The aim of this paper is to prove that the Dedekind zeta function for a number field has a meromorphic continuation to the complex plane, obtaining the analytic class number formula with it, and to present some of the applications of these results. In particular, It will be shown its use in proving Dirichlet’s prime number theorem and in calculating the class number of quadratic fields. The pro...

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Whaley’s Theorem on the existence of large proper sublattices of infinite lattices is extended to ordered sets and finite lattices. As a corollary it is shown that every finite lattice L with |L| ≥ 3 contains a proper sublattice S with |S| ≥ |L| 3 . It is also shown that that every finite modular lattice L with |L| ≥ 3 contains a proper sublattice S with |S| ≥ |L| 2 , and every finite distribut...

OBJECTIVES To investigate the relationship between accumulated health-related problems (deficits), which define a frailty index in older adults, and mortality in population-based and clinical/institutional-based samples. DESIGN Cross-sectional and cohort studies. SETTING Seven population-based and four clinical/institutional surveys in four developed countries. PARTICIPANTS Thirty-six tho...

Twenty-five (10 ruminally cannulated) Holstein cows averaging 82 +/- 34 d in milk were assigned to 5 x 5 Latin squares (21-d periods) and fed diets supplemented with one of four different proteins to assess effects on production, ruminal metabolism, omasal flow of N fractions, and degradation rates of protein supplements. Total mixed diets contained (dry matter basis) 44% corn silage, 22% alfal...

Effects of corn silage crop processing and chop length on intake, digestion, and milk production were evaluated. Corn silage treatments were harvested at one-half milkline stage of maturity (65% whole-plant moisture content) and at 0.95-cm theoretical length of cut without processing (control) or 0.95-, 1.45-, or 1.90-cm theoretical length of cut with processing at a 1-mm roll clearance. Twenty...

Two trials were conducted to evaluate the effects of corn in protein supplements fed to cattle receiving low-quality forages. In Trial 1, four ruminally cannulated steers (avg BW 500 kg) and four intact steers (avg BW 270 kg) were used in a replicated latin square to determine intake and digestibility fo a low-quality meadow hay (4.3% CP) when fed no supplement (NS), 1.12 g CP/kg BW (PS), 1.12 ...