Baker’s method in the theory of transcendental numbers

Author(s):  
Gregory Chudnovsky
Blood ◽  
1950 ◽  
Vol 5 (1) ◽  
pp. 79-88 ◽  
Author(s):  
MARVIN L. BLOOM ◽  
GEORGE B. WISLOCKI

Abstract By Baker’s method for the staining of lipids, various types of cytoplasmic rods and granules in leukocytes give a positive acid hematein test but are negative after the pyridine-extraction test. According to Baker,8 this combination of results indicates the presence of phospholipids. Some of the rods and granules which are stained appear to be mitochondria, judging from their shape, size and distribution. as well as from the fact that they coincide in all respects with mitochondria demonstrable by supravital staining. Baker8 pointed out that in a great variety of cells his method stained mitochondria, although he did not report having examined leukocytes. In addition to mitochondria, Baker’s method stains the specific granules of the three varieties of granular leukocytes. In the neutrophilic leukocytes, the granules appear gray or black; in some of the eosinophilic leukocytes, they are intensely black, while in others they are unstained although surrounded by stained ground substance; and in the basophilic leukocytes, some of the granules of individual cells are stained while others are not. Provided the method is chemically specific, it follows that the specific granules of these leukocytes contain phospholipids. The appearance of the eosinophils indicates that many of them possess granules composed of protein surrounded by phospholipid, although some of them contain granules with lipid in their interiors. These differences in the eosinophils suggest possible functional stages, related perhaps to the age of the cells or some other metabolic factors. In the monocytes, besides a general stippling of the cytoplasm corresponding to mitochondria, the region of the attraction sphere often contains numerous deeply stained granules. Sudan black B stains both the mitochondria and the specific leukocytic granules in a manner similar to Baker’s method. However, certain slight differences were noticed. Platelets were brought out moderately distinctly by the acid hematein procedure, whereas in sudan black preparations they were barely visible. A variability was noted in the staining of the granules of the eosinophilic leukocytes in Baker’s method, in contrast to the uniformity of their staining with sudan black. This difference should perhaps be interpreted as indicating that the eosinophilic granules contain other lipid substances besides phospholipids. Mitochondnia, in whatsoever type of leukocyte they are observed, appear to stain more deeply and distinctly by Baker’s method than with sudan black, a circumstance suggesting that mitochondria are very rich in phospholipids.


2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].


1962 ◽  
Vol 58 (2) ◽  
pp. 229-234 ◽  
Author(s):  
L. Mirsky

Throughout this note we shall consider a fixed polynomial with complex coefficients and of degree n ≥ 2. Its zeros will be denoted by ξ1, ξ2, …, ξn where the numbering is such that Making use of Jensen's integral formula, Mahler (4) showed that, for l ≥ k < n, A slightly weaker result had been established by Feldman in an earlier publication (2). Mahler's inequality (1) is of importance in the study of transcendental numbers, and our first object is to sharpen his bound by proving the following result.


2014 ◽  
pp. 159-233
Author(s):  
Yuri F. Bilu ◽  
Yann Bugeaud ◽  
Maurice Mignotte
Keyword(s):  

1968 ◽  
Vol 14 (1) ◽  
pp. 73-88 ◽  
Author(s):  
K. Ramachandra

1968 ◽  
Vol 14 (1) ◽  
pp. 65-72 ◽  
Author(s):  
K. Ramachandra

Math Horizons ◽  
2017 ◽  
Vol 24 (4) ◽  
pp. 22-23
Author(s):  
Eisuke Chikayama ◽  
Shori Chikayama

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