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The paper provides a new test of convergence and divergence of positive series. In particular, it extends the known test by Margaret Martin [Bull. Amer. Math. Soc. 47 (1941), 452-457]. [ABSTRACT FROM AUTHOR]

Reinhold Baer asked the relationship between certain properties in a nonempty set ρ with a partial operation (called an "add" by Baer [1]). The first paper in our sequence [Paper I] answered his question for a special type of an add called a pregroup by Stallings [12]. This paper [Paper II] answers an analogous question for a wider class of adds. [ABSTRACT FROM AUTHOR]

In the paper Free biassociative groupoids, the variety of biassociative groupoids (i.e., groupoids satisfying the condition: every subgroupoid generated by at most two elements is a subsemigroup) is considered and free objects are constructed using a chain of partial biassociative groupoids that satisfy certain properties. The obtained free objects in this variety are not canonical. By a canonical groupoid in a variety V of groupoids we mean a free groupoid (R, *) in V with a free basis B such that the carrier R is a subset of the absolutely free groupoid (TB, ·) with the free basis B and (tu ∊ R ⇒ t, u ∊ R & t*u = tu). In the present paper, a canonical description of free objects in the variety of biassociative groupoids is obtained. [ABSTRACT FROM AUTHOR]

We consider a class of functional equations with one operational symbol which is assumed to be a quasigroup. Equations are quadratic, level and have four variables each. Therefore, they are of the form x1x2 · x3x4 = x5x6 · x7x8 with xi ∊ {x, y, u, v} (1 ⩽ i ⩽ 8) with each of the variables occurring exactly twice in the equation. There are 105 such equations. They separate into 19 equivalence classes defining 19 quasigroup varieties. The paper (partially) generalizes the results of some recent papers of Föorg-Rob and Krapež, and Polonijo. [ABSTRACT FROM AUTHOR]

There are not many examples of complete analytical classification of specific families of singularities, even in the case of plane algebraic curves. In 1989, Kang and Kim published a paper on analytical classification of plane curve singularities yn+a(x)y+b(x) = 0, or, equivalently, yn+xαy+xβA(x) = 0 where A(x) is a unit in Ct{x}, α and β are integers, α ⩾ n - 1 and β ⩾ n. The classification was not complete in the most difficult case α / n-1 = β / n. In the present paper, the classification is extended also in this case, the proofs are improved and some gaps are removed. [ABSTRACT FROM AUTHOR]

Let {Xi, i ⩾ 1} denote a sequence of variables that take values in {0, 1} and suppose that the sequence forms a Markov chain with transition matrix P and with initial distribution (q, p) = (P(X1 = 0), P(X1 = 1)). Several authors have studied the quantities Sn, Y (r) and AR(n), where Sn = Σn i=1 Xi denotes the number of successes, where Y (r) denotes the number of experiments up to the r-th success and where AR(n) denotes the number of runs. In the present paper we study the number of singles AS(n) in the vector (X1,X2, … , Xn). A single in a sequence is an isolated value of 0 or 1, i.e., a run of length 1. Among others we prove a central limit theorem for AS(n). [ABSTRACT FROM AUTHOR]