Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

[[abstract]]In this paper, the vectorial Sturm-Liouville operator L Q =−d 2 dx 2 +Q(x) is considered, where Q(x) is an integrable m×m matrix-valued function defined on the interval [0,π] . The authors prove that m 2 +1 characteristic functions can determine the potential function of a vectorial Sturm-Liouville operator uniquely. In particular, if Q(x) is real symmetric, then m(m+1) 2 +1 characteristic functions can determine the potential function uniquely. Moreover, if only the spectral data of self-adjoint problems are considered, then m 2 +1 spectral data can determine Q(x) uniquely.[[notice]]補正完畢[[incitationindex]]SCI[[cooperationtype]]國外[[booktype]]電子

Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

<p>Abstract</p> <p>In this paper, the vectorial Sturm-Liouville operator <inline-formula> <m:math name="1687-2770-2011-40-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>Q</m:mi> </m:mrow> </m:msub> <m:mo class="MathClass-rel">=</m:mo> <m:mo class="MathClass-bin">-</m:mo> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">d</m:mtext> </m:mstyle> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">d</m:mtext> </m:mstyle> <m:msup> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo class="MathClass-bin">+</m:mo> <m:mi>Q</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:math> </inline-formula> is considered, where <it>Q</it>(<it>x</it>) is an integrable <it>m </it>&#215; <it>m </it>matrix-valued function defined on the interval [0,<it>&#960;</it>] The authors prove that <it>m</it> ²+1 characteristic functions can determine the potential function of a vectorial Sturm-Liouville operator uniquely. In particular, if <it>Q</it>(<it>x</it>) is real symmetric, then <inline-formula> <m:math name="1687-2770-2011-40-i2" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mfrac> <m:mrow> <m:mi>m</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo class="MathClass-bin">+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mo class="MathClass-bin">+</m:mo> <m:mn>1</m:mn> </m:math> </inline-formula> characteristic functions can determine the potential function uniquely. Moreover, if only the spectral data of self-adjoint problems are considered, then <it>m</it> ²+ 1 spectral data can determine <it>Q</it>(<it>x</it>) uniquely.</p

[[alternative]]在有限區間向量型Sturm-Liouville方程式的唯一性定理

博士[[abstract]]關於定義在區間的非對稱形Sturm-Liouville 微分方程式的反問題研究及學習，Yurko ( [24] , 2006)利用Weyl矩陣，提出了矩陣邊界值問題的反問題有唯一性的定理。 在本篇論文，首先；對於Sturm-Liouville矩陣微分方程式含有一般的邊界條件的反問題，我們將証明ㄧ般的h1 , H1，亦可得到Q(x)有唯一性。利用矩陣型式邊界值反問題的唯一性，我們主要工作是在向量微分方程式邊界值反問題上，探求向量頻譜(spectral sets)與位階函數Q(x)唯一性的關係。 對於h1 = H1 = In ,我們找出某些個頻譜就可以決定Q(x)了。而若為一對稱矩陣或對角化矩陣，則個別僅需某些頻譜集合即可。 對於一般的h1 , H1，我們也獲得了一些相關的結果。[[abstract]]Inverse spectral problems are studied for the non-self-adjoint matrix Sturm-Liouville differential equation on a finite interval. Using Weyl function, Yurko([24],2006) solved the inverse spectral problem for the matrix Sturm-Liouville operator on a finite interval with the boundary value problem L(Q(x), h, H ). At first, in this thesis, we try to solve the uniqueness theorem of the matrix-valued boundary value problem for arbitrary matrices h1 , h0 , H1 , H0 with the general boundary conditions. By the uniqueness theorem of L(Q(x),h1 , h0 , H1 , H0) described as above, our main work is to find those relations between spectra and potential Q(x) for the vectorial Sturm-Liouville differential equation. For h1 = H1 = In , we will give some characteristic functions corresponding to spectra to determine the Weyl matrix and to prove the uniqueness theorem. Furthermore, we also prove the uniqueness theorems for the vectorial Sturm-Liouville operators with real symmetric potential or real diagonal potential by given some spectra, respectively. We also obtain some results for arbitrary matrices h1 and H1.[[tableofcontents]]Chapter 1. Introduction 1.1 Sturm-Liouville operators on a finite interval .......... 3 1.2 Vectorial Sturm-Liouville equation on a finite interval ...........6 1.3 The Weyl matrix .............. 9 1.4 Sturm-Liouville equation on a graph ...... 11 Chapter 2. Uniqueness of the potential function for vectorial Sturm-Liouville equation on a finite interval 2.1 Preliminaries ............19 2.2 Main results ..............21 Chapter 3. Uniqueness theorem for the vectorial Sturm-Liouville equation with general boundary conditions 3.1 Introduction .................... 31 3.2 Preliminaries ............... 32 3.3 Main results .................... 38 References .............................50[[note]]學號: 892150052, 學年度: 10

Development and Testing of X-Ray Imaging-Enhanced Poly-L-Lactide Bone Screws.

Nanosized iron oxide particles exhibit osteogenic and radiopaque properties. Thus, iron oxide (Fe3O4) nanoparticles were incorporated into a biodegradable polymer (poly-L-lactic acid, PLLA) to fabricate a composite bone screw. This multifunctional, 3D printable bone screw was detectable on X-ray examination. In this study, mechanical tests including three-point bending and ultimate tensile strength were conducted to evaluate the optimal ratio of iron oxide nanoparticles in the PLLA composite. Both injection molding and 3D printing techniques were used to fabricate the PLLA bone screws with and without the iron oxide nanoparticles. The fabricated screws were implanted into the femoral condyles of New Zealand White rabbits. Bone blocks containing the PLLA screws were resected 2 and 4 weeks after surgery. Histologic examination of the surrounding bone and the radiopacity of the iron-oxide-containing PLLA screws were evaluated. Our results indicated that addition of iron oxide nanoparticles at 30% significantly decreased the ultimate tensile stress properties of the PLLA screws. The screws with 20% iron oxide exhibited strong radiopacity compared to the screws fabricated without the iron oxide nanoparticles. Four weeks after surgery, the average bone volume of the iron oxide PLLA composite screws was significantly greater than that of PLLA screws without iron oxide. These findings suggested that biodegradable and X-ray detectable PLLA bone screws can be produced by incorporation of 20% iron oxide nanoparticles. Furthermore, these screws had significantly greater osteogenic capability than the PLLA screws without iron oxide

Hinokitiol Exerts Anticancer Activity through Downregulation of MMPs 9/2 and Enhancement of Catalase and SOD Enzymes: In Vivo Augmentation of Lung Histoarchitecture

Melanoma is extremely resistant to chemotherapy and the death rate is increasing hastily worldwide. Extracellular matrix promotes the migration and invasion of tumor cells through the production of matrix metalloproteinase (MMP)-2 and -9. Evidence has shown that natural dietary antioxidants are capable of inhibiting cancer cell growth. Our recent studies showed that hinokitiol, a natural bioactive compound, inhibited vascular smooth muscle cell proliferation and platelets aggregation. The present study is to investigate the anticancer efficacy of hinokitiol against B16-F10 melanoma cells via modulating tumor invasion factors MMPs, antioxidant enzymes in vitro. An in vivo mice model of histological investigation was performed to study the patterns of elastic and collagen fibers. Hinokitiol inhibited the expression and activity of MMPs-2 and -9 in B16-F10 melanoma cells, as measured by western blotting and gelatin zymography, respectively. An observed increase in protein expression of MMPs 2/9 in melanoma cells was significantly inhibited by hinokitiol. Notably, hinokitiol (1–5 μM) increased the activities of antioxidant enzymes catalase (CAT) and superoxide dismutase (SOD) from the reduction in melanoma cells. Also, hinokitiol (2–10 µM) concentration dependently reduced in vitro Fenton reaction induced hydroxyl radical (OH·) formation. An in vivo study showed that hinokitiol treatment increased elastic fibers (EF), collagens dispersion, and improved alveolar alterations in the lungs of B16/F10 injected mice. Overall, our findings propose that hinokitiol may be a potent anticancer candidate through down regulation of MMPs 9/2, reduction of OH· production and enhancement of antioxidant enzymes SOD and CAT

A typical example of the histologic image at the screw/bone interface 4 weeks after implanting the screws.

<p>The leached debris of the nano-Fe₃O₄/poly-L-lactide (PLLA) composite was surrounded by newly formed bone and bone cells (black arrow). BC: bone cell; BV: blood vessel; CT: connective tissue; DB: leached debris; MB: mature bone; NB: new bone. Scale bar, 50 μm.</p

Histologic examination of bone tissue at the screw/bone interface 2 weeks after implanting the screws.

<p>Neat poly-L-lactide (PLLA) screws were fabricated by injection molding (a) and 3D printing (c) methods. Screws made of PLLA mixed with 20% Fe₃O₄ nanoparticles fabricated by injection molding (b) and 3D printing (d). New bone (black arrows) was visible between more of the threads and occupied a greater area with the 20% nano-Fe₃O₄/PLLA screw compared to the neat PLLA screw. Scale bar, 1.0 mm.</p

Images of the four fabricated poly-L-lactide (PLLA) bone screws.

<p>The screws containing 0% (a and c) and 20% (w/w) Fe₃O₄ nanoparticles (b and d). Injection molding (a and b) and 3D printing (c and d) methods were used to produce the screws.</p

The micro computed tomographic images of the 3D printed PLLA screws implanted in rabbit bone.

<p>The boundary and locations of the neat PLLA screws cannot be distinguished from surrounding tissues after 2 (a) and 4 (c) weeks of healing. With the addition of 20% Fe₃O₄ nanoparticles to the PLLA, the screws exhibited a significant improvement in radiopacity for clinical visibility after 2 (b) and 4 (d) weeks of healing.</p

Micro computed tomographic images of injection molded PLLA screws implanted in rabbit bone.

<p>The boundary and locations of the neat PLLA screws cannot be distinguished from surrounding tissues after 2 (a) and 4 (c) weeks of healing. With the addition of 20% Fe₃O₄ nanoparticles to the PLLA, the screws exhibited a significant improvement in radiopacity for clinical visibility after 2 (b) and 4 (d) weeks of healing.</p