This paper study about the analytical behaviour of Concrete beam encased with Steel castellated beam as composite member with various web opening section of the castellated beam as optimization of section by its maximum Load carrying capacity and deflection. The modelling and Finite Element Analysis was done using Ansys Workbench 16.2. The Concrete beam having section size of 150mm x 170 mm x 1500mm encased with Structural Steel ISMB100 of span 1400mm is used for castellated beam for various shape of web opening are provided. The parametric study has shown the Deflection and Load carrying capacity of the various cross sectional beams with Hexagonal opening (inscribed in the circle 25mm radius) which has high load carrying capacity and the less deflection while compared to the other sections of circular web opening (25mm radius), hexagonal wide web opening (25mm with 1:1:1 web ratio), and rectangular web opening of (25mm x 50mm). Alternate incremental loading is applied by using Ansys workbench 16.2 and results and graphs are plotted.

This paper presents a bi-port, single-layered (planar), dual circularly polarized (CP) patch antenna with improved interport isolation for S-band satellite telemetry and telecommand applications. The dual-port, planar antenna is based on a square-shaped radiator with trimmed corner to achieve CP characteristics (RHCP and LHCP) for excitations from respective ports. However, the RF isolation between the two ports is very low due to strong power leakage from transmit (Tx) to transmit (Rx) port. The externally employed tunable self-interference cancellation (SIC) circuit achieves high isolation between orthogonal ports while axial ratio (AR) is ≤3 dB for both right and left handed circular polarization modes. The employed single-tap SIC circuit/loop attains the high interport isolation through signal inversion mechanism. The proposed antenna design achieves ≥72 dB peak interport isolation in addition to ≥30 dB and 15 dB port to port isolation over the isolation or SIC bandwidths of 15 MHz and 90 MHz (−10 dB bandwidth for both ports), respectively. The port to port isolation performance is improved without significant degradation in antenna radiation characteristics. The validation model of the presented planar antenna based on single element characterizes much better measured interport isolation performance compared to those dual CP printed antennas reported earlier as endorsed through detailed comparison

Abstract. Posner [1] has proved the following theorem: Let R be a prime ring1 and d a derivation of R such that, for all a S R, ada -daa 6 Z (centre of R). Then, if d is not a zero derivation, R is commutative.Two proofs of this theorem are known-one by Posner himself and another by Ram Awtar [2]. It is natural to expect that the proof would be short and simple in the case when R is of characteristic 2, but, by chance, these proofs are somewhat contrary to this expectation. Although, in this case, the condition ada -daa G Z is the same as d(a2) E Z, which implies d(ab + ba) G Z for all a, b G R.The object of this paper is to give a very simple, short and direct proof of this theorem in the case when R is of characteristic 2, then to generalize this technique to prove a lemma which is similar to the result that, if d(ab -ba) E Z, when R is not necessarily of characteristic 2, then either d is zero or R is commutative, and lastly to prove a generalized form of the main theorem, i.e. Replacing b by ba in (i) and using (i), we have(ii) a (ab + ba)da = (ab + ba)daa.Replacing b by da in (ii), since ada -daa E Z, and R is of characteristic 2, we have (ada -daaf= 0.Since R is prime, this implies ada -daa = 0. Consequently, we have d(ab + ba) = 0.Replacing b by ba, we have (ab + ba)da = 0.

Letbe a polynomial of degree n with real and non-negative zeros x1 ≤ x2 ≤ … xn. The zeros xj. will be said to have extent 1 ifLet ξ1 ≤ ξ2 ≤ … ξn-1 be the zeros of the derived polynomial p'(x). The zeros ξ1, ξ2, …, ξn-1 are real and non-negative, and moreover their extent can be at most equal to the extent of the zeros x1, x2, …, xn. The two can indeed be equal. For if the extent of the zeros xj. is 1 and 1 is a multiple zero of p'(x) then ξn-1 = 1. However it is not quite clear how small ξn-1 = 1 can be if Xn = 1. The extent ξn-1 = 1. of the zeros of p'(x) is less than 1 only if 1 is not a multiple zero of p(x). So let us suppose that p(x) has a simple zero at x = 1. Consequently xn-1 is the largest zero of p(x)/(x-l) or equivalently the largest zero of p(x) in 0 < x < 1 and it follows by Rolle's theorem that p'(x) n as a zero in the interval (xn-1,1). Thus the extent ξn-1 of the zeros of p'(x) is greater than xn-1 and it remains to see how small it can be.

In this paper derivations on Lie and Jordan ideals of a prime ring R are studied. The following results are proved, (i) Let R be a prime ring of characteristic not 2, and let U be a Lie or Jordan ideal of R. If d is a derivation defined on U, and if a is an element of the subring T(U), generated by U, or a is an element of R, according as U is a Lie or Jordan ideal of R, such that adu-0, for all u e U, then either a = 0 or du-0. (ii) Let a\, d2 be derivations defined for all u G U, and also for u2 and u3 if U is a Lie ideal of R, such that the iterate dxd2 is also a derivation, satisfying the same conditions as dx, d2. Let dx (u) G U, whether U is a Lie or Jordan ideal of R. Then, at least, one of dx(u) and d2(u) is zero, for all u G U. Introduction. Lemma 1 of Posner [1] states that if d is a derivation of prime ring R and a an element of R, such that ad(r) = 0, for all r G R, then either a = 0 or d is zero. Theorem 1 of Posner [1], which is a direct consequence of Lemma 1, states that if R is a prime ring of characteristic not 2 and if dx, d2 are derivations of R such that the iterate dx d2 is also a derivation, then at least one of dx, d2 is zero. The object of this paper is to generalize these results to Lie and Jordan ideals of R. All rings considered in this paper are associative. For definitions, see [2]. We prove the following results: Lemma. Let R be a prime ring of characteristic not 2 and let U be a Lie or Jordan ideal of R. If d is a derivation defined on U, and if a is an element of the subring T(U), generated by U, or a is an element of R, according as U is a Lie or