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In this article, two types of contractive conditions are introduced, namely extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction. For the case of two mappings and their coincidence point theorems, a variant of (ϰ , Ω - Ϝ) -contraction has been introduced, which is called (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction. In the end, the applications of an extended integral Ϝ -contraction and (ϰ , Ω - Ϝ) -contraction are given by providing an existence result in the solution of a fractional order multi-point boundary value problem involving the Riemann–Liouville fractional derivative. An interesting existence result for the solution of the nonlinear Fredholm integral equation of the second kind using the (ϰ , Γ 1 , 2 , Ω - Ϝ) -contraction has been proven. Herein, an example is established that explains how the Picard–Jungck sequence converges to the solution of the nonlinear integral equation. Examples are given for almost all the main results and some graphs are plotted where required. [ABSTRACT FROM AUTHOR]

In this manuscript, the concept of ϝ-contraction is applied to extend the notion of Jaggi-Suzuki-type hybrid contraction in the framework of G-metric space, which is termed Jaggi-Suzukitype hybrid ϝ-(G-α-φ)-contraction, and invariant point results which cannot be inferred from their cognate ones in metric space are established. The results obtained herein provide a new direction and are generalizations of several well-known results in fixed point theory. An illustrative, comparative example is constructed to give credence to the results obtained. Furthermore, sufficient conditions for the existence and uniqueness of solutions of certain nonlinear polynomial and integral equations are established. For the purpose of future research, an open problem is highlighted regarding discretized population balance model whose solution may be investigated from the techniques proposed herein. [ABSTRACT FROM AUTHOR]

Hybrid contractions serve as a flexible and versatile framework for establishing fixed-point Theorems and analyzing the convergence of iterative algorithms. This paper demonstrates the adapted form of the admissible hybrid fuzzy Z -contraction in the perspective of £-fuzzy set-valued maps for extended ♭-metric spaces. Sufficient criteria for obtaining £-fuzzy fixed points for this contraction have been established. In addition, the hypotheses of its main result are endorsed by some nontrivial supportive examples featuring graphical illustrations. Consequently, the concept of graphical extended ♭-metric spaces is introduced and a £-fuzzy fixed point result in the context of newly defined space is derived. Illustrative examples, incorporating relevant graphs, are provided with the support of a computer simulation to validate the established results, enhancing the understanding of the underlying notions and investigations. The concepts presented here not only considerably improve, enrich, and extend a number of well-known pre-existing fixed-point results but also assemble and merge several ones in the corresponding domain. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the advancement of fixed-point results for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces that are endowed with a graph. The famous fixed-point theorems of Goebel and Kirk, Khamsi and Khan, along with other recent results in the literature can be obtained as corollaries of these main results. A nice graph and an interesting example are also provided in support of the hypothesis of the main results. [ABSTRACT FROM AUTHOR]

In this article, a Green's function for a fractional boundary value problem in connection with modified analytic kernel has been constructed to study the existence of multiple solutions of a type of characteristic fractional boundary value problems. It is done here by using a well-known result: Krasnoselskii fixed point theorem. Moreover, a practical example is created to understand the importance of main results regarding the existence of solution of a boundary value fractional differential problem with homogeneous conditions. This example analytically and graphically, explains circumstances under which the Green's functions with different types of differential operator are compatible. [ABSTRACT FROM AUTHOR]

This paper presents a novel concept of G -Hardy-Rogers functional operators on metric spaces endowed with a graph. It investigates sufficient circumstances under which such a mapping becomes a Picard operator. As applications of the principal idea discussed herein, a few important corresponding fixed point results in ordered metric spaces and cyclic operators are pointed out and analyzed. For upcoming research papers in this field, comparative graphical illustrations are created to highlight the pre-eminence of proposed notions with respect to the existing ones. [ABSTRACT FROM AUTHOR]

Fractional Differential inclusions, the multivalued version of fractional differential equations, yellow play a vital role in various fields of applied sciences. In the present article, a class of q-rung orthopair fuzzy (q-ROF) set valued mappings along with q-ROF upper/lower semi-continuity have been introduced. Based on these ideas, existence theorems for a numerical solution of a distinct class of fractional differential inclusions have been achieved with the help of Schaefer type and Banach contraction fixed point theorems. A physical example is also provided to validate the hypothesis of the main results. The notion of q-rung orthopair fuzzy mappings along with the use of fixed point techniques and a new-fangled Caputo type fractional derivative are the principal novelty of this article. [ABSTRACT FROM AUTHOR]

The focus of this paper is to establish a new concept of b-hybrid fuzzy contraction regarding the study of fuzzy fixed-point theorems in the setting of b-metric spaces. This idea harmonizes and refines several well-known results in the direction of point-valued, multivalued, and fuzzy-set-valued maps in the comparable literature. To attract new researchers to this field, some important results are shown to be corollaries. Moreover, a result is presented to establish sufficient conditions for the existence of solutions of integral inclusion of Fredholm type. Lastly, illustrations are presented to validate the suppositions of the given theorems. [ABSTRACT FROM AUTHOR]

In this note, by using basic properties of the recently introduced concepts of generalized metric spaces, new conditions for the existence of a fixed point for weakly type contractive operator which sends a closed subset into the ambient space under consideration are examined. Our obtained result extends and unifies its corresponding ideas in metric and modular spaces. A comparative nontrivial example is provided to show the novelty and preeminence of our proposed notion. [ABSTRACT FROM AUTHOR]

A group G is said to be (l, m, n)-generated if it is a quotient of the triangle group T (l , m , n) = ⟨ x , y , z ∣ x l = y m = z n = x y z = 1 ⟩. Moori posed in 1993 the question of finding all the triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated. We partially answer this question for the Fischer sporadic simple group Fi23. In particular, we investigate all (2, q, r)-generations for the Fischer sporadic simple group Fi23, where q and r are distinct prime divisors of |Fi23|. [ABSTRACT FROM AUTHOR]

A group G is said to be (l, m, n)-generated if it is a quotient of the triangle group T (l , m , n) = ⟨ x , y , z ∣ x l = y m = z n = x y z = 1 ⟩. Moori posed in 1993 the question of finding all the triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated. We partially answer this question for the Fischer sporadic simple group Fi23. In particular, we investigate all (2, q, r)-generations for the Fischer sporadic simple group Fi23, where q and r are distinct prime divisors of |Fi23|. [ABSTRACT FROM AUTHOR]

A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer's largest sporadic simple group Fi′24 by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi′24| with 3 < q < r. [ABSTRACT FROM AUTHOR]

A group G is said to be (l,m,n)-generated if it can be generated by two suitable elements x and y such that o(x) = l, o(y) = m and o(xy) = n. In [J. Moori, (p,q,r)-generations for the Janko groups J1 and J2, Nova J. Algebra Geom. 2 1993, 3, 277–285], J. Moori posed the problem of finding all triples of distinct primes (p,q,r) for which a finite non-abelian simple group is (p,q,r)-generated. In the present article, we partially answer this question for Fischer's largest sporadic simple group Fi′24 by determining all (3,q,r)-generations, where q and r are prime divisors of |Fi′24| with 3 < q < r. [ABSTRACT FROM AUTHOR]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G: X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks for the Fischer's simple group Fi'24 and the baby monster group B. [ABSTRACT FROM AUTHOR]

A study was conducted under the laboratory conditions in the Department of Zoology, Wildlife and Fisheries, University of Agriculture, Faisalabad, Pakistan during 2014 to investigate the effect of supplementation of lysine on nutrient digestibility in the juveniles of Labeo rohita. Seven experimental diets were formulated by supplementing 0, 0.5, 1, 1.5, 2, 2.5 and 3% lysine and designated as D1, D2, D3, D4, D5, D6 and D7, respectively. To test each diet, two replicates with 20 fish each (average weight 3.526±0.056g) were assigned. The experiment was performed within the time duration of 60 days. Water quality parameters like dissolved oxygen (5.8-7.3 mg/L), temperature (24.9-28.7°C) and pH (7.4-8.6) were maintained throughout this period. Results showed that lysine supplementation in fish diet increased the digestibility of crude protein and dry matter. Maximum digestibility of crude protein and dry matter was observed at lysine level of 1.5 and 2.5%, respectively. However, it was observed that digestibility of crude fat decreased in response to lysine supplementation. In conclusion, it was found that lysine supplementation improved the nutrient digestibility performance of juveniles of L. rohita. [ABSTRACT FROM AUTHOR]

IfGis a finite group andXa conjugacy class of elements ofG, then we define rank(G:X) to be the minimum number of elements ofXgeneratingG. Moori investigated rank(Fi22:X), whereXis a conjugacy class of involutions in the Fischer groupFi22. In the present article, we completely determine the ranks of the Fischer groupFi22by computing rank(Fi22:X), whereXis any conjugacy class ofFi22. [ABSTRACT FROM AUTHOR]

A group G is said to be (2,3,t)-generated if it can be generated by two elements x and y such that o(x) = 2, o(y) = 3 and o(xy) = t. In the present article, we investigate (2,3,t)-generations for the Held's sporadic simple group He, where t is a divisor of ∣He∣. Most of the computations were carried out with the aid of computer algebra system GAP [23]. [ABSTRACT FROM AUTHOR]

The Fischer group $Fi_{24}={\rm Aut} (Fi_{24}^{\prime})$ is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA. [ABSTRACT FROM AUTHOR]

The Fischer group $Fi_{24}={\rm Aut} (Fi_{24}^{\prime})$ is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA. [ABSTRACT FROM AUTHOR]

The Harada-Norton group HN is a sporadic simple group of order 273030912000000 equal to 214 x 36 x 56 x 7 x 11 x 19. It has 14 conjugacy classes of maximal subgroups and in the present paper we construct the Fischer-Clifford matrices and hence character table for one of these maximal subgroups, that is the non-split extension 26·U4(2), of index 164587500. There is not that many known examples of the application of Fischer-Clifford theory to non-split extensions, and hence this maximal subgroup of Hard-Norton's group serves as a good candidate. Most of the computations were carried out using the computer algebra systems GAℙ [10] and MAGMA [4]. [ABSTRACT FROM AUTHOR]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji, where i = 1; 2; 3; 4. [ABSTRACT FROM AUTHOR]

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 픸핋핃픸핊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He. In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 픾픸ℙ. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group and X a conjugacy class of G. We denote to be the minimum number of elements of X generating G. In the present article, we determine the ranks for two sporadic simple groups, namely the O’Nan sporadic simple group O’N and the Lyons sporadic simple group, Ly. Computations were carried with the aid of the computer algebra system [The GAP Group, —Groups, Algorithms and Programming, Version 4.3, Aachen, St Andrews, 2003, http://www.gap-system.org ]. [Copyright &y& Elsevier]

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co2 and Co3. Computations were carried with the aid of computer algebra system 픾픸ℙ [18]. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co2 and Co3. Computations were carried with the aid of computer algebra system 픾픸ℙ [18]. [ABSTRACT FROM AUTHOR]

Abstract: Exploration and cultivation of salt tolerant plants is a very effective strategy for utilization of salt affected soils. In this investigation, physiological traits that are conducive for salt tolerance of the ornamental plant Alternanthera bettzickiana, Amaranthaceae, were explored. A. bettzickiana was grown on soil substrate having six salinity levels (2.86, 10, 20, 30, 40 and 50dSm−1). It was observed that this plant can grow even at a salinity level of 40dSm−1. The survival rate of this plant was 75, 42 and 0% at salinity levels of 30, 40 and 50dSm−1, respectively. A. bettzickiana plants produced 30.3% less biomass than controls at the salinity level of 20dSm−1 and even less under still higher salt stress. Photosynthesis continued even at the salinity level of 40dSm−1, though its rate was reduced to 59% in plants exposed to such salinity relative to plants not affected by salinity. Total soluble proteins values in leaf and stem showed a gradual increase when plants were exposed to increasing salt stress. Plants growing at the high salinity level showed highest decrease in leaf nitrate reductase activity. A. bettzickiana plants accumulated less Na+ in shoot as compared to root when grown under salt stress. It can be characterized as a salt-tolerant glycophyte that could be used for greening of salt affected soils. [Copyright &y& Elsevier]