A viscosity solution of the Hamilton–Jacobi equation with exponential dependence of Hamiltonian on the momentum

The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.

Renormalon structure in compactified spacetime

We point out that the location of renormalon singularities in theory on a circle-compactified spacetime $\mathbb{R}^{d-1} \times S^1$ (with a small radius $R \Lambda \ll 1$) can differ from that on the non-compactified spacetime $\mathbb{R}^d$. We argue this under the following assumptions, which are often realized in large-$N$ theories with twisted boundary conditions: (i) a loop integrand of a renormalon diagram is volume independent, i.e. it is not modified by the compactification, and (ii) the loop momentum variable along the $S^1$ direction is not associated with the twisted boundary conditions and takes the values $n/R$ with integer $n$. We find that the Borel singularity is generally shifted by $-1/2$ in the Borel $u$-plane, where the renormalon ambiguity of $\mathcal{O}(\Lambda^k)$ is changed to $\mathcal{O}(\Lambda^{k-1}/R)$ due to the circle compactification $\mathbb{R}^d \to \mathbb{R}^{d-1} \times S^1$. The result is general for any dimension $d$ and is independent of details of the quantities under consideration. As an example, we study the $\mathbb{C} P^{N-1}$ model on $\mathbb{R} \times S^1$ with $\mathbb{Z}_N$ twisted boundary conditions in the large-$N$ limit.

Testing the Four Factors of the Carhart Model Against Excess Return of Shares in Companies Registered in the Kompas 100 Index for the 2014-2016 Period

The researcher intends to test the four carhart factor model of stock excess return in companies incorporated in Kompas 100 for the 2014-2016 period. Regression analysis was performed on four carhart factor models, namely market returns, firm size, book to market, and momentum towards excess return. The results of this study indicate that in the partial hypothesis testing market return, firm size, and book to market equty variables significantly influence the excess return, while the momentum variable does not significantly influence the magnitude of excess return.Keywords: Four factors, market returns, firm size, book to market equity, momentum, excess stock returns

Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation

We study the Cauchy problem for the first-order evolutionary Hamilton–Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax–Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.

The tunneling effect for a class of difference operators

We analyze a general class of self-adjoint difference operators [Formula: see text] on [Formula: see text], where [Formula: see text] is a multi-well potential and [Formula: see text] is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30–35]).Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs.The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian [Formula: see text] is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance [Formula: see text] induced by [Formula: see text], we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first [Formula: see text] eigenvalues of [Formula: see text] converge to the first [Formula: see text] eigenvalues of the direct sum of harmonic oscillators on [Formula: see text] located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions or quasimodes for the operator [Formula: see text], acting on [Formula: see text], via restriction to the lattice [Formula: see text].Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted [Formula: see text]-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two “wells” (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrödinger operator in [22].

QUANTUM HAMILTON MECHANICS AND THE THEORY OF QUANTIZATION CONDITIONS

A formulation of quantum mechanics in terms of complex canonical variables is presented. It is seen that these variables are governed by Hamilton's equations. It is shown that the action variables need to be quantized. By formulating a quantum Hamilton equation for the momentum variable, the energies for two different systems are determined. Quantum canonical transformation theory is introduced and the geometrical significance of a set of generalized quantization conditions which are obtained is discussed.

Nonplanar graphs and anomalies in chiral noncommutative gaugetheories

The AV (n) one-loop graphs are examined in a 2n-dimensional massless noncommutative gauge model in which both a U(1) axial gauge field A and a U(1) vector gauge field V have adjoint couplings to a Fermion field. A possible anomaly in the divergence of the n + 1 vertices is examined by considering the surface term that can possibly arise when shifting the loop momentum variable of integration. It is shown that despite the fact that the graphs are nonplanar, surface terms do arise in individual graphs, but that in 4n dimensions, a cancellation between the surface term contribution coming from pairs of graphs eliminates all anomalies, while in 4n + 2 dimensions such a cancellation cannot occur and an anomaly necessarily arises.PACS No.: 11.30.Rd

INFINITE NUCLEAR MATTER ON THE LIGHT FRONT: A MODERN APPROACH TO BRUECKNER THEORY

Understanding an important class of experiments requires that light-front dynamics and the related light cone variables k+, k⊥ be used. If one uses k+=k0+k3 as a momentum variable the corresponding canonical spatial variable is x-=x0-x3 and the time variable is x0+x3. This is the light front (LF) approach of Dirac. A relativistic light front formulation of nuclear dynamics is developed and applied to treating infinite nuclear matter in a method which includes the correlations of pairs of nucleons. This is light front Brueckner theory.