Eigenmode-based capacitance calculations with applications in passivation layer design

The design of high-speed metallic interconnects such as microstrips requires the correct characterization of both the conductors and the surrounding dielectric environment, in order to accurately predict their propagation characteristics. A fast boundary integral equation approach is obtained by modeling all materials as equivalent surface charge densities in free space. The capacitive behavior of a finite dielectric environment can then be determined by means of a transformation matrix, relating these charge densities to the boundary value of the electric potential. In this paper, a new calculation method is presented for the important case that the dielectric environment is composed of homogeneous rectangles. The method, based on a surface charge expansion in terms of the Robin eigenfunctions of the considered rectangles, is not only more efficient than traditional methods, but is also more accurate, as shown in some numerical experiments. As an application, the design and behavior of a microstrip passivation layer is treated in some detail

Mean-field solution of the parity-conserving kinetic phase transition in one dimension

A two-offspring branching annihilating random walk model, with finite reaction rates, is studied in one-dimension. The model exhibits a transition from an active to an absorbing phase, expected to belong to the $DP2$ universality class embracing systems that possess two symmetric absorbing states, which in one-dimensional systems, is in many cases equivalent to parity conservation. The phase transition is studied analytically through a mean-field like modification of the so-called {\it parity interval method}. The original method of parity intervals allows for an exact analysis of the diffusion-controlled limit of infinite reaction rate, where there is no active phase and hence no phase transition. For finite rates, we obtain a surprisingly good description of the transition which compares favorably with the outcome of Monte Carlo simulations. This provides one of the first analytical attempts to deal with the broadly studied DP2 universality class.Comment: 4 Figures. 9 Pages. revtex4. Some comments have been improve

Morphisms from P2 to Gr(2,C4)

In this note we study morphisms from P2 to Gr(2,C4) from the point of view of the cohomology class they represent in the Grassmannian. This leads to some new result about projection of d-uple imbedding of P2 to P5

The fundamental group of reduced suspensions

We classify pointed spaces according to the first fundamental group of their reduced suspension. A pointed space is either of so-called totally path disconnected type or of horseshoe type. These two camps are defined topologically but a characterization is given in terms of fundamental groups. Among totally path disconnected spaces the fundamental group is shown to be a complete invariant for a notion of topological equivalence weaker than that of homeomorphism

Small-Scale Markets for Bilateral Resource Trading in the Sharing Economy

We consider a general small-scale market for agent-to-agent resource sharing, in which each agent could either be a server (seller) or a client (buyer) in each time period. In every time period, a server has a certain amount of resources that any client could consume, and randomly gets matched with a client. Our target is to maximize the resource utilization in such an agent-to-agent market, where the agents are strategic. During each transaction, the server gets money and the client gets resources. Hence, trade ratio maximization implies efficiency maximization of our system. We model the proposed market system through a Mean Field Game approach and prove the existence of the Mean Field Equilibrium, which can achieve an almost 100% trade ratio. Finally, we carry out a simulation study motivated by an agent-to-agent computing market, and a case study on a proposed photovoltaic market, and show the designed market benefits both individuals and the system as a whole

Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles

In a recent paper, Buckmaster & Vicol (arXiv:1709.10033) used the method of convex integration to construct weak solutions $u$ to the 3D incompressible Navier-Stokes equations such that $\| u(t) \|_{L^2} =e(t)$ for a given non-negative and smooth energy profile $e: [0,T]\to \mathbb{R}$. However, it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI)), Leray-Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing. In this paper we are concerned with weak solutions to the Navier-Stokes inequality on $\mathbb{R}^3$, that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier-Stokes equations). Given $T>0$ and a nonincreasing energy profile $e\colon [0,T] \to [0,\infty )$ we construct weak solution to the Navier-Stokes inequality that are localised in space and whose energy profile $\| u(t)\|_{L^2 (\mathbb{R}^3 )}$ stays arbitrarily close to $e(t)$ for all $t\in [0,T]$. Our method applies only to nonincreasing energy profiles. The relevance of such solutions is that, despite not satisfying the Navier-Stokes equations, they satisfy the partial regularity theory of Caffarelli, Kohn & Nirenberg (Comm. Pure Appl. Math., 1982). In fact, Scheffer's constructions of weak solutions to the Navier-Stokes inequality with blow-ups (Comm. Math. Phys., 1985 & 1987) show that the Caffarelli, Kohn & Nirenberg's theory is sharp for such solutions. Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer's. Namely, we obtain weak solutions to the Navier-Stokes inequality with both blow-up and a prescribed energy profile.Comment: 26 pages, 4 figures. arXiv admin note: text overlap with arXiv:1709.0060