Abstract:
Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M,
a, b are constants and . Then there exist harmonic functions on M .

Abstract:
The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.

This paper is different from the existing
macroeconomic theory. It derives the relationship between the various
macroeconomic variables from the Cobb-Douglas function, and uses statistical data in United States to verify these
equations. We find that the Cobb-Douglas function is obtained in the case that the
marginal output is not 0, therefore it is a function with dynamic properties. On this basis, we use the difference
between the compound output and simple output to determine the incremental
output of a period, and its relationship with capital, interest rate, investment and other variables. These relations
not only deny the premise of Keynesian theory, but also because the proof of
the investment is not equal to the incremental capital, leaving the investment,
output growth, business cycle and other neoclassical theory crisis.

The Cobb-Douglas function not only leads to a
long-term relationship between the rate of output change and the interest rate, but also analyzes why they fluctuate in the short-term. This paper first divides the fluctuation cycle of the interest rate in
the statistical data of the past 45 years by using the mathematical phase
diagram method, and draws the phase diagram of the rate of output change on the
interest rate according to the cycle equation of output. From this phase
diagram, we explain the reason that the phase difference between the interest rate
and the rate of output change in the fluctuation. Then, according to the
optimal relation between L and K in the Cobb-Douglas function, we
further derive the employment equation and its relation to the real interest
rate and the rate of real output change, and verify the theoretical speculation
with statistical data. Finally, it is concluded that the business cycle is a
kind of endogenous production phenomenon.

The difficulty of inflation theory is to explain the fluctuations of the price
level in short-term. First, we use the relationship
between price and money in the traditional quantity equation to derive the
inflation equation that can explain changes of the price level in the long-term
and short-term. Then,
by analyzing the phase diagram of the single variable and the complex variable in the inflation
equation, we find that the fundamental reason of the periodic change of the
short-term price is the periodic change of the real interest rate. Because
fluctuations of the inflation rate and interest rate are the same in
phase, so there is no difference
in the business cycle. Finally, this paper analyzes
the rise and fall of core price under the influence of money and real interest
rate respectively. It lays the foundation for further discussion of the
relationship between inflation and unemployment in Phillips curve.

Officials always want to use the replacement relationship between the unemployment rate and the price of the Phillips curve to reduce the unemployment rate, but things go crisscross. The reason is that there is no causal relationship between price and unemployment. In this paper, it demonstrates the Phillips curve is derived from the unemployment cycle equation and the inflation equation. And then, it discusses the influence of core variables and fluctuating variables on Phillips curve and the cause of distortion or complete disappearance of Phillips curve. Finally, this paper discusses monetary neutrality and related monetary policy issues. The policy that is conducive to the smooth operation of the system is consistent with the constant money supply of the economic cycle, rather than counter-cyclical regulation.

Traditional macroeconomic theory is difficult to
analyze the long-term growth and short-term decisions of output in a unified
model. In this paper, the concept of “unit resource output” is proposed by using
the difference of production factors combination on different rays in Cobb-Douglas
function, and its maximization condition is derived according to algebraic
principle. And then we use this condition to
explain the reason why the distribution parameter α in the Cobb-Douglas function is growing continuously in the US
statistical data and predict the evolution path of the factor combination in
the growth of the output. Finally, this paper compares the important
differences between our model and the Solow model.

In traditional
macroeconomics argues that the
decision and fluctuation of output level is short-term theory, and the growth of output is a
long-term theory. The former is determined by the demand; the latter is
determined by the production. No one
has questioned why the former is determined by production and the latter is
determined by demand. This
paper argues that the factors that affect the output are the same in the short
and long term, but there is no need to analyze the problem of fluctuation in
the long term. Based on the analysis of
the growth path in our previous paper, this paper first examines whether our
model is applicable to the Pontryagin maximum principle, and then analyzes the
difference of the Cobb-Douglas function in the exogenous growth model and the
endogenous growth model. Reveal the special role of parameter A in Cobb-Douglas function: as long as A is considered as output-related
variables, there is no substantial difference between the so-called exogenous
model and endogenous mode. Finally, according to the trend of the growth path and the nature
of A, the paper derives the final
state of Cobb-Douglas function.

There are two reasons for the financial crisis. One is the spontaneous specula-tion in the capital market. The P/E ratio of the speculative object is much high-er than the ratio of capital to output in the factual economic system. The other is the crisis triggered by the bubble environment. In addition, inflation has a strong inhibitory effect on asset bubbles, and high inflation and asset bubbles will not occur at the same time. These can be verified by statistical data and theoretical explanation from the marginal state equation of Cobb-Douglas function.

Clinic Appointment Registration System is an important
way to see a doctor, and it’s also a preliminary tool for storage and
management of clinic medical records. This new system was developed using
Visual Studio 2008 and C#.NET as the development environment and tools and
Microsoft Access 2003 as the database to store the medical data based on Browse/Server
(B/S) model. The system consists of several data operation functions including
appointment registration, data management (e.g. addition, deletion and
searching), data backup and recovery, etc., thus, achieve key research ob-jectives.