Kohtaro Watanabe

The Best Constant of Sobolev Inequality Corresponding to Clamped Boundary Value Problem

Greens function of the clamped boundary value problem for the differential operator on the interval is obtained. The best constant of corresponding Sobolev inequality is given by . In addition, it is shown that a reverse of the Sobolev best constant is the one which appears in the generalized Lyapunov inequality by Das and Vatsala (1975).

Algorithms to solve the knapsack constrained maximum spanning tree problem

The knapsack problem and the minimum spanning tree problem are both fundamental in operations research and computer science. We are concerned with a combination of these two problems. That is, we are given a knapsack of a fixed capacity, as well as an undirected graph where each edge is associated with profit and weight. The problem is to fill the knapsack with a feasible spanning tree such that the tree profit is maximized. We prove this problem 𝒩𝒫-hard, present upper and lower bounds, develop a branch-and-bound algorithm to solve the problem to optimality and propose a shooting method to accelerate computation. We evaluate the developed algorithm through a series of numerical experiments for various types of test problems.

Riemann zeta function and Lyapunov-type inequalities for certain higher order differential equations

This note generalizes the well known Lyapunov-type inequalities for second-order linear differential equations to certain 2M-th order linear differential equations with five types of boundary conditions. The usage of the best constant of some Sobolev-type inequalities clarify the process for obtaining such inequality and sharpen the result of Cakmak [2].

Reproducing Kernels of Hm (a, b) (m = 1, 2, 3) and Least Constants in Sobolev’s Inequalities

In [J.T. Marti (1983). Evaluation of the least constant in Sobolev’s inequality for H l(0,s). SIAM J. Numer. Anal., 20(6), 1239–1242.], Marti proves that the least constant c l in Sobolev’s inequality for the embedding of the Sobolev space H l(a, b) into the space C[a, b] of bounded continuous functions on the interval [a, b] is . In this article, we compute the least constants in Sobolev’s inequalities for the spaces Hm (a,b) (m = 1, 2, 3) and Hm (ℝ), by the construction of the reproducing kernels of the Sobolev spaces.

Plane domains which are spectrally determined II

This paper studies the trace of the heat kernelZ(t) ≔ ∑j=1∞ exp (λjt), where{λj} are the eigenvalues of atwo-dimensional Dirichlet or Neumann Laplace operator. FromZ(t), a sequence of invariants (geometrical invariants)such as area, boundary measure, Euler characteristics, etc., can bedetermined. Using these invariants, the existence of the nondisk domainswhich are determined from the information of Dirichlet and Neumannspectrum, can be shown. In addition, we prove that the number of suchdomains is infinite (uncountable) and these domains are not similar eachother.

Sobolev type inequalities of time-periodic boundary value problems for Heaviside and Thomson cables

We consider a time-periodic boundary value problem of n th order ordinary differential operator which appears typically in Heaviside cable and Thomson cable theory. We calculate the best constant and a family of the best functions for a Sobolev type inequality by using the Green function and apply its results to the cable theory. Physical meaning of a Sobolev type inequality is that we can estimate the square of maximum of the absolute value of AC output voltage from above by the power of input voltage.MSC:46E35, 41A44, 34B27.

Listing all the minimum spanning trees in an undirected graph

Efficient polynomial time algorithms are well known for the minimum spanning tree problem. However, given an undirected graph with integer edge weights, minimum spanning trees may not be unique. In this article, we present an algorithm that lists all the minimum spanning trees included in the graph. The computational complexity of the algorithm is O(N(mn+n 2 log n)) in time and O(m) in space, where n, m and N stand for the number of nodes, edges and minimum spanning trees, respectively. Next, we explore some properties of cut-sets, and based on these we construct an improved algorithm, which runs in O(N m log n) time and O(m) space. These algorithms are implemented in C language, and some numerical experiments are conducted for planar as well as complete graphs with random edge weights.

Symmetrization of Functions and the Best Constant of 1-DIM Sobolev Inequality

The best constants of Sobolev embedding of into for and are obtained. A lemma concerning the symmetrization of functions plays an important role in the proof.

Time-Frequency Analysis of the Blood Flow Doppler Ultrasound Signal

Power spectral analysis is extensively used to interpret ultrasound data. However, the technique is useful only when the data can be treated as stationary. Ultrasound data are mostly nonstationary. Thus, a short time Fourier transform (STFT or spectrogram) is widely used to analyze spectral components which change with time. However, the STFT has a low accuracy in both time and frequency domains. Currently, Cohens class time-frequency (TF) analysis is popular for analyzing nonstationary signals. The authors recently proposed a new kernel (named a figure eight kernel). In order to apply the TF analysis with the new kernel to a blood flow signal, experimental data were obtained from the carotid artery by an ultrasound Doppler monitor (Toitsu, Japan). To analyze the data, three kernels were used: (1) a Wigner kernel, (2) a Choi-Williams kernel, and (3) a figure eight kernel. Using our new figure eight kernel, the demodulation accuracy was improved and blood flow components were observed.

Representation formula for the critical points of the Tadjbakhsh-Odeh functional and its application

In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic cn(·) function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.