Four Projects in Applied Mathematics

该项目包括将数学应用于四个重要的物理问题：（1）开发从冷冻血液中去除甘油的最佳方案;（2）开发将一种形状弯曲和拉伸成另一种形状的最佳策略（例如，将金属板弯曲成弯曲的结构表面或在数字成像中变形）;（3）平板上水流波动模式的稳定性;（4）现代物理学两体问题（即两个物体的运动产生波的运动理论，波以有限的速度传播，影响系统中另一个物体的运动）。 甘油必须添加到全血中，以防止冷冻时细胞受损，并且必须在输血前去除。 目前的清除战略需要的程序对于紧急情况或军事应用来说太长。 利用数学建模（包括耦合流体流动，对流扩散和跨半透膜传输的模型），优化理论和数值方法，该项目将扩大实验室测试的医疗用途去除协议。 最优变形的研究是利用微分几何和弹性理论，得到最小应变变形的代价泛函。 最小值超过容许的变形集（紧超曲面之间的同构）将被证明存在使用变分法，和数值方法将开发近似这些最小值。 利用流体动力学模型和自由边界问题理论证明平板绕流产生的表面波的存在性和稳定性。 此外，数学分析将与物理系统的数值模拟相结合。 将推导出两体相互作用的数学模型，其中力以有限的速度传播。 这种模型由双曲型偏微分方程与常微分方程耦合的系统或微分方程与延迟方程的混合系统组成。 将确定这两种类型的模型的适定性和定性动力学，并将这种类型的模型应用于理解声表面波传感器的同步现象。brbr该研究项目有两个主要目的：应用数学促进对物理科学四个领域的理解，并培养四名博士。学生在这些重要的应用数学领域。 第一个项目直接应用于健康科学和医学。 活细胞，例如血细胞，可以通过冷冻保存很长一段时间;并且，它们可以在以后解冻用于输血。 冷冻和解冻活细胞需要一项复杂的技术，包括在细胞内部添加称为冷冻保护剂的化学物质，以便在此过程中不会损坏细胞。 这些化学物质必须在输血前清除。 有方法可以做到这一点，但目前的方法需要很长的时间来实现。 因此，冷冻血液不能用于紧急情况（例如军事冲突或自然灾害）。 冷冻保护剂的快速去除已被证明在单细胞水平上是可能的;研究的目的是设计（最佳）方法，可用于在医学实际应用所需的更大规模上快速去除。 第二个项目在材料科学中有应用。 一张金属板要被弯曲和拉伸成所需的形状。 所需的形状可以通过许多不同的变形工艺来实现（例如，片材可以首先拉伸然后弯曲，或者片材可以在一个方向上拉伸同时在另一个方向上弯曲，以及许多其他可能性）。 哪种方法使用的能量最少？这个问题和其他同类型的问题将使用数学分析来解决。 第三个项目将确定通常由流体（例如水）在板块上移动形成的表面波的稳定性。 表面波是已知存在的，并且具有特定的形状。 此外，实验表明，当波浪受到干扰时，它们会迅速恢复到特定的形状。 但是，这种显着的稳定性的机制目前还不清楚。 该研究项目将确定观察到的稳定性的原因。 第四个研究项目涉及物理学和工程学中的两体相互作用理论。 一个实际的例子发生在应用声学。 施加到嵌入在硅晶片表面中的电极的电流引起声表面波在晶片上传播。 第二组电极（传感器）嵌入晶片中，可以将这些波转换回电流。 当电极和换能器连接到适当的电路时，这种表面波可以不断地产生并被电路放大。 如果第二组电极和换能器嵌入在同一晶片上的表面中，则可以产生第二组波。 这两个电路可以调谐，使波振荡在完美的同步。 一个非常灵敏的探测器可以通过覆盖一半的晶片（用一对电极和传感器）并将另一半暴露在环境中来构建。 如果后半部分的表面受到干扰（例如，由于有害生物制剂的沉积），同步被破坏，并且这种情况可以立即被连接到电路的计算机检测到。 本研究的目的是提供理解这些设备的行为的数学理论。

The project encompasses the application of mathematics to four important physical problems: (1) the development of optimal protocols for the removal of glycerol from frozen blood; (2) the development of optimal strategies for bending and stretching one shape into another (for example, bending metal plates into curved structural surfaces or morphing in digital imaging); (3) the stability of water wave patterns for flow over plates; and (4) the modern physics two-body problem (that is, the theory of motion for two bodies whose motions produce waves, which travel with finite speeds, that influence the motion of the other body in the system). Glycerol must be added to whole blood to prevent cell damage when it is frozen and must be removed before the blood is transfused. Current removal strategies require procedures that are too lengthy for emergencies or military applications. Using mathematical modeling (which includes models that couple fluid flow, convection diffusion, and transport across semipermeable membranes), optimization theory and numerical methods, the project will scale up laboratory tested removal protocols for medical use. The research on optimal morphing uses differential geometry and elasticity theory to obtain cost functionals for minimal strain morphing. Minima over admissible sets of deformations (diffeomorphisms between compact hypersurfaces) will be proved to exist using the calculus of variations, and numerical methods will be developed to approximate these minima. The existence and stability of surface waves produced by flow over flat plates will be proved using fluid dynamical modeling and the theory of free boundary problems. In addition, the mathematical analysis will be complemented with numerical simulations of a physical system. Mathematical models of two-body interaction, where forces propagate with finite speeds, will be derived. Such models consist of systems of hyperbolic partial differential equations coupled with ordinary differential equations, or hybrid systems of differential equations and delay equations. The well-posedness and qualitative dynamics of the two types of models will be determined and models of this type will be applied to understand synchronization phenomena for acoustical surface wave sensors.brbrThe research project has two main purposes: the application of mathematics to advance understanding in four fields of physical science and the training of four Ph.D. students in these important areas of applied mathematics. The first project has direct applications in the health sciences and medicine. Living cells, for example blood cells, can be preserved for long periods of time by freezing; and, they can later be thawed for use in transfusions. To freeze and thaw living cells requires a complex technology including the addition of chemicals called cryoprotectants to the insides of the cells so they will not be damaged during this process. These chemicals must be removed before blood is transfused. Methods are available for doing this, but current methods require a long time to implement. Thus, frozen blood is not available for emergency use (for example, in military conflicts or natural disasters). Fast removal of cryoprotectants has been proved to be possible at the level of single cells; the purpose of the research in to design (optimal) methods that can be used for fast removal on the larger scale necessary for practical application in medicine. The second project has applications in materials science. A sheet of metal is to be bent and stretched into a desired shape. The desired shape can be achieved by many different deformation processes (for example, the sheet can first be stretched then bent or it can be stretched in one direction while being bent in another, among many other possibilities). Which method uses the least energy? This problem and others of the same type will be solved using mathematical analysis. The third project will determine the stability of the surface waves often formed by a fluid (for example water) moving over a plate. Surface waves are known to exist and have a particular shape. Also, experiments show that when the waves are disturbed, they return rapidly to that particular shape. But, the mechanism for this remarkable stability property is not known at present. The research project will determine the reason for the observed stability. The fourth research project is concerned with the theory of two-body interactions in physics and engineering. A practical example occurs in applied acoustics. Electrical currents applied to electrodes embedded in the surface of a silicon wafer causes acoustic surface waves to propagate across the wafer. A second set of electrodes (transducers), embedded in the wafer, can transform these waves back to electrical currents. When the electrodes and transducers are connected to an appropriate electrical circuit, such surface waves can be constantly produced and amplified by the circuit. If a second set of electrodes and transducers are embedded in the surface on the same wafer, a second set of waves can be produced. The two circuits can be tuned so that the waves oscillate in perfect synchrony. A very sensitive detector can be constructed by covering half of the wafer (with one pair of electrodes and transducers) and exposing the other half to the environment. If the surface of the second half is disturbed (for example, by the deposition of a hazardous biological agent) the synchrony is broken and this occurrence can be instantly detected by a computer connected to the circuits. A goal of this research is to provide the mathematical theory for understanding the behavior of these devices.