We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.

Abstract:
In this
paper, a new viewpoint of the division by zero z/0 = 0 in matrices is introduced and the results will show that
the division by zero is our elementary and fundamental mathematics. New and
practical meanings for many mathematical and physical formulas for the
denominator zero cases may be given. Furthermore, a new space idea for the
point at infinity for the Eucleadian plane is also introduced.

Abstract:
We consider the usual wave equation utt(x,t)=c2uxx(x,t) on the real line with some typical initial and boundary conditions. In each case, we establish a natural isometrical identity and inverse formula between the sourse function and the response function.

Abstract:
For the solution u(x,t)=u(f)(x,t) of the equations {u ￠ € 2(x,t)= ”u(x,t),u(x,0)=f(x),u(x,t)=0, ￠ € x ￠ ,t>0x ￠ x ￠ ￠ ,t>0} where ￠ ￠ r,2 ￠ ‰ ¤r ￠ ‰ ¤3 is a bounded domain with C2-boundary and for an appropriate subboundary “ of we prove a Lipschitz estimate of ￠ € –f ￠ € –L2( ) : For ￠ (1,54) and for a positive constant CC ￠ ’1 ￠ € –f ￠ € –L2( ) ￠ ‰ ¤ ￠ € – ￠ u(f) ￠ v ￠ € –B ( “ —(0, ￠ )) ￠ ‰ ￠ “{ ￠ ‘n=0 ￠ 1n! “(n+2 +1) ￠ 0 ￠ |(p ￠ pn+1+n ￠ pn)p ￠ ’32 ￠ u(f) ￠ v(x,14p)|2p2n+2 ￠ ’1dp}ds. The norm ￠ € – ￠ … ￠ € –B ( “ —(0, ￠ )) is involved and strong, but it is a natural one in our situation relating to a typical and simple norm for analytic functions. Furthermore, it is acceptable in the sense that ￠ € – ￠ u(f) ￠ v ￠ € –B ( “ —(0, ￠ )) ￠ ‰ ¤C ￠ € –f ￠ € –H2( ) holds.

Abstract:
For the solution of the equations where , is a bounded domain with -boundary and for an appropriate subboundary of we prove a Lipschitz estimate of : For and for a positive constant The norm is involved and strong, but it is a natural one in our situation relating to a typical and simple norm for analytic functions. Furthermore, it is acceptable in the sense that holds.

Abstract:
We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.

Abstract:
This article describes an easily constructed apparatus for an experiment on a rolling ball a rotating board.From the experimental and theoretical results, we can find that the ball leaves the board at the instant boththe ball and the board are released from hand. Then, the ball can’t roll down the board until the balltouches the board again.

Abstract:
We present an apparatus for “Monkey and Hunter” demonstration in the student lab. We can easily construct it at a low cost and the blow gun has a high hit rate.

Abstract:
In this article I shall describe an easily constructed apparatus for an experiment on winding motion in a spiral-liketrajectory in three dimensions. The experimental results show how the total time of the process depends on the initialspeed, and the total time has its maximum value of 16.3s for a speed of 2.67m/s. The experimental results were ingood agreement with the theoretical predictions. The analytical solution of the problem is original

Abstract:
We present a new model of equipotential surfaces. We can easily construct it at a low cost. Using the model, students come to realize that the gradient of the electric potential is directly proportional to the density of lines of force, and hence the electric field intensity.