Analytic hierarchy process (AHP) has been considerably criticized for possible rank reversal phenomenon caused by the addition or deletion of an alternative. This paper looks into the cause of rank reversal phenomenon and finds that rank reversal is caused by change of local priorities before and after an alternative is added or deleted. An approach is therefore proposed to keep the local prior...

It is known from work by H.Abels and P.Abramenko that for a classical Fqgroup G of rank n the arithmetic lattice G(Fq[t]) of Fq[t]-points is of type Fn−1 provided that q is large enough. We show that the statement is true without any assumption on q and for any isotropic, absolutely almost simple group G defined over Fq. Let k be a global function field and let G be a connected, noncommutative,...

Using Serre’s proposed complement to Shih’s Theorem, we obtain PSL2(Fp) as a Galois group over Q for at least 614 new primes p. Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3 8 of the primes that were not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture conc...

Such identities arise in Rankin-Selberg integral representations of L-functions. For GL2, a naive, direct computation is sufficient. However, for general GLn and other higher-rank groups, direct computation is inadequate. Further, connecting local Rankin-Selberg computations to Schur functions usefully connects these computations to the Shintani-Casselman-Shalika formulas for spherical p-adic W...

Background The purpose of this study is to review the results of radiation therapy (RT) for hepatocellular carcinoma (HCC) with portal venous tumor thrombus (PVTT) in a Western patient population. Methods Thirty-four patients with HCC PVTT treated from 2007 to 2014 with RT were identified. Biologically effective dose (BED) was calculated for each patient, and greater than the median dose deli...

It is well-known that tensor decompositions show separations, is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here we many of these separations disappear the approximate case. Specifically, for every approximation error $\varepsilon$ and norm, define rank minimum element $\varepsilon$-ball with respect to norm. For positive semide...

Let k be an algebraically closed field. We list the finitely many isomorphism types of rank n commutative k-algebras for n ≤ 6. There are infinitely many types for each n ≥ 7. All algebras are assumed to be commutative, associative, and with 1 (except briefly in Remark 1.1). We assume that k is an algebraically closed field, except in Section 2. By the rank of a k-algebra, we mean its dimension...

We show that Uno’s refinement of the projective conjecture of Dade holds for every block whose defect groups intersect trivially modulo the maximal normal p-subgroup. This corresponds to the block having p-local rank one as defined by Jianbei An and Eaton. An immediate consequence is that Dade’s projective conjecture, Robinson’s conjecture, Alperin’s weight conjecture, the Isaacs– Navarro conje...

Let p > 2 be prime. We prove the weight part of Serre’s conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard–Diamond–Jarvis conjecture for unitary groups), by proving that any Serre weight which occurs is a predicted weight. Our methods are purely local, using the theory of (φ, Ĝ)-modules to determine the possible reductions of certain tw...