answer:Yes, there are some matrix norms that do not satisfy the submultiplicative property. One example is the max-norm, defined as the maximum absolute value of the entries of a matrix. For example, consider the matrices: A = begin{bmatrix}1 & 1 1 & 1end{bmatrix} quad text{and} quad B = begin{bmatrix}2 & 2 2 & 2end{bmatrix} Then, |A| = |B| = 2, quad text{but} quad |AB| = 4 > |A| |B| = 4. Therefore, the max-norm does not satisfy the submultiplicative property. Another example is the p-norm, defined as: |A|_p = left(sum_{i,j}|a_{ij}|^pright)^{1/p}, quad 1 leq pleqinfty for p>2, the p-norm does not satisfy the submultiplicative property. For example, consider the matrices: A = B = begin{bmatrix}1 & 1 1 & 1end{bmatrix} Then, |A|_p = |B|_p = 2^{1/p}, quad text{but} quad |AB|_p = 2cdot 2^{1/p} > |A|_p |B|_p. Therefore, the p-norm for p>2 does not satisfy the submultiplicative property.

answer:The Schwarz inequality is actually stated as: |langle x,y rangle| leq |x|cdot|y| or, in your notation: |x_1y_1 + x_2y_2| leq sqrt{x_1^2+x_2^2}sqrt{y_1^2+y_2^2} This is a stronger statement than what you originally wrote, as we always have x_1y_1 + x_2y_2 leq |x_1y_1 + x_2y_2|. Now, let's consider the case where x_1=lambda y_1 and x_2 = lambda y_2. Substituting these values into the Schwarz inequality, we get: |lambda (y_1^2+ y_2^2)| leq |lambda|(y_1^2+y_2^2) Since lambda is a scalar, we can factor it out of the absolute value signs: lambda |y_1^2+ y_2^2| leq lambda (y_1^2+y_2^2) Dividing both sides by lambda (assuming lambda neq 0), we get: |y_1^2+ y_2^2| leq y_1^2+y_2^2 This inequality holds for any lambda, regardless of its sign. Therefore, equality holds in the Schwarz inequality whenever x_1=lambda y_1 and x_2 = lambda y_2.

answer:Partitions of unity are a powerful tool in many areas of mathematics and physics, including: * **Image processing:** Partitions of unity can be used to blend images together seamlessly, create panoramas, and remove unwanted objects from images. * **Computer graphics:** Partitions of unity can be used to create smooth transitions between different materials or textures on a 3D model, and to generate realistic shadows and reflections. * **Medical imaging:** Partitions of unity can be used to combine data from different medical imaging modalities, such as MRI and CT scans, to create a more complete picture of a patient's anatomy. * **Computational fluid dynamics:** Partitions of unity can be used to create smooth transitions between different flow regimes, such as laminar and turbulent flow. * **Machine learning:** Partitions of unity can be used to create smooth transitions between different models or features in a machine learning algorithm, which can improve the accuracy and robustness of the algorithm. These are just a few examples of the many applications of partitions of unity. Their versatility and power make them a valuable tool in a wide range of fields.

answer:The Jacobian matrix of mathbf{r}(x, y, z) is given by: J(mathbf{r}(x, y, z)) = begin{bmatrix} frac{partial f}{partial x} & frac{partial f}{partial y} & frac{partial f}{partial z} frac{partial g}{partial x} & frac{partial g}{partial y} & frac{partial g}{partial z} frac{partial h}{partial x} & frac{partial h}{partial y} & frac{partial h}{partial z} end{bmatrix} Evaluating each partial derivative, we get: frac{partial f}{partial x} = 0, quad frac{partial f}{partial y} = -1, quad frac{partial f}{partial z} = 1 frac{partial g}{partial x} = 0, quad frac{partial g}{partial y} = 0, quad frac{partial g}{partial z} = -frac{1}{sqrt{1-z^2}} frac{partial h}{partial x} = 1, quad frac{partial h}{partial y} = -1, quad frac{partial h}{partial z} = 0 Therefore, the Jacobian matrix of mathbf{r}(x, y, z) is: J(mathbf{r}(x, y, z)) = begin{bmatrix} 0 & -1 & 1 0 & 0 & -frac{1}{sqrt{1-z^2}} 1 & -1 & 0 end{bmatrix}