The almost square rectangle packing problem involves packing all rectangles with sizes 1 × 2 to n × (n + 1) (almost squares) into an enclosing rectangle with minimal area. This extends the previously studied square packing problem by adding an additional degree of freedom for each rectangle, deciding in which orientation the item should be packed. We show how to extend the model and search stra...

Below a (comparatively large) measure of non-Hermiticity Z = Z (crit) 0 > 0 of a PT symmetrically complexified square well, bound states are constructed non-numerically. All their energies prove real and continuous in the (Hermitian) limit Z → 0. Beyond the threshold Z (crit) 0 (and, in general, beyond Z (crit) m at m = 0, 1, . . .) the lowest two real energies (i.e., E2m and E2m+1) are shown t...

A A-linked design is an abstract system of v 'points' and b lines' such that every two points are joined by exactly A lines. A partial classification is given of those extremal examples with b = v, and necessary and sufficient conditions are proved for them to be of each of the only two non-trivial types known, namely square BIBDs and point-complemented square BIBDs. For each A > 1 the number o...

Three algorithm implementations for square root computation are considered in this paper. Newton-Raphson's, iterative, and binary search algorithm implementations are completely designed, verified and compared. The algorithms, entire system-on-chip realisations and their functioning are described.

A ring R for which x = 0 for all x e R is called a zerosquare ring. Zero-square rings are easily seen to be locally nilpotent. This leads to two problems: (1) constructing finitely generated zero-square rings with large index of nilpotence, and (2) investigating the structure of finitely generated zerosquare rings with given index of nilpotence. For the first problem we construct a class of zer...

Identify the domain and range of each. Then sketch the graph.

A permutation is square-free if it does not contain two consecutive factors of length more than one that coincide in the reduced form (as patterns). We prove that the number of square-free permutations of length n is nn(1−εn) where εn → 0 when n → ∞. A permutation of length n is crucial with respect to squares if it avoids squares but any extension of it to the right, to a permutation of length...

Graph products are characterized by the existence of non-trivial equivalence relations on the edge set of a graph that satisfy a so-called square property. We investigate here a generalization, termed RSP-relations. The class of graphs with non-trivial RSP-relations in particular includes graph bundles. Furthermore, RSP-relations are intimately related with covering graph constructions. For K2,...

The algorithm of Tonelli and Shanks for computing square roots modulo a prime number is the most used, and probably the fastest among the known algorithms when averaged over all prime numbers. However, for some particular prime numbers, there are other algorithms which are considerably faster. In this paper we compare the algorithm of Tonelli and Shanks with an algorithm based in quadratic fiel...

We show that the Lusin area function and essentially all of its realvariable generalizations are pointwise dominated by an “intrinsic” square function, and that this latter function is, for all practical purposes, no larger than a “generic” square function. 0. Introduction. The Lusin area (or “square”) function is a familiar object. If f : R 7→ R is such that u(x, y) ≡ Py ∗ f(x), the Poisson in...