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1581Worst-Case Curvature of a Coupled Penalty 1Consider q(x,y) = 3x 2 + 4xy + 8y 2. What is the largest real number m such that q(x,y) >= m(x 2 + y 2) for every pair (x,y)?数学简单数值题未尝试免费1582Worst-Case Curvature of a Coupled Penalty 2Consider q(x,y) = 5x 2 - 6xy + 5y 2. What is the largest m such that q(x,y) >= m(x 2 + y 2) for all (x,y)?数学中等derivation未尝试免费1583Cross-Term Budget Under a Curvature Floor 3For q(x,y) = 4x 2 + 2kxy + 7y 2, what is the largest absolute value |k| such that q(x,y) >= 2(x 2 + y 2) for all (x,y)?数学中等数值题未尝试免费1585One-Sided Diagonal Repair 5For what smallest u does the quadratic form (1+u)x 2 + 4xy + 5y 2 stay nonnegative for all (x,y)?数学困难derivation未尝试面试订阅1586Eliminate One Coordinate of a Quadratic 1Consider q(x,y) = 4x 2 + 6xy + 5y 2. If y is chosen optimally for each fixed x, what coefficient multiplies x 2 in the reduced one-variable penalty?数学简单derivation未尝试免费1587Eliminate One Coordinate of a Quadratic 2Consider q(x,y) = 7x 2 + -4xy + 3y 2. If y is chosen optimally for each fixed x, what coefficient multiplies x 2 in the reduced one-variable penalty?数学简单数值题未尝试免费1589Pick the Residual Variance in a Schur Reduction 4A coupled penalty is q(x,y) = 10x 2 + 12xy + cy 2. After optimizing out y, the desk wants the remaining x 2 coefficient to be exactly 6. What value of c is required?数学困难数值题未尝试面试订阅1590Cross-Term Budget After Eliminating One Coordinate 5For q(x,y) = 5x 2 + 2bxy + 4y 2, what is the largest absolute value |b| such that, after optimizing out y, the remaining x 2 coefficient is at least 3.75?数学困难derivation未尝试面试订阅1591Recover a Cholesky Pivot 1A positive definite matrix A = [[9, 6], [6, 13]] has Cholesky factor L = [[l11, 0], [l21, l22]] with positive diagonal. What is l22?数学简单数值题未尝试免费1593Choose a Diagonal Entry to Hit a Cholesky Pivot 3For q(x,y) = 5x 2 + 4xy + ky 2, the Cholesky factor of the associated matrix is required to have second diagonal entry equal to 2. What value of k achieves that?数学中等derivation未尝试免费1594Residual Coefficient After Completing the Square 4Rewrite 8x 2 + 12xy + 7y 2 in the form 8(x + 3y/4) 2 + cy 2. What is c?数学困难数值题未尝试面试订阅1595Coupling Consistent With a Target Cholesky Pivot 5A matrix A = [[9, k], [k, 10]] has a Cholesky factor whose second diagonal entry is exactly 1. What values of k are possible?数学困难derivation未尝试面试订阅