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Let $\mathcal X$ be a Banach space and let T be a bounded linear operator on $\mathcal {X}$ . We denote by S(T) the set of all complex $\lambda \in \mathcal {C}$ such that T does not have the single-valued extension property. In this paper it is shown that if MC is a 2 × 2 upper triangular operator matrix acting on the Banach space $\mathcal {X} \oplus \mathcal {Y}$ , then the passage from σLD(A) ∪ σLD(B) to σLD(MC) is accomplished by removing certain open subsets of σd(A) ∩ σLD(B) from the former, that is, there is the equality σLD(A) ∪ σLD(B) = σLD(MC) ∪ℵ, where ℵ is the union of certain of the holes in σLD(MC) which happen to be subsets of σd(A) ∩ σLD(B). Generalized Weyl’s theorem and generalized Browder’s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how generalized Weyl’ theorem, generalized Browder’s theorem, generalized a-Weyl’s theorem and generalized a-Browder’s theorem survive for 2 × 2 upper triangular operator matrices on the Banach space.

Background and Objective: Sulphur and boron are found as most critical nutrient elements for the better growth and yield of mustard crop however no such concrete information for their uses in field production of this crop is yet to be reported. Therefore, an experiment was conducted to investigate the effect of agronomic biofortification of sulphur and boron nutrients on the growth and yield of mustard crop. Study Design: The experiment was laid out in a Randomized Complete Block Design (RCBD) with three replicates. Place of Study: The experiment was carried out in the Crop Botany Field Laboratory, Bangladesh Agricultural University, Mymensingh. Methodology: Three doses of sulphur (S) viz. 0, 20 and 40 kg ha-1 and three doses of boron (B) viz. 0, 0.5 and 1.0 kg ha-1 and their possible combinations were used as basal doses. Field data were collected from periodic destructive samplings on the plant height, number of leaves and branches per plant, total dry matter accumulation and finally yield components and yield. Results: Sulphur and boron fertilizations significantly influence the plant height, production of branches and leaves per plant, dry matter accumulation and yield attributes and yield of mustard crop. The mustard crop fertilized with 40 kg S ha-1 in combination with 1 B kg ha-1 produced taller plant, higher number of branches and leaves in each plant and higher amount of dry matter per plant while these plant traits were found as minimum when the growing the mustard crops in control plots i.e. the plants received neither sulphur nor boron. Application of sulphur @ 40 kg ha-1 along with boron @ 1 kg ha-1 produced the highest seed yield (2.73 t ha-1) whereas the lowest seed yield (1.08 t ha-1) was found where no sulphur and boron were applied. Conclusion: The result conclude that combine application of sulphur and boron @ 40 and 1 kg per hectare, respectively was found to be most effective dose in enhancing growth and yield of mustard crop.

Strawberry is a very popular, refreshing and nutritious fruit having powerful antioxidant and a safe, delicious addition to any diet. An experiment was conducted at the Landscaping section of the Department of Horticulture, Bangladesh Agricultural University, Mymensingh during the period from October 2018 to April 2019 to develop a sustainable technology for optimum growth, yield and quality of strawberry cv. RU-1 (Festival) through vertical farming. The experiment comprised three vertical staking viz., 1.5, 3.0 and 4.5 feet (ft) above ground staking, and five organic manures viz., control (no manures), cowdung @ 500 g/plant, mustard oilcake (MOC) @ 100 g/plant, poultry manure @ 500 g/plant and cowdung+MOC+poultry manure @ (500+100+500) g/plant. The two-factor experiment was conducted in randomised complete block design with three replications. Significant variations were observed due to vertical staking and organic manures on all the parameters studied. Highest plant height (15.95 cm), number of leaves (27.62), number of flowers (24.86), number of runner (4.73) and number of fruits per plant (22.90), fruit length (3.70 cm) and diameter (3.04 cm), individual fruit weight (13.87 g), yield per plant (318.82 g) and TSS (8.23%) were observed at 4.5 ft above ground staking and application of cowdung+MOC+poultry manure @ (500+100+500) g/plant, while the lowest plant height (9.61 cm), number of leaves (15.02), number of flowers (12.40), number of runner (2.18) and number of fruits per plant (22.90), fruit length (3.70 cm) and diameter (3.04 cm), individual fruit weight (13.87 g), yield per plant (124.48 g) and TSS (6.40%) were recorded at 1.5 ft above ground staking with control. Thus, combined treatment of 4.5 ft above ground staking with cowdung+MOC+poultry manure @ (500+100+500) g/plant was found to be better in respect of optimum growth, yield and quality of strawberry. [J Bangladesh Agril Univ 2020; 18(2.000): 307-315]

In this paper we define convex, strict convex and normal structures for sets in fuzzy cone metric spaces. Also, existence and uniqueness of a fixed point for non-self mappings with nonlinear contractive condition will be proved, using the notion of strictly convex structure. Moreover, we give some examples illustrate our results.

An experiment was conducted to study the application of alternative treatments to control postharvest fungal infection and shelf life extension of papaya under different maturity stages at the Laboratories of the Department of Horticulture and Agricultural Chemistry, Bangladesh Agricultural University, Mymensingh during the period from August to October, 2018. The two factor experiment (maturity stages and postharvest treatments) was laid out in a completely randomized design (CRD) with three replications. Four postharvest treatments viz., control, hot water treatment (Fruits treated at 50±2ºC for 10 minutes), chitosan coating (2000ppm chitosan solution) and hot water+chitosan coating were assigned to papaya fruits which were maturity stage I or maturity stage II. Significant variations were observed between maturity stage I (green mature) and maturity stage II (0-10% yellowing of fruits) in relation to most of the parameters studied. Results showed that most of the parameters studied were significantly influenced by the afore-mentioned factors. Longer shelf life (12.64 days) was observed in maturity stage I fruits than maturity stage II fruits (12.18 days). Significant reduction in disease incidence and severity was noticed in the maturity stage I fruits as compared to maturity stage II fruits. The longest shelf life (15.20 days) was found in hot water treated and chitosan coated fruits of maturity stage I, whereas the shortest shelf life (8.52 days) was observed in control of maturity stage II fruits. Progressive Agriculture 30 (3): 298-304, 2019

An experiment was conducted at the Landscaping Section of the Department of Horticulture, Bangladesh Agricultural University, Mymensingh during the period from November 2017 to May 2018 to investigate the effect of varieties and inorganic fertilizers on growth and flowering of gerbera. The experiment consisted of two factors viz., Factor A: varieties (V1= Mini Daisy, V2= Aladin) and Factor B: Different levels of inorganic fertilizers (T0 = 0, T1 = N 300 kg/ha, T2 = P 275 kg/ha, T3 = K 225 kg/ha, T4= N+P+K 300+275+225 kg/ha). The experiment was laid out in Randomized Complete Block Design with three replications. Varieties and different levels of inorganic fertilizers had significant effect on plant height, number of leaves per plant, spread of leaves, days to first flower bud emergence, days to harvest maturity, total number of flowers produced and spread of flowers and field life of gerbera. Total number of flowers per plant (10.89) was recorded from Aladin (V2), which was higher than that of Mini Daisy (V1) (10.16). The highest number of flowers per plant(12.44), while, the lowest number of flowers (9.10) was obtained from T0 (control). Considering the combined effect, the maximum number of flowers per plant (12.87) was obtained from the treatment combination of V2T4 (Aladin+N+P+K), while, the lowest number of flowers per plant (8.67) was recorded from V1T0 (Control). Therefore, combined application of N, P and K fertilizers along with variety Aladin was found to be better in respect of growth and flowering of gerbera.

Inefficient use of existing scarce resources has adverse effect on food production and production cost hence, low income among the farmers across the nation. Strawberry farming is profitable but costly. So the study was designed to explore resource use efficiency of strawberry cultivation in Rajshahi and Joypurhat district of Bangladesh. Primary data were collected from 100 strawberry growing farmers. Data were analyzed by using descriptive statistics, Cobb-Douglass production function and marginal productivity analysis. The results showed that 41% farmers were 18 to 30 years old. Highest 26% farmers had primary level of education, 61% farmers had small farm, and 69% farmers had 1 to 3 years’ experience of strawberry cultivation. The double log function showed the best fit with adjusted R2 of 61%. Production inputs such as sapling and fertilizer had positive and labour and chemicals had negative and significant effect on outputs. Sapling, land and water were underutilized and labour, cowdung, fertilizer and chemicals were over used. About 37% farmers reported that strawberry plants were attacked by many more diseases. Approximately, 48% farmers reported that transportation facilities of the study areas were poor. Efficiency ratio of the inputs indicates, farmers of the study area were not efficient in using inputs and it is needed to adjust resource use in order to improve farm profit at this level of technology used. The study also recommends that the farmers need training to be efficient. SAARC J. Agri., 17(1): 189-200 (2019)

Mussaendas are increasingly popular for the showy colour they provide during much of the year in garden landscapes. However, root formation in stem cuttings is a major challenge for cultivation of Mussaenda species. Therefore, the experiment was carried out at the Landscaping section of the Department of Horticulture, Bangladesh Agricultural University, Mymensingh during the period from August to November 2018 to study the effect of phytohormone naphthalene acetic acid (NAA) on sprouting and rooting of stem cuttings in Mussaenda species. The two-factor experiment consisted of three Mussaenda species viz., Mussaenda frondosa (White flag bush), Mussaenda philippica (Pink flag bush) and Mussaenda erythrophylla (Red flag bush), and five concentrations of NAA viz., 0 (control), 0.1%, 0.2%, 0.3% and 0.4%. The experiment was laid out in randomised complete block design (RCBD) with three replications. The results revealed that all the traits related to sprouting and rooting potential of stem cuttings were significantly influenced by combined effects of Mussaenda species and various NAA concentrations. The maximum sprouted bud per cutting (3.57), highest number of roots per cutting (5.88) and the longest root length (4.32 cm) were recorded from M. philippica except number of leaves per stem cutting (4.42) in M. erythrophylla. The increasing concentrations of NAA application responded positively on sprouting and rooting characters in all three Mussaenda species used for this experiment. The best results obtained from the treatment where NAA was applied at the concentration of 0.3% with 3.92 sprouted buds per cutting, 4.77 leaves per cutting, 7.15 roots per cutting and 5.26 cm length of roots. The treatment combination of M. philippica (Pink flag bush) and 0.3% NAA, followed by NAA at 03% with M. erythrophylla (Red flag bush) was found to be best in terms of sprouting and rooting of stem cuttings in Mussaenda. Progressive Agriculture 30 (1): 48-54, 2019

Turmeric is a good source of income for hilly people of Bangladesh. The study estimated the profitability and technical efficiency of turmeric cultivation in Khagrachhari district. In total 150 turmeric farms located in Khagrachari Sadar, Panchari and Matiranga Upazilas of Khagrachari district, were surveyed. Data were collected, using a pre-tested questionnaire during January, 2015. The study revealed that turmeric farming is a profitable farming with some dominating variable costs like seed (rhizome) and sowing, harvesting and carrying. As the net return was Tk. 112139 per hectare and the BCR of sampled farmers was 2.20, this indicates that turmeric farms with a BCR greater than 1 have greater benefits than costs as well as positive net benefits. Seed (rhizome) and fertilizer showed significant positive effects on the turmeric production in the stochastic frontier production model. Turmeric farming displayed a mean technical efficiency of 82%, which suggested a substantial 18% of potential output of turmeric can be recovered by removing inefficiency. Besides improving technical efficiency, potential also exists for raising turmeric production through higher education and extension services. For a land scarce country like Bangladesh this gain could help increase income and ensure better livelihood for the hilly farmers. The policy implication of the analysis is that investment in education and extension service would greatly improve technical efficiency that contribute to income of the hilly people. Bangladesh J. Agril. Res. 44(1): 43-58, March 2019

When $A\in \mathscr{L}(\mathbb {X})$ and $B\in \mathscr{L}(\mathbb {Y})$ are given, we denote by MC an operator acting on the Banach space $\mathbb {X}\oplus \mathbb {Y}$ of the form $M_{C}=\left (\begin {array}{cccccccc} A & C \\ 0 & B \\ \end {array}\right ) $ . In this paper, first we prove that σw(M0) = σw(MC) ∪{S(A∗) ∩ S(B)} and $\mathbf {\sigma }_{aw}(M_{C})\subseteq \mathbf {\sigma }_{aw}(M_{0})\cup S_{+}^{*}(A)\cup S_{+}(B)$ . Also, we give the necessary and sufficient condition for MC to be obeys property (w). Moreover, we explore how property (w) survive for 2 × 2 upper triangular operator matrices MC. In fact, we prove that if A is polaroid on $E^{0}(M_{C})=\{\lambda \in \text {iso}\sigma (M_{C}):0

Gladiolus is an excellent cut flower grown throughout the world for its spikes with florets of massive form, brilliant colours, attractive shapes, varying size and long shelf life. However, major constraint for gladiolus cultivation is the corm dormancy. Plant growth regulators (PGRs) play an important role in breaking dormancy and promote more number of quality corm and cormel productions in gladiolus. Therefore, an experiment was conducted to study the influence of corm size and plant growth regulators on corm and cormel production of gladiolus during the period from October 2017 to April 2018 at the Landscaping section of the Department of Horticulture, Bangladesh Agricultural University, Mymensingh. The two-factor experiment included two corm sizes viz., 3-4 cm and 4.1-5 cm, and two PGRs viz., GA3 @ 250, 500 and 750 ppm; and NAA @ 100, 200 and 300 ppm along with tap water as control. The experiment was laid out in a randomised complete block design (RCBD) with three replications. The results revealed that the corm size and PGRs at different concentrations significantly influenced the corm and cormel producing attributes of gladiolus. Maximum number of corms and cormels per plant, maximum weight of single corm, maximum weight of corms and cormels per plant, biggest size of single corm, highest yield of corms and cormels per hectare were recorded from 4.1-5 cm sized corms and GA3 @ 500 ppm, compared to rest of the treatments. It was observed that combined treatments had significant influence on all the parameters studied. The treatment combination of 4.1-5 cm sized corms and GA3 @ 500 ppm was found to be best in terms of corm and cormel production of gladiolus.Progressive Agriculture 29 (2): 91-98, 2018

It is known that if $$A\in \mathscr {L}(\mathscr {X})$$ and $$B\in \mathscr {L}(\mathscr {Y})$$ are Banach operators with the single-valued extension property, SVEP, then the matrix operator $$M_\mathrm{{C}}=\begin{pmatrix} A &{} C \\ 0&{} B \\ \end{pmatrix} $$ has SVEP for every operator $$C\in \mathscr {L}(\mathscr {Y},\mathscr {X}),$$ and hence obeys generalized Browder’s theorem. This paper considers conditions on operators A, B, and $$M_0$$ ensuring generalized Weyl’s theorem and property (Bw) for operators $$M_\mathrm{{C}}$$. Moreover, certain conditions are explored on Banach space operators T and S so that $$T\oplus S$$ obeys property (gw).

The study was conducted in four districts namely Rajshahi, Chuadanga, Comilla and Rangpur during 2015-2016 to estimate the effect of drought in maize production, technical efficiency and adaptation strategy of maize farmers, and explore related problems of maize cultivation in the study areas.Rajshahi and Chuadanga were selected as drought prone areas whereas Rangpur and Comilla were selected as favorable environment. A total of 200 farmers taking 50 from each district were selected randomly for the study. Per hectare total cost of maize cultivation in drought prone areas was found Tk. 92,582, whereas in normal environment it was Tk. 79,594. Per hectare average yield in drought prone and normal areas were 7576 kg and 8729kg, respectively. Per hectare net return of maize in drought prone and normal areas were Tk. 28,062 and Tk. 59, 871, respectively.On full cost basis benefit cost ratio (BCR) in drought prone and normal areas were 1.31 and 1.75, respectively. Result of semi-logarithmic regression model indicated that maize production was decreased by 22.4 percent in drought prone areas than normal environment. Loss of yield (70%), loss of plant growth (55%) and problem in flowering stage (43%) were reported to be the major effects of drought.The major adaptation strategies in the drought prone areas were increase number of irrigation (77%), increase amount of fertilizer (42%) and seed (31%) and change of planting date (30%). Higher price of irrigation (70%), lack of drought tolerant variety (48%) and lack of quality seed (31%) were the major problem in drought prone areas. Again,disease infestation (64%), lack of quality seed (41%) and high price of seed (36%) were the major problem in normal environment. Scientists should take attempt to develop drought tolerant maize variety.Bangladesh J. Agril. Res. 43(2): 219-234, June 2018

Let $$A=U|A|$$ be the polar decomposition of A on a complex Hilbert space $${\mathscr {H}}$$ and $$00$$ with $$s+t=1$$ . Also, we prove that $$({\widetilde{A}}_{s, t},{\widetilde{B}}_{s, t})$$ has the FP-property if and only if $$(({\widetilde{A}}_{s, t})^{*},({\widetilde{B}}_{s, t})^{*})$$ has the FP-property, where A, B are invertible and $$ 0 < s, t $$ with $$ s + t =1$$ . Moreover, we prove that if $$0 < s, t$$ and $${\widetilde{A}}_{s, t}$$ is positive and invertible, then $$\left\| {\widetilde{A}}_{s, t}X-X{\widetilde{A}}_{s, t}\right\| \le \left\| A\right\| ^{2t}\left\| ({\widetilde{A}}_{s, t})^{-1}\right\| \left\| X\right\| $$ for every operator X. Also, if $$ 0 0$$ .

In this paper, we prove that if T∈B(H) is w-hyponormal, then the quasinilpotent part H0(T) of T is given by ker(Tp)={xH:rT(x)=0}=⋂λ≠0(T-λ)pH for all sufficiently large integers p, where rT(x)=limn→∞supTnx1n . We prove that if T is w-hyponormal and the spectrum σ(T) is finite, then T is algebraic. In addition, we prove that if T∈B(H) is w-hyponormal and has decomposition property (δ) , then T has a non-trivial invariant closed linear subspace. Also, we obtain that such an operator with rich spectrum has a nontrivial invariant subspace. Moreover, we consider invariant and hyperinvariant subspaces for w-hyponormal operators.

Present study aspires at the design of an immediate release formulation with prospective use of fexofenadine hydrochloride by exploring the effect of sodium starch glycolate as super disintegrant. Fexofenadine hydrochloride immediate release tablets (Formulations F-1, F-2, F-3, F-4 and F-5) using different ratios of sodium starch glycolate as a disintegrant were prepared by direct compression method. Standard physicochemical tests were performed for all the formulations. Dissolution studies of the formulations were done in phosphate buffer, pH 6.8 using USP apparatus II (paddle apparatus) at 50 rpm. Percent release of fexofenadine hydrochloride of formulations F-1, F-2, F-3, F-4 and F-5 were 89.98%, 90.98%, 92.95, 96.92% and 99.85%, respectively after 1 h and the release pattern followed the zero order kinetics. The release rate in the formulation F-5 was higher compared to other formulations and the studied market products. Sodium starch glycolate speed up the release of the drug from the core tablets, and the release of fexofenadine hydrochloride from tablets was directly proportional to the amount of sodium starch glycolate present in the formulations and there by produced immediate action.

In this paper, we study Weyl type theorems for $f(T)$ , where $T$ is algebraically class $p$ - $wA(s, t)$ operator with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$ and $f$ is an analytic function defined on an open neighborhood of the spectrum of $T$ . Also we show that if $A , B^{*} \in B(\mathcal{H}) $ are class $p$ - $wA(s, t)$ operators with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$ , then generalized Weyl's theorem , a- Weyl's theorem, property $(w)$ , property $(gw)$ and generalized a- Weyl's theorem holds for $f(d_{AB})$ for every $f \in H(\sigma(d_{AB})$ , where $ d_{AB}$ denote the generalized derivation $\delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\delta_{AB}(X)=AX-XB$ or the elementary operator $\Delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\Delta_{AB}(X)=AXB-X$ .

In this paper, we introduces the property (a B w), a variant of generalized a-Weyl’s theorem for a bounded linear operator T on an infinite-dimensional Banach space $\mathbb {X}$ . We establish several sufficient and necessary conditions for which property (a B w) holds. Also, we prove that if $T\in \mathbf {L(\mathbb {X})}$ satisfies property (a B w) then T satisfies property (B w). Certain conditions are explored on Hilbert space operators T and S so that T ⊕ S obeys property (a B w).

A Banach space operator satisfies property (b) if the complement of its essential Weyl approximate point spectrum in its approximate point spectrum is the set of all poles of the resolvent of finite rank. Property (b) does not transfer from operators A and B to their tensor product $$A\otimes B;$$ we give necessary and/or sufficient conditions ensuring the passage of property (b) from A and B to $$A\otimes B.$$ Perturbations by Riesz operators are considered.

To develop a database on growth promoters and feed additives in cattle fattening a purposive survey was carried out. It is done among 30 cattle fattening farmers covering 1 Upazilla under 3 union in Mymensingh district of Bangladesh, who were involved in cattle fattening activities before Eid-Ul-Azha (August to November2015). The survey was done covering farmer’s information, livestock population and production system, feed sources, manpower involvement, availability, types of hormones, feed additives use with negative opinion and probable suggestions for controlling whereas the researcher had no control or could not manipulate the variables as the appeared. All the farmers interviewed during the survey period were literates, and about 52% of them had primary education, 27% had secondary school certificates, 14% obtained higher secondary certificate and the rest were graduates and post-graduates, 7%. Irrespective of literacy about 10% of the interviewed farmer had training on cattle fattening and the rest fattened cattle without having any sort of training. About 35% respondents use skilled personal for cattle fattening. In the study areas, most of the farmers (55%) were using roughage and 45% of them use concentrate. The fattened cattle were marketed by farmers is 98% and rest 2% by butcher or others. About 72% of the farmer use Tracoll and 28% use Ketofox as feed additives and as hormone 30% farmers use Betamethasone, 55% use Decasone and 15% use Periactin. In public perception 89% farmers use growth promoters in wrong period and 80% use harmful feed additives, feeding natural feed in low amount is (65%). Consumers are less concern to buy fattened bull (93%). The mean of the suggestions like veterinary services (1.26), doses in proper ages (2.06), feeding balanced feed (1.62), organic growth promoters program (0.60) and knowledge through ICT program (0.03). From the above results it may be concluded that, major number of cattle are fattened by small farmers in the survey area. For this fattening purpose different types of hormones and feed additives available in the market are used by them. It can be concluded that, the percentages of different negative opinions on using hormones and feed additives is higher.J. Environ. Sci. & Natural Resources, 9(2): 109-113 2016

Mango has emerged as an important area for diversification and as an alternative cropping pattern due to higher returns and productivity. The study was conducted in three mango growing districts, namely Chapai Nawabganj, Natore, and Rajshahi during 2014-2015 to estimate the financial benefit of shifting cereal lands to mango production, factors influencing shifting decision, and explore related problems of mango cultivation in the study areas. A total of 180 farmers taking 60 farmers from each district were selected through using multistage stratified random sampling for the study. About 49% lands were shifted to mango cultivation from cereal crops which was higher in Chapai Nawabganj (55%) followed by Natore (48%) district. The main reason of this shifting was reported to be higher profit compared to other crops. The average total cost of mango cultivation was Tk. 1, 33,889 per hectare. Higher cost was observed in the 16th-20th year of garden (Tk. 1, 52,010) followed by 11th -15th year (Tk. 1, 48,952). The average yield of mango was found to be the highest in 16th – 20th year (26.48 ton/ha) followed by 11-16th year (19.38 ton/ha). Per hectare net return from mango cultivation was Tk. 1, 75,244. Total cost of mango cultivation was 10% higher than Boro-Fallow-T.Aman cultivation. On the other hand, total cost was about 40% lower than Wheat- Jute- T.Aman, Wheat-Aus-T.Aman and Potato-Fallow-T.Aman. The net return from mango cultivation was 75% higher than other cropping patterns. The shifting of cereal lands to mango cultivation was found to be a profitable since the BCR (2.89), net present value (Tk. 33, 71,166) and internal rate of return (39%) were very high. Relative income, farm size and education turned out to be positively significant, whereas age was negatively significant for shifting decision from cereal crops to mango cultivation. Therefore, Farmers should be motivated to cultivate mango in the fallow lands or areas where other crops are not grown well.Bangladesh J. Agril. Res. 42(1): 137-158, March 2017

A minimax theorem is a theorem providing conditions which guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neumann’s minimax theorem, which was considered the starting point of game theory. Since then, several alternative generalizations of von Neumann’s original theorem have appeared in the literature. Variational inequality and minimax problems are of fundamental importance in modern non-linear analysis. They are widely applied in mechanics, differential equations, control theory, mathematical economics, game theory, and optimization. The purpose of this paper is first to establish a minimax theorem for mixed lower–upper semi-continuous functions in fuzzy metric spaces which extends the minimax theorems of many von Neumann types. As applications, we utilize this result to study the existence problems of solutions for abstract variational inequalities and quasi-variational inequalities in fuzzy metric spaces and to study the coincidence problems and saddle problems in fuzzy metric spaces.

A Banach space operator T obeys property (gm) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T − λI is a left Drazin invertible. In this article, we study the stability of property (gm), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, by quasi-nilpotent operators, or more generally by algebraic operators commuting with T.

We investigate several properties of Aluthge transform of an operator . We prove (i) if T is (s, p)-w-hyponormal operator and is quasinormal (resp., normal), then T is quasinormal (resp., normal), (ii) if T is (s, p)-w-hyponormal operator and is a partial isometry, then T is quasinormal partial isometry, (iii) if T and are (s, p)-w-hyponormal operator, then T is normal, and (iv) Fuglede–Putnam type theorem holds for a class p-w-hyponormal operator T with if T satisfies a kernel condition

We study the transformation versions of the Weyl-type theorems for operators T and S and their tensor product T ⊗ S in the infinite-dimensional space setting.

A Banach space operator is said to be obeys property (m) if the isolated points of the spectrum σ(T) of T which are eigenvalues of finite multiplicity are exactly those points λ of the spectrum for which T - λ is an upper semi-Browder. In this article, we study the stability of property (m), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, quasi-nilpotent operators, Riesz operator or algebraic operators commuting with T.

Tumor is a pre-stage of cancer which has become a serious problem in this era. Researchers are trying to develop methods and treatments to round it. Brain tumor is an exceptional cell enhancement in brain tissue and may not always be seen in imaging tricks. Magnetic Resonance Imaging (MRI) is a technique which is applied to display the detailed image of the attacked brain location. The medical imaging trick plays a significant behavior in identification of the disease. In this paper, the brain MRI image is chosen to investigate and a method is targeted for more clear view of the location attacked by tumor. An MRI abnormal brain images as input in the introduced method, Anisotropic filtering for noise removal, SVM classifier for segmentation and morphological operations for separating the affected area from normal one are the key stages if the presented method. Attaining clear MRI images of the brain are the base of this method. The classification of the intensities of the pixels on the filtered image identifies the tumor. Experimental result showed that the SVM has obtained 83% accuracy in segmentation. Finally, the segmented region of the tumor is put on the original image for a distinct identification.

For a positive integer k, an operator \(T\in \mathscr {B}(\mathscr {H})\) is called k-quasi-\(*\)-paranormal if \(\Vert T^*T^kx\Vert ^2\le \Vert T^kx\Vert \Vert T^{k+2}x\Vert \) for all \(x\in \mathscr {H}\), which is a common generalization of \(*\)-paranormal and quasi-\(*\)-paranormal. In this paper, firstly we prove some inequalities of this class of operators; secondly we give a necessary and sufficient condition for T to be k-quasi-\(*\)-paranormal. Using these results, we prove that: (1) if \(\Vert T^*T^n\Vert =\Vert T^n\Vert \Vert T\Vert \) for some positive integer \(n\ge k,\) then a k-quasi-\(*\)-paranormal operator T is normaloid; (2) if E is the Riesz idempotent for an isolated point \(\mu _0\) of the spectrum of a k-quasi-\(*\)-paranormal operator T, then (i) if \(\mu _0\ne 0\), then \(\mathscr {R}(E)=\ker (T-\mu _0)\) (ii) if \(\mu _0=0\), then \(\mathscr {R}(E)=\ker (T^k)\).

A Hilbert space operator is M-hyponormal if there exists a positive real number M such that for all . Let be M-hyponormal and let denote either the generalized derivation or the elementary operator . We prove that if are M-hyponormal, then satisfies the generalized Weyl's theorem and satisfies the generalized a-Weyl's theorem for every f that is analytic on a neighborhood of .

In this paper we introduce the notion of $\mathcal{I}$-convergence and $\mathcal{I}$ -Cauchyness of sequences in 2-fuzzy 2-normed spaces and established some basic results related to these notions. Further, we de fine $\mathcal{I}$ -limit and $\mathcal{I}$ -cluster points of a sequence in a 2-fuzzy 2-normed linear space and investigate the relations between these concepts.

Let \(\mathcal {H}\) be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on \(\mathcal {H}\) to satisfy that f(T) obeys property (gw) for each function f analytic on some neighborhood of σ(T). Also, we investigate the stability of property (gw) under (small) compact perturbations.

An operator $T$ acting on a Banach space $\X$ obeys property $(R)$ if $\pi_a^0(T)=E^0(T),$ where $\pi_a^0(T)$ is the set of all left poles of $T$ of finite rank and $E^0(T)$ is the set of all isolated eigenvalues of $T$ of finite multiplicity. In this paper we introduce and study two new properties $(S)$ and $(gS)$ in connection with Weyl type theorems. Among other things, we prove that if $T$ is a bounded linear operator acting on a Banach space, then $T$ satisfies property $(R)$ if and only if $T$ satisfies property $(S)$ and $\pi^0(T)=\pi_a^0(T),$ where $\pi^0(T)$ is the set of poles of finite rank. Also we show that Weyl's theorem holds for $T$ if and only if $T$ satisfies property $(S)$ and $\sigma_w(T)=\sigma_b(T)$, where $\sigma_w(T)$ is the Weyl spectrum and $\sigma_b(T)$ is the the Browder spectrum. Analogous results for property $(gS)$ are given. Moreover, these properties are also studied in the frame of polaroid operator.

For a bounded linear operator $T$ acting on acomplex infinite dimensional Hilbert space $\h,$ we say that $T$is $m$-quasi-class $A(k)$ operator for $k>0$ and $m$ is apositive integer (abbreviation $T\in\QAkm$) if$T^{*m}\left((T^*|T|^{2k}T)^{\frac{1}{k+1}}-|T|^2\right)T^m\geq0.$ The famous {\it Fuglede-Putnam theorem} asserts that: the operator equation$AX=XB$ implies $A^*X=XB^*$ when $A$ and $B$ are normal operators.In this paper, we prove that if $T\in \QAkm$ and $S^*$ isan operator of class $A(k)$ for $k>0$. Then $TX=XS$, where $X\in\bh$ is an injective with dense range implies $XT^*=S^*X$.

Highly diversified tropical tree species could be a potential source of allelopathic compounds, which could lead to development of natural bio-herbicides for sustainable weed management. The present study was therefore designed to investigate into the allelopathic potential of sawdust obtained from eleven tropical tree species available in Bangladesh viz., Azadirachta indica, Swietenia macrophylla, Acacia auriculiformis, Tamarindus indica, Eucalyptus camaldulensis, Syzygium cumini, Mangifera. indica, Albizia. saman, Artocarpus. heterophyllus, Diospyros discolor and Tectona grandis. The aqueous sawdust extract of these plant species were diluted into four different concentrations viz. 1:5, 1:10, 1:15, 1:20 (w/v) and tested for their potentiality in inhibiting seedling growth of allelopathic sensitive plant Raphanus sativus under laboratory condition. A control (distilled water without extract) was also maintained in every cases and the bioassay experiment was replicated thrice. The results of this experiment showed that S. macrophylla, E. camaldulensis, M. indica and A. saman inhibited more than 90% shoot and root growth of R. sativus. The sawdust of these four plant species were selected to evaluate their potentiality in controlling paddy field weeds under filed condition. A total of 16 weed control treatments were considered in the field experiment viz., sawdust of selected four tree plant species at three application rates (1, 2 and 3 t ha-1), manual weeding (three times), chemical control (pre- + post- emergence herbicides), chemical + manual control and season long weedy (control). The results showed that the effect of different sawdust on the weed control varied significantly. Weed growth suppression by the sawdust was increased with the increase in application rate. The results revealed that manual, chemical weed control and application of E. camaldulensis saw dust @ 3 t ha-1 reduced the weed density by 79, 77 and 72%, respectively, and weed biomass by 86, 84 and 79%, respectively. On the other hand, manual weed control offered 100% rice yield increase while, chemical control and E. camaldulensis saw dust @ 3 t ha-1 both resulted in 92% rice yield increase over control. Although, manual and chemical weed control offered higher weed control efficiency and resulted in better rice productivity, but from environmental viewpoint application of E. camaldulensis sawdust @ 3 t ha-1 may be considered for sustainable weed management in rice. [Fundam Appl Agric 2019; 4(3.000): 906-915]

This study determines the impact of homestead agro-forestry on livelihood of rural households in Mymensingh district. In total 100 homestead agro-forestry practicing farmers from three upazilas namely Mymensingh Sadar, Bhaluka and Muktagachha of Mymensingh district were randomly selected for this study following a purposive sampling technique. Analysis was done considering the pre (before) and post (after) homestead agro-forestry practicing condition of farmers. The major findings of the study showed that per hectare net returns for vegetables and fruits cultivation were Tk. 6,703.62 and Tk. 14,532.61 respectively considering all farms. On an average, the contribution of vegetables and fruits in total homestead income was 20.23 per cent in before and 22.46 per cent in the after homestead agro-forestry practicing situation indicating enough potentiality to generate income from homestead agro-forestry. It was found that vegetables cultivation was much better for small farmers compared to medium and large farmers. For fruits and vegetables enterprises, the performance of large farmers was better than small and medium farmers. Large farmers were more efficient than medium and small category in case of fruit production. Most of the variables included in the Cobb-Douglas production function model had significant impact on homestead agro-forestry. The study revealed that homestead agro-forestry had positive impact on improving the status of rural households and women empowerment.DOI: http://dx.doi.org/10.3329/pa.v19i1.17367 Progress. Agric. 19(1): 169 - 178, 2008

Let $$T=U|T|$$ and $$S=V|S|$$ be the polar decompositions of $$T\in {\fancyscript{L}}({\fancyscript{H}})$$ and $$S\in {\fancyscript{L}}({\fancyscript{K}})$$ and let $$Com (T,S)$$ stand for the set of operators $$X\in \fancyscript{L}(\fancyscript{K},\fancyscript{H})$$ such that $$TX=XS.$$ A pair $$(T,S)$$ is said to have the Fuglede–Putnam property if $$Com(T,S)\subseteq Com(T^*,S^*).$$ Let $$\widetilde{Z}(s,t)$$ denote the generalized Aluthge transform of a bounded operator $$Z$$ . We show that (i) if $$T$$ is invertible class $$A(s,t)$$ operator with $$s+t=1$$ and $$S^*$$ is a class $${\mathcal {Y}}$$ , then $$Com(T,S)\subseteq Com(T^*,S^*);$$ (ii) if the pair $$(T,S)$$ have the Fuglede–Putnam property, then the range of $$\delta _{T,S}$$ is orthogonal to the kernel of $$\delta _{T,S}$$ ; (iii) if the pair $$(T,S)$$ have the Fuglede–Putnam property, then $$Com(T,S)\subseteq Com(\widetilde{T}(s,t),\widetilde{S}(s,t))$$ , furthermore, if $$T$$ is invertible, then $$Com(T,S)= Com(\widetilde{T}(s,t),\widetilde{S}(s,t))$$ . Finally, if $$Re(U|T|^s)\ge a>0$$ and $$Re(V|S|^s)\ge a>0$$ and $$X$$ is an operator such that $$U^*X=XV,$$ then we prove that $$\left\| (\widetilde{T}(s,s))^*X-X\widetilde{S}(s,s)\right\| _p\ge 2a\left\| |T|^sX-X|S|^s\right\| _p$$ for any $$1\le p\le \infty $$ such that $$0

Rural women are silent workers and major contributors to rural economy. Their economic contributions are often appreciated but not widely recognized. The purpose of the study was to make the participation of rural women in farm and non-farm activities visible and their contribution in generating household income. Data were collected from a purposively-drawn sample of 60 rural households consisting of 20 low, 20 medium and 20 high income households. The obtained data were analyzed by using tabular and statistical techniques. Women of these three categories contributed to Tk. 19398, 24180 and 20917.50 which were 35.18, 25.11 and 14.20 per cent of total household income respectively. It was evident that, female participation of low and medium income households is markedly higher in non-farm activities than the high income households. The results of the study showed clear evidences of greater extent of women’s participation in farm and non-farm activities as well as in various household decision making events. The findings of the study suggest that, in order to enhance women participation in farm and non-farm activities and their contribution to household income, the quality and status of support service like input supply, credit facilities, extension and motivation, need-based training should be improved. A successful government and non-government collaborative programme can best address these issues in an integrated manner.DOI: http://dx.doi.org/10.3329/pa.v20i1-2.16878 Progress. Agric. 20(1 & 2): 245 – 251, 2009