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Erdős proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erdős property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if $m\in [0,\infty )$ , then there exist $2^{\mathfrak {c}}$ dense subsets W of S each of Lebesgue measure m such that W has the Erdős property and no two of these W are homeomorphic. It is also proved that there are $2^{\mathfrak {c}}$ dense subsets W of S each of full Lebesgue measure, which have the Erdős property. Finally, it is proved that there are $2^{\mathfrak {c}}$ dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic.

For any Liouville number $\alpha $ , all of the following are transcendental numbers: ${e}^\alpha $ , $\log _{e}\alpha $ , $\sin \alpha $ , $\cos \alpha $ , $\tan \alpha $ , $\sinh \alpha $ , $\cosh \alpha $ , $\tanh \alpha $ , $\arcsin \alpha $ and the inverse functions evaluated at $\alpha $ of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if ‘Liouville number’ is replaced by ‘U-number’, where U is one of Mahler’s classes of transcendental numbers.

In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$ , the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$ . Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$ .

A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on $\mathbb {R}$ maps a thin set onto a fat set; in fact the fat set is all of $\mathbb {R}$ . Our argument depends on the theorem of Paul Erdős that every real number is a sum of two Liouville numbers. Our thin set is the set $\mathcal {L}^{2}$ , where $\mathcal {L}$ is the set of all Liouville numbers, and the fat set is $\mathbb {R}$ itself. Finally, it is shown that $\mathcal {L}$ and $\mathcal {L}^{2}$ are both homeomorphic to $\mathbb {P}$ , the space of all irrational numbers.

This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2ℵ0 countably infinite topological transcendental fields and each is homeomorphic to the space Q of rational numbers with its usual topology. It is also shown that there is a class of 22ℵ0 of topological transcendental fields of the form Q(T) with T a set of Liouville numbers, no two of which are homeomorphic.

About 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However, Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Nemes, Windschitl, and Chen. Here it is shown (i) how all these asymptotic results can be easily verified; (ii) how Hirschhorn and Villarino's argument allows tweaking of Ramanujan's result to give a better approximation; and (iii) that new asymptotic formulae can be obtained by further tweaking of Ramanujan's result. Tables are calculated displaying how good each of these approximations is for $n$ up to one million.

Myocardial infarction (MI) can result in a maladaptive remodeling of the heart that leads to heart failure. The post-MI inflammation is necessary for wound healing. Unfortunately it is also a key component of subsequent heart failure pathology. Thus, understanding the mechanism of post-MI inflammation and identifying targets for intervention are of translational interests. To understand the role of FoxO4 in post-MI remodeling, we induced MI in WT and FoxO4 global knockout ( gKO ) mice by surgical ligation of the left anterior descending coronary artery. FoxO4- inactivation resulted in a significantly higher post-MI survival, better cardiac function, and reduced fibrosis and infarct size compared to WT mice. FoxO4 gKO mouse hearts had significantly less mount of neutrophils, and reduced expressions of cytokines, MMP9, and Arginase 1 (Arg1). To determine the contribution of cellular FoxO4 to the post-MI phenotype, we generated cardiomyocyte and endothelial cell specific FoxO4 knockout mice ( FoxO4 cKO and eKO ). While FoxO4 cKO mice showed similar post-MI phenotype to those of WT littermates, FoxO4 eKO mice had better cardiac function with reduced inflammation, neutrophil infiltration, and Arg1 expression similar to those of FoxO4 gKO mice. Since Arg1 is a competitive inhibitor of nitric oxide (NO) synthase and NO inhibits lymphocyte adhesion and transmigration across the endothelial barrier, downregulation of neutrophils in post-MI FoxO4- null mouse hearts could be due to reduced endothelial Arg1 and thus increased NO. Consistent with this hypothesis, we show that Arg1 is a direct transcriptional target of FoxO4. Endothelial FoxO4 is necessary and sufficient to activate Arg1 expression in response to hypoxia. Knockdown of FoxO4 in human aortic endothelial cells suppressed leukocyte adhesion. Furthermore, chemical inhibition of Arg1 in WT mice had similar cardioprotection following MI as FoxO4- inactivation and administration of NOS inhibitor to FoxO4 gKO mice reversed the beneficial effects of FoxO4 -deletion on post-MI cardiac function. Our studies showed that FoxO4 promotes early inflammatory response via endothelial Arg1 and suggest a potential therapy to reduce the early post-MI inflammation and enhance cardiac function in MI-injured hearts.

The human X chromosome has a unique biology that was shaped by its evolution as the sex chromosome shared by males and females. We have determined 99.3% of the euchromatic sequence of the X chromosome. Our analysis illustrates the autosomal origin of the mammalian sex chromosomes, the stepwise process that led to the progressive loss of recombination between X and Y, and the extent of subsequent degradation of the Y chromosome. LINE1 repeat elements cover one-third of the X chromosome, with a distribution that is consistent with their proposed role as way stations in the process of X-chromosome inactivation. We found 1,098 genes in the sequence, of which 99 encode proteins expressed in testis and in various tumour types. A disproportionately high number of mendelian diseases are documented for the X chromosome. Of this number, 168 have been explained by mutations in 113 X-linked genes, which in many cases were characterized with the aid of the DNA sequence.

Human chromosome 12 contains more than 1,400 coding genes and 487 loci that have been directly implicated in human disease. The q arm of chromosome 12 contains one of the largest blocks of linkage disequilibrium found in the human genome. Here we present the finished sequence of human chromosome 12, which has been finished to high quality and spans approximately 132 megabases, representing approximately 4.5% of the human genome. Alignment of the human chromosome 12 sequence across vertebrates reveals the origin of individual segments in chicken, and a unique history of rearrangement through rodent and primate lineages. The rate of base substitutions in recent evolutionary history shows an overall slowing in hominids compared with primates and rodents.

(1965). PHILOSOPHY OF EDUCATION AND THE TRAINING OF TEACHERS. Educational Review: Vol. 17, No. 3, pp. 181-188.

This paper is a sequel to one entitled “Varieties of topological groups”. The variety V of topological groups is said to be full if it contains every group which is algebraically isomorphic to a group in V For any Tychonoff space X, the free group F of V on X exists, is Hausdorff and disconnected, and has X as a closed subset. Any subgroup of F which is algebraically fully invariant is a closed subset of F. If X is a compact Hausdorff space, then F is normal. Let V be a full Schreier variety and X a Tychonoff space, then all finitely generated subgroups of F are free in V.A β-variety V is one for which the free group of V on each compact Hausdorff space exists and is Hausdorff. For any β-variety V and Tychonoff space X, the free group of V exists, is Hausdorff and has X as a closed subset. A necessary and sufficient condition for V to be a β-variety is given.The concept of a projective (topological) group of a variety V is introduced. The projective groups of V are shown to be precisely the summands of the free groups of V. A finitely generated Hausdorff projective group of a Schreier variety V is free in V.

The modern concept of hierarchy extends and deepens the dictionary definition of "graded division" by stressing the element of the "hieros." This holy secrecy lies at the root of hierarchy. Division based upon difference of function is rational and consonant with democracy; division based upon mystique-supported status, rank, position is irrational and fundamentally anti-democratic. Western civilization inherited from the late Roman Empire a tradition of hierarchical political and social organi? zation. Sufficiently strengthened by the Medieval Church to contain the attacks of the Renaissance and the Reformation in the intellectual and