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The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the $\belr$ monoid and the $\bell$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the $\belr$ monoid consists of elements that have the same number of each of its composing symbols. In this paper, with the aid of the computational tool SageMath, we study the diameter of the connected components from the cyclic shift graph of the $\belr$ monoid.

Within the theory of monoids, the cyclic shift relation, among other relations, generalizes the relation of conjugacy for groups. We examine several of these relations for both the $\belr$ and the $\bell$ monoids.

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Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the q-analogue of the general linear Lie algebra gln, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson–Schensted–Knuth correspondence. The authors previously constructed an analogous ‘quasi-crystal’ structure for the related hypoplactic monoid (the monoid of quasi-ribbon tableaux), which has similarly neat combinatorial properties. This paper constructs an analogous ‘crystal-type’ structure for the sylvester and Baxter monoids (the monoids of binary search trees and pairs of twin binary search trees, respectively). Both monoids are shown to arise from this structure just as the plactic monoid does from the usual crystal graph. The interaction of the structure with the sylvester and Baxter versions of the Robinson–Schensted–Knuth correspondence is studied. The structure is then applied to prove results on the number of factorizations of elements of these monoids, and to prove that both monoids satisfy non-trivial identities.

There have been several attempts to extend the notion of conjugacy from groups to monoids.
The aim of this paper is study the decidability and independence of conjugacy problems
for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in
certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids,
$p$-conjugacy is ``almost'' transitive, $\sim_c$ is strictly included in $\sim_p$, and
the $p$- and $c$-conjugacy problems are decidable with linear compexity.
For other classes of monoids, the situation is more complicated.
We show that there exists a monoid $M$ defined by a finite complete
presentation such that the $c$-conjugacy problem for $M$ is undecidable, and
that for finitely presented monoids, the $c$-conjugacy problem and the word
problem are independent, as are the $c$-conjugacy and $p$-conjugacy problems.

This study intends to analyze the perspectives of teachers of different levels regarding proof and its functions in Mathematics and Mathematics teaching. Adopting a methodology of a qualitative nature, and based on interviews, the perspectives of teachers of upper secondary, higher education and training teachers of Mathematics were collected. The conclusions reached suggest that teachers seem to share a formal conception of mathematical proof, recognizing the need to introduce some simplification when considering proof in Mathematics teaching as well as the importance of their functions of validation, contribution to learning and even a cultural function.

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Abstract This paper addresses the durability of bond between concrete and carbon fibre reinforced polymer (CFRP) strips installed according to the near-surface mounted (NSM) technique (NSM CFRP-concrete systems) under the effects of two main groups of environmental conditions: (i) laboratory-based ageing conditions; (ii) real outdoor ageing conditions. The bond degradation was evaluated by carrying out direct pullout tests on aged specimens that were previously subjected to distinct environmental conditions for different periods of exposure. Moreover, the degradation of the mechanical properties of the involved materials was investigated. The digital image correlation (DIC) method was used to document the evolution of the deformation fields at the surface over the whole region of interest consisting of concrete and epoxy adhesive at the ligament region. This information supported the discussion about the evolution of the bond resistant mechanism developed in \{NSM\} CFRP-concrete specimens during testing, as well as the assessment of the bond quality of the system. In general, the results obtained from the durability tests conducted have shown that the different exposure environments, which may be considered as quite severe, did not result in significant damage on \{NSM\} CFRP-concrete system. The maximum decrease of about 12% on bond strength was obtained for real outdoor environments. Conversely, a maximum increase of 8% on bond strength was obtained on the specimens exposed to the temperature cycles between -15��C and +60��C. \{DIC\} allowed to document the stress transfer mechanisms established between the \{CFRP\} and the concrete substrate, revealing the crack patterns and the influence widths of the \{CFRP\} reinforcement strips, which were shown to be important for avoiding group effect when using multiple parallel strengthening \{CFRP\} strips.

This paper deals with the experimental and theoretical evaluation of punching shear capacity of steel fiber reinforced concrete (SFRC) slab–column connections. Five experimental specimens with a thickness of 160 mm, different fiber volume contents (0, 1.0, and 1.5%) and different flexural reinforcement ratios (0.75 and 1.5%) have been tested. The experimental results were evaluated using a physical–mechanical model based on the critical shear crack theory (CSCT). The model has given a good approximation of experimental punching shear strengths. In general, tests have highlighted a significant increase in load and deformation capacity of fiber reinforced concrete slab–column connections in comparison with reinforced concrete connections.

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