The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

Solving large sparse linear systems over the field GF(2)

Nowadays problem of solving sparse linear systems over the field GF(2) remain as a challenge. The popular approach is to improve existing methods such as the block Lanczos method (the Montgomery method) and the Wiedemann-Coppersmith method. Both these methods are considered in the thesis in details: there are their modifications and computational estimation for each process. It demonstrates the most complicated parts of these methods and gives the idea how to improve computations in software point of view. The research provides the implementation of accelerated binary matrix operations computer library which helps to make the progress steps in the Montgomery and in the Wiedemann-Coppersmith methods faster.

A Business Incubator, Accelerator, or Coworking Space? Case Health Innovation Village at GE

Health Innovation Village at GE is one of the new communities targeted for startup and growth-oriented companies. It has been established at the premises of a multinational conglomerate that will promote networking and growth of startup companies. The concept combines features from traditional business incubators, accelerators, and coworking spaces. This research compares Health Innovation Village to these concepts regarding its goals, target clients, source of income, organization, facilities, management, and success factors. In addition, a new incubator classification model is introduced. On the other hand, Health Innovation Village is examined from its tenants’ perspective and improvements are suggested. The work was implemented as a qualitative case study by interviewing GE staff with connections to Health Innovation Village as well as startup entrepreneurs and employees’ working there. The most evident features of Health Innovation Village correspond to those of business incubators although it is atypical as a non-profit corporate business incubator. Strong network orientation and connections to venture capitalists are common characteristics of these new types of accelerators. The design of the premises conforms to the principles of coworking spaces, but the services provided to the startup companies are considerably more versatile than the services offered by coworking spaces. The advantages of Health Innovation Village are that there are first-class premises and exceptionally good networking possibilities that other types of incubators or accelerators are not able to offer. A conglomerate can also provide multifaceted special knowledge for young firms. In addition, both GE and the startups gained considerable publicity through their cooperation, indeed a characteristic that benefits both parties. Most of the expectations of the entrepreneurs were exceeded. However, communication and the scope of cooperation remain challenges. Micro companies spend their time developing and marketing their products and acquiring financing. Therefore, communication should be as clear as possible and accessible everywhere. The startups would prefer to cooperate significantly more, but few have the time available to assume the responsibility of leadership. The entrepreneurs also expected to have more possibilities for cooperation with GE. Wider collaboration might be accomplished by curation in the same way as it is used in the well-functioning coworking spaces where curators take care of practicalities and promote cooperation. Communication issues could be alleviated if the community had its own Intranet pages where all information could be concentrated. In particular, a common calendar and a room reservation system could be useful. In addition, it could be beneficial to have a section of the Intranet open for both the GE staff and the startups so that those willing to share their knowledge and those having project offers could use it for advertising.