1,069 research outputs found
A fully-discrete Semi-Lagrangian scheme for a first order mean field game problem
In this work we propose a fully-discrete Semi-Lagrangian scheme for a {\it
first order mean field game system}. We prove that the resulting discretization
admits at least one solution and, in the scalar case, we prove a convergence
result for the scheme. Numerical simulations and examples are also discussed.Comment: 28 pages,16 figure
A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations
We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the
dependence of the coefficients is nonlinear and nonlocal in time with respect
to the unknowns. We extend the numerical scheme proposed and studied recently
by the authors for a single FPK equation of this type. We analyse the
convergence of the scheme and we study its applicability in two examples. The
first one concerns a population model involving two interacting species and the
second one concerns two populations Mean Field Games
On the dimensions of secant varieties of Segre-Veronese varieties
This paper explores the dimensions of higher secant varieties to
Segre-Veronese varieties. The main goal of this paper is to introduce two
different inductive techniques. These techniques enable one to reduce the
computation of the dimension of the secant variety in a high dimensional case
to the computation of the dimensions of secant varieties in low dimensional
cases. As an application of these inductive approaches, we will prove
non-defectivity of secant varieties of certain two-factor Segre-Veronese
varieties. We also use these methods to give a complete classification of
defective s-th Segre-Veronese varieties for small s. In the final section, we
propose a conjecture about defective two-factor Segre-Veronese varieties.Comment: Revised version. To appear in Annali di Matematica Pura e Applicat
Square Root Actions, Metric Signature, and the Path-Integral of Quantum Gravity
We consider quantization of the Baierlein-Sharp-Wheeler form of the
gravitational action, in which the lapse function is determined from the
Hamiltonian constraint. This action has a square root form, analogous to the
actions of the relativistic particle and Nambu string. We argue that
path-integral quantization of the gravitational action should be based on a
path integrand rather than the familiar Feynman expression
, and that unitarity requires integration over manifolds of both
Euclidean and Lorentzian signature. We discuss the relation of this path
integral to our previous considerations regarding the problem of time, and
extend our approach to include fermions.Comment: 32 pages, latex. The revision is a more general treatment of the
regulator. Local constraints are now derived from a requirement of regulator
independenc
A semi-Lagrangian scheme for Hamilton–Jacobi–Bellman equations with oblique derivatives boundary conditions
We investigate in this work a fully-discrete semi-Lagrangian approximation of second order possibly degenerate Hamilton–Jacobi–Bellman (HJB) equations on a bounded domain O⊂ RN (N= 1 , 2 , 3) with oblique derivatives boundary conditions. These equations appear naturally in the study of optimal control of diffusion processes with oblique reflection at the boundary of the domain. The proposed scheme is shown to satisfy a consistency type property, it is monotone and stable. Our main result is the convergence of the numerical solution towards the unique viscosity solution of the HJB equation. The convergence result holds under the same asymptotic relation between the time and space discretization steps as in the classical setting for semi-Lagrangian schemes on O= RN. We present some numerical results, in dimensions N=1,2, on unstructured meshes, that confirm the numerical convergence of the scheme
New examples of defective secant varieties of Segre-Veronese varieties
We prove the existence of defective secant varieties of three-factor and
four-factor Segre-Veronese varieties embedded in certain multi-degree. These
defective secant varieties were previously unknown and are of importance in the
classification of defective secant varieties of Segre-Veronese varieties with
three or more factors.Comment: 10 page
Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain
We give analytical solutions for the time-optimal synthesis of entangling
gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three
qubits subject to an Ising Hamiltonian interaction with equal coupling plus
a local magnetic field acting on the intermediate qubit. The energy available
is fixed, but we relax the standard assumption of instantaneous unitary
operations acting on single qubits. The time required for performing an
entangling gate which is equivalent, modulo local unitary operations, to the
between the indirectly coupled qubits 1 and 3 is
, i.e. faster than a previous estimate based on a similar
Hamiltonian and the assumption of local unitaries with zero time cost.
Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space
of qubit 3 shows that the time-optimal synthesis of the (which acts as the identity when the control qubit 1 is in the state
, while if the control qubit is in the state the target
qubit 3 is flipped as ) also requires the same
time .Comment: 9 pages; minor modification
Decoupling the Spread of Grasslands from the Evolution of Grazer-type Herbivores in South America
The evolution of high-crowned cheek teeth (hypsodonty) in herbivorous mammals during the late Cenozoic is classically regarded as an adaptive response to the near-global spread of grass-dominated habitats. Precocious hypsodonty in middle Eocene (~38 million years (Myr) ago) faunas from Patagonia, South America, is therefore thought to signal Earth’s first grasslands, 20 million years earlier than elsewhere. Here, using a high-resolution, 43–18 million-year record of plant silica (phytoliths) from Patagonia, we show that although open-habitat grasses existed in southern South America since the middle Eocene (~40 Myr ago), they were minor floral components in overall forested habitats between 40 and 18 Myr ago. Thus, distinctly different, continent-specific environmental conditions (arid grasslands versus ash-laden forests) triggered convergent cheek–tooth evolution in Cenozoic herbivores. Hypsodonty evolution is an important example where the present is an insufficient key to the past, and contextual information from fossils is vital for understanding processes of adaptation
Dynamical Determination of the Metric Signature in Spacetime of Nontrivial Topology
The formalism of Greensite for treating the spacetime signature as a
dynamical degree of freedom induced by quantum fields is considered for
spacetimes with nontrivial topology of the kind , for varying . It is shown that a dynamical origin for the Lorentzian
signature is possible in the five-dimensional space with small torus radius (periodic boundary conditions), as well as in
four-dimensional space with trivial topology. Hence, the possibility exists
that the early universe might have been of the Kaluza-Klein type, \ie
multidimensional and of Lorentzian signature.Comment: 10 pages, LaTeX file, 4 figure
Decoupling the spread of grasslands from the evolution of grazer-type herbivores in South America
The evolution of high-crowned cheek teeth (hypsodonty) in herbivorous mammals during the late Cenozoic is classically regarded as an adaptive response to the near-global spread of grass-dominated habitats. Precocious hypsodonty in middle Eocene (∼38 million years (Myr) ago) faunas from Patagonia, South America, is therefore thought to signal Earth's first grasslands, 20 million years earlier than elsewhere. Here, using a high-resolution, 43-18 million-year record of plant silica (phytoliths) from Patagonia, we show that although open-habitat grasses existed in southern South America since the middle Eocene (∼40 Myr ago), they were minor floral components in overall forested habitats between 40 and 18 Myr ago. Thus, distinctly different, continent-specific environmental conditions (arid grasslands versus ash-laden forests) triggered convergent cheek-tooth evolution in Cenozoic herbivores. Hypsodonty evolution is an important example where the present is an insufficient key to the past, and contextual information from fossils is vital for understanding processes of adaptation.Facultad de Ciencias Naturales y Muse
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