University of Ngaoundéré

This paper investigates the impact of the incorporated reflector on the dynamic performance of a flat-plate solar collector. To this end, a mathematical model was developed based on the energy balance of the collector’s various components. The equations were then solved using a numerical algorithm. The results show that incorporating the reflective surface into the collector envelope, and despite the energy losses that may result, improves performance by increasing the energy received by the absorber. Within the scope of this study the heat gain between configurations with and without reflection results in a temperature rise of 10.85°C. In particular, temperatures fall with the mass flow rate of the fluid, while they increase with the number of tubes. When the number of tubes increases from 5 to 10, outlet temperatures rise from 38.8–53.95°C for lengths of 1–2.5 m. When the number of tubes increases from 5 to 10, thermal efficiency drops from 60.0–41.6

We study when, and how compactly, a finite connected graph (G) embeds isometrically into a Cayley graph of a finite abelian group. The classical theory of partial cubes answers this for isometric subgraphs of hypercubes through the Djokovic-Winkler relation (θ); we extend the question to the full family of abelian Cayley graphs, whose hosts may carry composite generators and cyclic factors of any order. We introduce an involutive edge relation (φ), defined by two simultaneous distance equalities, which coincides with (θ) exactly on partial cubes and remains informative beyond them, together with an oriented relation (Φ) for non-involutive hosts, where generator classes are constrained to be partial permutations rather than matchings.The central result is a quotient labeling theorem: for any partition of the edge set into candidate generator classes, the most generic consistent vertex labeling is the quotient of the free module on the classes by the lattice of signed cycle-class incidences, computed by the Smith normal form; the binary case is its reduction modulo two. We prove that the finest partition always yields an isometric labeling, that compactifying the resulting universal group is itself an instance of the same quotient construction, and that the whole construction is algorithmic and certifiable. Worked examples include the triangle, the Petersen graph (embedding into the Clebsch graph of order 16), the Pappus graph (a 1024-fold compaction), and the diamond (a non-diagonal fold). Sharp dimension bounds and an exhaustive census of small graphs are developed in a companion paper. 2020 MSC: 05C12, 05C25, 20K01, 05C50

We investigate the minimum size of finite abelian Cayley graphs that admit an isometric embedding of a finite connected graph. While every connected graph on n vertices embeds isometrically into a binary Cayley graph of dimension at most n-1, the smallest possible abelian host has remained largely unexplored. We establish fundamental lower bounds showing that every binary host has dimension at least max(diam(G), floor(log2 n)), whereas every finite abelian host has order at least max(n, 2^diam(G)). Moreover, we prove that the minimum host order equals n if and only if G is itself an abelian Cayley graph. Exact binary dimensions are obtained for several important graph families. Hypercubes, complete graphs of order 2^k, and even cycles attain the lower bound. For stars we prove k_min(K1,q)=floor(log2 q)+1 using maximum sum-free sets, yielding an exponential improvement over the naive and isometric dimensions. For odd cycles we prove k_min(Cm)=m-1 for all m<17 and reduce the general case to a cyclic-interval lemma, showing that the universal upper bound is tight. Our computational contribution is a certified exhaustive census of all 995 connected graphs with 2<=n<=7 vertices under general abelian compactifications. The data reveal an "abelian dividend": 569 graphs (57 percent) admit a strictly smaller abelian host than the best binary host, 707 (71 percent) admit an optimal host containing a cyclic factor Zm with m>2, and only 17 graphs attain the theoretical order floor max(n,2^diam(G)). These results demonstrate that compact non-binary abelian hosts are typical rather than exceptional, while binary hosts remain the universal worst-case construction. 2020 MSC:05C12, 05C25, 05C30, 11B75, 20K01

The search for efficient methods for the removal of organic dyes in textile industry effluents is a permanent quest. The objective of the present work is to determine the effective conditions for the removal by adsorption of two azo dyes, Orange Acid 7 (OA7) and Indigo Carmine (IC) regularly contained in the effluents of certain textile industries. To achieve this, the adsorbent was produced from the shell of the Detarium Microcarpum kernel and then characterized by determining its surface function, structure, morphology and specific surface area. The adsorption efficiency was determined by following certain parameters determining the adsorption capacity such as contact time, pH of the effluent, initial concentration of pollutants. The kinetics and adsorption mechanisms were also studied. The results obtained reveal that the adsorbent produced is micro and macroporous with a specific surface area equal to 905.76 m2/g. The main chemical functions identified on its surface are C--O, N-H, O-H, C--C. The equilibrium time for the adsorption of OA7 and IC is 10 min with a maximum reduction rate of 97 %. The adsorption is optimal at pH = 4 of the solution. The adsorption capacity increases with the initial concentration of the dyes. Acid Orange 7 is adsorbed in a monolayer and Indigo Carmine in a multilayer. The adsorption is generally governed by the pseudo-second-order kinetic model. The maximum adsorption capacity of both dyes on the adsorbent is approximately 27 mg/g.

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given G with n vertices and m edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order n on the same vertex set? This defines two invariants, the completion number γ^+ (additions only) and the Cayley edit distance γ_ (both), each normalized by m. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is n+m-2k when it realizes a longest path with k edges; the optimal cost is m-n+2pp(G), bounded in polynomial time by the matching number. We prove that irregularity alone forces γ^+(G)≥ nΔ^*/(2m)-1, where Δ^* is the least d with nd even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where γ^+(K_1,q)=(q-1)/2 and the star maximizes γ^+, while γ_ stays bounded by an absolute constant. We determine paths and grids exactly, γ^+(P_n)=γ^+(P_n □ P_n)=1/(n-1), and show γ_(K_1,q)→ 2, not the 3/2 suggested by the additive case. We report an exhaustive certified census of all 995 connected graphs on at most seven vertices. The degree bound is attained on 89.4% and the two invariants separate strictly on 84.7%, though both rates vary sharply with order: attainment 100%,100%,84.8%,89.7% and separation 0%,61.9%,73.2%,87.7% for n=4,5,6,7, dominated by the 853 graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.