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.

We announce a rigorous theory of optimal transport for signed (real) measures on Rd in the separated-support regime: the supports of the positive and negative parts of measures are assumed spatially disjoint. Under this structural hypothesis, together with regularity and non-degeneracy conditions on the absolutely continuous, discrete, and fractal singular components of the underlying Jordan and Lebesgue decompositions, we obtain existence and uniqueness of an optimal transport map for a cost that distinguishes same-sign and opposite-sign transports via a positional penalty, a coupled Monge-Ampere system together with a double Legendre transform characterising the solution, and exact preservation of Hausdorff dimension on the fractal singular components, the latter via a novel adaptive regularisation kernel and a direct measure-theoretic argument requiring no continuity of the transport map. This note states the results and outlines the two main proof techniques, full proofs, uniform estimates, and technical lemmas appear in the complete version of this work. We further report, in full and exhaustive summary, a complete numerical validation of this theory in ambient dimension one, two, and three: an entropic Sinkhorn solver for the single combined transport problem underlying the existence proof, applied to an explicit test case in which inter-sign transport is forced by mass imbalance and calibrated to occur with a large, robust cost margin, recovers exactly the predicted four-region structure with zero measured crossing of fractal mass between signs, and an exact-assignment refinement confirms pointwise-exact preservation of Hausdorff dimension under transport, with box-counting estimates converging to the theoretical dimension as resolution increases. The case of overlapping supports and the fully general unbalanced setting are the subject of Part II, in preparation.