Houssem Haddar
email: houssem.haddar@inria.fr

Jing-Rebecca Li
email: jingre-becca.li@inria.fr

Simona Schiavi
email: simona.schiavi@polytechnique.edu

UNDERSTANDING THE TIME-DEPENDENT EFFECTIVE DIFFUSION COEFFICIENT MEASURED BY DIFFUSION MRI: THE INTRA-CELLULAR CASE *

Keywords: Diffusion MRI, time-dependent ADC, homogenization, effective medium, AMS subject classifications. 35B27, 35Q99, 65M32, 65Z05

Diffusion Magnetic Resonance Imaging (dMRI) can be used to measure a timedependent effective diffusion coefficient that can in turn reveal information about the tissue geometry. Recently a mathematical model for the time-dependent effective diffusion coefficient was obtained using homogenization techniques after imposing a certain scaling relationship for the time, the biological cell membrane permeability, the diffusion-encoding magnetic field gradient strength, and a periodicity length of the cellular geometry. With this choice of the scaling of the physical parameters, the effective diffusion coefficient of the medium can be computed after solving a diffusion equation subject to a time-dependent Neumann boundary condition, independently in the biological cells and in the extra-cellular space. In this paper, we analyze this new model, which we call the H-ADC model, in the case of finite domains, which is relevant to diffusion inside biological cells. We use both the eigenfunction expansion and the single layer potential representation for the solution of the above mentioned diffusion equation to obtain analytical expressions for the effective diffusion coefficient in different diffusion time regimes. These expressions are validated using numerical simulations in two dimensions.

Introduction. Diffusion Magnetic Resonance Imaging (dMRI) encodes water displacement due to diffusion via the application of diffusion-encoding gradient pulses

and is a powerful tool to obtain information on the tissue microstructure. A major application has been in detecting acute cerebral ischemia minutes after stroke [START_REF] Moseley | Diffusion-weighted MR imaging of acute stroke: correlation with T2-weighted and magnetic susceptibility-enhanced MR imaging in cats[END_REF][START_REF] Warach | Fast magnetic resonance diffusion-weighted imaging of acute human stroke[END_REF]. DMRI has been used to detect and differentiate a wide range of physiological and pathological conditions in the brain, including tumors [START_REF] Maier | Diffusion imaging of brain tumors[END_REF][START_REF] Sugahara | Usefulness of diffusion-weighted MRI with echo-planar technique in the evaluation of cellularity in gliomas[END_REF][START_REF] Tsushima | Magnetic resonance (MR) differential diagnosis of breast tumors using apparent diffusion coefficient (ADC) on 1.5-t[END_REF] and myelination abnormalities (for a review, see [START_REF] Le Bihan | Diffusion MRI at 25: Exploring brain tissue structure and function[END_REF]). It also has been used to study brain connectivity (for a review, see [START_REF] Lazar | Mapping brain anatomical connectivity using white matter tractography[END_REF]) and in functional imaging [START_REF] Lebihan | Direct and fast detection of neuronal activation in the human brain with diffusion mri[END_REF] as well as in cardiac applications [START_REF] Chen | Regional ventricular wall thickening reflects changes in cardiac fiber and sheet structure during contraction: quantification with diffusion tensor MRI[END_REF][START_REF] Chen | Remodeling of cardiac fiber structure after infarction in rats quantified with diffusion tensor MRI[END_REF][START_REF] Rohmer | Reconstruction and visualization of fiber and sheet structure with regularized tensor diffusion MRI in the human heart[END_REF].

In particular, we are interested in an important quantity measured by dMRI called the "Apparent Diffusion Coefficient" (ADC) which is based on a measure of the mean diffusion displacement inside an imaging voxel. The mean squared displacment of spins during a diffusion time t D is defined as:

(1)

M SD(t D ) ≡ 1 x0 ρ(x 0 )dx 0 x0 x ρ(x 0 ) ((x -x 0 ) • u g ) 2 u(x, x 0 , t D )dx dx 0 .
where u(x, x 0 , t D ) is the proportion of spins starting at x 0 when t = 0 ending up at position x after a time t D , ρ(x 0 ) is the density of spins at x 0 , and u g is a unit vector in R 3 (u g is called the diffusion-encoding direction). In the same vein, we can define an effective diffusion coefficient in the direction u g by the following expression:

(2) 1 2 t D M SD(t D ).

Since the mean squared displacement in a heterogeneous medium is not necessarily linear in t D , effective diffusion coefficient typically depends on t D .

The M SD can be measured by dMRI using a sequence of magnetic field gradient pulses (called a diffusion-encoding sequence). A commonly used sequence is the Pulse Gradient Spin Echo (PGSE) experiment [START_REF] Stejskal | Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient[END_REF]. In this paper, we will focus on the PGSE time profile to simplify the presentation.

The PGSE sequence contains two rectangular pulses of the diffusion-encoding gradient magnetic field B diff = gu g • x, where g is the strength of the gradient and u g is the gradient direction. Each pulse has a duration δ, with the delay between the start of the two pulses denoted ∆, and there is also a radio-frequency (RF) pulse to affect a 180 degree spin reversal between the pulses (see Figure 1). In the ideal case, where the pulse duration is very short compared to the delay between the pulses, δ ∆, called the narrow pulse case, it is easy to relate the magnetization of spins to the diffusion propagator u(x, x 0 , t D ). Let us consider spins initially located at x 0 . After the first pulse, the complex phase of these spins is

e iδγg•x0
, where γ = 42.576 MHz/Tesla is the gyro-magnetic ratio of the water proton.

Because the gradient magnetic field is turned off after the first pulse, the spins move but the phase of the spins does not change. The phase remains e iδγg•x0 , until the application of the RF pulse resulting in a 180 degree spin reversal. After the 180 degree RF pulse, the complex phase becomes e -iδγg•x0 . Again, spins move but the phase of the spins stays the same until the application of the second pulse, after which the complex phase due to spins ending up at position x f becomes e iδγg•(x f -x0) . So the magnetization at position x and the echo time T E, which is the time that the MRI signal is acquired, some time after the end of the second pulse ( i.e. T E ≥ ∆ + δ), is:

(3) M (x, T E) ≈ x0 ρ(x 0 ) u(x, x 0 , ∆) e iδγgug•(x-x0) dx 0 ,

where we used the assumption that δ ∆. The dMRI signal S is the total water

This manuscript is for review purposes only.

proton magnetization in an imaging voxel V :

(4) S = x∈V M (x, T E) dx.
Because the diffusion displacement is usually much shorter than the size of the imaging voxel size we can ignore spins that enter and leave the voxel during the signal acquisition and thus take domain of integration in Eq. ( 4) to be R 3 . Using properties of the Fourier transform, we obtain

(5) ∂ S S0 ∂ (δγg) 2 δγg=0 ≈ M SD(∆),
in the case of the narrow pulse PGSE sequence, where S 0 is the signal at g = 0 (a derivation of this statement can be found in [START_REF] Li | Numerical simulation of diffusion MRI signals using an adaptive time-stepping method[END_REF]).

Without the narrow pulse assumption, Eq. ( 3) does not hold exactly. Rather, M (x, t) is governed by the Bloch-Torrey equation, which is a complex-valued diffusive PDE:

(6)          ∂ ∂t M (x, t) = -ιγgu g • xf (t)M (x, t) + div(D 0 (x)∇M (x, t)) in Ω j × [0, T E] D 0 ∇M • ν Γij = 0 on Γ ij × [0, T E] D 0 ∇M • ν| Γij = κ M Γij on Γ ij × [0, T E] M (•, 0) = ρ in Ω j
where Ω 0 is the extra-cellular space and each of Ω

j , j = 1, • • • , N , is a biological cell.
The vector ν is the exterior normal to the biological cells;

• Γij , i, j = 0, • • • , N , i = j,
is the jump (the limit value in compartment i minus the limit value in compartment j) on Γ ij , the interface between Ω i and Ω j ; κ is the membrane permeability coefficient;

ι is the imaginary unit. The function f (t) gives the normalized time profile of the diffusion-encoding magnetic field gradient pulses. For the classic Pulsed Gradient Spin Echo (PGSE) sequence [START_REF] Stejskal | Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient[END_REF], simplified to include only the parameters relevant to diffusion (the imaging gradients are ignored), ( 7)

f (t) =          1 t s < t ≤ t s + δ, 0 t s + δ < t ≤ t s + ∆, -1 t s + ∆ < t ≤ t s + ∆ + δ, 0 elsewhere,
where t s is the start of the first pulse and we made f (t) negative in the second pulse to include the effect of the 180 degree spin reversal between the pulses. For simplicity, since t s does not play a role in the results of this paper, we set t s → 0. For the same reason, we set T E = δ + ∆ in this paper.

We note that the Bloch-Torrey equation needs to be supplemented by additional boundary conditions on the sides of the imaging voxel. For example, periodic boundary conditions on the boundary of the voxel would be an acceptable choice.

In the case of unrestricted diffusion in a homogeneous medium with a diffusion coefficient D 0 , the integral of the solution of the Bloch-Torrey equation, in other words, the total magnetization, takes the following exponential form [START_REF] Callaghan | Diffraction-like effects in nmr diffusion studies of fluids in porous solids[END_REF][START_REF] Kenkre | Simple solutions of the Torrey-Bloch equations in the NMR study of molecular diffusion[END_REF] :

(8) S = S 0 e -D0b ,
This manuscript is for review purposes only.

with the b-value defined as:

(9) b ≡ γ 2 g 2 T E 0 F (t) 2 dt, where (10)
F (t) ≡ t 0 f (s)ds.
In particular, for the PGSE sequence, ( 11)

F (t) =          t t s < t ≤ t s + δ, δ t s + δ < t ≤ t s + ∆, ∆ + δ -t t s + ∆ < t ≤ t s + ∆ + δ, 0 elsewhere.
To adapt the definition of the effective diffusion coefficient to the non-narrow pulse case, we make the following mathematical definition:

(12) D eff ug ≡ - 1 γ 2 T E 0 F (t) 2 dt ∂ ∂g 2 ln S S 0 g=0 .
In the dMRI community, the above quantity is fitted using the experimental MRI signal at several b values and the obtained value is called the "apparent diffusion coefficient" (ADC). The ADC is widely used medical applications, for instance, ADC maps of brain have been used to identify tumours (see [START_REF] Le Bihan | Diffusion MRI at 25: Exploring brain tissue structure and function[END_REF][START_REF] Tuch | Diffusion MRI of complex neural architecture[END_REF]). The D eff ug defined in the above formula depends on the gradient direction u g and the temporal profile f (t) but not on the gradient amplitude. In this paper, with the phrase "diffusion time-dependent" we actually mean dependent on ∆ and δ.

The motivation of our work is the experimentally observed phenomenon (see [START_REF] Pyatigorskaya | Relationship between the diffusion time and the diffusion MRI signal observed at 17.2 tesla in the healthy rat brain cortex[END_REF] and the references contained there) that the ADC depends on ∆ (and δ in the nonnarrow pulse case), leading to the need to characterize the time-dependent ADC in terms of tissue-related quantities over a wide range of diffusion time regimes. The ultimate goal is of course the estimation of these tissue-related quantities from the measured dMRI signal.

In this paper, we focus on the case of finite domains, where the membrane permeability is small enough to have negligible effect on the effective diffusion coefficient, which is related to the first order moment of the dMRI signal in the b-value (Eq. 12).

We note that this does not exclude the permeability from having an effect on the higher order moments of the signal. For the case where the permeability does affect the D eff ug , the analysis is more difficult and we refer the reader to [START_REF] Cheng | Effective conductivity of periodic arrays of spheres with interfacial resistance[END_REF][START_REF] Hasselman | Effective thermal conductivity of composites with interfacial thermal barrier resistance[END_REF][START_REF] Latour | Time-dependent diffusion of water in a biological model system[END_REF][START_REF] Szafer | Theoretical model for water diffusion in tissues[END_REF][START_REF] Torquato | Effect of the interface on the properties of composite media[END_REF] for results on periodic media and to [START_REF] Burcaw | Mesoscopic structure of neuronal tracts from time-dependent diffusion[END_REF][START_REF] Fieremans | In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter[END_REF][START_REF] Novikov | Random walks with barriers[END_REF][START_REF] Novikov | Revealing mesoscopic structural universality with diffusion[END_REF][START_REF] Novikov | Effective medium theory of a diffusion-weighted signal[END_REF] on more general heterogeneous media, not necessarily periodic. Now we summarize some existing results concerning the effective diffusion coefficient for finite domains where the membrane permeability is negligible. In the short time regime, the effective diffusion coefficient is reduced from the free diffusion coefficient D 0 by the presence of the cell membranes that affects only the molecules in the adjacent layer. The thickness of this layer is of the order of the diffusion length √ D 0 t [START_REF] Kiselev | The cumulant expansion: an overarching mathematical framework for understanding diffusion nmr[END_REF], where D 0 is the bulk diffusion coefficient. Calculations in [START_REF] Mitra | Short-time behavior of the diffusion coefficient as a geometrical probe of porous media[END_REF][START_REF] Mitra | Diffusion propagator as a probe of the structure of porous media[END_REF] show

(13) D eff short (t) = D 0 1 - 4 3d √ π S V D 0 t ,
This manuscript is for review purposes only.

where d is the spatial dimension and S/V is the surface to volume ratio. This result was extended to include higher order terms accounting for permeable membranes, surface relaxation and mean curvature [START_REF] Latour | Time-dependent diffusion coefficient of fluids in porous media as a probe of surface-to-volume ratio[END_REF][START_REF] Novikov | Random walks with barriers[END_REF]. It was also shown that, in the case of anisotropic media subjected to a linear gradient with direction u g , one should replace S dV above by ∂Ω (ug•ν) 2 dx |Ω| [1,[START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF]. In the long time limit, the spins explore the whole available space of the finite domain and then their mean square displacement saturates while the effective diffusion coefficient decreases as ∆ increases. For an isolated cell of a typical size R the diffusion becomes Gaussian as was shown in [START_REF] Neuman | Spin echo of spins diffusing in a bounded medium[END_REF][START_REF] Robertson | Spin-echo decay of spins diffusing in a bounded region[END_REF]. In the case of the PGSE sequence in the narrow pulse limit one gets ( 14)

D eff long (∆) ≈ C R 2 ∆ ,
where C is a geometrical constant (for example, C = 1/4 for the reflecting cylinder and C = 1/12 for a 1D configuration [START_REF] Callaghan | Diffraction-like effects in nmr diffusion studies of fluids in porous solids[END_REF][START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF]).

Finally, an approach that is closely related to the work of this paper is the "matrix formalism" approach used to describe restricted diffusion in bounded domains [START_REF] Barzykin | Theory of spin echo in restricted geometries under a step-wise gradient pulse sequence[END_REF][START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF][START_REF] Grebenkov | Laplacian eigenfunctions in NMR. I. A numerical tool[END_REF][START_REF] Grebenkov | Laplacian eigenfunctions in NMR. II. Theoretical advances[END_REF]. There one considers the applied diffusion-encoding magnetic field as a perturbation of the Laplace operator and the magnetization is decomposed on the basis of Laplacian eigenfunctions.

In contrast to the "matrix formalism" approach, the homogeneous model [START_REF] Haddar | A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity[END_REF] which we call the H-ADC model and which is the focus of this paper, was derived using a certain scaling of the membrane permeability with respect to other physical parameters and thus is not limited to impermeable domains. Our derivation of the H-ADC model justifies neglecting the membrane permeability for the choice of scaling that we have made. In addition, since we have formulated the time-dependent effective diffusion coefficient as the solution of a diffusion equation rather than directly in the eigenfunction basis, we have the freedom to analyze the solution of the resulting diffusion equation using both the eigenfunction representation as well as the layer potential representation according the relevant time regime under consideration.

Finally, we note that we preferred the term "apparent" to the term "effective" in naming the H-ADC model due to the more common usage of the term ADC in the MRI community.

This paper is organized as follows. In Section 2 we describe the H-ADC model derived in [START_REF] Haddar | A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity[END_REF]. In Section 3 we represent the solution of the relevant diffusion equation of the H-ADC model using the eigenfunction basis as well as by single layer potentials and discuss the regime where each representation is advantageous. In Section 4 we provide formulas for the effective diffusion coefficient that is averaged over diffusion-encoding gradient directions that are uniformly distributed in the unit sphere. In Section 5 we validate our analytical results with numerical simulations on two-dimensional geometries. Section 6 contains our conclusions.

Effective diffusion coefficient in finite domains.

In a previous work [START_REF] Haddar | A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity[END_REF],

we obtained an homogenized model by starting from the Bloch-Torrey equation using the following scaling relationship between the time (∆ and δ), the biological cell membrane permeability (κ), the diffusion-encoding magnetic field gradient strength (g), and a periodicity length of the cellular geometry (L):

L = O( ), κ = O( ), g = O( -2 ), {∆, δ} = O( 2 ),
where is a non-dimensional parameter. It was shown that with this choice, there is no coupling between the different geometrical compartments in the g 2 term which

This manuscript is for review purposes only.

gives rise to the effective diffusion coefficient. The total effective diffusion coefficient is the sum of the effective diffusion coefficient in each geometrical compartment weighted by its volume fraction. Thus, in this paper we are justified in considering each compartment separately.

According to [START_REF] Haddar | A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity[END_REF], with the definitions of F (t) given in [START_REF] Grebenkov | Laplacian eigenfunctions in NMR. II. Theoretical advances[END_REF], the effective diffusion coefficient in the compartment Ω can be obtained in the following way:

(15)

D eff ug = D 0 - D 0 T E 0 F (t) 2 dt T E 0 F (t) h(t) dt,
where ( 16)

h(t) = 1 |Ω| Ω u g • ∇ω(x, t)
is a quantity related to the directional gradient of a function ω that is the solution of the homogeneous diffusion equation with Neumann boundary condition and zero initial condition:

∂ ∂t ω(x, t) -∇ (D 0 ∇ω(x, t)) = 0, x ∈ Ω, D 0 ∇ω(x, t) • ν(x) = D 0 F (t)u g • ν(x), x ∈ ∂Ω, ω(x, t) = 0, x ∈ Ω, (17)
ν is the outward normal and t ∈ [0, T E]. We can see that if h(t) is close to F (t), then
D eff ug is close to 0. The above set of equations, ( 15)-( 17), comprise the homogenized model that we call the H-ADC model.

In our previous work [START_REF] Haddar | A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity[END_REF], we imposed periodic boundary conditions on the boundary of the voxel. In this paper, we are interested in analyzing ( 15)-( 17) for spatially finite compartments, which is relevant to diffusion inside biological cells. It will not be necessary to impose periodic boundary conditions on the sides of the imaging voxel if we consider only cells that do not touch the sides.

3. Solution of the model. Defining the right hand side of the Neumann boundary condition as

(18) β(y, t) := D 0 F (t) u g • ν(y),
we will use the following two equivalent expressions for h(t):

(19) h(t) = 1 |Ω| Ω u g • ∇ω(x, t)dx = 1 |Ω| Γ ω(y, t) (u g • ν(y)) ds y ,
where the second expression can be obtained by applying the divergence theorem to [START_REF] Kenkre | Simple solutions of the Torrey-Bloch equations in the NMR study of molecular diffusion[END_REF]. We observe that the first expression uses values of the gradient of ω inside the domain while the second uses the values of ω on the boundary. Each expression will have advantages depending on whether we use the eigenfunctions of the Laplace operator or layer potentials to represent ω.

Eigenfunctions representation.

Writing ω as the sum

(20) ω(x, t) = ω(x, t) + F (t) x • u g , x ∈ Ω, t ∈ [0, T E]
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where ω(x, t) satisfied the diffusion equation with a forcing term and homogeneous boundary condition:

∂ ∂t ω(x, t) -∇ (D 0 ∇ ω(x, t)) = -f (t)x • u g , x ∈ Ω, t ∈ [0, T E], (21)
D 0 ∇ ω(x, t) • ν(x) = 0, x ∈ Γ, t ∈ [0, T E], (22)
ω(x, t) = 0,

x ∈ Ω, [START_REF] Li | Numerical simulation of diffusion MRI signals using an adaptive time-stepping method[END_REF] it is well-known that ω(x, t) can be expanded in the basis of Laplace eigenfunctions.

Let φ n (x) and λ n be the L 2 normalized eigenfunctions and eigenvalues associated to the Laplace operator with homogeneous Neumann boundary conditions:

-∇D 0 (∇φ n (x)) = λ n φ n (x), x ∈Ω, D 0 ∇φ n (x) • ν(x) = 0,
x ∈Γ.

We can write ω(x, t) in the basis of the eigenfunctions as

(24) ω(x, t) = -a 0 φ 0 (x)F (t) + ∞ n=1 (-a n )φ n (x) t 0 e -D0λn(t-s) f (s)ds,
where the coefficients are

a 0 = 1 |Ω| Ω x • u g dx, a n = Ω x • u g φ n (x)dx, (25)
which coincide with the first moments of the eigenfunctions in the u g direction.

Finally, the solution to the diffusion equation is

(26) ω(x, t) = x • u g - 1 |Ω| a 0 F (t) + ∞ n=1 (-a n )φ n (x)
t 0 e -D0λn(t-s) f (s)ds and using properties of the eigenfunctions:

(27) Ω φ n (x)dx = |Ω|, n = 0 0, n ≥ 1,
and the divergence theorem:

(28) Ω λ n φ n (u g • x) dx = Ω ∇φ n (x)•u g dx- Γ D 0 ∇φ n (x)•ν u g •νds x = Ω ∇φ n (x)•u g dx we obtain (29) h(t) = F (t) + ∞ n=1 - (a n ) 2 λ n |Ω| t 0 e -D0λn(t-s) f (s)ds.
This leads to the final formula:

(30) D eff ug = ∞ n=1 (a n ) 2 D 0 λ n |Ω| T E 0 F 2 (t)dt T E 0 F (t) t 0 e -D0λn(t-s) f (s)ds dt.
This manuscript is for review purposes only.

We remark that this formula is the same as the one obtained with the matrix formalism in [START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF]. In particular, if we consider PGSE sequence, we can rewrite (15) using the contribution of each of the three intervals as (31)

D eff ug = D 0 -     D 0 A δ 0 t h(t) dt I + D 0 A ∆ δ δ h(t) dt II + D 0 A ∆+δ ∆ (∆ + δ -t) h(t) dt III     , where (32)
A = T E 0 F 2 (t)dt = δ 2 ∆ - δ 3 ,
and in the first pulse

I = D 0 δ 3 3A + 1 |Ω|A ∞ n=1 (a n ) 2 - δ 2 2 - δe -D0λnδ D 0 λ n - e -D0λnδ -1 (D 0 λ n ) 2 , ( 33
)
between the pulses

II = D 0 δ 2 (∆ -δ) A + 1 |Ω|A ∞ n=1 -δ(a n ) 2 D 0 λ n e -D0λn∆ -e -D0λn(∆-δ) -e -D0λnδ + 1 , (34)
and in the second pulse

(35) III = D 0 δ 3 3A + ∞ n=1 -(a n ) 2 D 0 λ n A|Ω| δ - δ 2 D 0 λ n 2 -δe -D0λn∆ + δe -D0λn(∆-δ) + 2e -D0λn∆ -1 + e -D0λnδ -e -D0λn(∆+δ) -e -D0λn(∆-δ) D 0 λ n .
In the end, we obtain

(36) D eff ug = ∞ n=1 -(a n ) 2 D 2 0 λ 2 n δ 2 ∆ -δ 3 |Ω| e -D0λn(∆+δ) + e -D0λn(∆-δ) -2 D 0 λ n δ + e -D0λnδ + e -D0λn∆ -1 .
In the narrow pulse case (δ ∆), we obtain

(37) D eff ug ≈ ∞ n=1 (a n ) 2 ∆ 1 -e -D0λn∆ ,
which confirms that D eff ug approach its long time limit as proportional to 1/∆ inside finite domains. In particular, for a 1D configuration of length L, a 1 = L 2 12 and for a reflecting cylinder of radius R, a 1 = R 2 4 , which confirm the results in [START_REF] Callaghan | Diffraction-like effects in nmr diffusion studies of fluids in porous solids[END_REF][START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF].

This manuscript is for review purposes only.

3.2. Layer potential representation. The solution of the diffusion equation can be also represented using layer potentials [START_REF] Guenther | Partial differential equations of mathematical physics and integral equations[END_REF]. This representation is more efficient than the eigenfunction representation at short diffusion times. Since the PDE has a

Neumann boundary condition, we choose to represent the solution ω(x, t) = S[µ](x, t)

as a single layer potential with a density µ defined on Γ,

(38) S[µ](x, t) = t 0 Γ D 0 G(x -y, t -τ )µ(y, τ )ds y dτ,
where G(x, t) is the fundamental solution of the heat equation in free space given by ( 39)

G(x, t) = (4πD 0 t) -d/2 exp -x 2 4D 0 t
and d is the space dimension. At short times, there is an unavoidable square root singulary in t in the single layer potential, therefore in what follows, we separate out the integrand in [START_REF] Szafer | Theoretical model for water diffusion in tissues[END_REF] in the following way, ( 40)

S[µ](x, t) = t 0 1 4D 0 π(t -τ ) B S [µ](x, t, τ )dτ, where (41)
B S [µ](x, t, τ ) := Γ D 0 4D 0 π(t -τ ) G(x -y, t -τ )µ(y, τ )ds y is analytic in time if µ is. The single layer potential satisfies ∂ ∂t S[µ](x, t) -∇ D 0 ∇S[µ](x, t) = 0, x ∈ Ω, t ∈ [0, T E], (42)
S[µ](x, 0) = 0, x ∈ Ω. ( 43
)
Then the density µ is chosen to be a causal function and is determined by imposing the Neumann boundary conditions:

lim x→x 0 ∈Γ D 0 ∇S[µ](x, t) • ν(x) = β(x 0 , t), x 0 ∈ Γ, t ∈ [0, T E],
where β(x 0 , t) is defined in [START_REF] Latour | Time-dependent diffusion coefficient of fluids in porous media as a probe of surface-to-volume ratio[END_REF]. Using the jump properties of the traces of double layer potentials, the integral equation to be solved for µ is then the following:

(44) D 0 2 µ(x 0 , t) + D 0 K * [µ](x 0 , t) = β(x 0 , t), x 0 ∈ Γ, t ∈ [0, T E],
where

K * [µ](x 0 , t) = t 0 1 4πD 0 (t -τ ) B K [µ](x 0 , t, τ )dτ, (45) with B K [µ](x 0 , t, τ ) ≡ Γ -2(x 0 -y) • ν(y) 4D 0 (t -τ ) D 0 4πD 0 (t -τ )G(x 0 -y, t -τ )µ(y, τ )ds y (46)
This manuscript is for review purposes only.

being analytic in time if µ is.

To compute h(t) from [START_REF] Latour | Time-dependent diffusion of water in a biological model system[END_REF] we only need to evaluate ω(x 0 , t) = S[µ](x 0 , t) on the boundary Γ. We write the density µ as the solution of (44):

(47) µ(x 0 , t) = 2 D 0 (1 + 2K * ) -1 β(x 0 , t), x 0 ∈ Γ, t ∈ [0, T E],
and expand the operator (1 + 2K * ) -1 for short time t (corresponding with 2K * being a contraction) as

(48) µ(x 0 , t) = 2 D 0 1 -2K * + 4(K * ) 2 + . . . β(x 0 , t), to obtain (49) µ(x 0 , t) = 2 D 0 β(x 0 , t) - 4 D 0 K * [β](x 0 , t) + higher order terms.
This means in particular that

S[µ](x 0 , t) = S 2 D 0 β (x 0 , t) + S - 4 D 0 K * [β] (x 0 , t)
+ higher order terms.

(50)

We will now compute the first term on the right hand side of the above equation to get an approximate expression for ω(x 0 , t) = S[µ](x 0 , t) and we will compute the second term to get an expression for the error.

We note here that asymptotic results for small values of (t -τ ) concerning B S [µ] and B K [µ] for a density µ that has a Taylor expansion in both the space and the time variables have been obtained in [START_REF] Greengard | A fast algorithm for the evaluation of heat potentials[END_REF][START_REF] Li | High order accurate methods for the evaluation of layer heat potentials[END_REF], but only in two dimensions. However, we expect that the generalization to three dimensions should be straightforward. In particular, the curvature term and higher order spatial derivatives need to be generalized to analogous quantities in three dimensions.

Here, we summarize the results derived in [START_REF] Greengard | A fast algorithm for the evaluation of heat potentials[END_REF][START_REF] Li | High order accurate methods for the evaluation of layer heat potentials[END_REF] for two dimensions. Suppose a local parametrization for Γ around x 0 of the form

Γ = {(s, y(s)) , s = -∞ • • • ∞},
having translated and rotated the x and y-axes so that the origin is at x 0 and the tangent of Γ at x 0 is parallel to the x-axis, with the end points extended to ±∞ because such an approximation facilitates the computation of the integrals and the resulting error is exponentially small. Let y(s) be oriented counter-clockwise, with the following expansion in terms of the curvilinear abscissa s :

(51)

y(s) = 1 2 γ ss s 2 + 1 6 γ sss s 3 + 1 24 γ ssss s 4 + . . . .
For the operator S, it was shown in [START_REF] Greengard | A fast algorithm for the evaluation of heat potentials[END_REF][START_REF] Li | High order accurate methods for the evaluation of layer heat potentials[END_REF] that:

B S [µ](x 0 , t, τ ) = µ(x 0 , t) + γ ss (x 0 ) 2 4 µ(x 0 , t) + µ t (x 0 , t) -µ ss (x 0 , t) (t -τ ) + O (t -τ ) 2 (52)
This manuscript is for review purposes only.

and for the operator K * it was shown that:

B K [µ](x 0 , t, τ ) = γ ss (x 0 ) 2 µ(x 0 , t) + 1 8
16γ sss (x 0 )µ s (x 0 , t) + 12γ ss (x 0 )µ ss (x 0 , t)

+ 4γ ss (x 0 )µ t (x 0 , t) + 6γ ssss (x 0 ) -15γ ss (x 0 ) 3 µ(x 0 , t) (t -τ ) + O (t -τ ) 2 .
(53) Specifically γ ss is the curvature of Γ at the point x 0 and in what follows we will indicate it as k(x 0 ). In three dimensions, it should be easy to see that the constant term in B S [µ](x 0 , t, τ ) would not change, and the constant term in B K [µ](x 0 , t, τ ) would contain spatial derivatives on a two dimensional manifold.

For the PGSE sequence, β(x 0 , t) assumes the following three expressions in the three time intervals:

(54) β(x 0 , t) = D 0 u g • ν(x 0 )      t 0 < t ≤ δ, δ δ < t ≤ ∆, ∆ + δ -t ∆ < t ≤ ∆ + δ.
First, using the definition [START_REF] Tsushima | Magnetic resonance (MR) differential diagnosis of breast tumors using apparent diffusion coefficient (ADC) on 1.5-t[END_REF] and the result (52) we obtain

S 2 D 0 β (x 0 , t) = 4(D 0 ) 1/2 3 √ π u g • ν(x 0 )      t 3/2 t 3/2 -(t -δ) 3/2 t 3/2 -(t -δ) 3/2 -(t -∆) 3/2 + O        t 5/2 if 0 < t ≤ δ, t 5/2 -(t -δ) 5/2 if δ < t ≤ ∆, t 5/2 -(t -δ) 5/2 -(t -∆) 5/2 if ∆ < t ≤ ∆ + δ. (55)
Similarly, using the definition (45) and the result (53) we obtain

(56) K * [β](x 0 , t) = α 1 (x 0 )α 2 (t) + O(α 3 (t))
where

α 1 (x 0 ) = (D 0 ) 3/2 k(x 0 ) 3 √ π u g • ν(x 0 ) , α 2 (t) =      t 3/2 if 0 < t ≤ δ, t 3/2 -(t -δ) 3/2 if δ < t ≤ ∆, t 3/2 -(t -δ) 3/2 -(t -∆) 3/2 if ∆ < t ≤ ∆ + δ, α 3 (t) =      t 5/2 if 0 < t ≤ δ, t 5/2 -(t -δ) 5/2 if δ < t ≤ ∆, t 5/2 -(t -δ) 5/2 -(t -∆) 5/2 if ∆ < t ≤ ∆ + δ. (57)
To compute S -4 D0 K * [β] (x 0 , t), we first observe that S -4

D0 K * [β] (x 0 , t) = -4 D0 S [K * [β]] (x 0 , t).
Moreover, to compute S [K * [β]] (x 0 , t) we cannot use the result (52) because α 2 and

This manuscript is for review purposes only.

α 3 do not have a Taylor expansion in t. Following the idea in [START_REF] Greengard | A fast algorithm for the evaluation of heat potentials[END_REF], we explicitly compute the lowest order terms of S[K * [β]](x 0 , t) in two dimensions by

(58) S[K * [β]](x 0 , t) = t 0 1 4πD 0 (t -τ ) B S [K * [β]](x 0 , t, τ )dτ,
where

B S [K * [β]](x 0 , t, τ ) = +∞ -∞ D 0 e - s 2 +y(s) 2 4D 0 (t-τ ) 4πD 0 (t -τ ) (α 1 (s)α 2 (τ ) + O(α 3 (τ ))) 1 + (y (s)) 2 ds. (59)
Note for simplicity, we replaced α 1 (x 0 ) by α 1 (s) to indicate the local parametrization of Γ around x 0 , as described previously.

To compute the above spatial integral we note the dominant contribution of the Gaussian e to obtain

B S [K * [β]](x 0 , t, τ ) = +∞ -∞ D 0 √ π e -r 2 e -y(rv) 2 v 2 α 1 (rv)α 2 t - v 2 4D 0 + O α 3 t - v 2 4D 0 1 + (y (rv)) 2 dr,
where v = 4D 0 (t -τ ). We would like an asymptotic expansion of the above integral in v. We note y(rv)

= 1 2 k(x 0 )r 2 v 2 + O(r 3 v 3 ) and α 1 (0) = (D0) 3/2 k(x 0 ) 3 √ π u g • ν(x 0 ) .
The order O(v) term only occurs in α 1 and we do not need to take it into account due to the anti-symmetry of e -r 2 r. So we compute the space integral to obtain

= +∞ -∞ D 0 √ π e -r 2 α 1 (0) + O(r 2 v 2 ) α 2 t - v 2 4D 0 + O α 3 t - v 2 4D 0 dr = α 2 t - v 2 4D 0 D 0 α 1 (0) + O(v 2 ) + O α 3 t - v 2 4D 0 .
We now take the above expression and put it into the time integral to get

S[K * [β]](x 0 , t): t 0 1 4πD 0 (t -τ ) (α 2 (τ ) (D 0 α 1 (0) + O(t -τ )) + O(α 3 (τ ))) dτ.
Using the following property of the beta function (the Euler integral of the first kind)

t 0 (t -τ ) w τ p dτ = t p+w+1 Γ(p + 1)Γ(w + 1) Γ(w + p + 2) ,
This manuscript is for review purposes only.

we can compute t 0 1 (t -τ ) α 2 (τ )dτ exactly:

t 0 τ 3 2 √ t -τ dτ = 3π 8 t 2 , δ 0 τ 3 2 √ t -τ dτ + t δ τ 3 2 -(τ -δ) 3 2 √ t -τ dτ = 3π 8 (t 2 -(t -δ) 2 ) δ 0 τ 3 2 √ t -τ dτ + ∆ δ τ 3 2 -(τ -δ) 3 2 √ t -τ dτ + t ∆ τ 3 2 -(τ -δ) 3 2 -(τ -∆) 3 2 √ t -τ dτ = 3π 8 (t 2 -(t -δ) 2 -(t -∆) 2 ). ( 60
)
Therefore, the dominant asymptotic terms are:

S - 4 D 0 K * [β] (x 0 , t) = - 4 D 0 D 2 0 16 k(x 0 )(u g • ν(x 0 ))      t 2 t 2 -(t -δ) 2 t 2 -(t -δ) 2 -(t -∆) 2 . + O      t 3 if 0 < t ≤ δ, t 3 -(t -δ) 3 , if δ < t ≤ ∆, t 3 -(t -δ) 3 -(t -∆) 3 if ∆ < t ≤ ∆ + δ, (61)
where we computed the error term by evaluating

t 0 1 (t -τ ) α 2 (τ )(t -τ )dτ and t 0 1 (t -τ ) α 3 (
τ )dτ , again using property of the beta function. Namely, for

t 0 1 (t -τ ) α 2 (τ )(t -τ )dτ , we have t 0 τ 3 2 √ t -τ (t -τ )dτ = π 16 t 3 , δ 0 τ 3 2 √ t -τ (t -τ )dτ + t δ τ 3 2 -(τ -δ) 3 2 √ t -τ (t -τ )dτ = π 16 (t 3 -(t -δ) 3 ) δ 0 τ 3 2 √ t -τ (t -τ )dτ + ∆ δ τ 3 2 -(τ -δ) 3 2 √ t -τ (t -τ )dτ + t ∆ τ 3 2 -(τ -δ) 3 2 -(τ -∆) 3 2 √ t -τ (t -τ )dτ = π 16 (t 3 -(t -δ) 3 -(t -∆) 3 ). ( 62
)
This manuscript is for review purposes only.

√ t -τ dτ = 5π 16 t 3 , δ 0 τ 5 2 √ t -τ dτ + t δ τ 5 2 -(τ -δ) 3 2 √ t -τ dτ = 5π 16 (t 3 -(t -δ) 3 ) δ 0 τ 5 2 √ t -τ dτ + ∆ δ τ 5 2 -(τ -δ) 5 2 √ t -τ dτ + t ∆ τ 5 2 -(τ -δ) 5 2 -(τ -∆) 5 2 √ t -τ dτ = 5π 16 (t 3 -(t -δ) 3 -(t -∆) 3 ). ( 63
)
Replacing the various expressions in (50) with the calculations we did above, we obtain the approximation with the error bound:

S[µ](x 0 , t) = 4(D 0 ) 1/2 3 √ π u g • ν(x 0 )      t 3/2 t 3/2 -(t -δ) 3/2 t 3/2 -(t -δ) 3/2 -(t -∆) 3/2 - D 0 16 k(x 0 )(u g • ν(x 0 ))      t 2 t 2 -(t -δ) 2 t 2 -(t -δ) 2 -(t -∆) 2
+ higher order terms.

(64)

Now using (64) we compute the approximate expressions of h(t) in each time-interval with the corresponding errors in time.

In the first interval, we obtain (65)

h(t) = 1 |Ω| Γ ω(x, t) (u g • ν(x)) ds x = P t 3/2 + O P err t 2 ,
where

P = 1 |Ω| Γ 4 3 √ π D 0 (u g • ν(x)) 2 ds x (66) P err = - D 0 4|Ω| Γ k(x)(u g • ν(x)) 2 ds x (67) and (68) I = D 0 δ 2 ∆ -δ 3 δ 0 t h(t) dt = 2D 0 P 7 ∆ -δ 3 δ 7/2 + O D 0 P err δ 2 4 ∆ -δ 3 .
Between the pulses, we obtain (69)

h(t) = 1 |Ω| Γ ω(x, t) (u g • ν(x)) ds x = P t 3/2 -(t -δ) 3/2 + O P err (t 2 -(t -δ) 2 )
and

II = D 0 δ 2 ∆ -δ 3 ∆ δ δ h(t) dt = - 2 5
D 0 P δ 7/2 -∆ 5/2 δ + (∆ -δ) 5/2 δ δ 2 ∆ -δ 3 + O D 0 P err ∆ 2 -δ∆ ∆ -δ 3 . ( 70
)
This manuscript is for review purposes only.

During the second pulse, we find

h(t) = 1 |Ω| Γ ω(x, t) (u g • ν(x)) ds x = P t 3/2 -(t -δ) 3/2 -(t -∆) 3/2 + O P err t 2 -(t -δ) 2 -(t -∆) 2 (71)
and

III = D 0 δ 2 ∆ -δ 3 ∆+δ ∆ (∆ + δ -t) h(t) dt = 2 35
D 0 P δ 2 ∆ -δ 3 2∆ 3 + ∆ 2 δ -8∆δ 2 + 5δ 3 √ ∆ -δ + 2 (∆ + δ) 7/2 -4∆ 7/2 -7∆ 5/2 δ -2δ 7/2 + O D 0 P err ∆δ -1 4 δ 2 ∆ -δ 3 . (72)
Finally, adding up the above expressions, we obtain that using the layer potentials representation,

D eff ug = D 0 1 - 4 35 P δ 2 ∆ -δ 3 (∆ + δ) 7/2 -2 δ 7/2 + ∆ 7/2 + (∆ -δ) 7/2 (73) + O D 0 P err ∆ 2 ∆ -δ 3 , (74)
with P and P err defined in (66) and (67), respectively.

We observe that, in the narrow pulse limit, δ ∆, the expression (73)

D eff ug = D 0 1 - 4 3 √ π D 0 ∆ Γ (u g • ν) 2 ds x |Ω| + O(D 0 P err ∆),
reduces to the formula given in [1,[START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF]. If Ω is an isotropic domain, then

(75) Γ (u g • ν) 2 ds x |Ω| = |Γ| d|Ω| ,
which is the ratio of surface to volume divided by the space dimension d, exactly the quantity contained in the formula in [START_REF] Mitra | Diffusion propagator as a probe of the structure of porous media[END_REF]. Hence our new formula in (73) is a correction of the results in [1,[START_REF] Grebenkov | NMR survey of reflected brownian motion[END_REF][START_REF] Mitra | Diffusion propagator as a probe of the structure of porous media[END_REF] because it takes into account the contribution of δ. This makes the new formula applicable for cases where the narrow pulse assumption δ ∆ does not hold. Of course, this description still hold only for short times due to the nature of the asympotic expansions for layer potentials.

Mixed approximation.

When the pulses are short but the delay between the pulses is not short (with respect to diffusion in Ω), we use the single layer potential representation in the first and third intervals and the eigenfunction representation between the pulses.

In the first pulse, t ∈ [0, δ], we have the same results as in the previous section, namely,

I = 8 21A|Ω| √ π D 3/2 0 δ 7/2 Γ (u g • ν) 2 ds x + O δ 2 ∆ -δ 3 . (76)
This manuscript is for review purposes only.

Between the pulses, t ∈ [δ, ∆], the Neumann boundary condition in ( 18) is

(77) D 0 ∇ω(x, t) • ν = D 0 δu g • ν, on Γ × [δ, ∆]
and the initial condition is

(78) ω(x, δ) = S [2δu g • ν] (x, t) + O(δ 2 ), x in Ω.
The function ω(x, t) = ω(x, t) -δx • u g satisfies homogeneous Neumann boundary condition and the initial condition

(79) ω(x, δ) = S [(2δu g • ν)] (x, δ) -δx • u g .
This means

(80) ω(x, t) = c 0 + ∞ n=1 c n e -λnD0(t-δ) φ n (x)
where

c 0 = -δa 0 + b 0 = - δ |Ω| Ω x • u g dx + 1 |Ω| Ω ω(x, δ)dx), (81)
c n = -δa n + b n = -δ Ω x • u g φ n (x)dx + Ω ω(x, δ)φ n (x)dx (82)
with again φ n and λ n the Neumann eigenfunctions and eigenvalues associated to the Laplace operator (n = 1, 2, . . . ). Thus, for t ∈

[δ, ∆], (83) ω(x, t)
= c 0 + ∞ n=1 c n e -λnD0(t-δ) φ n (x) + δx • u g + O(δ 2 ), and (84)
h(t) = ∞ n=1 c n λ n a n |Ω| e -λnD0(t-δ) + δ + O δ 2 A , and (85)
II = 1 A ∞ n=1 c n a n |Ω| 1 -e -λnD0(∆-δ) + D 0 δ 2 (∆ -δ) A + O δ(∆ -δ) ∆ -δ 3 .
During the second pulse, t ∈ [∆, ∆ + δ], we keep the solution from the previous interval in Eq. ( 83) which satisfies homogeneous boundary conditions and just add a single layer potential to match the Neumann boundary condition. We obtain

(86) ω(x, t) = c 0 + ∞ n=1 c n e -λnD0(t-δ) φ n (x) + δx • u g + S [(-2τ u g • ν)] (x, t -∆) + O((t -∆) 2 ), where t ∈ [∆, ∆ + δ].
The density in the single layer potential is -2τ u g • ν with τ ∈ [0, δ] from a shift in time τ = t -∆. Similar reasoning as in the previous sections gives

S [(-2τ u g • ν)] (x, t -∆) = - 4 3 √ π|Ω| D 0 (t -∆) 3/2 Γ (u g • ν) + O((t -∆) 5/2 ),
This manuscript is for review purposes only.

which leads to

(87) h(t) = ∞ n=1 c n |Ω| e -λnD0(t-δ) λ n a n - 4 3 √ π|Ω| D 0 (t -∆) 3/2 Γ (u g • ν) 2 ds x + δ + O (t -∆) 2 A , and
(88) III = D 0 2A δ 3 + ∞ n=1 c n a n D 0 λ n A|Ω| e -λnD0(∆-δ) (D 0 λ n δ -1) + e -λnD0∆ - 16 105 √ πA|Ω| D 3/2 0 Γ (u g • ν) 2 ds x δ 7/2 + O δ 2 ∆ -δ 3 .
The effective diffusion coefficient for the compartment Ω assumes thus the form

(89) D eff ug = D 0 δ 6(∆ -δ 3 ) - 8D 3/2 0 δ 3/2 35 √ π|Ω|(∆ -δ 3 ) Γ (u g • ν) 2 ds x + - ∞ n=1 -δ(a n ) 2 + a n b n |Ω|δ 2 (∆ -δ 3 ) δ - e -λnD0∆ (1 -e λnD0δ ) λ n D 0 + O max δ 2 ∆ -δ 3 , δ(∆ -δ) ∆ -δ 3
In the narrow pulse limit, we get

(90) D eff ug ≈ ∞ n=1 (δa n -b n )a n |Ω|δ(∆ -δ 3 )
1 -e -λnD0∆ , which again tells us that D eff ug approaches its long time limit as 1/∆, because b n = O(δ 3/2 ) for all n ≥ 1 due to the maximum principle for heat equation applied to ω(x, t) in the first pulse: We recall that (93)

(91) ω(x, t) ≤ ω(x 0 , t) ≈ O(t 3/2 ) ∀x ∈ Ω, x 0 ∈ Γ, t ∈ [0, δ].
D eff ug = D 0 - D 0 A|Ω| T E 0 F (t) Ω u g • ∇ω ug (x, t)dx dt
This manuscript is for review purposes only.

where ω ug (x, t) solves the problem [START_REF] Kiselev | The cumulant expansion: an overarching mathematical framework for understanding diffusion nmr[END_REF]. Because of the linearity of the Neumann problem, for every direction u g = [u 1 , . . . , u d ] we have that (94)

ω ug (x, t) = d i=1 u i ω ei (x, t),
where e i is the i-th vector of the canonical basis of R d . As a consequence We observe that, for all i, j = 1, . . . , d and i = j, We use the fact that (99)

D eff ug = D 0 - D 0 A|Ω| T E 0 F Ω (u 1 e 1 + • • • + u d e d ) • (u 1 ∇ω e1 + • • • + u d ∇ω ei ) dx dt = D 0 - D 0 A|Ω| T E 0 F     d i=1 u 2 i Ω e i • ∇ω ei dx + d i =j i,j=1 u i u j Ω e i •
d i=i Γ (u g i • ν) 2 ds x d = |Γ| d
and we define (100)

k n := d i=1 -(a i n ) 2 d|Ω| = d i=1 -Ω x • u g i φ n (x)dx 2 du g d|Ω| ,
i.e. the mean over d orthogonal directions of the square of the first moment along these directions, and

(101)

j n := d i=1 b i n a i n d|Ω| = d i=1 Ω ω ug i (x, δ)φ n (x)dx Ω x • u g i φ n (x)dx d|Ω| .
This manuscript is for review purposes only.

In summary, the eigenfunction expansion in [START_REF] Stejskal | Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient[END_REF] gives

(102) D eff ave = ∞ n=1 k n D 2 0 λ n 2 δ 2 ∆ -δ 3
e -D0λn(∆+δ) + e -D0λn(∆-δ)

-2 D 0 λ n δ + e -D0λnδ + e -D0λn∆ -1

The single layer potential representation gives

D eff ave = D 0 - 16 35
D 3/2 0 δ 2 (3 ∆ -δ) √ π (∆ -δ) 7/2 + (∆ + δ) 7/2 -2 δ 7/2 + ∆ 7/2 |Γ| d|Ω| + O(∆). ( 103
)
The mixed approximation in (89) gives

D eff ave = D 0 δ 6(∆ -δ 3 ) - 8D 3/2 0 δ 3/2 35 √ π(∆ -δ 3 ) |Γ| d|Ω| - ∞ n=1 δk n + j n δ 2 (∆ -δ 3 ) δ - e -λnD0∆ (1 -e λnD0δ ) λ n D 0 + O max δ 2 ∆ -δ 3 , δ(∆ -δ) ∆ -δ 3 .
(104) 5. Numerical results. In this Section we numerically validate the approximate formulas we derived in the previous sections. To compute the reference quantities we solved the diffusion equation in [START_REF] Kiselev | The cumulant expansion: an overarching mathematical framework for understanding diffusion nmr[END_REF] using the Matlab PDEToolbox. The eigenvalues and eigenfunctions of the Laplace equation with Neumann boundary conditions were also computed with the same software. The convergence between the H-ADC model and the Bloch-Torrey equation was shown previously in [START_REF] Haddar | A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity[END_REF].

First we show the three approximations of h(t). We consider a 2D geometry of one vertically orientated ellipse with semi-axes of 19µm and 9µm. The intrinsic diffusion coefficient is set to D 0 = 1e -3 mm 2 /s and we vary the values of δ, ∆ and u g . To compute the reference solution h(t) we solved the problem [START_REF] Kiselev | The cumulant expansion: an overarching mathematical framework for understanding diffusion nmr[END_REF] on the finite element mesh shown in Fig. 2. The eigenvalues and eigenfunctions are also computed on the same finite element mesh. The projections a n and b n are computed according the the formulas in ( 25) and (81).

For this particular geometry the first four non-zero eigenvalues are λ 1 = 0.0097, λ 2 = 0.0325, λ 3 = 0.0383, λ 4 = 0.0644, and their numerically calculated projections a i are reported in the table below.

Clearly, among the four eigenvalues, in the direction u g = [1, 0], all but λ 3 have negligible contribution, and in the direction u g = [0, 1], all but λ 1 have negligible contribution.

In the following plots we always indicate the reference quantity with a line, the single layer approximation with squares, the eigenfunction approximation with circles and the mixed approximation with asterisks.

In Figure 3 we considered δ = 5ms, ∆ = 10ms and u g = [1, 0]. As we can see the single layer approximation (squares) fits very well the reference quantity (continuous line) in all three time intervals. We also notice that the mixed approximation This manuscript is for review purposes only. This manuscript is for review purposes only.

h(t) potential approx eigenfunc approx with 1 eigenvalues mixed approx with 1 eigenvalues Fig. 4: h(t) and its approximation (using the three different formulas found) with respect to the gradient directions u g = [1, 0] for an ellipse of semiaxes 19µm and 9µm. Intrinsic diffusion coefficients D 0 = 1 × 10 -3 mm 2 /s, pulses duration δ = 5ms and time-delay between pulses ∆ = 50ms.

(asterisks) works sufficiently well during the two pulses but not between them. For the eigenfunctions approximation, the fit is far from the reference quantity.

In Figure 4 we considered δ = 5ms, ∆ = 50ms and u g = [1, 0]. As we can see the single layer approximation fits well the reference quantity during the first pulse and until t ≈ 25ms, but after that, the approximation is no longer good. The eigenfunction approximation is not good during the pulses but it becomes accurate at the end of the interval between them. The mixed approximation fits well during the pulses and is the same as the eigenfunction approximation between the pulses.

In Figures 5a and5b In Figure 5c we show the average of D eff ave /D 0 along the two perpendicular directions.

Clearly, the single layer formula works well for at short ∆ + δ, the eigenfunctions formula for long ∆ + δ.

To conclude, in Figure 6 we report the absolute error

|D eff -D approx |
for the same parameters as before in the two orthogonal directions. As we can see, the single layer approximation is better at short times and the eigenfunctions approximation is better at long times. The time at which the switch between the short and the long diffusion time approximations occurs at around t = 50ms in the direction [1,0] and it occurs at around t = 25ms in the direction [0, 1]. This manuscript is for review purposes only. (c) Average over both directions Fig. 5: D eff with respect to two different gradient directions as well as D eff ave , the average over both direction, compared to approximations using the three different formulas, for an ellipse of semiaxes 19µm and 9µm. Intrinsic diffusion coefficients D 0 = 1 × 10 -3 mm 2 /s, pulses duration δ = 5ms and thirty different values of the time-delay between pulses in the interval [START_REF] Fieremans | In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter[END_REF]80]ms.

In this paper, we analysed the H-ADC model in the case of finite sub-domains.

580

In particular, we obtained three representations of the effective diffusion coefficient 581 This manuscript is for review purposes only. that are appropriate in different diffusion time regimes. In the short time regime, we proposed using a representation based on the single layer potential. In the long time regime when the pulse duration is not short, we proposed using a representation based on the eigenfunctions expansion of the Neumann Laplace operator. In the long time regime when the pulse duration is short, we proposed a representation that combines the single layer potential during the pulses with the eigenfunction expansion between the pulses. In particular, in the short time regime, our representation corrects an existing formula by correctly accounting for the pulse duration. Our work helps to make more precise how parameters of the tissue micro-structure such as the surface to volume ratio or the dominant eigenvalue and its projection affect the effective diffusion coefficient.

Fig. 1 :

 1 Fig. 1: The Pulsed Gradient Spin Echo (PGSE) sequence. The effective sequence time profile shown here takes into account the 180 degree pulse.

-

s 2 4D 0

 20 (t-τ ) and make the change of variables r = s √ 4D0(t-τ )

4 .

 4 Averaging D eff over multiple gradient directions. If we average the effective diffusion coefficient D eff ug over all the possible gradient directions u g , we can obtain a new formula that is independent of the orientation of the biological cells. We define the orientionally averaged effective diffusion coefficient as (92) D eff ave := S d-1 D eff ug du S d-1 du .

S d- 1 u

 1 i u j du = 0. Therefore, what remains in the average are just the terms (97) d i=1 S d-1 u 2 i du S d-1 du Ω e i • ∇ω ei dx, i.e. simply the average over d perpendicular directions and then (98) D eff ave = D 0ug i (x, t)dx dt where u g i , i = 1, . . . , d are d orthogonal directions. In short, averaging over all the possible directions is equivalent to average only over d orthogonal normalized directions.

Fig. 2 :

 2 Fig.2: Finite elements mesh of an ellipse with semiaxes of 19µm and 9µm, orientated vertically along the y-axis.

Fig. 3 :

 3 Fig.3: h(t) and its approximation (using the three different formulas found) with respect to the gradient directions u g = [1, 0] for an ellipse of semi-axes 19µm and 9µm. Intrinsic diffusion coefficients D 0 = 1 × 10 -3 mm 2 /s, pulses duration δ = 5ms and time-delay between pulses ∆ = 10ms.

  we show the behaviour of D eff computed for two different directions of the gradient (u g = [1, 0] and u g = [0, 1]) but the same parameters (δ = 5ms and thirty different values of ∆ equally distributed in the interval [8, 80]ms).

6 .

 6 Conclusions. Diffusion Magnetic Resonance Imaging (dMRI) can be used to measure a time and direction dependent effective diffusion coefficient which can in turn reveal information about the tissue micro-structure. Recently a new mathematical model for the effective diffusion coefficient, the H-ADC model, was obtained using homogenization techniques after imposing a certain scaling relationship between the physical parameters. The resulting model depends on the solution of a diffusion equation subject to time-dependent Neumann boundary conditions.

  Gradient direction ug =[1, 0]

  Potential approx Eigenfunction approx with 1 eigenvalues Mixed approx with 1 eigenvalues (b) Gradient direction ug = [0, 1]

  Gradient direction ug =[1, 0]

  Potential approx Eigenfunction approx with 1 eigenvalues Mixed approx with 1 eigenvalues (b) Gradient direction ug = [0, 1]

Fig. 6 :

 6 Fig.6: Absolute error |D eff -D approx | using the three different formulas found with respect to two different gradient directions for an ellipse of semi-axes 19µm and 9µm. Intrinsic diffusion coefficients D 0 = 1 × 10 -3 mm 2 /s, pulses duration δ = 5ms and thirty different values of the time-delay between pulses in the interval[START_REF] Fieremans | In vivo observation and biophysical interpretation of time-dependent diffusion in human white matter[END_REF] 80]ms.

  ∇ω ej dx

					
					   dt
	and thus, if we want to average over all the possible directions, we are interested in
	the integrals
	(95)	S d-1 u 2 i du S d-1 du	, i = 1, . . . , d and	S d-1 u i u j du S d-1 du	, i, j = 1, . . . , d, i = j.

Table 1 :

 1 The first moments of the eigenfunction associated with the first four non-zero eigenvalues in the two directions u g = [1, 0] and u g = [0, 1].

			u g	a 1		a 2		a 3	a 4
			[1, 0]	38.9	-25.7 -4.62e +5 -0.97
			[0, 1] 1.07e +6	1.75		-4.49	-5.58
		2
		1.5
	(ms)
	h(t)	1				h(t)
						potential approx
						eigenfunc approx with 1 eigenvalues
		0.5				mixed approx with 1 eigenvalues
		0	0	2	4	6	8	10	12	14
						time (ms)

This manuscript is for review purposes only.


Christos Pelekis
email: pelekis.chr@gmail.com

Jan Ramon
email: jan.ramon@cs.kuleuven.be

Yuyi Wang
email: yuyiwang920@gmail.com

H ölder-type inequalities and their applications to concentration and correlation bounds

Keywords: fractional chromatic number, Finner's inequality, Janson's inequality, dependency graph, hypergraphs

come

Introduction and related work

The main purpose of this article is to illustrate that certain H ölder-type inequalities can be employed in order to obtain concentration and correlation bounds for sums of weakly dependent random variables whose dependencies are described in terms of graphs, or hypergraphs. Before being more precise, let us begin with some notation and definitions that will be fixed throughout the text.

A hypergraph H is a pair (V, E) where V is a finite set and E is a family of subsets of V . The set V is called the vertex set of H and the set E is called the edge set of H; the elements of E are called hyperedges or just edges. The cardinality of the vertex set will be denoted by |V | and the cardinality of the edge set by |E|. A hypergraph is called k-uniform if every edge from E has cardinality k. A 2-uniform hypergraph is a graph. The degree of a vertex v ∈ V is defined as the number of edges that contain v. A hypergraph will be called dregular if every vertex has degree d. A subset V ⊆ V is called independent if it does not contain any edge from E. A fractional matching of a hypergraph, H = (V, E), is a function φ : E → [0, 1] such that e:v∈e φ(e) ≤ 1, for all vertices v ∈ V . The fractional matching number of H, denoted ν * (H), is defined as max φ e∈E φ(e) where the maximum runs over all fractional matchings of H. The fractional chromatic number of a graph G is defined in the following way. A b-fold colouring of G is an assignment of sets of size b to the vertices of the graph in such a way that adjacent vertices have disjoint sets. A graph is (a : b)colourable if it has a b-fold colouring using a different colours. The least a for which the graph is (a : b)-colourable is the b-fold chromatic number of the graph, denoted χ b (G). The fractional chromatic number of a graph G is defined as

χ * (G) = inf b χ b (G)
b . Here and later, P[•] and E[•] will denote probability and expectation, respectively.

Let us also recall H ölder's inequality. Let (Ω, A, P) be a probability space. Let A be a finite set and let Y a , a ∈ A, be random variables from Ω into R. Suppose that w a , a ∈ A are non-negative weights such that a∈A w a ≤ 1 and each Y a has finite 1 wa -moment, i.e., E Y 1/wa a < +∞, for all a ∈ A. H ölder's inequality asserts that

E a∈A Y a ≤ a∈A E Y 1/wa a wa .
This is a classic result (see [START_REF] Bogachev | Measure Theory[END_REF]). In this article we shall be interested in applications of H ölder-type inequalities to concentration and correlation bounds for sums of weakly dependent random variables. We focus on two particular types of dependencies between random variables. The first one is described in terms of a hypergraph. Definition 1 (hypergraph-correlated random variables). Let H = (V, E) be a hypergraph. Suppose that {Y e } e∈E is a collection of real-valued random variables, indexed by the edge set of H, which satisfy the following: there exist independent random variables {X v } v∈V indexed by the vertex set V such that, for every edge e ∈ E, we have Y e = f e (X v ; v ∈ e) is a function that depends only on the random variables X v with v ∈ e. We will refer to the aforementioned random variables {Y e } e∈E as H-correlated, or simply as hypergraph-correlated, when there is no confusion about the underlying hypergraph.

As an example of hypergraph-correlated random variables the reader may think of the indicators of the triangles in an Erdős-Rényi random graph model G(n, p). Recall that such a model generates a random graph, G, on n labelled vertices by joining pairs of vertices, independently, with probability p ∈ (0, 1). To see that the indicators of the triangles in G are hypergraph-correlated, let H G be the hypergraph whose vertex set, V, has an element for every (potential) edge in G and whose edge set, E, has a hyperedge for every triplet of edges in G that form (potential) triangles. Let X v , v ∈ V, be independent Bernoulli Ber(p) random variables. For every triangle T in G, let E T be the corresponding hyperedge of H G . Now notice that the indicator of each triangle, T , in G can be written as a product v∈E T X v . More examples of hypergraph-correlated random variables can be found in Section 4. Hypergraph-correlated random variables are encountered in the theory of random graphs and machine learning (see [START_REF] Gavinsky | Bound for Read-k Families of Functions[END_REF][START_REF] Janson | Large Deviations for Sums of Partly Dependent Random Variables[END_REF][START_REF] Ramon | Learning from networked examples[END_REF][START_REF] Wolfovitz | A concentration result with application to subgraph count[END_REF]] and references therein); a quintessential question that motivates their study asks for upper bounds on the probability that a random graph G ∈ G(n, p) contains more triangles than expected. Notice that the indicators of the triangles in G ∈ G(n, p) are not mutually independent and therefore the standard Chernoff-Hoeffding bounds do not apply. Another type of "dependency structure" that plays a key role in probabilistic combinatorics and related areas involves the notion of a dependency graph (see also [START_REF] Alon | The Probabilistic Method[END_REF][START_REF] Janson | Random Graphs[END_REF]).

Definition 2 (Graph-dependent random variables).

A dependency graph for the random variables {Y v } v∈V , indexed by a finite set V , is any loopless graph, G = (V, E), whose vertex set V is the index set of the random variables and whose edge set is such that if V ⊆ V and v i ∈ V is not incident to any vertex of V , then Y v is mutually independent of the random variables Y v for which v ∈ V . We will refer to random variables {Y v } v∈V having a dependency graph G as G-dependent or as graph-dependent.

A difference between the notions of graph-dependent and hypergraph-correlated random variables is illustrated in Example 2.2 below. An example of graph-dependent random variables is the indicators of the triangles in a random graph G ∈ G(n, p). Let us remark that the previous notions of weakly dependent random variables arise in the study of the, so-called, Lovász Local Lemma. In this setting one is dealing with a (finite) number of events in some probability space, and is interested in lower bounds on the probability that none of the events will occur. The events need not be independent and their indicators are assumed to be graph-dependent, or hypergraph-correlated (see [START_REF] Erdős | Problems and results on 3-chromatic hypergraphs and some related questions[END_REF][START_REF] Moser | A constructive proof of the general Lovász local lemma[END_REF]). In this article we focus on upper bounds on the probability that none of the events will occur.

Notice that if {Y e } e∈E are hypergraph-correlated random variables then they induce a dependency graph whose vertex set is E and with edges joining any two sets e, e ∈ E such that e ∩ e = ∅. Hence a set of hypergraph-correlated random variables is graphdependent. The reader may wonder whether the converse holds true. We will show, using a particular generalisation of H ölder's inequality, that this is not the case (see Example 2.2 below) and so the aforementioned notions of dependencies between random variables are not equivalent.

In the present paper we provide a H ölder-type inequality for graph-dependent random variables. Our result can be seen as a dependency-graph analogue of the following theorem, due to Helmut Finner.

Theorem 1.1 (Finner [7]). Let H = (V, E) be a hypergraph and let {Y e } e∈E be H-correlated e) . See [START_REF] Finner | A generalization of H ölder's inequality and some probability inequalities[END_REF] for a proof of Theorem 1.1 which is based on Fubini's theorem and H ölder's inequality. Alternatively, see [START_REF] Ramon | Learning from networked examples[END_REF] for a proof that uses the concavity of the weighted geometric mean and Jensen's inequality. In other words, Theorem 1.1 provides a H ölder-type inequality for hypergraph-correlated random variables which is formalised in terms of a fractional matching of the underlying hypergraph.

random variables. If φ : E → [0, 1] is a fractional matching of H then E e∈E Y e ≤ e∈E E Y 1/φ(e)
In this article we illustrate how certain H ölder-type inequalities yield concentration and correlation bounds for sums of hypergraph-correlated random variables as well as for sums of graph-dependent random variables. In that regard, our contribution is two-fold. On the one hand, we show that a particular graph-dependent analogue of Theorem 1.1 yields a new proof of the following result, due to Svante Janson, which provides an estimate on the probability that the sum of graph-dependent random variables is significantly larger than its mean. Theorem 1.2 (Janson [11]).

Let {Y v } v∈V be [0, 1]-valued random variables having a dependency graph G = (V, E). Set q := 1 |V | E [ v Y v ]. If t = n(q + ε) for some ε > 0, then P v Y v ≥ t ≤ exp - 2ε 2 |V | χ * , where χ * = χ * (G) is the fractional chromatic number of G.
See [11, Theorem 2.1] for a proof of this result which is based on breaking up the sum into a particular linear combination of sums of independent random variables. On the other hand, we show that Finner's inequality applies to the following problem, that is interesting on its own. Problem 1.3. Fix a hypergraph H = (V, E) and let I be a random subset of V formed by including vertex v ∈ V in I with probability p v ∈ (0, 1), independently of other vertices. What is an upper bound on the probability that I is independent?

Here and later, given a set of parameters in (0, 1), say p = {p v } v∈V , indexed by the vertex set of a hypergraph H, we will denote by π(p, H) the probability that I is independent. Let us remark that Problem 1.3 has attracted the attention of several authors and appears to be related to a variety of topics (see [START_REF] Boppana | A useful elementary correlation inequality[END_REF][START_REF] Galvin | The number of independent sets in a graph with small maximum degree[END_REF][START_REF] Janson | The Lower Tail: Poisson Approximation Revisited[END_REF][START_REF] Krivelevich | On the probability of independent sets in random graphs[END_REF][START_REF] Wolfovitz | A concentration result with application to subgraph count[END_REF] and references therein). A particular line of research is motivated by question about independent sets and subgraph counting in random graphs. In this context, Problem 1.3 has been considered by Janson et al. [START_REF] Janson | The Lower Tail: Poisson Approximation Revisited[END_REF], Krivelevich et al. [START_REF] Krivelevich | On the probability of independent sets in random graphs[END_REF] and Wolfovitz [START_REF] Wolfovitz | A concentration result with application to subgraph count[END_REF]. It is observed in [START_REF] Krivelevich | On the probability of independent sets in random graphs[END_REF] that when H is k-uniform, d-regular and p v = p for all v ∈ V , an exponential estimate on π(p, H) can be obtained using the so-called Janson's correlation inequality. Additionally, it is shown in [15, Section 5] that under certain "mild additional assumptions" the bound provided by Janson's inequality can be improved to

π(p, H) ≤ exp -Ω p|E| (1 -p)kd .
See [START_REF] Krivelevich | On the probability of independent sets in random graphs[END_REF] for a precise formulation of the "mild additional assumptions" and a proof of this result which is based on a martingale-type concentration inequality.

The remaining part of our paper is organised as follows. In the Section 2 we collect our main results. In particular, we provide a H ölder-type inequality for graph-dependent random variables and we apply this result in order to deduce upper bounds on the probability that a sum of graph-dependent random variables is significantly larger than its mean. We also show that Finner's inequality yields an upper bound on π(p, H). The proofs of our main results are deferred to Section 3. Finally, in Section 4, we illustrate that Finner's inequality can be seen as an alternative to the, well-known, Janson's correlation inequality. Our paper ends with Section 5 in which we briefly sketch how one can apply similar ideas to yet another class of weakly dependent random variables.

Main results

In this section we collect our main results. Our first result provides an analogue of Theorem 1.1 for random variables having a dependency graph G. The corresponding H öldertype inequality is formalised in terms of the b-fold chromatic number of G and reads as follows.

Theorem 2.1. Let {Y v } v∈V be real-valued random variables having a dependency graph G = (V, E). Then, for every b-fold colouring of G using χ b := χ b (G) colours, we have

E v∈V Y v ≤ v∈V E Y χ b b v b χ b .
We prove Theorem 2.1 in Section 3. The proof employs the concavity of the weighted geometric mean and the definition of b-fold chromatic number. Recall that Finner's inequality applies to hypergraph-correlated random variables and that such an ensemble of random variables induces a dependency graph. However, as is shown in the following example, there exist graph-dependent random variables which are not hypergraph-correlated and therefore Theorem 2.1 provides a H ölder-type inequality can be applied to more general ensembles of "weakly dependent" random variables.

Example 2.2 (R.v.'s that are graph-dependent but not hypergraph-correlated). Let G be a cycle-graph on

5 vertices {v 1 , . . . , v 5 } such that v i is incident to v i+1 , for i ∈ {1, 2, 3, 4} and v 5 is incident to v 1 . Let Y = (Y 1 , . . . , Y 5
) be a vector of Bernoulli 0/1 random variables whose distribution is defined as follows. The vector Y takes the value (0, 0, 0, 0, 0) with probability 1 2 (2p)(1 -p) 2 , the value (1, 1, 1, 1, 1) with probability p 2 +p 3 2 , the values (0, 0, 0, 1, 1), (0, 0, 1, 1, 0), (0, 1, 1, 0, 0), (1, 1, 0, 0, 0), (1, 0, 0, 0, 1)

with probability p(1-p) 2 2
, the values (0, 0,

with probability p 2 -p 3

2

and the remaining values with probability 0. Elementary, though quite tedious, calculations show that E[Y j ] = p, for j = 1, . . . , 5 and that G is a dependency graph for {Y j } 5 j=1 . Now assume that {Y j } 5 j=1 are H-correlated, for some hypergraph H = (V, E). Notice that the cardinality of E must be equal to 5. If e i ∈ E is the edge corresponding to the random variable Y i , i = 1, . . . , 5, then the fact that {Y i } 5

i=1 have G as a depencency graph implies that e i ∩ e (i+2) mod 5 = ∅, for i ∈ {1, 2, 3, 4, 5}. This means that the fractional matching number of H is at least 2.5 and therefore Theorem 1.1 implies that P [Y = (1, 1, 1, 1, 1)] ≤ p 2.5 . However, the arithmetic geometric means inequality implies p 2 +p 3 2 > p 2.5 and therefore the random variables {Y j } 5 j=1 cannot be hypergraph-correlated. We also show that Theorem 2.1, combined with standard techniques based on exponential moments, yields, one the one hand, a new proof of Theorem 1.2 and, on the other hand, the following Bennett-type inequality for graph-dependent random variables.

Theorem 2.3. Let {Y v } v∈V be random variables having a dependency graph G = (V, E). For every v ∈ V let σ 2 v := Var (Y v ) and assume further that Y v ≤ 1 and E[Y v ] = 0. Set S = v σ 2 v and fix t > 0. Then P v Y v ≥ t ≤ exp - S χ * (G) ψ t S ,
where

ψ(x) = (1 + x) ln(1 + x) -x.
Let us remark that Theorem 2.3 improves on [11, Theorem 2.3] which contains the same bound but with ψ 4t 5S instead of our ψ t S . Note that we assume a one-sided bound on each Y v . The proof of Theorem 2.3 is based on Theorem 2.1 and can be found in Section 3. Notice also that Theorems 2.1 and 2.3 are concerned with graph-dependent random variables. Our final result concerns hypergraph-correlated indicators. In Section 3 we show that Finner's inequality yields the following upper bound on the probability that a random subset of the vertex set of a hypergraph is independent.

Theorem 2.4. Let H = (V, E) be a hypergraph and p = {p v } v∈V be a set of parameters from (0, 1). Then

π(p, H) ≤ e 1 - v∈e p v φ(e)
, where φ : E → [0, 1] is a fractional matching of H. In particular, if the hypergraph H is k-uniform, d-regular and p v = p, for all v ∈ V, then

π(p, H) ≤ 1 -p k ν * (H) ≤ exp -p k |E| d ,
where ν * (H) is the fractional matching number of H.

Notice that the first bound in the second statement of Theorem 2.4 has a monotonicity property, in the sense that if

H 1 is a superhypergraph of H 2 then 1 -p k ν * (H 1 ) ≤ 1 -p k ν * (H 2 ) .
In Section 4 we illustrate that Theorem 2.4 may be seen as an alternative to Janson's correlation inequality.

Let us end this section by noting that Theorem 1.1, combined with standard exponentialmoment ideas, yields concentration inequalities for sums of hypergraph-correlated random variables. This has been reported in prior work and so we only provide the statement without proof. In [START_REF] Ramon | Learning from networked examples[END_REF] one can find a proof of the following result. This inequality has also been obtained in Gavinsky et al. [START_REF] Gavinsky | Bound for Read-k Families of Functions[END_REF] using entropy ideas. As is mentioned in the introduction, the indicators, say {I T } T , of the triangles in an Erdős-Rényi random graph are both graph-dependent and hypergraph-correlated; their dependency graph is the graph which is induced from the random variables {I T } T when they are viewed as hypergraph-correlated. This means that both Theorem 1.2 and Theorem 2.5 provide an upper bound on the probability that G ∈ G(n, p) contains more triangles than expected. We invite the reader to verify that Theorem 2.5 provides a better bound. An intuitive explanation as to why Theorem 2.5 gives a better bound is that, while considering the induced dependency-graph of a set of hypergraph-correlated random variables, one loses information. This behaviour has also been reported in the context of Lovász Local Lemma (see Kolipaka et al. [START_REF] Kolipaka | STOC '11 Proceedings of the forty-third annual ACM symposium on Theory of computing[END_REF]).

Proofs

In this section we prove our main results. We begin with Theorem 2.1, whose proof is based on the concavity of the weighted geometric mean.

Lemma 3.1. Let β = (β 1 , . . . , β k ) be a vector of non-negative real numbers such that k i=1 β i = 1. Then the function g : R k → R defined by g(t) = k i=1 t β i i is concave.

Proof. This is easily verified by showing that the Hessian matrix is positive definite. See [START_REF] Boyd | Convex Optimization[END_REF], or [START_REF] Ramon | Learning from networked examples[END_REF] for details.

We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1. We show that

E v∈V {Y v } b χ b ≤ v∈V {E [Y v ]} b χ b .
The theorem follows by applying this inequality to the random variables

Z v = Y χ b b v .
For every colour i = 1, . . . , χ b let I i be the set consisting of the vertices that are coloured with colour i. Note that each I i is an independent subset of V and every vertex v ∈ V appears in exactly b independent sets I i . Therefore,

E v∈V {Y v } b χ b = E   χ b i=1 v∈I i {Y v } 1 χ b   = E    χ b i=1    v∈I i Y v    1 χ b    .
Lemma 3.1 and Jensen's inequality combined with the observation that the random variables {Y v } v∈I i are mutually independent yield

E    χ b i=1    v∈I i Y v    1 χ b    ≤ χ b i=1    E   v∈I i Y v      1 χ b = χ b i=1 v∈I i {E [Y v ]} 1 χ b .
Now, using again the fact that each vertex v appears in exactly b sets I i , we conclude

χ b i=1 v∈I i {E [Y v ]} 1 χ b = v∈V {E [Y v ]} b χ b
and the result follows.

Recall the following, well-known, result whose proof is included for the sake of completeness. Lemma 3.2. Let X be a random variable that takes values on the interval [0, 1]. Suppose that E[X] = p, for some p ∈ (0, 1), and let B be a Bernoulli 0/1 random variable such that

E[B] = p. If f : [0, 1] → R is a convex function, then E [f (X)] ≤ E [f (B)].
Proof. Given an outcome from the random variable X, define the random variable B X that takes the values 0 and 1 with probability 1 -X and X, respectively. It is easy to see that E [B X ] = p and so B X has the same distribution as B. Now Jensen's inequality implies

E [f (X)] = E f E B X X ≤ E f B X X = E [f (B X )] ,
as required.

We now proceed with the proof of Theorem 1.2.

Proof of Theorem 1.2. Fix h > 0 and let

q v = E [Y v ],
for v ∈ V . Using Markov's inequality and Theorem 2.1 we estimate

P v∈V Y v ≥ t ≤ e -ht E e h v∈V Yv = e -ht E v∈V e hYv ≤ e -ht v∈V E exp χ b b hY v b χ b
For v ∈ V let B v be a Bernoulli 0/1 random variable of mean q v . Lemma 3.2 implies

e -ht v∈V E exp χ b b hY v b χ b ≤ e -ht v∈V E exp χ b b hB v b χ b = e -ht v∈V (1 -q v ) + q v e χ b b h b χ b .
Using the weighted arithmetic-geometric means inequality we conclude

e -ht v∈V (1 -q v ) + q v e χ b b h b χ b ≤ e -ht v∈V 1 |V | (1 -q v ) + q v e χ b b h b χ b |V | = e -ht 1 -q + qe χ b b h b χ b |V |
.

If we minimise the last expression with respect to h > 0 we get that h must satisfy e q(|V |-t) and therefore, since t = |V |(q + ε), we conclude

P v∈V Y v ≥ t ≤ q q + ε q+ε 1 -q 1 -(q + ε) 1-(q+ε) b χ b |V | = e -b χ b |V |D(q+ε||q) ,
where D(q + ε||q) is the Kullback-Leibler distance between q + ε and q. Finally, using the standard estimate D(q + ε||q) ≥ 2ε 2 , we deduce

P v∈V Y v ≥ t ≤ e -b χ b |V |2ε 2
and the result follows upon minimising the last expression with respect to b.

The proof of Theorem 2.3 is similar.

Proof of Theorem 2.3. Fix h > 0 to be determined later. As in the proof of Theorem 1.2, Markov's inequality and Theorem 2.1 yield

P v Y v ≥ t ≤ e -ht v∈V E exp χ b b hY v b χ b .
Using an inequality proved in [START_REF] Janson | Large Deviations for Sums of Partly Dependent Random Variables[END_REF] (see inequality (3.7), page 240), we have

E exp χ b b hY v ≤ exp σ 2 v g χ b b h ,
where g(a) := e a -1 -a. Summarising, we have shown

P v Y v ≥ t ≤ exp -ht + b χ b e hχ b /b - b χ b -h S . Now choose h = b χ b • ln 1 + t S to deduce P v Y v ≥ t ≤ exp b χ b t -S b χ b 1 + t S ln 1 + t S = exp -S b χ b ψ t S .
The result follows upon minimising the last expression with respect to b.

We end this section with the proof of Theorem 2.4.

Proof of Theorem 2.4. Let X v , v ∈ V, be indicators of the event v ∈ I. For each e ∈ E set Y e = v∈e B v . Clearly, the random variables {Y e } e∈E are H-correlated. Now look at the probability P [ e Y e = 0]. Notice that if all Y e are equal to zero, then every edge e ∈ E contains a vertex, v, such that B v = 0 and vice versa. This implies that if e Y e = 0 then the set of vertices, v, for which B v = 1 is an independent subset of V and vice versa. Therefore

P e∈E Y e = 0 = E e∈E (1 -Y e ) = π(p, H).
From Theorem 1.1 we deduce

E e∈E (1 -Y e ) ≤ E e∈E (1 -Y e ) φ(e) ≤ e∈E (E [1 -Y e ]) φ(e)
and the first statement follows. To prove the second statement notice that E [Y e ] = p k , for all e ∈ E, and therefore

E e∈E (1 -Y e ) ≤ 1 -p k e φ(e) .
The result follows by maximising the exponent on the right hand side over all fractional matchings of H.

In the next section we provide several applications of Theorem 2.4 to the theory of random graphs.

Applications

In this section we discuss comparisons between Finner's and Janson's inequality. Janson's inequality (see Janson [START_REF] Janson | New versions of Suen's correlation inequality[END_REF] and Janson et al. [START_REF] Janson | Random Graphs[END_REF]Chapter 2]) is a well known result that provides upper estimates on the probability that a sum of dependent indicators is equal to zero. It is described in terms of the dependency graph corresponding to the indicators. More precisely, let {B v } v∈V be indicators having a dependency graph

G. Set µ = E [ v B v ] and ∆ = e={u,v}∈G E [B u B v ].
Janson's inequality asserts that

P v B v = 0 ≤ min e -µ+∆ , exp ∆ 1 -max v E[B v ] v (1 -E[B v ]) .
This inequality has been proven to be very useful in the study of the Erdős-Rényi random graph model G(n, p). For G ∈ G(n, p) let us denote by T G the number of triangles in G. A typical application of Janson's inequality provides the estimate

P [G ∈ G(n, p) is triangle-free] ≤ (1 -p 3 ) ( n 3 ) • exp ∆ 2(1 -p 3 )
,

where ∆ = 6 n 4 p 5 . In this section we juxtapose the previous bound with the bound provided by Finner's inequality.

P [T G = 0] ≤ 1 -p 3 1 n-2 ( n 3 ) .
Proof. We apply Theorem 2.4. Define a hypergraph H = (V, E) as follows. Let v i , i = 1, . . . , n 2 , be an enumeration of all (potential) edges in G and consider v 1 , . . . , v ( n 2 ) as the vertex set V of the hypergraph H. Let B v i , i = 1, . . . , n 2 be independent Bernoulli Ber(p) random variables, corresponding to the edges of G, and let E i , i = 1 . . . , n 3 , be an enumeration of all triplets of edges in G that form (potential) triangles in G. Define E = {E 1 , . . . , E ( n 3 ) } to be the edge set of H and let Z i be the indicator of triangle E i ; thus T G = i Z i . Now form a subset I of V by picking each vertex, independently, with probability p. Then the probability that I is independent equals P [T G = 0] and, in order to apply Theorem 2.4, have to find a fractional matching of H. Since every vertex of H belongs to n -2 edges in E = {E 1 , . . . , E ( n 3 ) }, we obtain a fractional matching, φ(•) of H by setting φ(E i ) = 1 n-2 , for i = 1, . . . , n 3 . The result follows.

Notice that the bound obtained from Janson's inequality is smaller than the bound given by Proposition 4.1 for values of p that are close to 0, but the bound of Proposition 4.1 does better for large values of p. Similar estimates can be obtained for the probability that a graph G ∈ G(n, p) contains no k-clique, for k ≥ 3. The details are left to the reader.

We now proceed with one more application of Finner's inequality. Let G ∈ G(n, p) be a random graph on n labelled vertices. Fix two vertices, say u and v. What is an upper bound on the probability that there is no path of length k between u and v?

A path of length k is a sequence of edges {v 0 , v 1 }, {v 1 , v 2 }, . . . , {v k-2 , v k-1 }, {v k-1 , v k } such that v i = v j , for i = j.
We assume k ≥ 3, otherwise the problem is easy. Let us remark that it may be cumbersome to apply Janson's inequality to the previous question. By contrast, Theorem 2.4 reduces the problem to a counting one. More explicitly, let {P i } i be an enumeration of all (potential) paths of length k between u and v. Clearly, there are n-2 k-1 • (k -1)! such paths. Define the hypergraph H = (V, E) as follows. The vertices of H correspond to the (potential) edges of G and the edges of H correspond to the sets of edges in G which form a path of length k between u and v. Hence the probability that there is no path of length k between u and v equals π(p, H). In order to apply Theorem 2.4 we have to find a fractional matching of H and, in order to do so, it is enough to find an upper bound on the maximum degree of H. To this end, fix an edge, e = {x, y}, in G. In case one of the vertices x or y is equal to either u or v, then there are n-3 k-2 • (k -2)! paths of length k from u to v that pass through edge e. If none of the vertices x, y is equal to u or v, then we count the paths as follows. We first create a path, P k-2 , of length k -2 from u to v that does not pass through any of the points x, y and then we place the edge e = {x, y} in one of k -2 available edges in the path P k-2 . Since there are two ways of placing the edge e in each slot of P k-2 it follows that the number of paths from u to v that go through edge e is equal to

2(k -2) • n-4 k-3 • (k -3)!. If k ≤ (n -1)/2 then the later quantity is smaller than n-3 k-2 • (k -2)!, otherwise it is larger than n-3 k-2 • (k -2)!. Therefore, if k ≤ (n -1)/2, the fractional matching number of H is at least ( n-2 k-1 )•(k-1)! ( n-3 k-2 )•(k-2)! = n -2. If k > (n -1)/2 then the fractional matching number of H is at least (n-2)(n-3) 2(k-2)
. We have thus proven the following.

Proposition 4.2. Let G ∈ G(n, p). Fix two vertices u, v in G and a positive integer k ≥ 3. Let E be the event "there is no path of length k between u and v". If k ≤ (n -1)/2 then

P [E] ≤ 1 -p k n-2 .
If k > (n -1)/2, then

P [E] ≤ 1 -p k (n-2)(n-3) 2(k-2)
.

We end this section with an estimate on the probability that a G ∈ G(n, p) contains no vertex of fixed degree.

Proposition 4.3. Let G ∈ G(n, p) and fix a positive integer d ∈ {0, 1, . . . , n -1}. Then the probability that there is no in G whose degree equals d is less than or equal to

1 - n -1 d p d (1 -p) n-1-d n 2
.

Proof. This is yet another application of Theorem 2.4, so we sketch it. Let v 1 , . . . , v n be an enumeration of the vertices of G. Let the hypergraph H = (V, E) be defined as follows.

The vertex set V corresponds to the (potential) edges of G. The edge set E = {E 1 , . . . , E n } corresponds to the vertices of G. That is, for i = 1, . . . , n the edge E i contains those u ∈ V for which the corresponding edges of G are incident to vertex v i . The result follows from the fact that |E| = n and the maximum degree of H is equal to 2.

Remarks

As mentioned in Janson [START_REF] Janson | Large Deviations for Sums of Partly Dependent Random Variables[END_REF], there exist collections of weakly dependent random variables that do not have a dependency graph. The dependencies between such collections of random variables can occasionally be described using an independence system. Recall that an independence system is a pair A = (V, I) where V is a finite set and I is a collection of subsets of V (called the independent sets) with the following properties (see [START_REF] Bondy | Graph Theory[END_REF]):

-The empty set is independent, i.e., ∅ ∈ I. (Alternatively, at least one subset of V is independent, i.e., I = ∅.)

-Every subset of an independent set is independent, i.e., for each A ⊂ A ⊂ A, if A ∈ I then A ∈ I. This is sometimes called the hereditary property.

Given a set of random variables {Y v } v∈V , we say that their joint distribution is described with an independence system, say A = (V, I), if for every A ∈ I the random variables {Y a } a∈A are mutually independent. Let us remark that this definition includes the case of k-wise independent random variables (see [START_REF] Alon | The Probabilistic Method[END_REF]Chapter 16], or [START_REF] Schmidt | Chernoff-Hoeffding bounds for applications with limited independence[END_REF]). Notice that if {Y v } v∈V are random variables whose joint distribution is described with an independence system A = (V, I) then {v} ∈ I, for all v ∈ V . It is easy to see that if the random variables {Y v } v∈V have a dependency graph then their joint distribution is described with an independence system. However, the converse need not be true (see [START_REF] Janson | Large Deviations for Sums of Partly Dependent Random Variables[END_REF]). In a similar way as in Section 1, one may define the fractional chromatic number of an independent system as follows. b . With these concepts by hand, one can prove a corresponding H ölder-type inequality using a similar argument as in Theorem 2.1. As a consequence one can obtain tail bounds similar to Theorem 1.2 and Theorem 2.3, the only difference being that the fractional chromatic number of the dependency graph, χ * (G), is replaced with the fractional chromatic number of the independence system, χ * (A). We leave the details to the reader.

.≤

  Notice that, by applying the previous result to the random variablesZ e = Y e∈E {E [Y e ]} φ(

Proposition 4 . 1 .

 41 Let G ∈ G(n, p) be an Erdős-Rényi random graph and denote by T G the number of triangles in G. Then

  A b-fold colouring of an independence system A = (V, I) is a function λ : I → Z + such that A:v∈A λ(A) = b, for all v ∈ V . The b-fold chromatic number of A is defined as χ b (A) := inf λ A∈I λ(A), where the infimum is over all b-fold colourings, λ(•), of A = (V, I). Finally, the fractional chromatic number of A is χ * (A) := inf b χ b (A)

Acknowledgements

The authors are supported by ERC Starting Grant 240186 "MiGraNT, Mining Graphs and Networks: a Theory-based approach". We are grateful to an anonymous referee for several suggestions that improved the presentation of the paper.