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Agricultural development in District Budaun

We consider the problem of finding a sparse multiple of a polynomial. Given a polynomial f ∈ F[x] of degree d over a field F, and a desired sparsity t = O(1), our goal is to determine if there exists a multiple h ∈ F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F = Q, we give a polynomial-time algorithm in d and the size of coefficients in h. For finding binomial multiples we prove a polynomial bound on the degree of the least degree binomial multiple independent of coefficient size. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t = 2.

We consider the problem of finding a sparse multiple of a polynomial. Given a polynomial f ∈ F[x] of degree d over a field F, and a desired sparsity t = O(1), our goal is to determine if there exists a multiple h ∈ F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F = Q, we give a polynomial-time algorithm in d and the size of coefficients in h. For finding binomial multiples we prove a polynomial bound on the degree of the least degree binomial multiple independent of coefficient size. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t = 2.

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Thesis (M.A. Sc.)--University of Toronto, 2006. / Source: Masters Abstracts International, Volume: 45-03, page: 1638.

Purpose: Demonstrate the utility of a robotic needle-guidance template as compared to a manual template for in-bore 3T transperineal MR guided prostate biopsy. Materials and Methods: This non-randomized two-arm trial included 99 cases of targeted transperineal prostate biopsies. The biopsy needles were aimed to suspicious foci in multiparametric 3T MRI using a manual template before a cut-off date or a robotic template after the date. The following data was obtained: the accuracy of average and closest needle placement to the focus, cancer yield, percentage of cancer volume in positive core samples, and time to complete the procedure. Results: 56 cases were performed using the manual template, and 43 cases were performed using the robotic template. The mean accuracy of the best needle placement attempt was higher in the robotic group (2.39 mm) than the manual group (3.71 mm, p<0.027). The mean core procedure time was shorter in the robotic (90.82 min) than the manual group (100.63 min, p<0.030). Percentage of cancer volume in positive core samples was higher in robotic group (p<0.001). Cancer yield was not statistically different between the two sub-groups. Conclusion: The robotic needle-guidance template was useful in MRI-guided core biopsy of prostate cancer compared to the manual approach.

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