Abstract:
Pointlike objects cause many of the divergences that afflict physical theories. For instance, the gravitational binding energy of a point particle in Newtonian mechanics is infinite. In general relativity, the analog of a point particle is a black hole and the notion of binding energy must be replaced by quasilocal energy. The quasilocal energy (QLE) derived by York, and elaborated by Brown and York, is finite outside the horizon but it was not considered how to evaluate it inside the horizon. We present a prescription for finding the QLE inside a horizon, and show that it is finite at the singularity for a variety of types of black hole. The energy is typically concentrated just inside the horizon, not at the central singularity.

Abstract:
This work revisits the classical concept of electric energy and suggests that the common definition is likely to generate large errors when dealing with nanostructures. For instance, deriving the electrostatic energy in semiconductors using the traditional formula fails at giving the correct electrostatic force between capacitor plates and reveals the existence of an extra contribution to the standard electrostatic energy. This additional energy is found to proceed from the generation of space charge regions which are predicted when combining electrostatics laws with semiconductor statistics, such as for accumulation and inversion layers. On the contrary, no such energy exists when relying on electrostatics only, as for instance when adopting the so-called full depletion approximation. The same holds for charged or neutral insulators that are still consistent with the customary definition, but which are in fact singular cases. In semiconductors, this additional free energy can largely exceed the energy gained by the dipoles, thus becoming the dominant term. Consequently, erroneous electrostatic forces in nanostructure systems such as for MEMS and NEMS as well as incorrect energy calculations are expected using the standard definition. This unexpected result clearly asks for a generalization of electrostatic energy in matter in order to reconcile basic concepts and to prevent flawed force evaluation in nanostructures with electrical charges.

Abstract:
The principle of an electrostatic accelerator is that when a charge e escapes from a conducting plane that supports a uniform electric field of strength E_0, then the charge gains energy e E_0 d as it moves distance d from the plane. Where does this energy come from? We that the mechanical energy gain of the electron is balanced by the decrease in the electrostatic field energy of the system.

Abstract:
In this paper we pay attention to the inconsistency in the derivation of the symmetric electromagnetic energy-momentum tensor for a system of charged particles from its canonical form, when the homogeneous Maxwell equations are applied to the symmetrizing gauge transformation, while the non-homogeneous Maxwell equations are used to obtain the motional equation. Applying the appropriate non-homogeneous Maxwell equation to both operations, we have revealed an additional symmetric term in the tensor, named as "compensating term". Analyzing the structure of this "compensating term", we suggested a method of "gauge renormalization, which allows transforming the divergent terms of classical electrodynamics (infinite self-force, self-energy and self-momentum) to converging integrals. The motional equation obtained for a non-radiating charged particle does not contain its self-force, and the mass parameter includes the sum of mechanical and electromagnetic masses. The motional equation for a radiating particle also contains the sum of mechanical and electromagnetic masses, and does not yield any "runaway solutions". It has been shown that the energy flux in a free electromagnetic field is guided by the Poynting vector, whereas the energy flux in a bound EM field is described by the generalized Umov vector, defined in the paper. The problem of "Poincare stresses" is also examined. It has been shown that the presence of the "compensating term" in the electromagnetic energy-momentum tensor allows a solution of the "4/3 problem", where the total observable mass of the electron is completely determined by the Poincare stresses and hence the conventional relativistic relationship between the energy and momentum is recovered.

Abstract:
This chapter focuses on vibration energy harvesting using electrostatic converters. It synthesizes the various works carried out on electrostatic devices, from concepts, models and up to prototypes, and covers both standard (electret-free) and electret-based electrostatic vibration energy harvesters (VEH).

Abstract:
It has been shown that to calculate the parameters of the electrostatic field of the ion crystal lattice it sufficient to take into account ions located at a distance of 1-2 lattice spacings. More distant ions make insignificant contribution. As a result, the electrostatic energy of the ion lattice in the alkaline halide crystal produced by both positive and negative ions is in good agreement with experiment when the melting temperature and the shear modulus are calculated. For fcc and bcc metals the ion lattice electrostatic energy is not sufficient to obtain the observed values of these parameters. It is possible to resolve the contradiction if one assumes that the electron density is strongly localized and has a crystal structure described by the lattice delta - function. As a result, positive charges alternate with negative ones as in the alkaline halide crystal. Such delta-like localization of the electron density is known as a model of nearly free electrons.

Abstract:
In this work we attempt to show in a clear and simple manner the fundamental ideas of the Renormalization Theory. With that intention we use two well-known problems of the Physic and Engeneering undergraduate students, the calculation of the electrostatic and vector potential of a infinite line charge density and current, respectively. We still employ different regularization methods (cut-off, dimensional and zeta function) and the arising of the scale parameter is consider.

Abstract:
We review the variational principle in the density matrix renormalization group (DMRG) method, which maximizes an approximate partition function within a restricted degrees of freedom; at zero temperature, DMRG mini- mizes the ground state energy. The variational principle is applied to two-dimensional (2D) classical lattice models, where the density matrix is expressed as a product of corner transfer matrices. (CTMs) DMRG related fields and future directions of DMRG are briefly discussed.

Abstract:
The disadvantage of `traditional' multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(2,Z)\SL(2,R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. We explicitely construct renormalization schemes for (a) the linearization of vector fields on tori of arbitrary dimension and (b) the construction of invariant tori for Hamiltonian systems.

Abstract:
The binding of clusters of metal nanoparticles is partly electrostatic. We address difficulties in calculating the electrostatic energy when high charging energies limit the total charge to a single quantum, entailing unequal potentials on the particles. We show that the energy at small separation $h$ has a strong logarithmic dependence on $h$. We give a general law for the strength of this logarithmic correction in terms of a) the energy at contact ignoring the charge quantization effects and b) an adjacency matrix specifying which spheres of the cluster are in contact and which is charged. We verify the theory by comparing the predicted energies for a tetrahedral cluster with an explicit numerical calculation.