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DERIVATION OF CYCLE INDEX FORMULAS OF SEMIDIRECT PRODUCT GROUPS
(2019-12) Muthoka Geoffrey Ngovi
The concept of the cycle index formulas of a permutation group was discovered in
the year 1937. Since then cycle index formulas of several groups have been studied
by different scholars. For instance the cycle index of the dihedral group Dn acting on
the set of vertices of a regular n−gon is known and has been applied in enumeration
of different mathematical structures. In this study the relationship between the cycle
index formula of a semidirect product group and the cycle index formulas of the two
subgroups which the group is a semidirect product of was established. In particular
the cycle index formula of the dihedral group Dn of order 2n is expressed in terms
of the cycle index formula of a cyclic group of order two C2 and the cycle index
formula of the cyclic group of order n, Cn; the cycle index formula of the symmetric
group Sn is expressed in terms of the cycle index formula of the alternating group
An and the cycle index formula of a group generated by a cycle of length two, h(ab)i.
The cycle index formula of an affine(p) group has been derived by considering the
different cycle types of elements of the group and expressed in terms of the cycle
index formula of Cp = {x + b, where b ∈ Zp} and the cycle index formula of
Cp−1 = {ax, where 0 6= a ∈ Zp}. We further extend this to affine(q) where q is a
power of a prime p and to the affine square(p) and affine square(q) groups. Finally,
the cycle index formula of a Frobenius group is expressed in terms of the cycle index
formula of the Frobenius complement H and the cycle index formula of the Frobenius
kernel M. The cycle index formulas which are known such as that of the dihedral
group and the symmetric group were used and the groups whose cycle index formulas
are not known such as the affine(p), affine square(p), affine(q) and affine square(q)
group were first derived as part of the research. It was noted that for semidirect
groups which are Frobenius such as the dihedral group Dn with an odd value of n,
the affine groups and the affine square groups, we can fully express the cycle index
of the group in terms of the cycle index formulas of the subgroups which the group
is a semidirect of. However, for semidirect product groups which are not Frobenius
such as the dihedral group Dn with an even value of n and the symmetric group Sn,
the cycle index formula of the group cannot be expressed fully in terms of the cycle
index formulas of the subgroups the group is a semidirect product of.
Horticulture Research Centre: Kibos
(2025-06-24) Timon K. Moi - Centre Director
Undertake research in all aspects of production, management, postharvest and value addition of Horticultural, Food & Industrial crops.
The outputs from research activities implemented are to support the national Horticultural, Food & Industrial crops industry.
The research programmes respond to sector specific value chains in cotton, rice, fruits, vegetables, roots & tubers, millets, sorghum and other horticultural crops (medicinal plants).
BIOEFFICACY OF ORGANIC EXTRACTS OF FISH POISON BUSH (GNIDIA GLAUCA, FRESEN) AGAINST COWPEA WEAVIL (CALLOSOBRUCHUS MACULATUS, FABRICIUS)
(2018-11) WILHELMY MARION JEBET
Cowpea weevil (Callosobruchus maculatus) is a major pests of stored cowpea in the
tropical region of the world. In Kenya, the damage caused by C. maculatus impacts
negatively on its economic and nutritional values and contributes to food insecurity. The
widely adopted use of chemical pesticides is marred with health and environmental
hazards. Global concern on synthetic chemicals has led to heightened restrictions and
limitations on their use. This, therefore, has prompted the search for alternatives to
synthetic pesticides. New studies are focusing on the use of botanicals as a novel approach
to the management of pests. Gnidia glauca has been exploited by local people in control
of post-harvest pests. However, no scientific research has been undertaken to evaluate its
potential anti-insect properties. In this study, four organic leaf extracts of G. glauca
(methanol, ethyl acetate, DCM and blend) were evaluated for contact toxicity, oviposition
deterrence, inhibition of progeny emergence and repellency against cowpea weevil. The
plant leaves were collected from Embu County, Kenya. The samples were prepared,
extracted and investigation carried out under ambient laboratory conditions. The
experimental design entailed five test concentrations (2g/100ml, 4g/100m, 6g/100ml,
8g/100ml, and 10g/100ml) of each extract, the untreated control, the solvent control and
the positive control-Actellic. Each bioassay had four replications. Adult weevils (1-3 days
old) were exposed to the extracts and mortality was monitored daily for the first four days.
Subsequently, oviposition deterrence was assessed on the 15th day while inhibition of
progeny emergence was evaluated on the 49th-day post-treatment. Extract repellency was
assessed for the first 6 hours after treatment. Screening for plants phytochemicals was
conducted using the standard recommended procedures. The results of this study revealed
all G. glauca extracts, to a varied extent, induced mortality on C. maculatus. Mortality
was concentration and exposure time dependent. Highest mortality of 89.74% was
recorded with 10g/100ml ethyl acetate extract 96 hours post-treatment. The extracts
significantly deterred oviposition with the 10g/100ml concentration of ethyl acetate, DCMand blend statistically (p>0.05) comparable to the activity of synthetic pesticide. All the
extracts were found effective in inhibition of progeny emergence. Ethyl acetate extract at
the test dose of 10g/100ml demonstrated the highest inhibition of 99.3% while the least
inhibition of 9.03% was exhibited by 2g/100ml methanol extract. G. glauca extracts
proved to be attractant of C. maculatus rather than repellant, none of the extract
concentration attained repellency greater than 50%. Results also showed that the extracts
had tannins, phenols, flavonoids, terpenoids, saponins, alkaloids, cardiac glycosides and
steroids which have been associated with insect control properties. It was therefore
concluded that the plant extracts, possess bioactivities against Callosobruchus maculatus
on the tested parameters of contact toxicity, oviposition deterrence, inhibition of progeny
emergence and repellency. Hence the studied extracts can further be purified and
developed into the plant-derived bio-pesticides to control C. maculatus.
Upscaling African Indigenous Vegetables Climate Smart Technologies for Food & Nutrition security in Kenya
(2025-11-27) Prof Mary Abukutsa-Onyango, Prof Anselimo Makokha, Dr Eucabeth Majiwa, Mr Patrick Amunavi, Ms Joyce Liti Njenga, Mr James Shikwati, Ms. Esperance Chesoli, Mr Muchiri Nyaggah, and Dr Anne Aswani
Goal:
To contribute to achieving food and nutrition security in the context of climate change challenges facing Kenya by promoting the sustainable production, commercialization and utilization of African Indigenous Vegetables (AIVs).
Overall Objective:
To upscale the production, preservation and utilization technologies of African indigenous vegetables for food and nutrition Security in Kenya.
Potential use of Kenyan Entomopathogenic Nematodes and Neem (Azadirachta indica) for the Sustainable Management of Tomato Leaf Miner (Tuta absoluta)
(2018-06) DAVID MUNYUA MUTEGI
The tomato leaf miner, Tuta absoluta (Meyrick) (Lepidoptera: Gelechiidae) was reported in
Kenya in 2014 and has become a devastating pest of tomato crop in both the field and in the
greenhouse. The objectives of this study were; to determine the pathogenicity of Heterorhabdities
species and Steinernema karii nematodes against tomato leaf miner in laboratory conditions; to
evaluate the insecticidal effects of neem biopesticide against tomato leaf miner in the greenhouse
and to evaluate the effectiveness of entomopathogenic nematodes and neem combined with
entomopathogenic nematodes as management options for tomato leaf miner in the greenhouse
conditions. Entomopathogenic nematodes (EPNs) used in the study were obtained from Kenya
Agricultural and Livestock Research Organization (KALRO) entomopathogenic nematodes
laboratories and Tuta absoluta larvae were obtained from a colony reared and maintained in a
greenhouse at Kabete Campus Field Station, Nairobi. Multiplication of the EPNs was done by invivo method or the insect-bait technique with the third instar of greater wax moth (Galleria
mellonella). Bioassays were conducted in petri dishes where the effects of EPNs concentrations;
at 100, 300and 500Ijs/ml on Tuta absoluta larvae exposed for 24-72 hours were evaluated.
Secondly, three different concentrations namely; 20ml/20L, 40ml/20L and 60ml/20L of
Nimbecidine® (Azadirachtin) were evaluated against Tuta absoluta populations in the greenhouse.
Thirdly, the following management options; Steinernema karii nematodes alone applied at a rate
of 1000 Ijs/ml with oil adjuvant Addit® as a wetting agent in water, Steinernema karii nematodes
and neem (Azadirachtin 0.03%) applied sequentially after one hour at a rate of 40ml/20L were
evaluated against Tuta absoluta populations. These were compared with Coragen® SC (20%
Chlorantraniliprole) synthetic pesticide applied at the rate of 3ml/20L and control where only water
was applied. A delta sticky trap supplied with Tuta absoluta pheromone was hanged at the centre
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of the greenhouse at a height of one metre for monitoring Tuta absoluta numbers to enable the
initiation of treatments. The results obtained showed that, the evaluated concentration rates of
Heterorhabditis species and Steinernema karii at 100Ijs/ml, 300 Ijs/ml and 500 Ijs/ml significantly
caused (p < 0.05) mortality to the Tuta absoluta larvae and the highest mortality was recorded at
500 Ijs/ml concentrations with an exposure period of 72 hours. Steinernema karii was more
pathogenic compared to Heterorhabditis species with respect to exposure time, having shown
100% and 91.5% larval mortality, respectively. Secondly, the evaluated neem concentrations
20ml/20L, 40ml/20L and 60ml/20L significantly (p < 0.05) reduced Tuta absoluta population in
the greenhouse tomato. The high concentration (60ml/20L) was more effective in reducing Tuta
absoluta population and fruit damage compared to the lower concentrations and control. High fruit
damage of 96.2% was recorded in the control compared to 30.0%, 23.4% and 20.0% for 20ml/20L,
40ml/20L and 60ml/20L dose rates, respectively. Lastly, EPNs alone and EPNs combined with
neem significantly (p < 0.05) reduced the population of Tuta absoluta. The number reduction of
Tuta absolutaby EPNs combined with neem did not differ with that which was achieved in
Coragen®. Tomato fruit damage was highest in the control with 91.5% compared to 10.8%, 7.9%
and 3.0% for EPNs alone, EPNs combined with neem and Coragen®, respectively. This study
demonstrates the potential of EPNs alone or in combination with neem applied successively at a
one hour interval, as an alternative strategy for the sustainable management of Tuta absoluta in
the greenhouse conditions.
