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# Microphase separation of a miscible binary liquid mixture under confinement at the nanoscale

## Abstract

Recent experimental works suggested that the confinement into a cylindrical nanopore induced the microphase separation of a binary liquid, despite the miscible character of its bulk counterpart. A core–shell organization was evidenced such that one of the liquids was strongly anchored to the solid surface whereas the other was confined at the center of the pore. At the same time, a study based on atomistic simulations suggested a strong heterogeneity and the absence of a separation. In this work, by refining the solid–liquid interactions to qualitatively reproduce the experimental adsorption isotherms of both single liquids, the microphase separation and the core–shell structure are captured. By tuning the surface chemistry of the nanopore to mimic hydrophilic and hydrophobic confinement, we show that it is possible to control the structural characteristics of the core–shell structure.The molecular origin of the microphase separation is then ascribed to the strong hydrogen bonds and a commensurate arrangement between the confining material and both liquids.

## Introduction

For three decades confinement effects at the nanoscale on the physics of fluids have been been intensively studied.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 Many new properties have been then discovered such as the giant diffusion of liquids,16,17,18 the apparition of new phases and new transitions,19,20,21,22 giant dielectric properties,11,12,23,24,25 the increase of optical properties,26 and the possible mixing of non-miscible mixtures.27 These observations indicated that the classical understanding of the physics of liquids should be revisited in confined geometry. Whereas these effects have been largely investigated in the case of confined single components1,2,3,4,5,6,7,8,9,10,11,12,13,14 and immiscible binary mixtures,28,29,30 less works have been devoted to the confinement of miscible liquid mixtures.15,31,32,33 Recently, Muthulakshmi et al.34 reported an experimental evidence of a partial phase separation of an ethanol–water mixture confined in mesoporous silica using positron annihilation lifetime spectroscopy. They showed that a small fraction of the ethanol molecules seemed to be anchored at the silica surface.34 A similar result was also established by Guo et al.14 who have studied confinement of the ethanol–water mixture between two planar silica walls. A partial and local separation between water and ethanol close to the silica surface was thus evidenced.14 At the same time, Schmitz et al.35 have exhibited that the glycol–water mixture could undergo an interfacial separation. This phenomenon was also numerically observed by You et al.36 who exhibited a local demixing of binary hard-core Yukawa mixtures in a slitlike pore. More recently, Krycka et al.32 displayed a separation between two confined apolar liquids. At same time, Harrach et al.33 have studied the phase behavior of a mixture of isobutyric acid (iBA) and water confined in mesoporous SBA-15 silica material and have shown that the iBA-rich phase is close to the pore wall and the water-rich phase is in the center of the pores. Whereas these works only suggest a partial separation of two hydrogen bonds forming liquids or two apolar liquids near the solid surface, Morineau and co-workers15 have recently provided a direct experimental structural evidence of the microphase separation of macroscopically miscible liquids consisting of hydrogen bonds forming liquid and an apolar one. Indeed, the structure of a mixture comprising toluene (TOL) and tert-butanol (TBA) molecules confined in a cylindrical silica nanopore (MCM-41) of radius 24 Å was explored by neutron scattering and compared with the miscible bulk one.15 Using a core–shell (CS) model, the authors established, for the first time, a molecular-scale phase-separated tubular structure with the TBA molecules forming a layer at the pore surface (shell), surrounding a TOL-rich phase at the center of the pore (core). This observation was later extended down to low temperature in the glassy states and to the larger pores sizes of SBA-15.37,38 A consistent picture, showing the highly selective segregation of polar molecules at the pore surface, was deduced from binary gas adsorption experiments.31

Although these experiments revealed the presence of a microphase separation, the local organization and microscopic processes controlling the CS structure has to be clarified. In a recent numerical work, Essafri et al.39 showed that the nano-confinement of TBA/TOL mixture in a silica nanopore only induced strong heterogeneity. This result highlights then a scenario based on the local segregation rather than a core–shell organization of the confined TBA/TOL mixture. As suggested by the authors, the difference between experiment and simulation would be probably due to the hydrophilic character of the membrane that differs between both experimental and numerical studies. Therefore, the core–shell microphase separation of confined mixtures, which was experimentally evidenced by different studies, emerged as an open and challenging question for numerical methods. The current study paves the way to a thorough understanding of the phenomenon from carefully designed simulations.15,31,37,38

Using atomistic simulations, we aim to capture the CS structure and to clarify the microscopic driving force ruling it. To do so, the solid–liquid interactions were refined to qualitatively reproduce the experimental adsorption isotherm of pure TBA and TOL components confined through the silica nanopore.31 An illustration of three TBA, TOL, and silica framework is provided in Fig. 1a. This refinement allowed us to yield new insights into the molecular understanding of microphase separation. Moreover, by tuning the surface chemistry of the nanopore to mimic hydrophilic and hydrophobic porous materials, we investigated the role of the surface chemistry on the demixing.

## Results

### Force field refinement

To be in line with experiment, the confining medium corresponds to a cylindrical silica nanopore of radius of 24 Å. The models and the computational procedure are given in the Methods section. Adsorption isotherms were modeled using Monte Carlo simulation in the grand canonical statistical ensemble (GCMC). Liquid–liquid and liquid–solid interactions (Utot) were taken into account by considering the electrostatic (Uelec) and the van der Waals interactions from the Lennard-Jones (LJ) model (ULJ) such that $$U_{{\mathrm{tot}}} = U_{{\mathrm{elec}}} + U_{{\mathrm{LJ}}} = \mathop {\sum}\nolimits_i^{N - 1} \mathop {\sum}\nolimits_{j \ne i + 1}^N q_iq_j/4\pi \varepsilon _0r_{ij} + 4\varepsilon _{ij}\left[ {\left( {\sigma _{ij}/r_{ij}} \right)^{12} - \left( {\sigma _{ij}/r_{ij}} \right)^6} \right],$$ where qi is the partial charge of i atom, N the number of particles, rij the distance between i and j atoms, εij the depth of the potential well between i and j, and σij refers to the Van der Waals radius. To reproduce the adsorption isotherms of the pure components the solid–liquid interactions were refined. The OPLS (Optimized Potentials for Liquid Simulations) force field was used to model the TOL and TBA molecules. Indeed, it was shown that the thermodynamic properties of TBA/TOL mixtures were closely reproduced from the OPLS model.39 Whereas the intramolecular contributions (bonds, bending, and dihedral angles) were conserved as original, the partial charges were calculated using ab-initio calculations. Silica material was modeled by considering the ClayFF force field.40 All details are provided in the Methods section. To refine the LJ parameters ruling the solid–liquid interactions, we began to predict the adsorption isotherm by considering the initial force fields. Concerning the TBA adsorption through nanoporous silica, Fig. 2 shows a qualitative agreement between simulation and experiment that highlights the quality of ClayFF and OPLS force fields and their combining using the Lorentz-Berthelot mixing rules ($$\varepsilon _{ij} = \sqrt {\varepsilon _{ii}\varepsilon _{jj}}$$ and σij = (σii + σjj)/2). As shown in Fig. 2 the predicted adsorption isotherm of toluene strongly differs with experiment. To improve the silica–TOL interactions we adapted the procedure discussed in ref. 41 initially developed to derive the coarse-grained parameters from atomistic simulations. The LJ parameters have been optimized by minimizing the value of the function $$F = \frac{1}{n}\mathop {\sum}\limits_{i = 1}^n {\frac{{f_i^{sim} - f_i^{exp}}}{{s_i^2}}}$$ according to a procedure detailed in ref. 41. In this relation, si is the estimated statistical uncertainty, $$f_i^{sim}$$ and $$f_i^{exp}$$ are the values of the ith properties on n. In this work n = 2 and corresponds to the enthalpy of adsorption and adsorbed amount at low partial pressure. The minimum condition of F is that every partial derivative must be zero.41 Optimized LJ parameters are provided in Table 1. Figure 2 shows that the predicted adsorption isotherms are in fair agreement with those obtained from experiments.31 Interestingly, Fig. 2 exhibits that the capillary condensation pressure is qualitatively reproduced for both TOL and TBA components. These results allowed us to be confident in these refined interactions.

### Molecular dynamics

From GCMC simulation the numbers of TOL and TBA molecules were calculated for 7 compositions (xTBA = 0, 0.24, 0.49, 0.51, 0.71, 0.83, and 1.0). Molecular dynamics (MD) simulations were subsequently carried out. As shown in Fig. 1b, for both pure components, a layering structure was observed due to the confinement effect at the nanometric scale.5 Whereas three layers are observed for both cases, the interfacial TBA layer seems closer to the silica surface, suggesting that the attractive interactions between the silica nanopore and TBA molecules are stronger. As shown in Fig. 3, the hydroxide groups (OH) of the TBA molecules point towards the silica surface. The driving force of this interfacial anchoring is probably the result of the strong hydrogen bonding (HB) network between porous silica and TBA molecules. This point will be discussed later. Interestingly, as shown in Fig. 3, this HB anchoring leads to a peculiar organization such that hydrophobic and hydrophilic (HB) domains are disposed in alternating succession. These results should probably impact the molecular structure of confined mixture. The density profile of the SiOH groups is also reported in Fig. 3. As shown in Fig. 3, the SiOH profile begins from 11.5 Å and presents a peak around 14 Å, highlighting the roughness of the surface. Calculation of the accessible volume by probing the porous volume by an atom of Argon allowed us to approximatively evaluate the pore radius at 12 Å.

From xTBA = 0.24 to 0.83, Fig. 4 (from Fig. 4a–d) shows that TBA molecules were adsorbed close to the interface constituting a first layer, whereas the toluene molecules are located right after. For xTBA = 0.83, the second and third layers correspond to a homogeneous mixture phase reminiscent of the bulk medium. Very interestingly, Fig. 4 highlights the progressive development of two phases as the TBA concentration decreases. Indeed, for xTBA = 0.71, the TBA molecules are almost only located close to the silica surface in the first adsorbed layer without TOL molecules. For xTBA = 0.49 that corresponds to the concentration where the core–shell organization was observed experimentally, the microphase separation is nearly complete. Interestingly, the distance between peaks of TOL and TBA seems to decrease as a function of the decrease in molar fraction in TBA. Indeed, whereas in Fig. 4a, b the distance between the interfacial TOL and TBA density (dTBA−TOL) is ~4 Å, the distance is greatly reduced to just ~1 Å for xTBA = 0.49 and xTBA = 0.24. The difference in the dTBA−TOL as a function of the TBA molar fraction is due to the progressive increase in TOL. Indeed, for the TBA fraction lower than 0.5 corresponding to an unsaturated solid interface in TBA molecules, the SiOH surface can then adsorb the TOL molecules. The progressive adsorption in TOL molecules at the silica surface is then at the origin of the decrease of dTBA−TOL with the increase in TOL fraction and then is at the origin of the interpenetration of TOL molecules. That can be observed from the atomic density in Fig. 5 where this progressive interpenetration as a function of xMeOH is well evidenced. Indeed, from Fig. 5a (xTBA = 0.24) to Fig. 5d (xTBA = 0.83), the decrease in dTBA−TOL and the increase in the interpenetration region (dashed circles in Fig. 5) are well established.

We report in Fig. 6a, b the two-dimensional densities according to x and y directions for xTBA = 0.71, for TBA and TOL. As shown in Fig. 6, the demixing follows the cylindrical symmetry and a core–shell organization is highlighted with the TBA close to the interface (shell) and TOL at the center of the nanopore (core) that is in good agreement with the experiment.15 To ensure the microphase separation from a thermodynamics point of view we computed the local surface tension (γ(r))42 along the radial distance. Local surface tension expresses the surface tension through the radial r-dependent local components of the pressure tensor. Local surface tension is a local interfacial property related to the difference between the normal (Pn) and tangential (Pt) pressure components such that γ(r) = Pn(r) − Pt(r) with $$\gamma = {\int} \gamma (r)dr = {\int} {P_n} (r) - P_t(r)dr$$. The mechanical equilibrium involves that the normal pressure has to be constant even through the interface, whereas the tangential component is similar than the normal one in bulk phase and presents a negative peak at the interface highlighting a region under tension. Therefore, the difference Pn(r) − Pt(r) is null in bulk phase and presents a positive peak at the interface. Even if its definition is arbitrary (how the energy between a pairwise interaction is distributed between both particles), its calculation allows us to highlight the presence of an interface. Calculation of γ(r) is detailed in the Methods section. As shown in Fig. 6c, the local surface tension presents two peaks. Actually, a peak in surface tension exhibits an interface under tension and corresponds to an interface between two phases. From 0 to 8 Å no tension is observed and the local surface tension is close to 0 mN m−1 as expected for a homogeneous liquid phase. Figure 6c emphasizes then that two interfaces identified by the first peak in γ(r) located at 10.5 Å corresponds to the tension radius of surface of tension connected to the TBA–TOL liquid–liquid interface, and the another at 11.5 Å related to the liquid–solid interface.

Figure 7a reports the profiles of the surface tension and the density of TBA, TOL, and the atoms of silica material. As shown in Fig. 7a, both peaks of tension are located close to the liquid–liquid and liquid–solid Gibbs dividing surfaces. The Gibbs model for the boundary between two phases consists of considering the finite interfacial region as a dividing surface which acts as a third phase of zero volume in which some magnitudes of physical properties such as density change abruptly. Local surface tension and density profile calculations then unambiguously highlight the phase separation of two confined TBA and TOL liquids at the nanoscale. The formation of a first TBA layer anchored at the silica surface is induced by the strong hydrogen bonds between the silanol groups SiOH of the silica material and the OH groups of the TBA molecules. Indeed, hydrogen bond numbers were computed by considering a geometrical criterium based on the quantum consideration43 where two molecules are chosen as being hydrogen bonded only if their inter-oxygen distance is less than 3.5 Å, and simultaneously the oxygen–hydrogen is less than 2.5 Å. The profile of the hydrogen bond number (nHB) per molecule was then reported in Fig. 7b for xTBA = 0.24. As shown in Fig. 7b, a peak of nHB was observed close to the silica surface beyond the bulk value. As observed in Fig. 7b, this peak is the result of two contributions, the HBs between silica wall and the OH groups of TBA and HB between TBA molecules. This highlights a strong hydrogen bonding network between TBA and silica material that is at the origin of the core–shell organization. To ensure this hypothesis, MD simulations through a weakly hydrophilic silica nanopore (WH) was carried out. Figure 8a shows that the weak ability of WH to establish interfacial hydrogen bonds between the silanol groups and the TBA molecules explains the absence of microphase separation. This bears out the fact that the interfacial anchoring from HB is the driving force to observe a microphase segregation. Interestingly, Fig. 8b exhibits that the carbon atoms of TBA and TOL are preferentially adsorbed in comparison with the hydroxide groups of TBA that underlines a hydrophobic anchoring related to the interactions between silicon and carbon atoms. It seems then that the strong interfacial hydrogen bonding network between TBA molecules and the silica material rules the preferential adsorption of TBA at the interface. To strengthen this result, MD simulation of confined TBA/TOL mixture into a pure hydrophilic nanopore for a TBA concentration of xTBA = 0.24 was carried out. The ideal hydrophilic nanopore corresponds to a water nanotube (WNT) and details on pore building and force field are provided in the Methods section. As shown in Fig. 7c, TBA molecules are preferentially adsorbed at the silica surface given the strong hydrogen bonds between TBA and water molecules leading to a core–shell structure with a homogeneous TBA/TOL mixture at the center of pore. These results indicate that the highly hydrophilic surface induces a microphase separation of TBA/TOL mixture and could also suggest that a hydrophobic surface is not able to discriminate TBA and TOL molecules because TOL is fully hydrophobic. However, Fig. 7d shows that a carbon nanotube (CNT) is also capable of separating TBA and TOL by forming a core–shell structure such that TOL and TBA molecules are respectively preferentially located at the CNT surface and at the center of pore. As shown in Fig. 9, this separation is the result of a commensurate organization due to stacking interactions between benzenic cycles of CNT and TOL molecules. To go a little further, the curvature effect was also investigated. Indeed, as often shown, the curvature induced by the cylindrical geometry can drastically impact the physical properties in relation to the slitlike confinement.17,18 We then carried out MD simulation of a TBA/TOL mixture with a TBA concentration of xTBA = 0.5 confined through two silica slitlike pores such that the confinement in term of confined volume was conserved (two silica walls separated by a distance of 24 Å as illustrated in Fig. 10a). As observed in Fig. 10b, segregation between both liquids is observed highlighting that the separation is rather ruled by the interactions between the silica surface and TBA than the pore geometry. Eventually, pore size effect was also investigated and MD simulations of confined TBA/TOL mixture with a TBA concentration of xTBA = 0.5 confined in a silica nanopore of radius of 6 Å and 18 Å were carried out. As shown in Fig. 11, the microphase separation and the core–shell organization are still observed that corroborates that the driving force is connected to the interfacial anchoring by hydrogen bonds allowing that preferential adsorption of TBA molecules. Additionally, Morineau and colleagues38 have recently shown, from Neutron scattering, that the core–shell organization was also observed with a pore radius of 43 Å. Therefore, the solid–liquid interactions and more especially the solid–liquid hydrogen bonds seem to overcome the bulk ones whatever the pore diameter leading to the microphase separation and the core–shell structure. Eventually, by comparing the structures obtained from the silica nanotube and from the CNT material (smoothness nanopore), we show that the core–shell organization is independent of the surface roughness. Although the core–shell structure is indirectly correlated to the capillary condensation, the main mechanism is related to the wetting on the silica surface and then to the interfacial interactions between TBA/TOL and the SiOH. Therefore, the microphase separation at the nanoscale cannot be related to any putative de-wetting phenomenon, nor it could be understood by considering solely the capillary condensation of the gas. Indeed, a vision of the intermolecular interactions and then the strength of wetting is needed to capture the origin of the microphase, what was performed in this work.

## Discussion

In this work, the TBA/TOL mixture that is homogeneous in the bulk liquid phase was confined at the nanoscale into cylindrical pores. We showed that the nano-confinement can induce, under a few conditions, microphase separation of binary mixture with a core–shell organization in line with Neutron scattering experiments. Phase separation was evidenced by highlighting both liquid–liquid and liquid–solid interfaces by means of the profiles of the density of center of mass and the local surface tension. By computing the radial profile of the hydrogen bonds number and by tuning the hydrophilicity of the surface of the silica nanopore, we established that the phase separation was ruled at the molecular scale by the strong hydrogen bonds between TBA molecules and the silica material that favors the HB anchoring. Confinement through a carbon nanotube showed an inverse core–shell structure with TOL anchored at the solid surface highlighting an hydrophobic anchoring due to a commensurate organization implying the benzenic cycles stacking. These results shed light on the possibility to control the nanostructure of multicomponent fluids under confinement at the nanoscale by tuning the surface chemistry of nanopore. The nanoconfined phase separation seems to be independent to the pore size and is essentially connected to the strength of pore–fluid interactions. As shown in Fig. 8a, the microphase separation has not been established from a weakly hydrophilic pore, whereas the core–shell structure was also recovered from a hydrophobic matrix (CNT), highlighting that the phase separation is unambiguously connected to the liquid–solid interactions. Determination of a theoretical model predicting the phase separation as a function of the strength of the solid–liquid interactions could be possible by tuning progressively (thermodynamic integration) the pore–liquid interactions (Lennard-Jones parameters and partial charges). However, it will be difficult to connect this model with the realistic materials where the interactions are too complexes. We think that the so-developed strategy consisting of the refining of the pore–wall interactions by a comparison with experiment is probably the most relevant approach. Qualitatively, the microphase separation can be then predicted if one considers a mixture of an amphiphilic component and a hydrophobic one confined in pure hydrophilic and hydrophobic pores. Although the development of a theoretical model to predict the phase separation in a nanoconfined pore seems to be difficult to connect with realistic materials, it is possible to predict the optimal conditions to observe the core–shell organization that means the maximal number of TBA molecules anchored at the surface. Indeed, the pore volume can be calculated as $${V}_{{\mathrm{pore}}} = \pi R_{{\mathrm{pore}}}^2L_z$$ where Rpore is the pore radius of the nanopore while Lz is the height of the nanotube, the volume of the core region is $${V}_{{\mathrm{core}}} = \pi R_{{\mathrm{core}}}^2L_z$$ such that Rcore is the radius of the core region. The volume of the interfacial zone, i.e., the shell can be then evaluated as $${V}_{{\mathrm{shell}}} = \pi L_z(R_{{\mathrm{pore}}}^2 - R_{{\mathrm{core}}}^2)$$ such that Rcore = Rpore − eshell with eshell corresponding to the thickness of the interfacial shell. eshell is computed from the density profile. From the density profile calculations, we found that eshell = 4 Å. If we consider the bulk density as the interfacial density, the number of TBA molecules interfacially anchored can be calculated as NTBA,s = Vshell.ρTBA,bulk. With Rpore = 12 Å and eshell = 4 Å, an interfacial saturation of NTBA,s = 57 was predicted. Therefore, for the TBA molar fraction corresponding to a number of TBA molecules less than NTBA,s, the TBA will be fully anchored and a full core–shell organization will be observed (i.e., xTBA = 0.71, 0.49, and 0.24). Beyond NTBA,s, TBA molecules will also be located in the center of pore and in the intermediary layer. From GCMC simulations, we found that the solid interface is saturated for NTBA,s = 56 molecules that bears out our previous structural model. In Fig. 4a, the molar fraction xTBA = 0.83 corresponds to 79 TBA and 16 toluene molecules. Therefore, other TBA molecules (79–57) are distributed at the center of pore and in the intermediate layer. As shown in Fig. 1b, the first layer is located between 0 and 3 Å, the second between 3 and 7.5 Å, while the last is located between 7.5 and 12 Å. As the layer size corresponds to the molecule size and that TOL molecule is higher than TBA one, 3.5 Å vs 3.0 Å, it appears that TOL molecules have no place at the center of pore contrary to the intermediate layer where the layer size is 4.5 Å.

Eventually, the microphase separation raises exciting questions about its impact on the MD and the liquid flow in nanochannel, as suggested by the recent report on multiple glassy dynamics.37 Further insight into this aspect could be attained from ongoing experimental spectroscopy and molecular simulation studies.

## Methods

### Models

Silica cylindrical nanopore with a hydrophilic surface was managed by applying the procedure proposed by Bródka and Zerda.44 We generated a cylindrical cavity along the z axis of the cubic silica cell of 35.7 Å by removing the atoms within a cylinder of diameter (D) 24 Å. From their coordination numbers, we distinguished bridging oxygen (Ob) bonded to two silicon atoms from non bridging oxygens (Onb) bonded to only one silicon and bonded to one hydrogen atom (Hnb). An iterative procedure of atom (O and Si) removal was applied until only tetra-coordinated silicon atoms, bonded to a maximum of two Onbs, were present in the structure. Finally, non-bridging oxygens were saturated with hydrogen atoms to form surface hydroxyl groups. This procedure leads to a realistic description of the irregular inner surface of the porous silicate and of the interfacial interactions between the fluid and the matrix (see Fig. 2). The inner surface coverage of silanol groups was about 7.5 nm−2, which corresponds to highly hydrated protonated silica pore.44,45 Although the silica matrix was subsequently kept rigid, rotation around the Si-O bond of the hydroxyl groups was allowed from the SHAKE constraints algorithm,10,41,46 where the distance between the oxygen and hydrogen atoms are kept fixed at 1.09 Å.

Intermolecular interactions are the sum of both electrostatic and dispersive–repulsive Lennard-Jones contributions. Silica framework was modeled using the ClayFF force field developed by Cygan et al.40 Indeed, it has been shown that the potential developed by Bródka and Zerda44 was too weak and underestimated the adsorbed amount at low pressure.39,47 TBA and TOL were modeled by means of the non-polarizable flexible OPLS all atoms (AA) force field.48 Indeed, the OPLS-AA models of TBA and TOL were found to well reproduce density and structure in liquid bulk phase.48 Whereas the original Lennard-Jones parameters were initially conserved, the partial charges were calculated from ab-initio calculations. Calculation of the partial charges was carried out on the basis of 6-311G(d,p) Gaussian-type basis set. In the first time, geometry was optimized, whereas in the second stage the partial charges were calculated from the CHELPG (CHarges from ELectrostatic Potentials using a Grid-based) method.49 These calculations were performed using the Gaussian code.50

Force field parameters of silica nanopore, TOL, and TBA molecules are provided in the FIELD.txt file, the input file of DL_POLY software.51 Crossed van der Waals interactions between TOL and TBA were calculated using the Lorentz-Berthelot mixing rules. The LJ parameters between the silica nanopore and TOL were optimized to qualitatively reproduce the experimental isotherm by following the procedure developed in ref. 41. The so-optimized interactions between TOL molecules and silica material are provided in Table 1. Force field parameters of silica material, TOL, and TBA and crossed interactions can be found in the FIELD.txt file. Armchair CNT of radius 12 Å with a pore length of 100 Å was modeled by considering the uncharged force field developed by Werder et al.52 WNT was built by carving a cylindrical nanopore of radius 12 Å into an equilibrated cubic water box with a length box of 59 Å. Water molecules were modeled by considering the TIP4P/2005 model53 and were considered as frozen. Weakly hydrophilic silica membrane was also modeled by combining the Universal force field (UFF)54 for describing oxygen and hydrogen atoms, while silicon atoms were described using the DREIDING force field.55 All details are provided in ref. 39. Crossed Lennard-Jones parameters were calculated by considered the Lorentz-Berthelot mixing rules.

### MD simulation

MD simulations were performed using a time step of 0.002 ps to sample 10 ns (acquisition phase). The equilibration time corresponds to 10 ns. All MD simulations have been carried out with the DL_POLY package51 using the combination of the velocity-Verlet algorithm and the Nose-Hoover thermostat.56 Numbers of confined molecules of pure TBA and TOL and binary TBA/TOL were calculated using GCMC simulation at the saturation pressure vapor and 308 K. Compositions in TBA and TOL are provided in Table 2.

### Monte Carlo simulation

Monte Carlo simulations were performed in the grand canonical statistical ensemble. Periodic boundary conditions were applied in the three directions. Silica nanopore was considered as rigid and only the hydrogen of silanol can rotate from an angular move implying SiOH groups.44 Each cycle consisted of N randomly selected moves with fixed probabilities: translation of the center of mass of a randomly chosen, rotation of a randomly selected around its center of mass, change of the internal conformation using the configurational bias regrowth move57 and insertion/deletion trial move. The frequencies of each type of move are 0.20 for translation, 0.20 for rotation, 0.20 for the change of the conformation, and 0.4 for insertion/deletion. GCMC simulations consisted of 700,0000 cycles.

### Surface tension calculation

Surface tension was computed using the non-exponential test-area method TA258 based upon a thermodynamic route.59 This method expresses the surface tension as a change in the free energy (F) for an infinitesimal change in the surface area performed in the constant-NVT ensemble. This infinitesimal change in the area is performed throughout a perturbation process for which the perturbed system (state A + ΔA and noted 1) is obtained from an infinitesimal change ΔA of the area A of the reference system (noted 0). The box dimensions $$( {L_x^{(1)},L_y^{(1)},L_z^{(1)}} )$$ in the perturbed systems are changed using the following transformations $$L_x^1 = L_x^{(0)}\sqrt {1 \pm \xi } ,L_y^{(1)} = L_y^{(0)}\sqrt {1 \pm \xi } ,L_z^{(1)} = L_z^{(0)}/(1 \pm \xi )$$ where ξ → 0. The area of a cylindrical interface is A = 2πReLz and ΔA = A [(1 ± ξ)−1/2− 1], where Re is the radius equimolar dividing surface. These transformations conserve the volume of the box in the perturbed state. This means that it is possible to express the surface tension as a difference between the perturbed and reference states. U(0)(rN) and U(1)(rN) are the configurational energies of the systems with an area A and a configurational space rN, and an area A + ΔA and a configurational space rN, respectively.

$$\gamma = \left( {\frac{{\partial F}}{{\partial A}}} \right)_{N,V,T} = \mathop {{\lim }}\limits_{\xi \to 0} \left\langle {\left( {\frac{{(U^{(1)}({\mathbf{r}}^{\prime N}) - U^{(0)}({\mathbf{r}}^N))}}{{{\mathrm{\Delta }}A}}} \right)} \right\rangle _0,$$
(1)

where <···> 0 indicates that the average is carried out over the reference state. A local version of Eq. (1) can be obtained by assuming a decorrelation of the cylindrical slabs.60

$$\gamma (R_k) = \mathop {{\lim }}\limits_{\xi \to 0} \left\langle {\mathop {\sum}\limits_{i = 1}^N {\mathop {\sum}\limits_{j > i}^N H } (R_{ik})\left( {\frac{{(u_{R_k}^{(1)}({\mathbf{r}}_{ij}) - u_{R_k}^{(0)}({\mathbf{r}}_{ij}))}}{{{\mathrm{\Delta }}A}}} \right)} \right\rangle _0,$$
(2)

where k is the index of the cylindrical slab, Rk the radius of the cylindrical shell, $$u_{R_k}$$ is the energy of the kth element, H(Rik) is the Heaviside function with H(Rik) = 1 for Ri = Rk and 0 otherwise and rij is the distance between i and j molecules.

## Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

## References

1. 1.

Granick, S. Motions and relaxations of confined liquids. Science 253, 1374 (1991).

2. 2.

Klein, J. & Kumacheva, E. Confinement-induced phase transitions in simple liquids. Science 269, 816 (1995).

3. 3.

Bruni, F., Ricci, M. A. & Soper, A. K. Water confined in vycor glass. I. A neutron diffraction study. J. Chem. Phys. 109, 1478 (1998).

4. 4.

Soper, A. K., Bruni, F. & Ricci, M. A. Water confined in vycor glass. ii. excluded volume effects on the radial distribution functions. J. Chem. Phys. 109, 1486 (1998).

5. 5.

Morineau, D. & Alba-Simionesco, C. Liquids in confined geometry: how to connect changes in the structure factor to modifications of local order. J. Chem. Phys. 118, 9389 (2003).

6. 6.

Alcoutlabi, M. & McKenna, G. B. Effects of confinement on material behaviour at the nanometre size scale. J. Phys. Condens. Matter 17, R461 (2005).

7. 7.

Coasne, B., Jain, S. K. & Gubbins, K. Freezing of fluids confined in a disordered nanoporous structure. Phys. Rev. Lett. 97, 105702 (2006).

8. 8.

Alba-Simionesco, C. et al. Effect of confinement on freezing and melting. J. Phys. Condens. Matter 18, R15 (2006).

9. 9.

Ghoufi, A. et al. Molecular simulations of confined liquids: an alternative to the grand canonical Monte Carlo simulations. J. Chem. Phys 134, 074104 (2011).

10. 10.

Ghoufi, A., Hureau, I., Morineau, D. & Lefort, R. Hydrogen-bond-induced supermolecular assemblies in a nanoconfined tertiary alcohol. J. Phys. Chem. C 115, 17761 (2011).

11. 11.

Ghoufi, A., Szymczyk, A., Renou, R. & Ding, M. Calculation of local dielectric permittivity of confined liquids from spatial dipolar correlations. Europhys. Lett. 99, 37008 (2012).

12. 12.

Zhu, H., Ghoufi, A., Szymczyk, A., Balannec, B. & Morineau, D. Anomalous dielectric behavior of nanoconfined electrolytic solutions. Phys. Rev. Lett. 109, 107801 (2012).

13. 13.

Ghoufi, A., Hureau, I., Morineau, D., Renou, R. & Szymczyk, A. Confinement of tert-butanol nanoclusters in hydrophilic and hydrophobic silica nanopores. J. Phys. Chem. C 117, 15203 (2013).

14. 14.

Guo, X.-Y., Watermann, T. & Sebastiani, D. Local microphase separation of a binary liquid under nanoscale confinement. J. Phys. Chem. B 118, 10207 (2014).

15. 15.

Abdel Hamid, A. R. et al. Microphase separation of binary liquids confined in cylindrical pores. J. Phys. Chem. C 120, 9245 (2016).

16. 16.

Hummer, G., Rasaiah, J. C. & Noworyta, J. P. Water conduction through the hydrophobic channel of a carbon nanotube. Nature 414, 188 (2014).

17. 17.

Falk, K., Joly, L., Sedlmeier, F., Netz, R. R. & Bocquet, L. Molecular origin of fast water transport in carbon nanotube membranes: superlubricity versus curvature dependent friction. Nano Lett. 10, 4067 (2010).

18. 18.

Ghoufi, A., Szymczyk, A. & Malfreyt, P. Ultrafast diffusion of ionic liquids confined in carbon nanotubes. Sci. Rep. 6, 28518 (2016).

19. 19.

Kityk, A. V. et al. Continuous paranematic-to-nematic ordering transitions of liquid crystals in tubular silica nanochannels. Phys. Rev. Lett. 18, 187801 (2008).

20. 20.

Gruener, S. & Huber, P. Spontaneous imbibition dynamics of an n-alkane in nanopores: evidence of meniscus freezing and monolayer sticking. Phys. Rev. Lett. 103, 174501 (2009).

21. 21.

Grigoriadis, C. et al. Suppression of phase transitions in a confined rodlike liquid crystal. ACS Nano 5, 9208 (2011).

22. 22.

Nomura, K. et al. Evidence of low-density and high-density liquid phases and isochore end point for water confined to carbon nanotube. Proc. Natl Acad. Sci. USA 114, 4066 (2017).

23. 23.

Ballenegger, V. & Hansen, J.-P. Dielectric permittivity profiles of confined polar fluids. J. Chem. Phys. 122, 114711 (2005).

24. 24.

Bonthuis, D. J., Gekle, S. & Netz, R. R. Dielectric profile of interfacial water and its effect on double-layer capacitance. Phys. Rev. Lett. 107, 166102 (2011).

25. 25.

Zhang, C., Gygi, F. & Galli, G. Strongly anisotropic dielectric relaxation of water at the nanoscale. J. Phys. Chem. Lett. 4, 2477 (2013).

26. 26.

Emile, O., Emile, J. & Ghoufi, A. Influence of the interface on the optical activity of confined glucose films. J. Colloid. Interface Sci. 477, 103 (2016).

27. 27.

Palomarez-Baez, J. P., Panizon, E. & Ferrando, R. Nanoscale effects on phase separation. Nano Lett. 17, 5394 (2017).

28. 28.

Keblinsky, P., Ma, W. J., Maritan, A., Koplik, J. & Banavar, J. R. Molecular dynamics of phase separation in narrow channels. Phys. Rev. E 47, 2265 (1993).

29. 29.

Gelb, L. D. & Gubbins, K. E. Liquid-liquid phase separation in cylindrical pores: Quench molecular dynamics and Monte Carlo simulation. Phys. Rev. E 56, 3185 (1997).

30. 30.

Gelb, L. D. & Gubbins, K. E. Studies of binary liquid mixtures in cylindrical pores: phase separation, wetting and finite-size effects from Monte Carlo simulations. Phys. A Stat. Mech. Appl. 244, 112 (1997).

31. 31.

Dutta, S. et al. Thermodynamics of binary gas adsorption in nanopores. Phys. Chem. Chem. Phys. 18, 24361 (2016).

32. 32.

Krycka, K. L., Dura, J. A., Langston, L. J. & Burba, C. M. Nanoconfinement-induced phase segregation of binary benzene cyclohexane solutions within a chemically inert matrix. J. Phys. Chem. C 122, 7676 (2018).

33. 33.

Harrach, M. F., Drossei, B., Winschel, W., Gutmann, T. & Buntkowsky, G. Mixtures of isobutyric acid and water confined in cylindrical silica nanopores revisited: A combined solid-state nmr and molecular dynamics simulation study. J. Phys. Chem. C 119, 28961 (2015).

34. 34.

Muthulakshmi, T., Dutta, D., Maheshwaru, P. & Pujari, P. K. Evidence for confinement induced phase separation in ethanol–water mixture: a positron annihilation study. J. Phys. Condens. Matter 30, 025001 (2018).

35. 35.

Schmitz, R., Muller, N., Ullmann, S. & Vogel, M. A molecular dynamics simulations study on ethylene glycol-water mixtures in mesoporous silica. J. Chem. Phys. 145, 104703 (2016).

36. 36.

You, F., Yu, Y. & Gao, G. Structures and adsorption of binary hard-core yukawa mixtures in a slitlike pore: grand canonical monte carlo simulation and density-functional study. J. Chem. Phys. 123, 114705 (2005).

37. 37.

Abdel Hamid, A. R. et al. Multiple glass transitions of microphase separated binary liquids confined in mcm-41. J. Phys. Chem. C 120, 11049 (2016).

38. 38.

Mhanna, R. et al. More room for microphase separation: an extended study on binary liquids confined in sba-15 cylindrical pores. J. Chem. Phys. 146, 024501 (2017).

39. 39.

Essafri, I., Courtin, J. & Ghoufi, A. Numerical evidence of heterogeneity and nanophases in a binary liquid confined at the nanoscale. Mol. Sim. 44, 728 (2018).

40. 40.

Cygan, R. T., Liang, J. J. & Kalinichev, A. G. Molecular models of hydroxide, oxyhydroxide, and clay phases and the development of a general force field. J. Phys. Chem. B 108, 1255 (2004).

41. 41.

Ghoufi, A., Morineau, D., Lefort, R. & Malfreyt, P. Toward a coarse graining/all atoms force field (cg/aa) from a multiscale optimization method: an application to the mcm-41 mesoporous silicates. J. Chem. Theory Comput. 6, 3212 (2010).

42. 42.

Ghoufi, A. & Malfreyt, P. Importance of the tail corrections on surface tension of curved liquid-vapor interfaces. J. Chem. Phys. 146, 084703 (2017).

43. 43.

Luzar, A. & Chandler, D. Effect of environment on hydrogen bond dynamics in liquid water. Phys. Rev. Lett. 76, 928 (1996).

44. 44.

Bródka, A. & Zerda, T. W. Properties of liquid acetone in silica pores: molecular dynamics simulation. J. Chem. Phys. 104, 6319 (1996).

45. 45.

Puibasset, J. & Pellenq, R. J.-M. Water adsorption in disordered mesoporous silica (vycor) at 300k and 650k: a grand canonical Monte Carlo simulation study of hysteresis. J. Chem. Phys. 122, 094705 (2005).

46. 46.

Ryckaert, J.-P., Ciccotti, G. & Berendsen, H. J. C. Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of n-alkanes. J. Comp. Phys. 23, 327 (1977).

47. 47.

Pafong, E., Geske, J. & Drossel, B. On the influence of the intermolecular potential on the wetting properties of water on silica surfaces. J. Chem. Phys. 145, 114901 (2016).

48. 48.

Jorgensen, W. L., Maxwell, D. S. & Rives, J. Tirado Development and testing of the opls all-atom froce field on conformational energetics and properties of organic liquids. J. Am. Chem. Soc. 118, 11225 (1996).

49. 49.

Breneman, C. M. & Wiberg, KennethB. Determining atom-centered monopoles from molecular electrostatic potentials. the need for high sampling density in formamide conformational analysis. J. Comp. Chem. 11, 361 (1990).

50. 50.

Frisch, M. J. et al. Gaussian Development Version, Revision I.04+ (Gaussian Inc., Wallingford, 2014).

51. 51.

Forester, T. R. & Smith, W. DLPOLY, CCP5 Program Library (Daresbury Lab., Warrington, 2004).

52. 52.

Werder, T., Walther, J., Halicioglu, R., Halicioglu, T. & Koumoutsakos, P. On the water-carbon interaction for use in molecular dynamics simulations of graphite and carbon nanotubes. J. Phys. Chem. B 107, 1345 (2003).

53. 53.

Abascal, J. L. F. & Vega, C. A general purpose model for the condensed phases of water: Tip4p/2005. J. Chem. Phys. 123, 234505 (2005).

54. 54.

Rappé, A. K., Casewit, C. J., Colwell, K. S., Goddard, W. A. III & Skiff, W. M. UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations. J. Am. Chem. Soc. 114, 7863 (1992).

55. 55.

Mayo, S. L., Olafson, B. D. & Goddard, W. A. III Dreiding: a generic force field fo molecular simulations. J. Phys. Chem. 94, 8897 (1990).

56. 56.

Mechionna, S., Ciccotti, G. & Holian, B. L. Hoover NPT dynamics for systems varying in shape and size. Mol. Phys. 78, 533 (1993).

57. 57.

Chen, B., Potoff, J. J. & Siepmann, J. I. Monte carlo calculations for alcohols and their mixtures with alkanes. Transferable potentials for phase equilibria. 5. United-atom description of primary, secondary, and tertiary alcohols. J. Phys.Chem. B 105, 3093 (2001).

58. 58.

Ghoufi, A. & Malfreyt, P. Calculation of the surface tension and pressure components from a non-exponential perturbation method of the thermodynamic route. J. Chem. Phys. 136, 024104 (2012).

59. 59.

Gloor, G. J., Jackson, G., Blas, F. J. & de Miguel, E. Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials. J. Chem. Phys. 123, 134703 (2005).

60. 60.

Ibergay, C. et al. Molecular simulations of the n-alkane liquid-vapor interface: interfacial properties and their long range corrections. Phys. Rev. E 75, 051602 (2007).

## Author information

Authors

### Contributions

A.G. and D.M. elaborated the project and A.G. and I.E. performed molecular dynamics simulations. A.G., D.M., and I.E. contributed to the scientific discussions and the preparation of the manuscript.

### Corresponding author

Correspondence to Aziz Ghoufi.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Essafri, I., Morineau, D. & Ghoufi, A. Microphase separation of a miscible binary liquid mixture under confinement at the nanoscale. npj Comput Mater 5, 42 (2019). https://doi.org/10.1038/s41524-019-0179-y

• Accepted:

• Published:

• ### Extension and Limits of Cryoscopy for Nanoconfined Solutions

• Benjamin Malfait
• , Alban Pouessel
• , Aîcha Jani
•  & Denis Morineau

The Journal of Physical Chemistry Letters (2020)

• ### Dynamic Heterogeneities in Liquid Mixtures Confined in Nanopores

• Ramona Mhanna
• , Pierre Catrou
• , Sujeet Dutta
• , Ronan Lefort
• , Ilham Essafri
• , Aziz Ghoufi
• , Matthias Muthmann
• , Michaela Zamponi
• , Bernhard Frick
•  & Denis Morineau

The Journal of Physical Chemistry B (2020)

• ### Metallic n‐Type Mg 3 Sb 2 Single Crystals Demonstrate the Absence of Ionized Impurity Scattering and Enhanced Thermoelectric Performance

• Kazuki Imasato
• , Chenguang Fu
• , Yu Pan
• , Max Wood
• , Jimmy Jiahong Kuo
• , Claudia Felser
•  & G. Jeffrey Snyder


text:
Measure between the counting measure and the Lebegue measure - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:34:09Z http://mathoverflow.net/feeds/question/11641 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11641/measure-between-the-counting-measure-and-the-lebegue-measure Measure between the counting measure and the Lebegue measure Sune Jakobsen 2010-01-13T09:01:19Z 2010-01-13T09:24:32Z
There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is there a translation invariant measure m such that for some sets with Lebegue measure 0 the m-measure is infinite and for some sets with infinite counting measure, the m-measure is 0?

I have found one example: m(A)=0 if A is countable, and m(A)=infinite otherwise. So I will require that the measure can take the value 1.

If such a measure exist, can we find a measure between this and the counting measure? and between this and the Lebegue measure? and so on.
 http://mathoverflow.net/questions/11641/measure-between-the-counting-measure-and-the-lebegue-measure/11642#11642 Answer by Thorny for Measure between the counting measure and the Lebegue measure Thorny 2010-01-13T09:22:18Z 2010-01-13T09:22:18Z
Hausdorff measures of dimensions between 0 and 1 are a continuous spectrum of examples.
 http://mathoverflow.net/questions/11641/measure-between-the-counting-measure-and-the-lebegue-measure/11643#11643 Answer by Pete L. Clark for Measure between the counting measure and the Lebegue measure Pete L. Clark 2010-01-13T09:24:32Z 2010-01-13T09:24:32Z
I believe what you are looking for is the $\alpha$-dimensional Hausdorff measure for some $0 < \alpha < 1$.

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# ETech: I Just Don't Trust You: How the Tech Community Can Reinvent Risk Ratings

My favorite conference of the year, ETech kicked off its general sessions today and its looking as stimulating as ever! While the topics covered by the conference have become less hard-core geeky, they have become more green and more broad. Sustainable topics, the environment and becoming better global citizens are just a few of the topics that have been struck this morning. ETech continues to make me think, which is the primary reason I keep coming back for more.

The first session I’d like to share with you was Toby Segaran and Jesper Anderson’s “I Just Don’t Trust You: How the Tech Community Can Reinvent Risk Ratings” presentation. Toby and Jesper posited that the system for rating credit instruments is horribly broken. Right before Lehman Brothers crashed, Moody’s credit rating agency gave Lehman Brothers a AAA A2 credit rating. Moodys immediately down-rated Lehman Brothers after they crashed — a little too late! Jesper and Toby outlined four reasons why the current system fails to do its job:

1. Payments create bad ratings. NRSROs (Nationally Recognized Statistical Rating Organization), like Moody’s, S&P and Fitch all take payments to rate a given financial instrument and this incentivizes these companies to create false ratings. The structural problems that allow these conflict of interest transactions to occur beget a host of moral failures. This behavior amounts to bribery that doesn’t send anyone to jail.
2. Opacity creates bad ratings. Credit rating agencies enjoy patent-like protection in the US, but without the requirement of public disclosure. They can operate on whatever principles they choose, which leaves their rating system nearly meaningless. What does a AAA rating mean? The current system makes an AAA rating no more effective than a gold star that might be awarded in grade-school. A closed consensus model doesn’t explore alternative options and it gives to the question if the current risk models are suitable for doing their job.
3. Lack of ecosystem create bad ratings. Today’s market ignores good predictions only to celebrate them after the fact when the market starts crashing. We need to incorporate changing knowledge of the market into the models and constantly evolve them. Creating an ecosystem that can review financial information in broad daylight and encourage a greater accountability will create a more robust and accurate credit rating system. Its important to remove the incentives for secrecy.
4. Single source of information creates bad ratings. A single viewpoint of financial data is too narrow to do justice in the complex market of credit ratings. Any and all sources of information need to feed into the model for the greatest chance at success. The current model encourages blind thinking, when credit ratings should focus on analyzing as many sources of data as possible.
5. With the problems defined, Toby and Jesper propose the following requirements for a new credit rating system:

• Ratings need to be accessible. Credit ratings need to operate from an information commons. Debt is about trust — a Digg style system won’t work here.
• Ratings need to be open. Everyone needs to be able to see the complete view of the landscape.
• Ratings need to be diverse. Many people need to take part in credit rating. Any new system needs to be built as a Bazaar, not a Cathedral!
• Ratings need to be transparent. Transparency is a vital component in a new system — opacity only hides the shady practices of incumbents and prevents the system from really working.

And rather than ranting without substance, Jesper and Toby have been working on a new system at Freerisk.org that acts as a specialized front end to the FreeBase database. Based on Open Data principles, FreeBase contains public SEC records (among many other data portions, including the MusicBrainz data). FreeBase exposes the SEC records in RDF via a public API, which makes accessing SEC data much easier than before.

Jesper and Toby invite the general public to come and take part in this nascent project — they hypothesize that by exposing the data to more people, anyone can create a risk calculator. And of course, if everyone tried to create a new risk calculator, we’re bound to find new models that work better than the current models that allows financial catastrophes to happen. They conclude: “With a little help, we believe that we can beat the NRSROs!”

If the current financial crisis bothers you and you feel angry enough to help look for a solution, go visit freerisk.org. Thanks for the informative talk, Toby and Jesper!

tags: ,
• Falafulu Fisi

Some financial analysts have blamed Moody’s & other credit rating agencies for their system’s over-reliance on using Monte-Carlo based methods for risk management, which eventually didn’t assess the risks properly.

• Falafulu Fisi

Robert Kaye, note the comment by Nassim Taleb quoted by Toby Segaran:

Nassim Taleb:
“How many of you believe that Sharpe ratio is bull**? How many of you use portfolio theory, optimal portfolio, and all that cr*p?…If someone starts talking about Sharpe, VaR, and all that, throw them out of your office.”

I agree that financial risk management systems have been over-sold/hyped-up by the financial industries, but that doesn’t mean that these methods are crap as indicated by Toby Segaran.

I believe that Toby Segaran has no experienced in the domain of computational finance and I am surprised that a good author like him view financial engineering in this manner as a put down. The Modern Portfolio Theory (MPT) is an undisputed method that is in use today by analysts and although the original model had problems it has improved over the years with new methods that have been made available in the literatures. The inventors of MPT (Sharpe, Makowitz, et al), shared the Economic Nobel Prize of 1990.

These perfomance metrics (VaR, Sharpe, MPT, etc…) are not crap. Nothing in real-life is certain. Models are not 100% correct, because if they are, then there will be no research on financial & economic theory since we know everything. Models are approximations only and not the ultimate representative of reality.

ETech organizers should have invited some quant guys there talk to about what was the cause of the financial meltdown. Toby Segaran is a machine learn expert/researcher which is a completely different domain than computational finance (financial engineering), although machine learning methods have been applied in Technical analysis, it is still a different domain.

• SandipSen

This is a “Emperor’s new clothes” call. Credit Rating agencies & Auditors are still holy cow.However nobody still dares to touch credit rating agencies or their methodology adopted . The truth is that there is no easy risk measurement and management techniques today.To counter this deficiency for mega projects we are using a variance analysis methodology (Refer The Project Management Time Cycle : Time Cycle Module: TCM ISBN 1440493332 available at amazon.)

This enables day to day monitoring of data, risk analysis and measurement, and risk mitigation in projects. It is not difficult to transfer these concepts for the Risk Management of the Banking sector with requisite changes in data collected

Sandip

• Stephen Henderson

This is an interesting idea that has some merit– but I think you have to be careful here not to move from one extreme to another. There is a closed system relying on ‘experts’ and then there is the ‘madness of crowds’.

One might view the current financial catastrophe as a crisis of obfuscation and a lack of transparency (by the ratings agencies)– on the other hand you can view the problem as an asset price bubble created by consensual thinking (by the markets).

Actually the markets are swamped by new information and data about companies every day- and often react instantly and emotionally. This might be just another addition to this herd like moodswings– most hedge funds will analyse this data already.

My own opinion is not that investment banks have failed due to the business they have been doing off-balance sheet– i.e. not stuff in the SEC reports. This is essentially Enronisation of investment banking. Eliminate this shadow banking and then worry about the ratings model later.

• stephen Henderson

My own opinionis that investment banks have failed due to the business they have been doing off-balance sheet….

• http://www.snee.com/bobdc.blog Bob DuCharme

It would be very interesting to see Moody’s give anyone an AAA rating, because that’s a designation from the ratings scale of S&P, their chief competitor.

Before focusing on what is hidden at the ratings agencies, it might be a good idea to get the public parts of what they do straight.

• http://blog.kiwitobes.com Toby Segaran

Just a few clarifications –

Falafulu: Although we opened the abstract with the Taleb quote, the talk was not about VaR or the quality of specific risk-modeling techniques. It was about lack of transparency in models used and infrastructure ideas for opening up risk modeling. Also, if “some quant guys” had submitted an abstract about the crisis, there’s a good chance they would have been invited.

Bob: There’s actually a mistake in the writeup, we stated that Lehman had an “A2″ rating from Moody’s. However, your statement is actually incorrect, Moody’s does give “Aaa” ratings — see http://en.wikipedia.org/wiki/Moody%27s and the citation on that page.

• http://i2pi.com Joshua Reich

As a quant guy I have to say that Toby & Jesper’s work is certainly in the right direction.

It is a shame that that the Money:Tech conference was cancelled this year as it would have been a great venue for discussing these issues.

My primary take on the credit rating mess, and I’ll solely focus on MBS’s – but the arguments apply across the board, is that the core of credit rating technology dates back to the 1970s. While physical computing technology and mathematical techniques have updated, the underlying methods have not.

The entire point of the MBS market was to take uncorrelated risks and package them together to minimize the impact of non-systemic risk. The other point was to create fungible instruments. An individual whole mortgage is not fungible as there are too many parameters to allow for liquidity behind bid/asks. By pooling together loans and creating singular tradeable entities markets can attain depth and liquidity. Trading ensues.

However, people forget that behind these instruments lie whole loans, that while for ‘most of the time’ are dominated by idiopathic risks, can suffer from correlated systemic risks. Much of the mathematics that is used focuses on the system portions of the risk – e.g., prepayment risk as it relates to prevailing interest rates. Quants forgot about the real world and got jacked up on copulae and completely ignored the individuality of the component loans. For a toned down version of Taleb, read Paul Wilmott’s thoughts on this..

And even worse, if you read Frank Portnoy’s papers on the types of correlation models used inside the CRAs – you will see fundamentally stupid assumptions.

My point is that with current computer technology we can build new market designs that support liquidity of what are currently considered unfungible instruments. Look at weatherbill. Look at the work of Robin Hanson in combinatorial markets. New mechanisms allow us to escape the desire to model risk with a single number.

Yes, modern portfolio theory is awesome. But look at the assumptions. We are not homogeneous traders. We have our own risk appetites and world views. We should be able to take the entirety of the data that is available on risky instruments and draw our own conclusions of the maturity, convexity, prepayment risks, etc. with regard to our own cash flow needs and investment goals. And new market designs will allow us to find instruments that best allow us to take positions in line with our own situations to maximize our exposure to our own worldviews.

I do think technologists (as one myself) often take the viewpoint that there is 100% truth in data, and that if only we had all the data, we would be able to predict the future. The future is messy, fat-tailed and dangerous to predict but we can make great strides if we admit to this and stop trying to package ratings as a single one-size-fits-all number.

/rant

NB: Let me plug myself a bit here. For a related discussion, see my post Data Trades Inversely to Liquidity: http://blog.i2pi.com/index.cgi?p=186

• http://bexhuff.com bex

Freerisk won’t ever work…

If this system ever got off the ground, then it would just be one more attack vector for folks to game the system. CEOs always keep some stuff off the books to inflate the apparent value of their company. A system like this will only point them to exactly how to cheat.

Perhaps there needs to be a disincentive for the ratings agencies to cheat? Perhaps there should be laws that AAA bonds can only default 1% of the time… Anybody can be a “ratings agency,” and they can take as many bribes as they want… but if they fall below 1% then they can be sued to oblivion, and may never work in the financial industry again.

I say, let there be 50 ratings agencies, and give them an incentive to take each other out.

• http://i2pi.com Joshua Reich

(I guess my previous comment got lost in the system. Maybe someone can bail it out)

First up, congrats Toby & Jesper on tackling this issue. It is a shame that the Money:Tech conference was cancelled this year as that would be the perfect venue to address a good mix of quants and techs and spark some serious discussion. Barring that, I wanted to chime in with my own 2c as a techy/quant guy.

In my previous life I did a fair bit of equity research on the ratings agencies, and the views here are mine and probably not shared by others – even Roubini thought I was nuts… I have no positions in any CRAs right now.

While the talk focussed on corporate bond ratings, the largest growth area for these agencies was in structured finance. And much of this was mortgage backed securities and similar derivatives. So to understand the mess we are in now we need to look at the history of these instruments.

MBS’s were born for two key reasons. First off was the realization that in ‘normal’ times the dominant risks were idiosyncratic in nature and as such could be minimized through the application of portfolio theory and diversification, leading to pooled entities with smoothed cash flows and tranches providing for the needs of various risk profiles. In my opinion this story is primarily the sizzle. The real steak was the fact that by aggregating together whole loans new tradeable instruments could be formed.

The problem with whole loans was that their pricing was highly dependent on a large vector of unstandardised parameters whose diversity precluded the formation of any depth necessary to support liquidity in traditional market designs. By eliminating the idiosyncratic risk components these pooled instruments could theoretically be summarized by a small set of parameters and relatively simple models for prepayment risk meant that traders could respond to bids and asks against them.

Faced with these simple models quants took off in a great fantastical leap and applied ever more complex techniques to model out pricing – forgetting to properly model what happens in non-normal times. For one take, see Felix Salmon’s recent piece in Wired. Or look to Paul Wilmott’s take on the ever escalating departure into a mathematical wonderland that ignored the realities of the underlying loans and their associated risks.

Somewhere along the line practitioners forgot that the technology underpinning the frothy new market was based on 1970′s financial and computational technology. Back then a bank of associates armed with HP-12C’s could price out MBS’s using a small set of descriptive parameters.

Over the next 20 years more computing power was thrown at the problem, but the basic data was still confined in scope. Sure, some funds were taking apart these pools and doing a deep analysis of the components, but there wasn’t much reward in doing so as the market was moving at such an upward clip.

Even worse, if you look at the papers from Frank Partnoy, the credit ratings agencies – who were supposed to be taking a deeper look at these securities, without the demands of second by second trading – were using plainly silly assumptions. There was a huge amount of mathematical and financial stupidity going on. Not even going to mention the conflicts of interest and the regulatory arbitrage at play…

Anyway… To address Falafulu’s point about MPT – I agree, MPT is great stuff; a very powerful framework by which to understand finance. But just look at the assumptions. Sure, these assumptions make the math tractable, but modern computing power enables us to take a more nuanced view of the world. We no longer have to rely on single parameters of ‘default risk’ to price these instruments. The market would be far better served if all available data for the underlying components and use their own information about their own risk profile to come up with better measures of value. Just compare David Einhorn’s spreadsheet with a report put out by Moody’s. It is night and day. Give me the data, not some puff piece of pseudoscientific nonsense passing itself off as high finance.

The original problems with trading whole loans, namely that there were too many parameters to support liquid markets, is no longer an issue. Look at WeatherBill. Look at Robin Hanson’s work on combinatorial market mechanism design. Falafulu, sure some smart people were recognized for their ground breaking work of decades ago. But the most recent winner of the John Bates Clark medal in Economics went to Susan Athey, who is doing some fantastic work in mechanism design.

Computational power is such that we no longer need to pretend that all financial instruments have to be priced on with a slide rule. We have new marketplaces that can effectively support trade in financial instruments with high dimensionality. We have the computational power to let traders value these instruments. What we don’t have is the data.

Give us the data and we will trade.

For a less ranty take on my world view, check out my blog post at http://blog.i2pi.com/index.cgi?p=186

• Falafulu Fisi

Toby said…
Also, if “some quant guys” had submitted an abstract about the crisis, there’s a good chance they would have been invited.

Was the Etech conference announced at the Wilmott forum? That’s where all the quants hang out. Did you tip your ex colleague Prof. Andrew Lo at MIT financial engineering lab? I am sure that some quants out there from the industry would have love to attend if they knew about it, assuming that you didn’t announce the Etech conference to the communities of quants out there.

• http://jesperandersen.net Jesper Andersen

Thanks for all the feedback, this is great.

Bex:
Let’s be clear: we completely agree that there should be 50 ratings approaches; Freerisk is designed to do the heavy lifting of starting exploring quantitative risk measurement. There will never be any official Freerisk score, only a framework for others to create scores and publish them back into a single location, so that others can select the ratings approach they want with whatever strengths and weaknesses they may bring with them. The few models we presented aren’t specifically endorsed by us, only case studies to better understand the platform.

The punitive incentives idea is interesting; another approach that deserves consideration.

Falafulu:
I don’t think it’s fair to lay any fault with the Etech team or O’Reilly. There were a number of industry people at our presentation, with a diverse set of opinions. I’m sure some of the stronger ones will make it out here in the comments, or elsewhere on the web. We certainly welcome any constructive feedback.

There seems to be a confusion that we believe financial quants are the problem here. In fact that’s not true at all, Freerisk needs to appeal to quants for the algorithms that fuel it’s ecosystem of ratings. If anything, Freerisk raises the stakes in terms of depending on algorithmic and model-driven approaches to assessing risk. We simply introduce a more open arena for these approaches to compete, and provide a new set of market forces for existing credit rating agencies.

• http://bexhuff.com bex

@Jesper I see… so Freerisk is more of a system to help people “get started” doing risk analysis, and then they can fill in the quantitative analysis with their own “off the books” analysis.

That makes sense… I can see how this adds value.

• http://blog.kiwitobes.com Toby Segaran

@Falafulu I really like Andrew Lo’s work (in fact, I can see his book on my shelf from where I’m sitting) but we’ve never worked together. I would love to have a conversation with him about this.

Also, thanks for the tip about the Wilmott forum, it looks like a great resource.

• http://i2pi.com Joshua Reich

No idea why my comments aren’t being posted to this blog. I copied my last comment and ran it on my own blog. Discussion is beginning over there too.

• Falafulu Fisi

Toby Segaran, I am interested in your riskfree project perhaps I can contribute at some stage once its taken off. I am interested in the domain of Financial numerical modeling and I have implemented a number of different models (various derivatives & fix-interests instruments). I started developing these algorithms for a hobby, but I have decided to go further and turn it into a commercial product. So, now I have free time to concentrate on it.

The literatures are so huge that no one person in this domain can know everything from the left end of the spectrum to the far right. The best online free resources that I use are :

REPEC (research papers in economic)

SSRN (
Social Science Research Network)

There are other various economic/finance related journals available (many of those), if I see a title that I like, then I request a free copy from the author directly. I have always received a copy of the publication requested, since to buy an article from the publishers site they are expensive, ranging from $30 to$50 US to buy an article online.

Joshua Reich said…
modern computing power enables us to take a more nuanced view of the world.

Yes, but that is what algorithmic trading is meant to do, high-end intensive computation that trades in a blink of a second when the opportunity arises.

Josh said…
The market would be far better served if all available data for the underlying components and use their own information about their own risk profile to come up with better measures of value.

Who owns the data? It costs to aggregate the data. So, if whoever is paying to get all the available data, it is sure that it is not free or otherwise that vendor wouldn’t be in business in the first place. It would be good if the data is free, but that is wishful thinking and we have to understand that it is the nature of free-market. Someone produces a service or a product, and someone else buys that service of product. Moody, Fitch and the rest are private companies. If a vendor feels oblige to do a free service to replace those commercial vendors, that’s fine, but do you think that some Santa Claus company out there which is willing to step in and do that? I doubt it.

• http://i2pi.com Joshua Reich

Falafulu: Who owns the data – well, the SEC mandates timely distribution of data the the general public. Currently that data is disseminated in a difficult to manage format. Serious investors pay thousands of dollars per month to be able to access the data in computer readable formats. While the SEC has been transitioning to XRBL, it hasn’t happened yet. And that is just on the 10-K equity side.

For investors who don’t have that kind of cash, they pretty much are SOL when it comes to doing fundamental analysis, and in turn have to rely on the drivel you find in sell side reports.

It is even worse when you look at structured products. If you’ve ever read an MBS offering document you’d know that it is a bloody nightmare to consider the work required to consolidate and normalize the data. So most investors rely on the CRA’s and the CDO spreads to figure out the details. An echo chamber ensues.

I envision a world where by full disclosure in a standardized, normalized format is required to even get the attention of traders. At the moment the regulatory requirements are pretty thin.

The current structure gives the NRSO’s a way to bypass Reg-FD in their attempt to formula
te ratings. They have far more complete access to the companies books than the investing
public – with the understanding that they will not reveal sensitive details to the public
in a way that would violate the normal terms of FD. But they are allowed to use their in
formation to formulate better models of default risk.

But clearly they aren’t using their special regulatory status to make better models. So w
e have to question why we allow private companies access to data that is not used to benefit the public.

I don’t know what the solution is, but there are many low-hanging fruit between here and
there that can be addressed to lower the costs of analysis so that everyone can make thei
r own models and don’t have to rely on junk from the CRAs.

• mateo

tell the “Quant Guys” to start factoring “political corruption” “bailouts” and “too big to fail” variables – then you might have something worth looking at. seems like theyre only halfway done. nobel my ass.


text:
# Lambda calculus reduction in Mathematica

Are there any built-in functions for reducing lambda terms in Mathematica? If not, how might one do it? I've heard of Function, which is good for $$\beta$$-reduction, but it doesn't seem to properly handle $$\alpha$$-conversion. Regarding $$\eta$$-reduction, the best I could find was this snippet from A New Kind of Science (ref): rm[h_, v_] /; FreeQ[h, v] = k[h]

Specifically, what I need is the following:

• A function reduce[term_] which returns term after one step of reduction. It will do one of the following:
• $$\alpha$$-conversion (if necessary) followed by $$\beta$$-reduction
• $$\eta$$-reduction
• Abstraction and application are represented as functions, each taking two arguments:
• Abstraction: $$\lambda x.m$$ is lam[x_, m_]
• Application: $$n\ m$$ is app[n_, m_]
• Other details
• If no reduction is possible, reduce returns term unchanged
• The reduction strategy is normal order
• For $$\alpha$$-conversion, you can choose any method to change the names of variables, such as appending a character (xx´), incrementing a number (x1x2), or changing the name altogether (xa)

Examples of use:

$$\beta$$-reduction

reduce @ app[lam[x, lam[y, x]], a]

lam[y, a]

$$\alpha$$-conversion, $$\beta$$-reduction

reduce @ app[lam[x, lam[y, x]], y]

lam[y´, y]

More complex $$\alpha$$-conversion

reduce @ lam[y´, app[lam[x, lam[y, x]], app[y´, y]]]

lam[y´, lam[y´´, app[y´, y]]]  (* y became y´´ to avoid clashing with the first y´ *)

Normal order

reduce @ lam[x, app[lam[y, y], x]]

lam[y, y]  (* eta-reduction was done before beta-reduction *)

Lambda term in normal form

reduce @ app[a, b]

app[a, b]  (* Nothing can be reduced, so the term is left unchanged *)

If anything is difficult to understand, you're very welcome to ask about it.

• In order to get a good answer, you are going to have to be much clearer about what kinds of functions you are looking for. Can you give examples of what you expect such a function to do? – bill s Sep 8 '19 at 0:03
• @bills I have now clarified the specific requirements. – koro Sep 8 '19 at 8:59
• Find a copy of Gray's "Mastering Mathematica" - IIRC (away from that home so I can't check book), it has an implementation of the LC. – ciao Sep 9 '19 at 3:04
• @ciao Thank you, I'll be sure to look into that. – koro Sep 11 '19 at 17:44


text:
# A question about properties in modular arithmetic

Some time ago, I posted a question on six primes numbers forming an arithmetic progression, and someone commented the following: " Let $m$ and $n$ be positive integers with $\gcd(m,n)=1$. Then the sequence $m,2m,3m,…,nm$ when taken mod $n$ will give the integers from $1$ to $n$ , each occurring exactly once ". I wanted to know why so I tried to "prove" it, this what I did:

I supposed the contrary, that in the sequence $a_1 m \equiv a_2 m \pmod n$, with the condition that $a_1\not \equiv a_2 \pmod n$ (I do not know if it´s possible to use that notation but it just means that they are not congruent) , because $a_1 \neq a_2$ and because both $a_1$ and $a_2$ and have to be less or equal to $n$ .

So I did this:

$a_1 m \equiv a_2 m \pmod n$

$a_1 m - a_2 m \equiv 0 \pmod n$

$(a_1 - a_2)m\equiv 0 \pmod n$

This means that $(a_1 - a_2)m$ is a multiple of $n$, but this is a contradiction because $\gcd(m,n)=1$ and since $a_1\not \equiv a_2 \pmod n$ then $a_1 - a_2 \not \equiv 0 \pmod n$, which means that $a_1 - a_2$ doesn´t have all factors of $n$ and therefore it is not a multiple.

This may be in a textbook, so sorry if I made you lose your time.

Anyway, I would just like you to tell me if it´s ok what I did.

• That's all good. – Arnaud Mortier Jun 14 '18 at 22:04

Your argument is perfect. Note that the property $$\text{m,2m,3m,…,nm when taken mod n}\\ \text{will give the integers from 1 to n,} \\ \text{each occurring exactly once.}$$ is another way of saying that $$\text{m is a generator of the group \Bbb Z/n\Bbb Z}.$$
which is equivalent to $m,n$ being coprime like you said.
• Which is another way of saying $m$ is a unit in $\mathbb Z_n$; and there are $\phi (n)$ of them... – Chris Custer Jun 14 '18 at 22:14
• @Arnaud Mortier could you explain me the notations you are using here please? I also have another question. I know that the sign used in modular arithmetic ($\equiv$) behaves a lot like the equal sign in equations. The only operation you can't do is division but I heard that there was an exception to that rule, I can't remember how it was, but I was like if $ax \equiv m \pmod n$ then you could divide if $n$ and $x$ were coprime or something like that. Could you explain it to me and also tell me and explanation of why it happens please? – Vmimi Jun 16 '18 at 4:08
• I think that in my comment $m$ had to be $0$, but I can't remember, that's why I put $m$. – Vmimi Jun 16 '18 at 4:22
• @Vmimi If you have never seen the notation $\Bbb Z/n\Bbb Z$ it may be that you are a victim of the $\Bbb Z_n$ flu. – Arnaud Mortier Jun 16 '18 at 16:26
• @Vmimi As for your second question, you're right, you can divide by $x$ if and only if $x$ is coprime with $n$, and the reason is related to the OP: since $x,2x,\etc,nx$ represents once every integer modulo $n$, it means that for some integer $k$, you have $kx=1\bmod n$. You call that $k$ "$x$ inverse", and multiplying by $k$ is like dividing by $x$. – Arnaud Mortier Jun 16 '18 at 16:29


text:
# Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices

I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with respect to the spectral (operator) norm (actually I also have $a>0$, but I don't think that affects the answer).

One approach is with a volume argument, which goes something like this. Given any optimal $\epsilon$-packing $A_1,...,A_P$, defining $$S_i=\{A_i+B:\; B\in\mathbb{S}^d,\;\|B\|_2\leq\epsilon/2\}$$ and $$\Theta'=\{A+B:\;A\in\Theta,\;B\in\mathbb{S}^d,\;\|B\|_2\leq\epsilon/2\},$$ we have that $S_i \cap S_j=\emptyset$ for all $i\neq j$ and $\bigcup_i S_i \subseteq \Theta'$. So, given any measure $\mu$ on $\mathbb{S}^d$ (or even just on $\Theta'$), $$\sum_{i=1}^P \mu(S_i)\leq \mu(\Theta')$$ which can be used to upper bound $P$, especially if $\mu(S_i)=\mu(S_j)$ in which case $P\leq \mu(\Theta')/\mu(S_1)$. Now the question is how to define and compute such a "uniform" measure.

After some google and mathoverflow searches, I learned a little bit about Haar measures, Weyl integration formulas, and other such wonderful things, but every answer I found seemed to be not directly applicable to my problem because of the fact that I am dealing with symmetric matrices.

The approach I'm considering now is to separately define a uniform measure on onthonormal matrices, and one on eigenvalue matrices (equivalently, vectors with non-increasing entries), and combine the two as eigenvalues and eigenvectors of my symmetric matrices to give a "marginal" measure. But I'm not really sure what the best way to go about this is (I'm not even sure the resulting measure would, for instance, be absolutely continuous w.r.t. Lebesgue on the upper triangle, though perhaps that doesn't matter).

It may be worth noting that in my original problem I actually need a bound on the covering number of this set, and I initially considered trying to separately cover orthonormal matrices and the eigenvalues, and then combine them. But the obstacle was that I couldn't get a good upper bound on $\|U_1\Lambda_1U_1^T-U_2\Lambda_2U_2^T\|_2$ in terms of (some norm of) $U_1-U_2$ and $\Lambda_1-\Lambda_2$.

Any help would be appreciated.

• Martin, maybe the following similar question is sorta helpful (though it did not see much activity): mathoverflow.net/questions/122934/… --- also, by $a \preceq A$ do you mean $a I \preceq A$? Also, as for "good" upper bound on the last term you wrote, what does "good" mean? E.g., it can be bounded by $\|\Lambda_1 - \Lambda_2\|_2$, but that is not "good" enough I guess? – Suvrit Nov 7 '13 at 22:43
• @Suvrit Thanks. Yes, $a\preceq A$ is shorthand for $aI\preceq A$, and good means that the resulting upper bound should scale correctly in terms of $\epsilon$. So for instance if it were true that $\|U_1\Lambda_1 U_1^T-U_2\Lambda_2 U_2^T\|_2\leq\|U_1-U_2\|_2+\|\Lambda_1-\Lambda_2\|_2$, that would probably be good enough. On the other hand, $\|U_1\Lambda_1 U_1^T-U_2\Lambda_2 U_2^T\|_2\leq d \|U_1-U_2\|_{\infty,\infty}+d \|\Lambda_1-\Lambda_2\|_{\infty,\infty}$ (I'm just making that up, as an example) probably would be too loose. – martin Nov 8 '13 at 8:30
• what I meant was that $\|\Lambda_1^\downarrow-\Lambda_2^\uparrow\|$ is an upper bound (the arrows denote sorting)---this might be too loose, but with suitable choice of matrices, equality can be achieved in this....hence.... – Suvrit Nov 8 '13 at 13:59
• @Suvrit oh I see, yes that would be an upper bound, unfortunately in this example I think it's too loose because $a$ and $b$ are fixed, hence for small enough $\epsilon$ I can't guarantee $\|\Lambda_1^{\downarrow}-\Lambda_2^{\uparrow}\|\leq \epsilon$.. – martin Nov 8 '13 at 18:51


text:
Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki
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 Zh. Vychisl. Mat. Mat. Fiz.: Year: Volume: Issue: Page: Find

 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 7, Pages 1226–1236 (Mi zvmmf627)

The maximum principle for the transport equation in the case of Compton scattering

D. S. Konovalova

Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia

Abstract: The properties of solutions to the transport equation describing the Compton scattering of photons is investigated. The maximum and minimum principles are proved for this equation. According to them, the radiation density within a region cannot be greater than the maximum positive value of the incident radiation density and cannot be less than its minimum negative value. Furthermore, conditions under which a solution to the equation under consideration is constant are presented. The results of this work are obtained under the assumption that the properties of the medium change continuously with respect to the spatial and energy variables.

Key words: steady-state Compton scattering transport equation, maximum and minimum principles, mathematical modeling.

Full text: PDF file (1226 kB)
References: PDF file   HTML file

English version:
Computational Mathematics and Mathematical Physics, 2005, 45:7, 1185–1194

Bibliographic databases:
UDC: 519.634

Citation: D. S. Konovalova, “The maximum principle for the transport equation in the case of Compton scattering”, Zh. Vychisl. Mat. Mat. Fiz., 45:7 (2005), 1226–1236; Comput. Math. Math. Phys., 45:7 (2005), 1185–1194

Citation in format AMSBIB
\Bibitem{Kon05} \by D.~S.~Konovalova \paper The maximum principle for the transport equation in the case of Compton scattering \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 7 \pages 1226--1236 \mathnet{http://mi.mathnet.ru/zvmmf627} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2188414} \zmath{https://zbmath.org/?q=an:1116.78308} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 7 \pages 1185--1194


text:
# Finding Roots of Polynomials in C

## Homework Statement

Show the the equation f(z) = z^4 + iz^2 + 2 = 0 has two roots with |z|=1 and two roots with |z|=sqrt(2), without actually solving the equation.

## Homework Equations

Rouche's theorem, the argument principle?

## The Attempt at a Solution

This is what I have done so far: First show that no solutions lie outside of the circle |z|=sqrt(2). This is true because if we suppose there was a solution, say p, then 0 = |p^4 + ip^2 + 2| >= |p|^4 - |p|^2 - 2 > 0 since |p| > sqrt(2). This contradiction implies that all solutions must lie inside of the disk |z| <= sqrt(2). After this, I wanted to show that f(z) has only two solutions INSIDE the circle |z|=sqrt(2), which would imply that two solutions must exist ON the circle. I tried to use Rouche's theorem, but I could not get the strict inequality to hold in any case. Then I realized that in the statement of the theorem, we require that no zeros or poles lie on the circle, so I could not use Rouche's theorem anyway. Okay, now we can also show that no solutions exist inside the unit circle in exactly the same way as for outside of |z|=sqrt(2). Suppose there was a solution, say q. Then 0 = |q^4 + iq^2 + 2| >= 2 - |q|^4 - |q|^2 > 0 since |q| < 1. Hence, all solutions must lie on the set {z : 1 <= |z| <= sqrt(2)}. This is where I am stuck. I am not sure how to show that solutions cannot exist somewhere in between. Any hints would be greatly appreciated. I feel like I should be using some theorem in this part, because for the parts I have done, I have only used some basic properties of the absolute value. Thanks!!