Nested recursions with ceiling function solutions

Consider a nested, non-homogeneous recursion R(n) defined by R(n) = \sum_{i=1}^k R(n-s_i-\sum_{j=1}^{p_i} R(n-a_ij)) + nu, with c initial conditions R(1) = xi_1 > 0,R(2)=xi_2 > 0, ..., R(c)=xi_c > 0, where the parameters are integers satisfying k > 0, p_i > 0 and a_ij > 0. We develop an algorithm to answer the following question: for an arbitrary rational number r/q, is there any set of values for k, p_i, s_i, a_ij and nu such that the ceiling function ceiling{rn/q} is the unique solution generated by R(n) with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to R(n). The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form ceiling{n/q}$ is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if ceiling{rn/q} is the solution generated by any recursion R(n) of the form above, then r=1. We also use our ceiling function methodology to derive the first known connection between the recursion R(n) and a natural generalization of Conway's recursion.Comment: Published in Journal of Difference Equations and Applications, 2010. 11 pages, 1 tabl

FLOW PROPERTIES AND STRENGTH BEHAVIOUR OF MASONRY MORTAR INCORPORATING HIGH VOLUME FLY ASH

ABSTRACT: Masonry mortar is one of the most widely used building materials in construction as nearly all the walls made with bricks are finished with mortar. Masonry works with mortar are generally made with cement, sand and water. In this study, mortar mixes were made by replacing the masonry cement with up to 50% of fly ash at an incremental rate of 10%. Six mortar mixes with 1:4 volumetric cement-to-aggregate ratios using natural sand were made and tested for flow and strength properties. The results obtained from the tests showed that the masonry mortars manufactured with fly ash are better off as compared to the mix without any ash. In terms of flow, it was found that the addition of fly ash increased the flow, particularly when the replacement levels were above 40%. Mortar with high volume fly ash has also shown to attain a comparable strength with that of the control mix. Considering the better workability and acceptable strength properties, the results have demonstrated that high volume replacement of fly ash is a viable alternative for producing environmental friendly masonry mortar

Journal of Integer Sequences, Vol. 10 (2007), Article 07.7.1 On the Behavior of a Variant of Hofstadter’s Q-Sequence

We completely solve the meta-Fibonacci recursion V (n) = V (n − V (n − 1)) + V (n − V (n − 4)), a variant of Hofstadter’s meta-Fibonacci Q-sequence. For the initial conditions V (1) = V (2) = V (3) = V (4) = 1 we prove that the sequence V (n) is monotone, with successive terms increasing by 0 or 1, so the sequence hits every positive integer. We demonstrate certain special structural properties and fascinating periodicities of the associated frequency sequence (the number of times V (n) hits each positive integer) that make possible an iterative computation of V (n) for any value of n. Further, we derive a natural partition of the V-sequence into blocks of consecutive terms (“generations”) with the property that terms in one block determine the terms in the next. We conclude by examining all the other sets of four initial conditions for which this meta-Fibonacci recursion has a solution; we prove that in each case the resulting sequence is essentially the same as the one with initial conditions all ones.

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Dangerous graphs

Anomalies and faults are inevitable in computer networks, today more than ever before. This is due to the large scale and dynamic nature of the networks used to process big data and to the ever-increasing number of ad-hoc devices. Beyond natural faults and anomalies occurring in a network, threats proceeding from attacks conducted by malicious intruders must be considered. Consequently, there is often a need to quickly isolate and even repair a fault in a network when it appears. Furthermore, despite the presence in a network of faults stemming from malicious entities, we need to identify the latter and their behaviours, and develop protocols resilient to their attacks. Thus, defining models to capture the dangers inherent to various faults, anomalies and threats in a network and studying such threats, has become increasingly important and popular. Threats in networks can be of two kinds: either mobile or stationary. A malicious mobile process can move along the network, whereas a stationary harmful process resides in a host. One of the most studied models for stationary harmful processes is the black hole, which was introduced by Dobrev, Flocchini, Prencipe and Santoro in 2001. A black hole models a network node in which a destructive process deletes any visiting agent or incoming data upon arrival, without leaving any observable trace. Conversely, a network may face one or more malicious mobile processes infecting one or more nodes. Given both kinds of threats, a first crucial task consists in searching for and reporting as quickly as possible the location all faulty nodes while using a minimum number of mobile agents. In general, the main issue is to identify the minimal hypotheses under which faulty nodes can be found. This problem has been investigated in both asynchronous and synchronous networks. A corollary task is to make sure that the protocols designed for solving problems such as gathering and transferring data still work despite the presence of one or more faulty nodes. In this chapter, we review the state-of-the-art of research pertaining to the presence of faulty nodes in a network. We discuss different models in synchronous and asynchronous networks and for different communication and computation capabilities of the agents. We also address relevant computational issues and present algorithmic techniques and impossibility results