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In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

We introduce filtered cohomologies of differential forms on symplecticmanifolds. They generalize and include the cohomologies discussed in Paper Iand II as a subset. The filtered cohomologies are finite-dimensional and can beassociated with differential elliptic complexes. Algebraically, we show thatthe filtered cohomologies give a two-sided resolution of Lefschetz maps, andthereby, they are directly related to the kernels and cokernels of theLefschetz maps. We also introduce a novel, non-associative product operation ondifferential forms for symplectic manifolds. This product generates anA-infinity algebra structure on forms that underlies the filtered cohomologiesand gives them a ring structure. As an application, we demonstrate how the ringstructure of the filtered cohomologies can distinguish different symplecticfour-manifolds in the context of a circle times a fibered three-manifold.

Abstract: Frege’s theorem says that second-order Peano arithmetic is interpretable in Hume’s Principle and full impredicative comprehension. Hume’s Principle is one example of anabstraction principle, while another paradigmatic example is Basic Law V from Frege’sGrundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic (cf. Theorem 3.2).

AbstractFrege’s theorem says that second-order Peano arithmetic is interpretable in Hume’s Principle and full impredicative comprehension. Hume’s Principle is one example of anabstraction principle, while another paradigmatic example is Basic Law V from Frege’sGrundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic (cf. Theorem 3.2).

In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for nonhereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

This article is concerned with the infinite depth water wave equation in two space dimensions.We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove that small localized data leads to global solutions. This article is a continuation of authors' earlier paper [8].

We introduce filtered cohomologies of differential forms on symplecticmanifolds. They generalize and include the cohomologies discussed in Paper Iand II as a subset. The filtered cohomologies are finite-dimensional and can beassociated with differential elliptic complexes. Algebraically, we show thatthe filtered cohomologies give a two-sided resolution of Lefschetz maps, andthereby, they are directly related to the kernels and cokernels of theLefschetz maps. We also introduce a novel, non-associative product operation ondifferential forms for symplectic manifolds. This product generates anA-infinity algebra structure on forms that underlies the filtered cohomologiesand gives them a ring structure. As an application, we demonstrate how the ringstructure of the filtered cohomologies can distinguish different symplecticfour-manifolds in the context of a circle times a fibered three-manifold.

In the present paper we show that Hall algebras of finitary exact categories behave like quantum groups in the sense that they are generated by indecomposable objects. Moreover, for a large class of such categories, Hall algebras are generated by their primitive elements, with respect to the natural comultiplication, even for nonhereditary categories. Finally, we introduce certain primitively generated subalgebras of Hall algebras and conjecture an analogue of "Lie correspondence" for those finitary categories.

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [5] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.

A compact Riemannian homogeneous space G/H, with a bi-invariant orthogonal decomposition g=h+m is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in TeH(G/H) spanned by a linearly independent commuting pair in m. In this paper, we will prove that on the coset space Sp(2)/U(1), in which U(1) corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this Sp(2)/U(1) cannot be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.

Abstract In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that every normal semi-algebraic Ricci soliton also admits a k-dimensional Einstein extension for any k ≥ 2. We also prove converse theorems for these constructions and some geometric and topological structure results for homogeneous warped product Einstein metrics. In the appendix we give an alternative approach to semi-algebraic Ricci solitons which naturally leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous setting.

A compact Riemannian homogeneous space G/H, with a bi-invariant orthogonal decomposition g=h+m is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in TeH(G/H) spanned by a linearly independent commuting pair in m. In this paper, we will prove that on the coset space Sp(2)/U(1), in which U(1) corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this Sp(2)/U(1) cannot be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.

In a recent paper we found conditions for a nilpotent Lie group N to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel for almost all representations of N. The prototype for this sort of group is the group of upper triangular real matrices with 1’s down the diagonal. More generally, this class of groups contains the nilradicals of minimal parabolic subgroups of all (finite-dimensional) reductive real or complex Lie groups, in other words, all groups N in Iwasawa decompositions of reductive real or complex Lie groups. The construction of stepwise square integrable representations resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. Here we extend those results to direct limits of stepwise square integrable nilpotent Lie groups. There are two keys to this extension. The first is to set up the corresponding direct system so that it respects the construction at every finite level. In the case of simple (or more generally reductive) groups this means that the restricted root Dynkin diagrams increase in a particular manner. The second is to follow Schwartz space theory through the direct limit process, develop a Schwartz space theory for certain direct limit nilpotent groups, and use it to study stepwise square integrability for coefficients of direct limits of stepwise square integrable nilpotent Lie groups. This leads to the main result, an explicit Fourier inversion formula for that class of infinite-dimensional Lie groups. One important consequence is the Fourier inversion formula for nilradicals of classical minimal parabolic subgroups of finitary real reductive Lie groups such as GL(∞; ℝ), Sp(∞; ℂ) and SO(∞, ∞).

In a recent paper we found conditions for a nilpotent Lie group N to have a filtration by normal subgroups whose successive quotients have square integrable representations, and such that these square integrable representations fit together nicely to give an explicit construction of Plancherel for almost all representations of N. The prototype for this sort of group is the group of upper triangular real matrices with 1’s down the diagonal. More generally, this class of groups contains the nilradicals of minimal parabolic subgroups of all (finite-dimensional) reductive real or complex Lie groups, in other words, all groups N in Iwasawa decompositions of reductive real or complex Lie groups. The construction of stepwise square integrable representations resulted in explicit character formulae, Plancherel formulae and multiplicity formulae. Here we extend those results to direct limits of stepwise square integrable nilpotent Lie groups. There are two keys to this extension. The first is to set up the corresponding direct system so that it respects the construction at every finite level. In the case of simple (or more generally reductive) groups this means that the restricted root Dynkin diagrams increase in a particular manner. The second is to follow Schwartz space theory through the direct limit process, develop a Schwartz space theory for certain direct limit nilpotent groups, and use it to study stepwise square integrability for coefficients of direct limits of stepwise square integrable nilpotent Lie groups. This leads to the main result, an explicit Fourier inversion formula for that class of infinite-dimensional Lie groups. One important consequence is the Fourier inversion formula for nilradicals of classical minimal parabolic subgroups of finitary real reductive Lie groups such as GL(∞; ℝ), Sp(∞; ℂ) and SO(∞, ∞).

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type An-1 for c=mn, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SLm. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras Hmn(Sn) and Hnm(Sm). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case GCD(m, n)=1, and it has a natural interpretation in terms of invariants of torus knots.

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type An-1 for c=mn, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for SLm. Our third result is the construction of the Koszul-BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul-BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul-BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras Hmn(Sn) and Hnm(Sm). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case GCD(m, n)=1, and it has a natural interpretation in terms of invariants of torus knots.

Consider a vector bundle with connection on a p-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author's 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

The paper contains a new proof that a complete, non-compact hyperbolic 3–manifold with finite volume contains an immersed, closed, quasi-Fuchsian surface.

The paper contains a new proof that a complete, non-compact hyperbolic 3–manifold with finite volume contains an immersed, closed, quasi-Fuchsian surface.

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.

We analyse the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimensional incompressible Navier-Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which are represented by stochastic noise. Under these assumptions, the data-assimilation algorithm consists of a system of stochastically forced Navier-Stokes equations. The main result of this paper provides explicit conditions on the observation density (resolution) which guarantee explicit asymptotic bounds, as the time tends to infinity, on the error between the approximate solution and the actual solutions which is corresponding to these measurements, in terms of the variance of the noise in the measurements. Specifically, such bounds are given for the limit supremum, as the time tends to infinity, of the expected value of the L2-norm and of the H1 Sobolev norm of the difference between the approximating solution and the actual solution. Moreover, results on the average time error in mean are stated.

This paper is concerned with the cubic Szego{double acute} equationi ∂t u = Π (| u |2 u), defined on the L2 Hardy space on the one-dimensional torus T, where Π : L2 (T) → L+2 (T) is the Szego{double acute} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time t ∈ (- ∞, ∞). In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the ℓ1 norm of Fourier transforms (the Wiener algebra). © 2013 Elsevier Masson SAS. All rights reserved.

We compute the Heegaard Floer link homology of algebraic links in terms of the multi-variate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard Floer link homology of the local embedded link, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a generalized version of the Orlik-Solomon algebra of these local arrangements. In particular, the PoincarCrossed D sign polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic PoincarCrossed D sign series of the singularity.

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p-1] is "almost" finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p-1] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (ϕ, Γ)-modules associated with Artin Motives over Q. The (ϕ, Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.

This paper is concerned with the cubic Szego equationi {equation presented}, defined on the L2 Hardy space on the one-dimensional torus T, where π : L2(T)→L2+(T) is the Szego projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time t ∈(-infin;,∞). In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the l1 norm of Fourier transforms (the Wiener algebra).

We compute the Heegaard Floer link homology of algebraic links in terms of the multi-variate Hilbert function of the corresponding plane curve singularities. The main result of the paper identifies four homologies: (a) the Heegaard Floer link homology of the local embedded link, (b) the lattice homology associated with the Hilbert function, (c) the homologies of the projectivized complements of local hyperplane arrangements cut out from the local algebra, and (d) a generalized version of the Orlik-Solomon algebra of these local arrangements. In particular, the PoincarCrossed D sign polynomials of all these homology groups are the same, and we also show that they agree with the coefficients of the motivic PoincarCrossed D sign series of the singularity.

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H2 initial data. © 2014 Elsevier Inc. All rights reserved.

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H2 initial data.

In this paper, we consider a compact Kähler manifold with extremal Kähler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kähler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kähler metrics on the projectivization of the dual vector bundle. © 2014 Elsevier Inc.

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X=Cb1(R,PmH2) where Pm is the L2-projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes.

In this paper, we consider a compact Kähler manifold with extremal Kähler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kähler metric is liftable to the bundle and if the bundle is relatively stable with respect to the action of automorphisms of the manifold, then there exist extremal Kähler metrics on the projectivization of the dual vector bundle. © 2014 Elsevier Inc.

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation. © 2014 The Author(s).

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.

In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natural curvature conditions that characterize the rigid examples in higher dimensions.

In this paper, we consider the initial-boundary value problem of the 3D primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation. Global well-posedness of strong solutions are established with H2 initial data. © 2014 Elsevier Inc. All rights reserved.

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation. © 2014 The Author(s).

This paper is concerned with the cubic Szego{double acute} equationi ∂t u = Π (| u |2 u), defined on the L2 Hardy space on the one-dimensional torus T, where Π : L2 (T) → L+2 (T) is the Szego{double acute} projector onto the non-negative frequencies. For analytic initial data, it is shown that the solution remains spatial analytic for all time t ∈ (- ∞, ∞). In addition, we find a lower bound for the radius of analyticity of the solution. Our method involves energy-like estimates of the special Gevrey class of analytic functions based on the ℓ1 norm of Fourier transforms (the Wiener algebra). © 2013 Elsevier Masson SAS. All rights reserved.

In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1, 1) -forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides an alternate proof of a recent gap theorem of the first author. © © The Author(s) 2013.

In this paper, we develop a method of solving the Poincaré-Lelong equation, mainly via the study of the large time asymptotics of a global solution to the Hodge-Laplace heat equation on (1, 1) -forms. The method is effective in proving an optimal result when M has nonnegative bisectional curvature. It also provides an alternate proof of a recent gap theorem of the first author. © © The Author(s) 2013.

Near a charged surface, counterions of different valences and sizes cluster; and their concentration profiles stratify. At a distance from such a surface larger than the Debye length, the electric field is screened by counterions. Recent studies by a variational mean-field approach that includes ionic size effects and by Monte Carlo simulations both suggest that the counterion stratification is determined by the ionic valence-to-volume ratios. Central in the mean-field approach is a free-energy functional of ionic concentrations in which the ionic size effects are included through the entropic effect of solvent molecules. The corresponding equilibrium conditions define the generalized Boltzmann distributions relating the ionic concentrations to the electrostatic potential. This paper presents a detailed analysis and numerical calculations of such a free-energy functional to understand the dependence of the ionic charge density on the electrostatic potential through the generalized Boltzmann distributions, the role of ionic valence-to-volume ratios in the counterion stratification, and the modification of Debye length due to the effect of ionic sizes.

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w ∈ W a combinatorial module Tw whose support is the interval [1,w]R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.

Prediction markets provide a rare setting, where results of mathematical probability theory can be related to events of real-world interest and where theory can be compared to data. The paper discusses two simple mathematical results - the halftime price principle and the serious candidates principle - and corresponding data from baseball and the 2012 Republican Presidential nomination race. © The Mathematical Association of America.

We discuss the asjTnmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As appUcations, we derive various results on the existence of linear fanctionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duahty theorem. Most of the results are about afhne functions defined on convex subsets of vector spaces, rather than Unear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duaUty theorem) and also results that use Baire's theorem (as in the Robinson-Attouch-Brezis version of the Fendiel duaUty theorem). This paper also contams some new results about metrizable topological vector spaces that are not necessarily locally convex. © Heldermann Verlag.

We discuss the asjTnmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As appUcations, we derive various results on the existence of linear fanctionals that include bivariate, trivariate and quadrivariate generalizations of the Fenchel duahty theorem. Most of the results are about afhne functions defined on convex subsets of vector spaces, rather than Unear functions defined on vector spaces. We consider both results that use a simple boundedness hypothesis (as in Rockafellar's version of the Fenchel duaUty theorem) and also results that use Baire's theorem (as in the Robinson-Attouch-Brezis version of the Fendiel duaUty theorem). This paper also contams some new results about metrizable topological vector spaces that are not necessarily locally convex. © Heldermann Verlag.

Prediction markets provide a rare setting, where results of mathematical probability theory can be related to events of real-world interest and where theory can be compared to data. The paper discusses two simple mathematical results - the halftime price principle and the serious candidates principle - and corresponding data from baseball and the 2012 Republican Presidential nomination race. © The Mathematical Association of America.

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w ∈ W a combinatorial module Tw whose support is the interval [1,w]R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.

In this paper, we define a Banach SNL space to be a Banach space with a certain kind of linear map from it into its dual, and we develop the theory of linear L-positive subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis-Browder theorem. © 2012 Springer Science+Business Media B.V.