In this paper we present an English summary of al-Risāla al-muhītīyya ("The Treatise on the Circumference"). This treatise is one of the most significant mathematical achievements of Ghiyāth al-Dīn Jamshīd Mas'ūd al-Kāshī which he completed in July 1424 (Sha'bān 827 A.H.L.). Also, we present a complete English translation of the titles of all ten sections as well as the title of the introduction and the title of the conclusion. Additionally, we provide comments throughout the paper as necessary.

The choice of the formula for standard deviation is explained in elementary statistics textbooks in various ways. We give an explanation for this formula by representing the data as a vector in $\mathbb R^n$ and considering its distance from a central tendency vector. In this setting the "standard" formula represents a shortest distance in the standard metric. We also show that different metrics lead to different measures of central tendency.

A finite set of integers $D$ with $0\in D$ is basic for the base $b\in \mathbb Z$ if every $z\in\mathbb Z$ can be written uniquely as an integer in the base $b$ with digits from $D$. Since such a base representation is unsigned, basic sets must have a negative base $b$ or some negative integers in $D$. The fraction set for $(b,D)$ is the set of all representable numbers with integer part zero. We show that if $(b,D)$ is basic and $D$ consists of consecutive digits, then the fraction set is an interval of length one whose endpoints have redundant representations. Furthermore, we show that if $D$ does not consist of consecutive integers, then $F$ is disconnected.

We find that for any prime $P>5$ with digital sum equals 5, the number $21P$ is Smith, and so is $112P$.

We show that, for all odd natural numbers $N$, the $N$-torsion points on an elliptic curve may be placed in an $N \times N$ grid such that the sum of each column and each row is the point at infinity.

Deriving a suitable heterogeneous distance measure that mixes continuous and categorical attributes is a difficult problem with a variety of applications. We developed the Scaled Heterogeneous Euclidean Overlap Metric (SHEOM) and we adapted the Interpolated Value Difference Metric (IVDM) from Wilson and Martinez [2, 3]. Our adaptation of the IVDM utilizes output classes given by a continuous response variable. We applied both distance measures in a nearest-neighbor classification for an ecological assessment. Both of the distance measures we applied were shown to be improvements on the simpler Heterogeneous Euclidean Overlap Metric when used in this ecological assessment setting.

Given a polynomial with all real roots, the Polynomial Root Squeezing Theorem states that moving two roots an equal distance toward each other, without passing other roots, will cause each critical point to move toward $(r_i + r_j)/2$, or remain fixed. In this note, we extend the Polynomial Root Squeezing Theorem to the case where two roots are squeezed together a nonuniform distance.

In this note we present a simple new proof for a Lindelöf property for a normal map accessible to advanced undergraduate students. The proof extends the result to uniformly normal families: If $\{ f_n :D \to P^1(\mathbb{C})\}$ is a uniformly normal sequence from the unit disk $D$ in the complex plane ${\bf \mathbb{C}}$ into the Riemann Sphere $P^1(\mathbb{C})$ such that $\lim _{r_n \to 1}f_n(r_n)$ exists for all $ \{r_n\}\subset (0,1),$ then the sequence $\{f_n\}$ has non-tangential limit at $1$.

In this paper, using bitopological semi-open sets, an asymmetric generalization of Haworth-McCoy's well-known theorem [4, Theorem 3.1] about Baire spaces is established.