We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$ . We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.

We prove that most permutations of degree $n$ have some power which is a cycle of prime length approximately $\log n$. Explicitly, we show that for $n$ sufficiently large, the proportion of such elements is at least $1-5/\log \log n$ with the prime between $\log n$ and $(\log n)^{\log \log n}$. The proportion of even permutations with this property is at least $1-7/\log \log n$.

We study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$

$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$

$\unicode[STIX]{x1D70C}(n)$

$\unicode[STIX]{x1D706}_{f}(n)$

$f$

$\operatorname{SL}_{2}(\mathbb{Z})$

$A_{F}(n,1)$

$F$

$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$

$E_{1}(a)$

$a$

$\unicode[STIX]{x1D70C}(n)d(n-a)$

The technique of modified ultrafiltration is a more efficient application of the concept of ultrafiltration during cardiopulmonary bypass. It has been shown to be superior to the conventional method of ultrafiltration.

The method can save considerable quantities of donor blood by returning not only the red cells, but also the white cells, platelets, and clotting factors back to the patient, components which otherwise would be discarded. The elevation of the hematocrit made possible by this method after bypass will permit an “acceptable” hematocrit to be achieved when temperatures become normal after ultrafiltration. The return of platelets and clotting factors along with a high hematocrit will contribute towards significantly minimizing the postoperative loss of blood, and, consequently, the amount of blood requiring to be transfused. The possibility of elevation of the hematocrit allows for a lower hematocrit during bypass. This can achieve a considerable saving of donor blood, particularly in small children who invariably need blood added to the prime, making bloodless pediatric surgery a real possibility. The technique also minimizes the rise in total water in the body after cardiopulmonary bypass in children and promises favorably to alter the course of leaking capillaries as seen frequently in neonates and infants. It has been shown to improve hemodynamics at a crucial time when the patient has been weaned from cardiopulmonary bypass and needs optimum conditions for recovery. From the point of view of research, ultrafiltration offers a window to look at the inflammatory process induced by cardiopulmonary bypass.

It promises to be a valuable technique to investigate the elimination and possibly quantification of toxic metabolites produced during bypass. It also allows evaluation and assessment of different protocols for bypass, involving varying rates of flow and temperature, for their response to the production of these toxic metabolites, thus offering the potential to achieve a model of cardiopulmonary bypass which is as physiological as possible. With advances in the systems of perfusion, aided by computers, it should be possible to control the rate of ultrafiltration and precisely regulate the transfusion of the venous reservoir fluid with the aid of feedback loops from the venous and arterial pressures in the heart. This could make ultrafiltration an integral part of routine cardiopulmonary bypass.

The advent of cardiopulmonary bypass in 1953, and its subsequent continuous refinements, have securely established the specialty of cardiovascular surgery. In conjunction with the improved methods of myocardial preservation, this has allowed the safe repair of acquired and complex congenital cardiac defects. During the years, the many variables of the systems used for cardiopulmonary bypass have been altered to the point that most operations are now reproducible with, relatively, low morbidity and mortality. Although in its present form the system is quite reasonable for most routine surgery, it is far from perfect. There remains, therefore, a quest for alterations to improve the system as an ongoing process. Certain complications of open-heart surgery persist and many new and unique approaches to the concept of organ protection and the treatment of these complications must be developed and refined.

A weak form of faithfulness, depending on Green’s equivalence $\mathcal{D}$, is introduced for a ring $R$ graded by a semigroup $S$. Suppose that $R$ satisfies this condition. It is shown that if $e$ and $f$ are $\mathcal{D}$-equivalent idempotents of $S$ and $R_e$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_f$ is semiprime (respectively, prime, semiprimitive, right primitive). In addition, it is shown that if $G$ and $H$ are maximal subgroups of $S$ lying in the same $\mathcal{D}$-class and $R_G$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_H$ is semiprime (respectively, prime, semiprimitive, right primitive).

AMS 2000 Mathematics subject classification: Primary 16W50. Secondary 20M25