THE BEST CONSTANT OF L^p SOBOLEV INEQUALITY CORRESPONDING TO DIRICHLET-NEUMANN BOUNDARY VALUE PROBLEM

We have obtained the best constant of the following Lp Sobolev inequality sup 0≤y≤1| u(j)(y)| ≤C (∫ 01 | u(M)(x)| p dx)1/p , where u is a function satisfying u(M) ∈ Lp(0, 1), u(2i)(0) = 0 (0 ≤i ≤ [(M − 1)/2]) and u(2i+1)(1) = 0 (0 ≤ i ≤ [(M − 2)/2]), where u(i) is the abbreviation of (d/dx)iu(x). In [9], the best constant of the above inequality was obtained for the case of p = 2 and j = 0. This paper extends the result of [9] under the conditions p > 1 and 0 ≤ j ≤ M −1. The best constant is expressed by Bernoulli polynomials

Positivity and Hierarchical Structure of four Green Functions Corresponding to a Bending Problem of a Beam on a half line

We consider the boundary value problem for fourth order linear ordinary differential equation in a half line (0,∞), which represents bending of a beam on an elastic foundation under a tension. A tension is relatively stronger than a spring constant of elastic foundation. We here treat four self-adjoint boundary conditions, clamped, Dirichlet, Neumann and free edges, at x = 0. We show the positivity and the hierarchical structure of four Green functions

ADMM and Reproducing Sum-Product Decoding Algorithm Applied to QC-MDPC Code-based McEliece Cryptosystems

QC-MDPC (quasi cyclic moderate density parity check) code-based McEliece cryptosystems are considered to be one of the candidates for post-quantum cryptography. Decreasing DER (decoding error rate) is one of important factor for their security, since recent attacks to these cryptosystems effectively use DER information. In this paper, we pursue the possibility of optimization-base decoding, concretely we examine ADMM (alternating direction method of multipliers), a recent developing method in optimization theory. Further, RSPA (reproducing sum-product algorithm), which efficiently reuse outputs of SPA (sum-product algorithm) is proposed for the reduction of execution time in decoding. By numerical simulations, we show that the proposing scheme shows considerable decrement in DER compared to the conventional decoding methods such as BF (bit-flipping algorithm) variants or SPA

Racemization-free Monomer: α-Hydroxyisobutyric Acid from Bio-based Lactic Acid

In order to solve the important problem of the racemization of poly(L-lactic acid), a high yield of racemization-free monomer: a-hydroxyisobutyric acid (HIBA) was synthesized from bio-based lactic acid by methylation using specific bases with bulky side groups. Obtained HIBA can be converted into poly(tetramethylglycolide), which is racemization-free and has higher melting and glass transition points than poly(L-lactic acid)

A Simple Synthetic Route for the Preparation of Tetramethylglycolide from Lactic Acid

Poly(tetramethylglycolide), a chemical recyclable polymer, is currently attracting a lot of interest as a substitute for poly(L-lactic acid). We report here a simple and convenient route for the synthesis of tetramethylglycolide (TMG) from lactic acid. This method involves three steps: (1) one-step protection of lactic acid by cyclic acetalization employing acetone; (2) α-methylation of the obtained 2,2,5-trimethyl-1,3-dioxolan-4-one; and (3) one-pot synthesis of TMG including the hydrolysis of 2,2,5,5-tetramethyl-1,3-dioxolan-4-one. We found significant advantages of the incorporation of existing reactions in the synthesis of TMG

Poly(tetramethyl glycolide) from Renewable Carbon, a Racemization-Free and Controlled Depolymerizable Polyester

Racemization-free and depolymerization-controllable poly(tetramethyl glycolide) (PTMG) was synthesized from renewable resources: D/L-lactic acids and pyruvic acid. PTMG overcomes the undesirable properties of poly(lactic acid) such as low heat-resistance and racemization causing the decrease in crystallinity. PTMG was shown to have a higher melting point than 200 °C and to be a superior recyclable material capable of being depolymerized controllably into cyclic dimer tetramethyl glycolide (TMG) or methacrylic acid (MA) by using specific catalysts. PTMG can be reversibly synthesized from the depolymerized TMG. Moreover, biomass-based poly(methyl methacrylate) (PMMA) was prepared from the recovered MA