A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is p... more A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schro¨dinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.
The adhesion of lipid tubules to a rigid wall is governed by an equilibrium condition that holds ... more The adhesion of lipid tubules to a rigid wall is governed by an equilibrium condition that holds at the detachment points, where tubules and wall lose contact. Such an adhesion condition is derived in this paper for the most general elastic energy of the tubule and an arbitrarily curved wall. It also depends on the curvature of the wall, which thus plays a central rôle in determining whether the adhesion of a tubule is possible. This rôle is clearly illustrated by a specific equilibrium problem we solve here: the one where a tubule adheres to a rigid groove, which is modelled as a hollow half-cylinder.
Journal of Physics a Mathematical and General, Nov 30, 2003
Computing the second variation of the energy functional of a lipid vesicle is an issue that prece... more Computing the second variation of the energy functional of a lipid vesicle is an issue that precedes any local stability analysis. This issue becomes technically more delicate when the equilibrium configurations to be perturbed are partially adhered to a fixed substrate. Within a two-dimensional model, we introduce here a method that delivers the second variation of the energy functional for vesicles subject to adhesion. The formula thus obtained is then applied to estimate the maximum susceptibility to detachment of tubules, hollow vesicles that do not enclose a prescribed volume. In particular, we illuminate by example the role of the substrate curvature in enhancing or depressing the susceptibility of adhered tubules to a special class of detaching fluctuations.
European Journal of Applied Mathematics, Apr 1, 1999
Volume 9 (1998), pp. 485–506One of the author's names was misprinted in this article. EPIFIAN... more Volume 9 (1998), pp. 485–506One of the author's names was misprinted in this article. EPIFIANO G. VIRGA should read EPIFANIO G. VIRGA.In addition, on page 502, line 9, the argument leading to evaluate g(∞) := limb→+∞g(b) is not conclusive. The passage‘On the other hand, if it were g(∞)>0, taking the limit as b→+∞ in (9.9), we would get A=πh2, which by (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that g(∞)=0.’should be changed as follows:‘On the other hand, taking the limit as b→+∞ in (9.9), we would get A=πh2, by which (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that, in the limit as b→+∞, the equilibrium configuration must be singular whenever A>πh2, and so g(∞)=0 because, otherwise, by (9.7) there would be asymptotic regular configurations, as soon as h/2kc(τ0+w) is sufficiently small.’On page 502, line −7, ‘below’ should be replaced with ‘above’ and ‘regular’ with ‘singular’.We apologise for the errors in the above paper and hope that no inconvenience has been caused to readers.
In this paper we are concerned with the adhesion of lipid tubules to a plane wall: we treat both ... more In this paper we are concerned with the adhesion of lipid tubules to a plane wall: we treat both the statics and the dynamics of this phenomenon. Though we provide an exact solution to the equilibrium problem, our treatment of the dynamical problem is approximate, as it is based on a simplified model, which, nevertheless allows us to obtain quantitative information about the detachment dynamics of tubules.
Continuum Mechanics and Thermodynamics, Feb 1, 2002
We propose a two-dimensional model for the dynamics of the kinks created in a biological membrane... more We propose a two-dimensional model for the dynamics of the kinks created in a biological membrane by the interaction with a movable bead. We arrive at the evolution equations for both the bead and the membrane, whence we conclude that the force exerted on the bead by a fixed membrane points in the direction along which the curvature of the membrane is more concentrated. This is the first step towards understanding the basic mechanism behind the dynamics of protein aggregation which takes place on biological membranes.
Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, 2001
... direction of c, different in nature from the force proportional to the curvature of c, respon... more ... direction of c, different in nature from the force proportional to the curvature of c, responsible for shrinking disclination loops (see eg Sonnet and Virga [13 ... the local normal plane to c with oRo, and f is the length-measure (Figure 2), Likewise, we express the total torque Ko and the ...
ABSTRACT This paper, which is mainly based on unpublished material, focuses on the scientific inf... more ABSTRACT This paper, which is mainly based on unpublished material, focuses on the scientific influence that Felice Casorati exerted on Salvatore Pincherle. This influence can be traced, in particular, in Casorati’s work on the finite-difference calculus as conceived and published between 1879 and 1880 when Pincherle was living in Pavia. Casorati’s work has an interesting back story related to his entry to the 1880 Grand Prix of the French Académie des Sciences that helps us in understanding Casorati’s personality. Moreover, the correspondence that Casorati exchanged with other mathematicians on his work reveals that some of the results contained in Casorati (Annali di Matematica pura ed applicata 10(S. II):10-45, 1880b) had been obtained—though in a narrower context—in an early paper by Christoffel. Finally, the letters between Casorati and Pincherle contain a short unpublished note by Pincherle on a paper by Jules Tannery (Ann Sci. l’École Norm Super 4(S. II):113-182, 1875). This note offers the first evidence of the influence of Casorati (Annali di Matematica pura ed applicata 10(S. II):10-45, 1880b) on Pincherle’s work on the finite-difference calculus.
We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensiona... more We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensional (2D) toroidal shell. For the toroidal topology the total topological charge of TDs is equal to zero. We use a mesoscopic Landau-de Gennes approach which features a 2D nematic order tensor Q. We show that fat tori unbind TDs. If no extrinsic free energy couples Q with the Weingarten tensor of the torus, then defects and antidefects are assembled along the innermost and the outermost circles of the torus, respectively. In this case, we estimate the critical condition for the onset of TDs using an electrostatic analogy. If, on the other hand, an extrinsic free energy is present, then defects are repelled from these regions. 4 ref. 1 (see, in particular, § VII.E.3), for uniaxial liquid crystals the topological charge of a point defect in 3D is a relative integer, but all admissible defects are
Within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of... more Within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of a nematic cell with twist boundary conditions. Under the assumption that the order tensor Q be uniaxial on both bounding plates, we find three separate classes of solutions, one of which contains the absolute energy minimizer, a twist-like solution that exists for all values of the distance d between the plates. The solutions in the remaining two classes exist only if d exceeds a critical value d c. One class consists of metastable, twist-like solutions, while the other consists of unstable, exchange-like solutions, where the eigenvalues of Q are exchanged across the cell. When d = d c , the metastable solution relaxes back to the absolute energy minimizer, undergoing an order reconstruction somewhere within the cell. The critical distance d c equals in general a few biaxial coherence lengths. This scenario applies to all values of the boundary twist but π 2 , which thus appears as a very special case, though it is the one more studied in the literature. In fact, when the directors prescribed on the two plates are at right angles, two symmetric twist-like solutions merge continuously into an exchange-like solution at the critical value of d where the latter becomes unstable. Our analysis shows how the classical bifurcation associated with this phenomenon is unfolded by perturbing the boundary conditions.
Modeling and Simulation in Science, Engineering and Technology, 2007
This contribution is concerned with the mathematical modeling of biological membranes. In particu... more This contribution is concerned with the mathematical modeling of biological membranes. In particular, it explores the role played by some structures that make them inhomogeneous
We propose a general stability criterion for the wetting of solid substrates, both arbitrarily cu... more We propose a general stability criterion for the wetting of solid substrates, both arbitrarily curved and inhomogeneous. In addition to the classical surface tension, the adhering drops can also exhibit a tension along the contact line where three phases meet, namely, the solid, the liquid, and the environment fluid. Moreover, we show how some stability issues currently debated in the specialized literature of disparate fields could profit from the application of this general criterion.
Within a two-dimensional Landau-de Gennes tensorial formalism we study the equilibrium configurat... more Within a two-dimensional Landau-de Gennes tensorial formalism we study the equilibrium configurations of nematic shells, thin films of nematic liquid crystal deposited on the boundary of colloidal particles of arbitrary shape enforcing a degenerate tangential anchoring on the nematic molecules. In this formalism, defects appear wherever a scalar order parameter vanishes. Their total number is the colloidal valence, as there an assembly of colloids could be bound together through appropriate molecular bridges. We show how the defect organization on a colloidal particle is affected by the Gaussian curvature of the colloidal boundary, to the point of changing its valence.
We study the effects that a curvature correction to a bare value of the line tension has on the e... more We study the effects that a curvature correction to a bare value of the line tension has on the equilibrium and stability of droplets laid upon a rigid substrate. In the simple case of cylindric liquid bridges we prove that even a tiny curvature correction prevents the onset of wildly oscillating perturbations that would make the contact line unstable if a negative line tension were present alone. However, if the curvature correction is not large enough, unstable modes can persist that are not related to the classical Rayleigh instability.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2000
ABSTRACT We derive the evolution equation for a lipid vesicle in the presence of an adhering wall... more ABSTRACT We derive the evolution equation for a lipid vesicle in the presence of an adhering wall, under the assumption that the vesicle' membrane is permeable to the surrounding fluid that is permanently at rest. For vesicles in two dimensions, we study the asymptoti decay to a circle and the incipient adhesion to a flat wall with a piecewise adhesion potential. We illustrate several compute animations driven by the evolution equation obtained here. They are available in full at http://smmm.unipv.it/vesicles.html.
A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is p... more A representation of invariants in quantum mechanics for a general time-dependent Hamiltonian is provided. The main properties of these operators and their relevance for solving the Schro¨dinger equation are discussed in light of this representation. In particular, the well-known Lewis and Riesenfeld’s method is shown to be recovered quite directly by the present approach. A detailed analysis is concerned with Hamiltonians which at each instant are elements of a finite dimensional Lie algebra. Exploiting the Wei and Norman’s theory for such Hamiltonians, it is found that the evolution of a quantum system is always determined, in part or fully, by an invariant which belongs to the same algebra.
The adhesion of lipid tubules to a rigid wall is governed by an equilibrium condition that holds ... more The adhesion of lipid tubules to a rigid wall is governed by an equilibrium condition that holds at the detachment points, where tubules and wall lose contact. Such an adhesion condition is derived in this paper for the most general elastic energy of the tubule and an arbitrarily curved wall. It also depends on the curvature of the wall, which thus plays a central rôle in determining whether the adhesion of a tubule is possible. This rôle is clearly illustrated by a specific equilibrium problem we solve here: the one where a tubule adheres to a rigid groove, which is modelled as a hollow half-cylinder.
Journal of Physics a Mathematical and General, Nov 30, 2003
Computing the second variation of the energy functional of a lipid vesicle is an issue that prece... more Computing the second variation of the energy functional of a lipid vesicle is an issue that precedes any local stability analysis. This issue becomes technically more delicate when the equilibrium configurations to be perturbed are partially adhered to a fixed substrate. Within a two-dimensional model, we introduce here a method that delivers the second variation of the energy functional for vesicles subject to adhesion. The formula thus obtained is then applied to estimate the maximum susceptibility to detachment of tubules, hollow vesicles that do not enclose a prescribed volume. In particular, we illuminate by example the role of the substrate curvature in enhancing or depressing the susceptibility of adhered tubules to a special class of detaching fluctuations.
European Journal of Applied Mathematics, Apr 1, 1999
Volume 9 (1998), pp. 485–506One of the author's names was misprinted in this article. EPIFIAN... more Volume 9 (1998), pp. 485–506One of the author's names was misprinted in this article. EPIFIANO G. VIRGA should read EPIFANIO G. VIRGA.In addition, on page 502, line 9, the argument leading to evaluate g(∞) := limb→+∞g(b) is not conclusive. The passage‘On the other hand, if it were g(∞)>0, taking the limit as b→+∞ in (9.9), we would get A=πh2, which by (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that g(∞)=0.’should be changed as follows:‘On the other hand, taking the limit as b→+∞ in (9.9), we would get A=πh2, by which (3.11) is violated by all admissible values of A, but the limiting one. Thus we conclude that, in the limit as b→+∞, the equilibrium configuration must be singular whenever A>πh2, and so g(∞)=0 because, otherwise, by (9.7) there would be asymptotic regular configurations, as soon as h/2kc(τ0+w) is sufficiently small.’On page 502, line −7, ‘below’ should be replaced with ‘above’ and ‘regular’ with ‘singular’.We apologise for the errors in the above paper and hope that no inconvenience has been caused to readers.
In this paper we are concerned with the adhesion of lipid tubules to a plane wall: we treat both ... more In this paper we are concerned with the adhesion of lipid tubules to a plane wall: we treat both the statics and the dynamics of this phenomenon. Though we provide an exact solution to the equilibrium problem, our treatment of the dynamical problem is approximate, as it is based on a simplified model, which, nevertheless allows us to obtain quantitative information about the detachment dynamics of tubules.
Continuum Mechanics and Thermodynamics, Feb 1, 2002
We propose a two-dimensional model for the dynamics of the kinks created in a biological membrane... more We propose a two-dimensional model for the dynamics of the kinks created in a biological membrane by the interaction with a movable bead. We arrive at the evolution equations for both the bead and the membrane, whence we conclude that the force exerted on the bead by a fixed membrane points in the direction along which the curvature of the membrane is more concentrated. This is the first step towards understanding the basic mechanism behind the dynamics of protein aggregation which takes place on biological membranes.
Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, 2001
... direction of c, different in nature from the force proportional to the curvature of c, respon... more ... direction of c, different in nature from the force proportional to the curvature of c, responsible for shrinking disclination loops (see eg Sonnet and Virga [13 ... the local normal plane to c with oRo, and f is the length-measure (Figure 2), Likewise, we express the total torque Ko and the ...
ABSTRACT This paper, which is mainly based on unpublished material, focuses on the scientific inf... more ABSTRACT This paper, which is mainly based on unpublished material, focuses on the scientific influence that Felice Casorati exerted on Salvatore Pincherle. This influence can be traced, in particular, in Casorati’s work on the finite-difference calculus as conceived and published between 1879 and 1880 when Pincherle was living in Pavia. Casorati’s work has an interesting back story related to his entry to the 1880 Grand Prix of the French Académie des Sciences that helps us in understanding Casorati’s personality. Moreover, the correspondence that Casorati exchanged with other mathematicians on his work reveals that some of the results contained in Casorati (Annali di Matematica pura ed applicata 10(S. II):10-45, 1880b) had been obtained—though in a narrower context—in an early paper by Christoffel. Finally, the letters between Casorati and Pincherle contain a short unpublished note by Pincherle on a paper by Jules Tannery (Ann Sci. l’École Norm Super 4(S. II):113-182, 1875). This note offers the first evidence of the influence of Casorati (Annali di Matematica pura ed applicata 10(S. II):10-45, 1880b) on Pincherle’s work on the finite-difference calculus.
We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensiona... more We study nematic liquid crystal textures exhibiting topological defects (TDs) on a two-dimensional (2D) toroidal shell. For the toroidal topology the total topological charge of TDs is equal to zero. We use a mesoscopic Landau-de Gennes approach which features a 2D nematic order tensor Q. We show that fat tori unbind TDs. If no extrinsic free energy couples Q with the Weingarten tensor of the torus, then defects and antidefects are assembled along the innermost and the outermost circles of the torus, respectively. In this case, we estimate the critical condition for the onset of TDs using an electrostatic analogy. If, on the other hand, an extrinsic free energy is present, then defects are repelled from these regions. 4 ref. 1 (see, in particular, § VII.E.3), for uniaxial liquid crystals the topological charge of a point defect in 3D is a relative integer, but all admissible defects are
Within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of... more Within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of a nematic cell with twist boundary conditions. Under the assumption that the order tensor Q be uniaxial on both bounding plates, we find three separate classes of solutions, one of which contains the absolute energy minimizer, a twist-like solution that exists for all values of the distance d between the plates. The solutions in the remaining two classes exist only if d exceeds a critical value d c. One class consists of metastable, twist-like solutions, while the other consists of unstable, exchange-like solutions, where the eigenvalues of Q are exchanged across the cell. When d = d c , the metastable solution relaxes back to the absolute energy minimizer, undergoing an order reconstruction somewhere within the cell. The critical distance d c equals in general a few biaxial coherence lengths. This scenario applies to all values of the boundary twist but π 2 , which thus appears as a very special case, though it is the one more studied in the literature. In fact, when the directors prescribed on the two plates are at right angles, two symmetric twist-like solutions merge continuously into an exchange-like solution at the critical value of d where the latter becomes unstable. Our analysis shows how the classical bifurcation associated with this phenomenon is unfolded by perturbing the boundary conditions.
Modeling and Simulation in Science, Engineering and Technology, 2007
This contribution is concerned with the mathematical modeling of biological membranes. In particu... more This contribution is concerned with the mathematical modeling of biological membranes. In particular, it explores the role played by some structures that make them inhomogeneous
We propose a general stability criterion for the wetting of solid substrates, both arbitrarily cu... more We propose a general stability criterion for the wetting of solid substrates, both arbitrarily curved and inhomogeneous. In addition to the classical surface tension, the adhering drops can also exhibit a tension along the contact line where three phases meet, namely, the solid, the liquid, and the environment fluid. Moreover, we show how some stability issues currently debated in the specialized literature of disparate fields could profit from the application of this general criterion.
Within a two-dimensional Landau-de Gennes tensorial formalism we study the equilibrium configurat... more Within a two-dimensional Landau-de Gennes tensorial formalism we study the equilibrium configurations of nematic shells, thin films of nematic liquid crystal deposited on the boundary of colloidal particles of arbitrary shape enforcing a degenerate tangential anchoring on the nematic molecules. In this formalism, defects appear wherever a scalar order parameter vanishes. Their total number is the colloidal valence, as there an assembly of colloids could be bound together through appropriate molecular bridges. We show how the defect organization on a colloidal particle is affected by the Gaussian curvature of the colloidal boundary, to the point of changing its valence.
We study the effects that a curvature correction to a bare value of the line tension has on the e... more We study the effects that a curvature correction to a bare value of the line tension has on the equilibrium and stability of droplets laid upon a rigid substrate. In the simple case of cylindric liquid bridges we prove that even a tiny curvature correction prevents the onset of wildly oscillating perturbations that would make the contact line unstable if a negative line tension were present alone. However, if the curvature correction is not large enough, unstable modes can persist that are not related to the classical Rayleigh instability.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2000
ABSTRACT We derive the evolution equation for a lipid vesicle in the presence of an adhering wall... more ABSTRACT We derive the evolution equation for a lipid vesicle in the presence of an adhering wall, under the assumption that the vesicle' membrane is permeable to the surrounding fluid that is permanently at rest. For vesicles in two dimensions, we study the asymptoti decay to a circle and the incipient adhesion to a flat wall with a piecewise adhesion potential. We illustrate several compute animations driven by the evolution equation obtained here. They are available in full at http://smmm.unipv.it/vesicles.html.
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