The different representations of functions are assumed as central on the development of the concept of function. Being widely recognized the complexity of this concept, the different representations allow the student to understand in a representation what could not be understood in another representation. And the integration of technology into the teaching and learning process provides an easy and quick way to access different representations. This study intends to analyse the understanding of upper secondary students about the information transmitted by each of the representations of functions usually available on technology. Specifically, it intends to understand which transitions between representations are more easily understood by the students and which ones are more difficult to perform. It also intended to identify some aspects that may contribute to this. This study adopts a quantitative methodology in which the answers given by a class to a test focused on the transition from one representation to another are analysed; and a qualitative methodology based on interviews to three of the students in the class, as a way of seeking comprehension about their answers. The results achieved suggest a greater ease of understanding associated to the graphical representation and a greater difficulty associated to the tabular representation. The reasons for this seem to be related to the specific characteristics of each representation, but fundamentally with aspects related to the experiences lived by the students on the mathematics classes, being the integration of technology an influence not to neglect.

n/a

n/a

n/a

Abstract An experimental work on reinforced concrete flat slab specimens to test the efficiency of postinstalled bolts, as punching shear reinforcement in resisting vertical and cyclic horizontal loads, was conducted and is presented in this paper. The test protocol consisted in increasing horizontal drifts combined with constant vertical load until failure. Two different detailing solutions for the shear reinforcement were considered, one using a radial distribution around the column and another using a cross distribution, being the results compared with a previously tested reference specimen. The dimensions of the specimens were 4.25 x 1.85 x 0.15 m3. The test setup used for these tests was developed by the research team and simulates the boundary conditions with already recognized good results. Postinstalled steel bolts were proven to be an efficient solution for strengthening of existing structures, improving the structural behavior, and the punching resistance.

Abstract An experimental work on reinforced concrete flat slab specimens to test the efficiency of postinstalled bolts, as punching shear reinforcement in resisting vertical and cyclic horizontal loads, was conducted and is presented in this paper. The test protocol consisted in increasing horizontal drifts combined with constant vertical load until failure. Two different detailing solutions for the shear reinforcement were considered, one using a radial distribution around the column and another using a cross distribution, being the results compared with a previously tested reference specimen. The dimensions of the specimens were 4.25 x 1.85 x 0.15 m3. The test setup used for these tests was developed by the research team and simulates the boundary conditions with already recognized good results. Postinstalled steel bolts were proven to be an efficient solution for strengthening of existing structures, improving the structural behavior, and the punching resistance.

https://docentes.fct.unl.pt/lblm/files/bioelectrochem-2019-127-76.pdf

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of `plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same
evaluation (that is, contain the same number of each generating symbol). This paper focusses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank-$n$ case. For the hypoplactic monoid, this is $n-1$; for the sylvester and taiga monoids, at least $n-1$ and at most $n$; for the stalactic monoid, $3$ (except for ranks $1$ and $2$, when it is respectively $0$ and $1$); for the plactic monoid, at least $n-1$ and at most $2n-3$. The current state of knowledge, including new and previously-known results, is summarized in a table.

This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words and pairs of tableaux.

We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number $n$, the $n$-th Bell number is given by the number of standard tableaux containing $n$ symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a `hook length formula' that gives the number of standard tableaux of a given shape and deduce some consequences.

This paper constructs presentations via finite complete rewriting systems for plactic monoids of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$, using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right $\mathrm{FP}_\infty$. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic.