In this brief note, we give two partition statistics which explain the following partition congruences: b(5n + 4) ≡ 0 (mod 5), b(7n + a) ≡ 0 (mod 7), if a = 2, 3, 4, or 6. Here, b(n) is the number of 4-color partitions of n with colors r, y, o, and b subject to the restriction that the colors o and b appear only in even parts.

soils around mining sites play a significant role in the transport and control of heavy metals in the environment. therefore, understanding the adsorption properties of soil is essential in solving pollution problems. the objective of this paper was to study zn(ii) adsorption onto sarcheshmeh copper mine soils. the adsorption isotherms of zn(ii) were studied. the langmuir isotherm indicated an ...

Ethyl benzene used increasingly each year is the raw material of producing styrene monomer. This substanceis produced from benzene alkylation with ethylene or ethanol, depending on the availability and cost of rawmaterials. In this study benzene alkylation in the presence of ethanol in a catalytic fixed bed reactor inthree states of isotherm, adiabatic and non-isotherm-non-adiabatic is mathemat...