithm computes, in polynomial time, a approximation to the minimal size set that meets all the demands of the ground sets.

###### Proof:

Consider a partial solution . The service of to is the number of elements of the union of the edges of cover, formally where . Observe that , is clearly monotone, and its maximal value is . As for submodularity, consider sets , and an edge , and note that as potentially covers more new elements of when added to a smaller cover. For the given PCMS instance, for a given solution , the target function is

 f(Z)=m∑i=1fi(Z),

which is a sum of submodular functions. As such, is submodular itself. Observe that , and using the algorithm of Theorem 2.1 implies the result.

It is worth noting that one can also obtain a approximation for Problem 3.2 via LP rounding [KY05], which is particularly useful when is much smaller than . However, this does not change our final result, since the number of ground sets in our reduction is polynomial in (see Lemma 3.7).

### 3.2 Cutting a set into smaller pieces

We are given a set-system , where . A set of edges, induces a natural partition of , where two elements are in the same set of the partition if and only if and belongs to same set of edges in . Formally, , where, abusing notations, . The partition of induced by (i.e., the equivalence classes of ) is the arrangement of , denoted by . A set of is a face of . For an element , the face of that contains is denoted by .

###### Example 3.4.

For , and , we have .

###### Problem 3.5 (Reduce by half).

Given the above, find a minimal size set , such that every set in the partition is of size at most , where .

###### Problem 3.6 (Ptd: Partition to demand).

Given a set system , where , and an integral demand for each , compute a minimal size set , such that for every , we have .

Observe that the reduce by half question can be reduced to PTD immediately, by setting the demand of every vertex in the ground set to .

###### Lemma 3.7.

Given an instance of PTD, a greedy algorithm provides approximation to the optimal solution, where .

###### Proof:

Consider the complete graph , where . For every element , consider the associated cut . A set cuts if . In particular, let be the set of edges of that cuts.

Now, a set of edges meets the demand of , if the sets of cuts at least edges of . Put differently, the partial cover covers at least edges of . Thus, let be the universe set, and be the set of ground sets. Here a ground set has demand . The family of allowable sets to be used in the cover is

The triple is an instance of PCMS, and the greedy algorithm yields a approximation in this case, where , by Lemma 3.3.

### 3.3 Cutting a Ham-Sandwich into small pieces

###### Problem 3.8 (Rmc: Reduce measures via cuts).

The input is a triplet with . Here is a collection of ground sets that are not necessarily disjoint, and is a collection of edges. For every ground set , there is an associated target size . The problem is to compute a minimal set , such that, for all , we have

 ∀ψ∈A(K)|ψ∩Gi|≤μi.

The idea is to reduce the problem into “parallel” instances of PTD. The target function is the sum of the respective target functions to each instance, and is thus submodular and can be applied into the greedy algorithm.

###### Lemma 3.9.

Given an instance of RMC with and , one can compute, in polynomial time, a approximation to the smallest that satisfies the given instance.

###### Proof:

For a set , and an element , let if , and otherwise . The pair with the demand function form an instance of PTD (Problem 3.6), and its approximation algorithm Lemma 3.7 has an associated submodular function , that is non-negative, monotone, submodular and has maximum value .

Consider the submodular function , and let . Clearly, is submodular, monotone, and has maximum value . Furthermore, a subset such that is a valid solution to the given instance. As such, one can plug this into the algorithm of Theorem 2.1 and get the desired approximation.

###### Theorem 3.10.

Let be (not necessarily disjoint) point sets in , where . For each point set , we are given a constant . We would like to find a minimal set of hyperplanes such that for every face in the arrangement of , for all . One can -approximate, in time, the optimal solution.

###### Proof:

The reduction is straightforward and uses Lemma 3.9. Let the shared ground set be . Let be the set of ground sets. Finally, let be the (finite) number of combinatorially different hyperplanes. For each , we add the set to our collection of subsets . The values remain unchanged. This forms an instance of Problem 3.8, and thus we can apply Lemma 3.9 to obtain the desired separating hyperplanes.

As for the running time, computing the set system takes time by brute force. Indeed, unraveling the above reduction, the shared ground set is made of pairs of points of . Every point has up to different sets of such pairs that needs to be partially covered. Fortunately, there are only edges in the resulting set system. Evaluating the contribution of a new edge (in the set system) to the target function takes time. Since there edges in set system, it follows that evaluating all edges takes time. Finally, it is easy to verify that the algorithm performs at most iterations.

###### Remark.

No effort was made to improve the running time of the algorithm of Theorem 3.10.

## References

• [AMS13] P. K. Agarwal, J. Matoušek, and M. Sharir. On range searching with semialgebraic sets. II. SIAM Journal on Computing, 42(6):2039–2062, 2013.
• [BHJ08] I. Bárány, A. Hubard, and J. Jerónimo. Slicing convex sets and measures by a hyperplane. Discrete Comput. Geom., 39(1-3):67–75, 2008.
• [HJ18] S. Har-Peled and M. Jones. On separating points by lines. In Artur Czumaj, editor, Proc. 29th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 918–932. SIAM, 2018.
• [IV18] T. Inamdar and K. R. Varadarajan. On partial covering for geometric set systems. In Bettina Speckmann and Csaba D. Tóth, editors, Proc. 34th Int. Annu. Sympos. Comput. Geom. (SoCG), volume 99 of LIPIcs, pages 47:1–47:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2018.
• [KMS12] H. Kaplan, J. Matoušek, and M. Sharir. Simple proofs of classical theorems in discrete geometry via the guth-katz polynomial partitioning technique. Discrete Comput. Geom., 48(3):499–517, 2012.
• [KY05] S. G. Kolliopoulos and N. E. Young. Approximation algorithms for covering/packing integer programs. Journal of Computer and System Sciences, 71(4):495–505, 2005.
• [LMS94] C. Lo, J. Matoušek, and W. L. Steiger. Algorithms for ham-sandwich cuts. Discrete Comput. Geom., 11:433–452, 1994.
• [Mat94] J. Matoušek. Geometric range searching. ACM Comput. Surv., 26(4):421–461, 1994.
• [Ram96] E. A. Ramos. Equipartition of mass distributions by hyperplanes. Discrete Comput. Geom., 15(2):147–167, 1996.
• [ST42] A. H. Stone and J. W. Tukey. Generalized “sandwich” theorems. Duke Math. J., 9(2):356–359, 1942.
• [SZ10] W. Steiger and J. Zhao. Generalized ham-sandwich cuts. Discrete Comput. Geom., 44(3):535–545, 2010.
• [Wol82] L. A. Wolsey. An analysis of the greedy algorithm for the submodular set covering problem. Combinatorica, 2(4):385–393, 1982.


text:
# Why are derived features used in neural networks?

For example, one wants to predict house prices and have two input features the length and width of the house. Sometimes, one also includes ‘derived’ polynomial input features, such as area, which is length * width.

1) What is the point of including derived features? Shouldn’t a neural network learn the connection between length, width and price during the training? Why isn’t the third feature, area, redundant?

In addition, sometimes I also see that people run genetic selection algorithms on the input features in order to reduce their number.

2) What is the point of reducing the input features if they all contain useful information? Shouldn’t the neural network assign appropriate weights to each input feature according to its importance? What is the point of running genetic selection algorithms?

1): Including derived features is a way to inject expert knowledge into the training process, and so to accelerate it. For example, I work with physicists a lot in my research. When I’m building an optimization model, they’ll give me 3 or 4 parameters, but they usually also know certain forms that are supposed to appear in the equation. For example, I might get variables $n$ and $l$, but the expert knows that $n*l$ is important. By including it as a feature, I save the model the extra effort of finding out that $n*l$ is important. Granted, sometimes domain experts are wrong, but in my experience, they usually know what they’re talking about.