an algorithmic method for showing existence of nontrivial non classical symmetries of partial differential equations without solving determining equations abstract in this paper based on differential characteristic set theory and the associated algorithm also called wu s method an algorithmic method is presented to decide on the existence of a nontrivial non classical symmetry of a given partial differential equation without solving the corresponding nonlinear determining system the theory and algorithm give a partial answer for the open problem posed by p a clarkson and e l mansfield in 21 on non classical symmetries of partial differential equations as applications of our algorithm non classical symmetries and corresponding invariant solutions are found for several evolution equations

classification of reduction operators and exact solutions of variable coefficient newell whitehead segel equations abstract a class of the newell whitehead segel equations also known as generalized fisher equations and newell whitehead equations is studied with lie and nonclassical symmetry points of view the classifications of lie reduction operators and of regular nonclassical reduction operators are performed the set of admissible transformations the equivalence groupoid of the class is described exhaustively the criterion of reducibility of variable coefficient newell whitehead segel equations to their constant coefficient counterparts is derived wide families of exact solutions for such variable coefficient equations are constructed

analytical and numerical studying of the perturbed korteweg de vries equation the perturbed korteweg de vries equation is considered this equation is used for the description of one dimensional viscous gas dynamics nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves the integrability of this equation is investigated using the painleve approach the condition on parameters for the integrability of the perturbed korteweg de vries equation equation is established new classical and nonclassical symmetries admitted by this equation are found all corresponding symmetry reductions are obtained new exact solutions of these reductions are constructed they are expressed via trigonometric and airy functions stability of the exact solutions of the perturbed korteweg de vries equation is investigated numerically

lie symmetries of a coupled nonlinear burgers heat equation system a symmetry group analysis of a coupled burgers heat equation system of partial differential equations initially introduced in a study of non classical similarity solutions of the heat equation is carried out the point lie algebra of the system g is shown to possess a maximal solvable ideal a with quotient algebra b g a approximately sl2 r where sl2 r is the lie algebra of 2 2 matrices with zero trace analysis of the prolongation structure of the system yields the backlund transformation obtained previously by painleve analysis the backlund transformation can be expressed as a map onto two coupled linear heat equations or alternatively as a map onto burgers equation the role of sl2 r in a seven dimensional lie algebra l1 obtained by truncating the open ended algebraic prolongation structure is emphasised

generalized conditional symmetries of evolution equations we analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited this leads to a no go theorem on determining equations for operators of generalized conditional symmetry it is also shown that up to certain equivalences there exists a one to one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions

mitigation of loopback self interference in full duplex mimo relays full duplex relaying is more spectrally efficient than half duplex relaying as only one channel use is needed per two hops however it is crucial to minimize relay self interference to render full duplex feasible for this purpose we analyze a broad range of multiple input multiple output mimo mitigation schemes natural isolation time domain cancellation and spatial suppression cancellation subtracts replicated interference signal from the relay input while suppression reserves spatial dimensions for receive and transmit filtering spatial suppression can be achieved by antenna subset selection null space projection i e receiving and transmitting in orthogonal subspaces or joint transmit and receive beam selection to support more spatial streams by choosing the minimum eigenmodes for overlapping subspaces in addition minimum mean square error mmse filtering can be employed to maintain the desired signal quality which is inherent for cancellation and the combination of time and spatial domain processing may be better than either alone targeting at minimal interference power we solve optimal filters for each scheme in the cases of joint separate and independent design the performance of mitigation schemes is evaluated and compared by simulations the results confirm that self interference can be mitigated effectively also in the presence of imperfect side information

computing the p selmer group of an elliptic curve in this paper we explain how to bound the p selmer group of an elliptic curve over a number field k our method is an algorithm which is relatively simple to implement although it requires data such as units and class groups from number fields of degree at most p 2 1 our method is practical for p 3 but for larger values of p it becomes impractical with current computing power in the examples we have calculated our method produces exactly the p selmer group of the curve and so one can use the method to find the mordell weil rank of the curve when the usual method of 2 descent fails

explicit n descent on elliptic curves iii algorithms this is the third in a series of papers in which we study the n selmer group of an elliptic curve with the aim of representing its elements as genus one normal curves of degree n the methods we describe are practical in the case n 3 for elliptic curves over the rationals and have been implemented in magma one important ingredient of our work is an algorithm for trivialising central simple algebras this is of independent interest for example it could be used for parametrising brauer severi surfaces

conditional symmetry of a porous medium equation abstract we obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension it is proved that solving the determining equations for the conditional symmetries either yields classical lie symmetries or is equivalent to solving the porous medium equation itself the second possibility is shown to be typical for an arbitrary evolution type partial differential equation in one spatial variable we show that this very property is responsible for the well known phenomenon when using non classical symmetries yield exact solutions which prove to be invariant ones

reduction operators of burgers equation abstract the solution of the problem on reduction operators and nonclassical reductions of the burgers equation is systematically treated and completed a new proof of the theorem on the special no go case of regular reduction operators is presented and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case all possible nonclassical reductions of the burgers equation to single ordinary differential equations are exhaustively described any lie reduction of the burgers equation proves to be equivalent via the hopf cole transformation to a parameterized family of lie reductions of the linear heat equation

symmetry reductions and exact solutions of a class of nonlinear heat equations abstract classical and nonclassical symmetries of the nonlinear heat equation ut uxx f u are considered the method of differential grobner bases is used both to find the conditions on f u under which symmetries other than the trivial spatial and temporal translational symmetries exist and to solve the determining equations for the infinitesimals a catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of the nonlinear heat equation for cubic f u in terms of the roots of f u 0

group properties of generalized quasi linear wave equations abstract in this paper complete group classification of a class of 1 1 dimensional generalized quasi linear wave equations is performed by using the lie ovsiannikov method additional equivalent transformation and furcate split method lie reductions of some truly variable coefficient wave equations which are singled out from the classification results are investigated some classes of exact solutions of these variable coefficient wave equations are constructed by means of both the reductions and the additional equivalent transformations the nonclassical symmetries to the generalized quasi linear wave equation are also studied this enabled to obtain some exact solutions of the wave equations which are invariant under certain conditional symmetries

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

reduction operators of linear second order parabolic equations the reduction operators i e the operators of nonclassical conditional symmetry of 1 1 dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described this problem proves to be equivalent in some sense to solving initial equations the no go result is extended to the investigation of point transformations admissible transformations equivalence transformations lie symmetries and lie reductions of the determining equations for the nonclassical symmetries transformations linearizing the determining equations are obtained in the general case and under different additional constraints a nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented an observed connection between reduction operators and darboux transformations is discussed

a unified approach for calculating error rates of linearly modulated signals over generalized fading channels we present a unified analytical framework to determine the exact average symbol error rate ser of linearly modulated signals over generalized fading channels the results are applicable to systems employing coherent demodulation with maximal ratio combining multichannel reception the analyses assume independent fading paths which are not necessarily identically distributed in all cases the proposed approach leads to an expression of the average ser involving a single finite range integral which can be easily computed numerically in addition as special cases ser expressions for single channel reception are obtained these expressions reduce to well known solutions give alternative often simpler expressions for previous results or provide new formulas that are either closed form expressions or simple to compute numerically

on a reduction and solutions of nonlinear wave equations with broken symmetry a generalised definition for invariance of partial differential equations is proposed exact solutions of the equations with broken symmetry are obtained

singular reduction operators in two dimensions the notion of singular reduction operators i e of singular operators of nonclassical conditional symmetry of partial differential equations in two independent variables is introduced all possible reductions of these equations to first order odes are exhaustively described as examples properties of singular reduction operators of 1 1 dimensional evolution and wave equations are studied it is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in depth prior study of the corresponding singular vector fields

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

algorithms for the nonclassical method of symmetry reductions in this article the authors first present an algorithm for calculating the determining equations associated with so called nonclassical method of symmetry reductions a la bluman and cole for systems of partial differential equations this algorithm requires significantly less computation time than that standardly used and avoids many of the difficulties commonly encountered the proof of correctness of the algorithm is a simple application of the theory of grobner bases in the second part they demonstrate some algorithms which may be used to analyse and often to solve the resulting systems of overdetermined nonlinear pdes the authors take as their principal example a generalised boussinesq equation which arises in shallow water theory although the equation appears to be nonintegrable the authors obtain an exact two soliton solution from a nonclassical reduction

extended symmetry analysis of generalized burgers equations using enhanced classification techniques we carry out the extended symmetry analysis of the class of generalized burgers equations of the form ut uux f t x uxx 0 this enhances all the previous results on symmetries of these equations and includes the description of admissible transformations lie symmetries lie and nonclassical reductions hidden symmetries conservation laws potential admissible transformations and potential symmetries the study is based on the fact that the class is normalized and its equivalence group is finite dimensional

how to do a p descent on an elliptic curve in this paper we describe an algorithm that reduces the computation of the full p selmer group of an elliptic curve e over a number field to standard number field computations such as determining the p torsion of the s class group and a basis of the s units modulo pth powers for a suitable set s of primes in particular we give a result reducing this set s of bad primes to a very small set which in many cases only contains the primes above p as of today this provides a feasible algorithm for performing a full 3 descent on an elliptic curve over q but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory when the galois module structure of e p is favorable simplifications are possible and p descents for larger p are accessible even today to demonstrate how the method works several worked examples are included

on the performance of full duplex two way relay channels with spatial modulation in this paper the spatial modulation sm technique is employed at the source and relay nodes in a full duplex two way relay channel fd twrc to support spectral efficient bi directional communications while guaranteeing a low cost implementation maximum likelihood detectors are employed at each node that is subject to an intrinsic self loop interference we first propose a tight upper bound on the average bit error probability abep then based on the abep upper bound an asymptotic abep expression is derived in the high signal to noise ratio snr regime exploiting the asymptotic abep an exact snr threshold for the selection between fd twrc sm and half duplex hd twrc sm is derived in a closed form which sheds light on when it is beneficial to select the fd or hd mode in addition the power allocation pa among sources and relay is investigated through which an optimal pa factor in terms of abep is obtained all analytical results derived in this paper are verified by monte carlo simulations from which some new insights are obtained on the performance of fd twrc sm

the symmetry and some exact solutions of the nonlinear many dimensional liouville d alembert and eikonal equations multiparametrical exact solutions of the many dimensional nonlinear d alembert liouville sine gordon and eikonal equations are obtained the maximally extensive local invariance groups of the equations are determined and invariants of the extended poincare group are found

performance and distance spectrum of space time codes in fast rayleigh fading channels in this paper we analyze the performance of spacetime codes and propose a distance spectrum computation method in fast rayleigh fading channels we first derive a new fer upper bound using the union bound and the pep upper bound in the fast fading environment the derived fer upper bound is very accurate requires only the distance spectrum of the spacetime code and takes a closed form expression then we propose a distance spectrum computation method of space time codes in fast fading channels which exploits the symmetric property of the error state diagram in space time trellis coded mpsk modulation to reduce the computation complexity numerical results illustrate that the derived fer bound is tight enough to estimate the performance of space time codes in fast fading channels with sufficient accuracy

singular reduction modules of differential equations the notion of singular reduction modules i e of singular modules of nonclassical conditional symmetry of differential equations is introduced it is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in depth prior study of the associated singular modules of vector fields the form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations as examples singular reduction modules of evolution equations and second order quasi linear equations are studied reductions of differential equations to algebraic equations and to first order ordinary differential equations are considered in detail within the framework proposed and are related to previous no go results on nonclassical symmetries

the nonclassical group analysis of the heat equation abstract the nonclassical method of reduction was devised originally by bluman and cole in 1969 to find new exact solutions of the heat equation much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance however the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved although particular solutions have been given recently arrigo goard and broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical lie symmetry reduction solutions their arguments leave open the problem of what is the general nonclassical group action and its effect on the relevant solution of the heat equation in this article both these problems are solved in the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of burgers equation