Unilateral visuospatial neglect and neglect dyslexia are neuropsychological syndromes in which patients exhibit consistently lateralised perceptual deficits. However, there is little agreement surrounding whether neglect dyslexia is best understood as a consequence of a domain-general visuospatial neglect impairment or as an independent, content-specific cognitive deficit. Previous case studies have revealed that neglect dyslexia is an exceptionally heterogeneous condition and have strongly suggested that not all neglect dyslexia patient error patterns can be fully explained as a consequence of domain-general visuospatial neglect impairment. Additionally, theoretical models which attempt to explain neglect dyslexia as a consequence of domain-general unilateral visuospatial neglect fail to account for neglect dyslexia errors which occur when reading vertically presented words, lack of neglect errors when reading number strings, and neglect dyslexia which co-occurs with oppositely lateralised domain-general visuospatial neglect. Cumulatively, these shortcomings reveal that neglect dyslexia cannot always be accurately characterised as a side-effect of domain-general visuospatial unilateral neglect deficits. These findings strongly imply that neglect dyslexia may be better understood as a content-specific impairment.
Citation: Margaret Jane Moore, Nele Demeyere. Neglect Dyslexia in Relation to Unilateral Visuospatial Neglect: A Review[J]. AIMS Neuroscience, 2017, 4(4): 148-168. doi: 10.3934/Neuroscience.2017.4.148
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Unilateral visuospatial neglect and neglect dyslexia are neuropsychological syndromes in which patients exhibit consistently lateralised perceptual deficits. However, there is little agreement surrounding whether neglect dyslexia is best understood as a consequence of a domain-general visuospatial neglect impairment or as an independent, content-specific cognitive deficit. Previous case studies have revealed that neglect dyslexia is an exceptionally heterogeneous condition and have strongly suggested that not all neglect dyslexia patient error patterns can be fully explained as a consequence of domain-general visuospatial neglect impairment. Additionally, theoretical models which attempt to explain neglect dyslexia as a consequence of domain-general unilateral visuospatial neglect fail to account for neglect dyslexia errors which occur when reading vertically presented words, lack of neglect errors when reading number strings, and neglect dyslexia which co-occurs with oppositely lateralised domain-general visuospatial neglect. Cumulatively, these shortcomings reveal that neglect dyslexia cannot always be accurately characterised as a side-effect of domain-general visuospatial unilateral neglect deficits. These findings strongly imply that neglect dyslexia may be better understood as a content-specific impairment.
In the last several decades, the kinetic theory of polyatomic gases witnessed extensive interest due to its vigorous relation with a wide range of practical applications including spacecraft flights, hypersonic flights and aerodynamics [1], plasma physics [20], thermal sciences [13,23], combustion processes, and chemical reactors. In the context of polyatomic gases, Borgnakke and Larsen proposed a microscopic model [6]. Later on, an entropic kinetic model consistent with [6] has been derived [8]. This model originates from the Boltzmann equation, which was a breakthrough in the kinetic theory, and offered an accurate description of the gas flow.
However, it is usually expensive and cumbersome to solve the Boltzmann equation directly. As an alternative to the Boltzmann equation, kinetic theory provides macroscopic models for not too large Knudsen numbers. These models are derived as approximations to the Boltzmann equation and offer high computational speed and explicit equations for macroscopic variables, which are helpful for understanding and analyzing the flow behavior. Macroscopic models are classically obtained by Chapman-Enskog method [5] and moments method [22,18]. Using the Chapman-Enskog method, Nagnibeda and Kustova [19] studied the strong vibrational nonequilibrium in diatomic gases and reacting mixture of polyatomic gases, and derived the first-order distribution function and governing equations. Cai and Li [10] extended the NRxx model to polyatomic gases using the ES-BGK model of [2] and [9]. In [24], the existence result of the ES-BGK model was achieved in the case where the solution lies close to equilibrium.
Simplified Boltzmann models for mixtures of polyatomic gases have also been proposed in [3,12]. The authors of [4] developed a generalized macroscopic 14 field theory for the polyatomic gases, based on the methods of extended thermodynamics [18]. In the full non-linear Boltzmann equation, Gamba and Pavić-Čolić [15] established existence and uniqueness theory in the space homogeneous setting.
The relation of the kinetic theory with the spectral theory was initiated by Grad [17], who was behind the history of serious investigation of the spectral properties of the linearized Boltzmann operator for monoatomic gases. With his pioneering work, Grad showed that the linearized collision operator
In fact, diatomic gases gain a solid importance due to the fact that in the upper atmosphere of the earth, the diatomic molecules Oxygen (
The plan of the document is the following: In section 2, we give a brief recall on the collision model [8], which describes the microscopic state diatomic gases. In section 3, we define the linearized operator
For the sake of clarity, we present the model in [8] on which our work is mainly based. We start with physical conservation equations and proceed as follows.
Without loss of generality, we first assume that the particle mass equals unity, and we denote as usual by
v+v∗=v′+v′∗ | (1) |
12v2+12v2∗+I+I∗=12v′2+12v′2∗+I′+I′∗. | (2) |
From the above equations, we can deduce the following equation representing the conservation of total energy in the center of mass reference frame:
14(v−v∗)2+I+I∗=14(v′−v′∗)2+I′+I′∗=E, |
with
14(v′−v′∗)2=REI′+I′∗=(1−R)E, |
and
I′=r(1−R)EI′∗=(1−r)(1−R)E. |
Using the above equations, we can express the post-collisional velocities in terms of the other quantities by the following
v′≡v′(v,v∗,I,I∗,ω,R)=v+v∗2+√RETω[v−v∗|v−v∗|]v′∗≡v′∗(v,v∗,I,I∗,ω,R)=v+v∗2−√RETω[v−v∗|v−v∗|], |
where
14(v−v∗)2=R′EI+I∗=(1−R′)E, |
and
I=r′(1−R′)EI∗=(1−r′)(1−R′)E. |
Finally, the post-collisional energies can be given in terms of the pre-collisional energies by the following relation
I′=r(1−R)r′(1−R′)II′∗=(1−r)(1−R)(1−r′)(1−R′)I∗. |
The Boltzmann equation for an interacting single polyatomic gas reads
∂tf+v.∇xf=Q(f,f), | (3) |
where
Q(f,f)(v,I)=∫R3×R+×S2×(0,1)2(f′f′∗(I′I′∗)α−ff∗(II∗)α)×B×(r(1−r))α(1−R)2α×IαIα∗(1−R)R1/2dRdrdωdI∗dv∗, | (4) |
where we use the standard notations
Q(f,f)(v,I)=∫R3×R+×S2×(0,1)2(f′f′∗−ff∗)×B×(1−R)R1/2dRdrdωdI∗dv∗, | (5) |
The function
B(v,v∗,I,I∗,r,R,ω)=B(v∗,v,I∗,I,1−r,R,−ω),B(v,v∗,I,I∗,r,R,ω)=B(v′,v′∗,I′,I′∗,r′,R′,ω). | (6) |
Main assumptions on
Together with the above assumption (6), we assume the following boundedness assumptions on the collision cross section
C1φ(R)ψ(r)|ω.(v−v∗)|v−v∗||(|v−v∗|γ+Iγ2+Iγ2∗)≤B(v,v∗,I,I∗,r,R,ω), | (7) |
and
B(v,v∗,I,I∗,r,R,ω)≤C2φ˜α(R)ψ˜β(r)(|v−v∗|γ+Iγ2+Iγ2∗), | (8) |
where for any
ψp(r)=(r(1−r))p,and φp(R)=(1−R)p. |
In addition,
φ(R)≤φ˜α(R),and ψ(r)≤ψ˜β(r), | (9) |
and
We remark that the above assumptions (7) and (8) are compatible with Maxwell molecules, hard spheres and hard potentials in the monoatomic case.
We state first the H-theorem for diatomic gases which was initially established for polyatomic gases in [8]. Namely, suppose that the positivity assumption of
D(f)=∫R3∫R+Q(f,f)logfdIdv≤0, |
and the following are equivalent
1. The collision operator
2. The entropy production vanishes, i.e.
3. There exists
f(v,I)=n(2πkT)32kTe−1kT(12(v−u)2+I), | (10) |
where
Mn,u,T(v,I)=n(2πκT)32kTe−1κT(12(v−u)2+I), | (11) |
where
n=∫R3∫R+fdIdv,nu=∫R3∫R+vfdIdv,52nT=∫R3∫R+((v−u)22+I)fdIdv. |
Without loss of generality, we will consider in the sequel a normalized version
M(v,I)=M1,0,1(v,I)=1(2π)32e−12v2−I. |
We look for a solution
f(v,I)=M(v,I)+M12(v,I)g(v,I). | (12) |
The linearization of the Boltzmann operator (5) around
Lg=M−12[Q(M,M12g)+Q(M12g,M)], |
In particular,
Lg=M−12∫Δ[M′M′12∗g′∗−MM12∗g∗+M′12M′∗g′−M12M∗g]B(1−R)R1/2drdRdωdI∗dv∗. | (13) |
Thanks to the conservation of total energy (2) we have
L(g)=−∫ΔBM12M12∗g∗(1−R)R1/2drdRdωdI∗dv∗−∫ΔBM∗g(1−R)R1/2drdRdωdI∗dv∗+∫ΔBM12∗M′12g′∗(1−R)R1/2drdRdωdI∗dv∗+∫ΔBM12∗M′12∗g′(1−R)R1/2drdRdωdI∗dv∗. |
Here,
L=K−νId, |
where
Kg=∫ΔBM12∗M′12g′∗(1−R)R1/2drdRdωdI∗dv∗+∫ΔBM12∗M′12∗g′(1−R)R1/2drdRdωdI∗dv∗−∫ΔBM12M12∗g∗(1−R)R1/2drdRdωdI∗dv∗, | (14) |
and
ν(v,I)=∫ΔBM∗(1−R)R1/2drdRdωdI∗dv∗, | (15) |
which represents the collision frequency. We write also
K1=∫ΔBM12M12∗g∗(1−R)R1/2drdRdωdI∗dv∗, | (16) |
K2=∫ΔBM12∗M′12g′∗(1−R)R1/2drdRdωdI∗dv∗, | (17) |
and
K3=∫ΔBM12∗M′12∗g′(1−R)R1/2drdRdωdI∗dv∗. | (18) |
The linearized operator
kerL=M1/2span {1,vi,12v2+I}i=1,⋯,3. |
Since
Dom(ν Id)={g∈L2(R3×R+):νg∈L2(R3×R+)}, |
then
We give now the main result on the linearized Boltzmann operator based on the assumptions of the collision cross section (8) and (7). In particular, using (7) we prove that the multiplication operator by
We state the following theorem, which is the main result of the paper.
Theorem 4.1. The operator
We carry out the proof of the coercivity of
Proof. Throughout the proof, we prove the compactness of each
Compactness of
k1(v,I,v∗,I∗)=1(2π)32∫S2×(0,1)2Be−14v2∗−14v2−12I∗−12I(1−R)R1/2drdRdω, |
and therefore
K1g(v,I)=∫R3×R+g(v∗,I∗)k1(v,I,v∗,I∗)dI∗dv∗∀(v,I)∈R3×R+. |
If
Lemma 4.2. With the assumption (8) on
Proof. Applying Cauchy-Schwarz we get
||k1||2L2≤c∫R3∫R+∫R3∫R+(Iγ+Iγ∗+|v−v∗|2γ)e−12v2∗−12v2−I∗−IdIdvdI∗dv∗≤c∫R3e−12v2∗[∫|v−v∗|≤1e−12v2dv+∫|v−v∗|≥1|v−v∗|⌈2γ⌉e−12v2dv]dv∗≤c∫R3e−12v2∗[∫|v−v∗|≥1⌈2γ⌉∑k=0|v|k|v∗|⌈2γ⌉−ke−12v2dv]dv∗≤c⌈2γ⌉∑k=0∫R3|v∗|⌈2γ⌉−ke−12v2∗[∫R3|v|ke−12v2dv]dv∗<∞, |
where
This implies that
Compactness of
Lemma 4.3. Let
σ=Tω(v−v∗|v−v∗|)=v−v∗|v−v∗|−2v−v∗|v−v∗|.ωω, | (19) |
then the Jacobian of the
dω=dσ2|σ−v−v∗|v−v∗||. |
Proof. It's enough to assume that
dσω:R3⟼R3→ω⟶→σ=−2⟨v−v∗|v−v∗|,→ω⟩ω−2⟨v−v∗|v−v∗|,ω⟩→ω. | (20) |
Let
Gram=|→σ1|2|→σ2|2−⟨→σ1,→σ2⟩2, |
where
|→σ1|2=4(⟨v−v∗|v−v∗|,→ω1⟩2+⟨v−v∗|v−v∗|,ω⟩2)=4|v−v∗|v−v∗||2=4,|→σ2|2=4(⟨v−v∗|v−v∗|,→ω2⟩2+⟨v−v∗|v−v∗|,ω⟩2)=4⟨v−v∗|v−v∗|,ω⟩2, |
and
⟨σ1,σ2⟩=0. |
As a result,
Gram=16⟨v−v∗|v−v∗|,ω⟩2=4|σ−v−v∗|v−v∗||2. |
We thus write
K2g(v,I)=∫Δe−I∗2−12r(1−R)((v−v∗)24+I+I∗)−14v2∗−14(v+v∗2+√R(14(v−v∗)2+I+I∗)σ)2×g(v+v∗2−√R(14(v−v∗)2+I+I∗)σ,(1−R)(1−r)[14(v−v∗)2+I+I∗])1(2π)32(1−R)R12B|σ−v−v∗|v−v∗||−1drdRdσdI∗dv∗. | (21) |
We seek first to write
h:R3×R+⟼h(R3×R+)⊂R3×R+(v∗,I∗)⟼(x,y)=(v+v∗2−√R(14(v−v∗)2+I+I∗)σ,(1−R)(1−r)[14(v−v∗)2+I+I∗]), |
for fixed
v∗=2x+2√Rayσ−v,I∗=ay−I−(x−v+√Rayσ)2, |
and
v′=x+2√Rayσ,I′=r1−ry, |
where
J=|∂v∗∂I∗∂x∂y|=8(1−r)(1−R), |
and the positivity of
Hv,IR,r,σ=h(R3×R+)={(x,y)∈R3×R+:ay−I−(x−v+√Rayσ)2>0}. | (22) |
In fact,
Hv,IR,r,σ={(x,y)∈R3×R+:x∈Bv−√Rayσ(√ay−I) and y∈((1−r)(1−R)I,+∞)}. |
Therefore, equation (
K2g=1(2π)32∫(0,1)2×S2∫Hv,IR,r,σ(1−R)R12JB|σ−v−x−√Rayσ|v−x−√Rayσ||−1g(x,y)×e−ay−I−(x−v+√Rayσ)22−r2(1−r)y−14(2x+2√Rayσ−v)2−14(x+2√Rayσ)2dydxdσdrdR. | (23) |
We now point out the kernel form of
Hv,I:={(y,x,σ,r,R)∈Δ:R∈(0,1),r∈(0,1),σ∈S2,x∈Bv−√Rayσ(√ay−I), and y∈((1−r)(1−R)I,+∞)}. |
We remark that
Hv,I=Hv,Ix,y×R3×R+ which is equivalent to Hv,I=(0,1)×(0,1)×S2×Hv,IR,r,σ. |
In other words,
Hv,Ix,y={(r,R,σ)∈(0,1)×(0,1)×S2:(y,x,σ,r,R)∈Hv,I}. | (24) |
Then by Fubini theorem, it holds that
K2g(v,I)=1(2π)32∫Hv,I(1−R)R12JB|σ−v−x−√Rayσ|v−x−√Rayσ||−1g(x,y)×e−ay−I−(x−v+√Rayσ)22−r2(1−r)y−14(2x+2√Rayσ−v)2−14(x+2√Rayσ)2dydxdσdrdR=1(2π)32∫R3×R+∫Hv,Ix,y(1−R)R12JB|σ−v−x−√Rayσ|v−x−√Rayσ||−1g(x,y)×e−ay−I−(x−v+√Rayσ)22−r2(1−r)y−14(2x+2√Rayσ−v)2−14(x+2√Rayσ)2dσdrdRdydx. | (25) |
The kernel of
Lemma 4.4. With the assumption (8) on
k2(v,I,x,y)=1(2π)32∫Hv,Ix,y(1−R)R12JB|σ−v−x−√Rayσ|v−x−√Rayσ||−1×e−ay−I−(x−v+√Rayσ)22−r2(1−r)y−14(2x+2√Rayσ−v)2−14(x+2√Rayσ)2dσdrdR |
is in
Proof. Rewriting
‖k2‖2L2≤c∫R3∫R+∫R3∫R+∫(0,1)2×S2(1−R)2RJ2B2×e−[ay−I−(x−v+√RayTω(v−v∗|v−v∗|))2]−r(1−r)y−12(2x+2√RayTω(v−v∗|v−v∗|)−v)2e−12(x+2√RayTω(v−v∗|v−v∗|))2dωdrdRdydxdIdv. |
Writing back in
‖k2‖2L2≤c∫R3∫R+∫R3∫R+∫(0,1)2×S2e−I∗−12v2∗−r(1−R)((v−v∗)24+I)(1−R)2RJB2(v,v∗,I,I∗,r,R,ω)dωdrdRdI∗dv∗dIdv. |
Assumption (8) on
‖k2‖2L2≤c∫(0,1)2∫R3∫R+∫R3∫R+(1−R)2RJ(|v−v∗|2γ+Iγ+Iγ∗)(r(1−r))2˜β(1−R)2˜α×e−I∗−12v2∗−r(1−R)((v−v∗)24+I)dIdvdI∗dv∗drdR≤c∫(0,1)2r2˜β−52−γ(1−r)2˜β−1R(1−R)2˜α−32−γdrdR<∞. |
with
Remark 1. For any
∫R3∫R+∫R3∫R+IaIb∗|v−v∗|ce−I∗−12v2∗−r(1−R)(v−v∗)24−r(1−R)IdIdvdI∗dv∗≤C(∫R+Iae−r(1−R)IdI)(∫R3[∫R3|v−v∗|ce−r(1−R)(v−v∗)24dv]e−12v2∗dv∗)≤C[r(1−R)]−a−1[r(1−R)]−c+32, |
for some constant
The lemma is thus proved, which implies that
Compactness of
K3g(v,I)=∫Δe−I∗2−12(1−r)(1−R)((v−v∗)24+I+I∗)e−14v2∗−14(v+v∗2−√R(14(v−v∗)2+I+I∗)σ)2g(v+v∗2+√R(14(v−v∗)2+I+I∗)σ,r(1−R)[14(v−v∗)2+I+I∗])1(2π)32R12(1−R)B|σ−v−v∗|v−v∗||−1drdRdσdI∗dv∗, |
inherits the same form as
˜h:R3×R+⟼R3×R+(v∗,I∗)⟼(x,y)=(v+v∗2+√R(14(v−v∗)2+I+I∗)σ,r(1−R)[14(v−v∗)2+I+I∗]), |
is calculated to be
˜J=8r(1−R). |
The final requirement for the kernel of
∫(0,1)2(1−r)2˜β−52−γr2˜β−1R(1−R)2˜α−32−γdrdR<∞, |
which holds by the change of variable
To this extent, the perturbation operator
We give in this section some properties of
Proposition 1 (Coercivity of
ν(v,I)≥c(|v|γ+Iγ/2+1), |
for any
Proof. The collision frequency (15) is
ν(v,I)=∫ΔBe−I∗−12v2∗drdRdωdI∗dv∗, |
where by
ν(v,I)≥c∫S2∫R3(|v−v∗|γ+Iγ/2)e−12v2∗dωdv∗≥c(Iγ/2+∫R3||v|−|v∗||γe−12v2∗dv∗), |
where
ν(v,I)≥c(Iγ/2+∫|v∗|≤12|v|(|v|−|v∗|)γe−12v2∗dv∗)≥c(Iγ/2+|v|γ∫|v∗|≤12e−12v2∗dv∗)≥c(|v|γ+Iγ/2+1). |
For
ν(v,I)≥c(Iγ/2+∫|v∗|≥2(|v∗|−|v|)γe−12v2∗dv∗)≥c(Iγ/2+∫|v∗|≥2e−12v2∗dv∗)≥c(1+Iγ/2+|v|γ). |
The result is thus proved. We give now the following proposition, which is a generalization of the work of Grad [17], in which he proved that the collision frequency of monoatomic single gases is monotonic based on the choice of the collision cross section
Proposition 2 (monotony of
∫(0,1)2×S2(1−R)R12B(|V|,I,I∗,r,R,ω)drdRdω | (26) |
is increasing (respectively decreasing) in
In particular, for Maxwell molecules, where
B(v,v∗,I,I∗,r,R,ω)=Cφ(r)ψ(R)(|v−v∗|γ+Iγ/2+Iγ/2∗), |
the integral (26) is increasing, and thus
In fact, if
Proof. We remark first that
ν(|v|,I)=1(2π)32∫Δ(1−R)R12B(|V|,I,I∗,r,R,ω)e−12(v−V)2−I∗drdRdωdI∗dV, | (27) |
where
The partial derivative of
∂ν∂vi=1(2π)32∫(1−R)R12vi−v∗i|v−v∗|∂B∂|v−v∗|(|v−v∗|,I,I∗,r,R,ω)e−12v2∗−I∗drdRdωdI∗dv∗. | (28) |
Perform the change of variable
∂ν∂vi=1(2π)32∫(1−R)R12Vi|V|∂B∂|V|(|V|,I,I∗,r,R,ω)e−12(v−V)2−I∗drdRdωdI∗dV, |
and thus,
3∑i=1vi∂ν∂vi=1(2π)32∫(1−R)R12v.V|V|∂B∂|V|(|V|,I,I∗,r,R,ω) | (29) |
e−12(v−V)2−I∗drdRdωdI∗dV. | (30) |
Applying Fubini theorem, we write (29) as
3∑i=1vi∂ν∂vi=1(2π)32∫[∫(1−R)R12∂B∂|V|(|V|,I,I∗,r,R,ω)drdRdω]v.V|V| | (31) |
e−12(v−V)2−I∗dI∗dV. | (32) |
The partial derivative of
I∂ν∂I=1(2π)32∫(1−R)R12I∂B∂I(|V|,I,I∗,r,R,ω)e−12(v−V)2−I∗drdRdωdI∗dV=1(2π)32I∫[∫(1−R)R12∂B∂I(|V|,I,I∗,r,R,ω)drdRdω]e−12(v−V)2−I∗dI∗dV. | (33) |
When
∫(1−R)R12∂B∂|V|(|V|,I,I∗,r,R,ω)drdRdω. |
It's clear as well that the partial derivative of
∫(1−R)R12∂B∂I(|V|,I,I∗,r,R,ω)drdRdω. |
As a result, for a collision cross-section
∫(0,1)2×S2(1−R)R12B(|V|,I,I∗,r,R,ω)drdRdω |
is increasing (respectively decreasing) in
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