Function Space

In present paper, I intend to introduce our study of new normed function space type of Lorentz-Morrey ℒ<l> (s,G) associated parameters of many groups of variables started in works by A.Dj. Djabrailov. I must note that, this space belongs to spaces type of Lebesgue-Morrey type. As an application, we give some properties for these spaces again. In addition, I have given two needing lemmas and they have been proved. In view of the embedding theorems we study some properties of the functions, which are belonging to these spaces. Although I have dealt with a lot of measurable cases, differentiable function spaces are very difficult in general. The most important cases are Lebesgue-Morrey type spaces with many groups of variables. I begin with the general theory of Mathematical Analysis, I have constructed new normed spaces type of Lorentz-Morrey, gave and proved some characterization of these type of spaces. In addition, specific techniques for introducing some embedding theorems will be given late.

Je tiens à remercier Aziz El Kacimi pour tout ce que j'ai appris à ses côtés, à l'écouter et à le lire. Je le remercie pour m'avoir proposé ce sujet, pour tous les conseils qu'il a pu me donner, pour s'être intéressé à ce que je faisais, pour m'avoir souvent motivé, pour avoir réussi à me faire revenir sur une position (rien que cela est un exploit, j'espère que tu t'en rends bien compte !) et pour avoir toujours été disponible dès que j'ai eu besoin de lui parler. Je suis certain que les nombreuses connaissances qu'il m'a transmises me seront très utiles dans mes futures recherches. Abouquateb et Jean-Jacque Loeb se soient intéressés à mon travail et aient accepté de rapporter dessus. Qu'ils trouvent ici l'expression de toute ma gratitude. Un grand merci également à François Lescure et Luc Vrancken pour avoir bien voulu faire partie du jury en tant qu'examinateurs et à Raymond Barre pour avoir gentiment accepté d'en assurer la présidence. Je remercie le gouvernement iranien et le gouvernement français, qui m'ont permis financièrement de mener à bien la préparation de cette thèse. Bien sûr, je n'oublie pas de remercier Madame France Lamiscare qui a préparé et géré administrativement mon séjour en France. Je remercie chaleureusement les membres du Laboratoire de Mathématiques de Valenciennes et plus particulièment Cédric Rousseau et Abdellatif Zeggar avec qui j'ai eu beaucoup de discussions mathématiques ou autres. J'associe à ces remerciements l'ensemble de mes collègues de l'Université de Yazd en Iran. Merci encore à tous mes amis d'Iran, de France et d'ailleurs, entre autres Norbert Cazzadori, pour leur soutien et leurs encouragements constants. Durant mon itinérance, entre les universités de Yazd et Valenciennes, j'ai croisé beaucoup de personnes et je ne saurai tous les inclure ici. Elles m'ont toujours très bien accueilli ; je les en remercie. Enfin je remercie tous les membres de ma famille et de ma belle famille, qui m'ont toujours accompagné pendant ces années de préparation de ma thèse.

Generalizations of earlier negative results on Property (a) are proved and two questions on an (a)-version of Jones' Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions "2 ω is regular" and "2 ω < 2 ω 1 " the existence of a T 1 separable locally compact (a)-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants such as d to prove results in the class of locally compact spaces that strengthen, in such class, the negative results mentioned above.

Generalizations of earlier negative results on Property (a) are proved and two questions on an (a)-version of Jones' Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions "2 ω is regular" and "2 ω < 2 ω 1 " the existence of a T 1 separable locally compact (a)-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants such as d to prove results in the class of locally compact spaces that strengthen, in such class, the negative results mentioned above.

The present paper deals with the representation of the generalized K-function, which is an extension of the multi-index Mittag-Leffler function defined by Kiryakova [9], the topic has been introduced and studied by the author in terms of some special functions. it investigates the relations that exists between the generalized K-function and the operators of Riemann-Liouville fractional integrals and derivatives.

Assume that K is a complete non-Archimedean valued field. We prove that every infinite-dimensional Fréchet-Montel space over K which is not isomorphic to K N has a nuclear Köthe quotient. If the field K is non-spherically complete, we show that every infinite-dimensional Fréchet space of countable type over K which is not isomorphic to the strong dual of a strict LB-space has a nuclear Köthe quotient.

Let K be a non-archimedean field and let X be an ultraregular space. We study the non-archimedean locally convex space C p (X, K) of all K-valued continuous functions on X endowed with the pointwise topology. We show that K is spherically complete if and only if every polar metrizable locally convex space E over K is weakly angelic. This extends a result of Kiyosawa-Schikhof for polar Banach spaces. For any compact ultraregular space X we prove that C p (X, K) is Fréchet-Urysohn if and only if X is scattered (a non-archimedean variant of Gerlits-Pytkeev's result). If K is locally compact we show the following: (1) For any ultraregular space X the space C p (X, K) is K-analytic if and only if it has a compact resolution (a non-archimedean variant of Tkachuk's theorem); (2) For any ultrametrizable space X the space C p (X, K) is analytic if and only if X is σ-compact (a non-archimedean variant of Christensen's theorem).

Within the framework of the development of the theory of hidden oscillations, the problem of determining the boundary of global stability and revealing its hidden parts corresponding to the nonlocal birth of hidden oscillations is considered. For a phase-locked loop with a proportional-integrating filter and a piecewise-linear phase detector characteristic, effective methods for determining bifurcations of global stability loss, for obtaining analytical formulas for bifurcation values, and for constructing trivial and hidden parts of the global stability boundary are suggested.

Let y , i s 1, . . . , n, be independent observations with the density of Ž . ing the estimate, 4 numerical algorithms for the calculations and, finally, Ž . 5 public software. In this paper we carry out this program, relying on earlier work and filling in important gaps. The overall scheme is applied

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau' theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau's theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau's theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau's theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau's community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau's theory: a new branch of functional analysis which naturally generalizes the Schwartz * Work supported by START-project Y237 of the Austrian Science Fund MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05). is the non-standard extension of Ω. 8.2 Proposition. * E(Ω) is a differential algebra over the field * C. 8.3 Definition (Sup and Support). Let f i ∈ * E(Ω) and let K ⊂⊂ Ω. Then

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau' theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau's theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau's theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau's theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau's community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau's theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differential equations, differential geometry, relativity theory and other areas of mathematics and physics. MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05).

In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its applications. The main purpose of our non-standard approach to Colombeau' theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau's theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau's theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau's theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau's community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau's theory: a new branch of functional analysis which naturally generalizes the Schwartz * Work supported by START-project Y237 of the Austrian Science Fund MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05). is the non-standard extension of Ω. 8.2 Proposition. * E(Ω) is a differential algebra over the field * C. 8.3 Definition (Sup and Support). Let f i ∈ * E(Ω) and let K ⊂⊂ Ω. Then

In this work, the general form of all normal quasi-differential operators for first order in the weighted Hilbert spaces of vector-functions on right semi-axis in term of boundary conditions has been found. Later on, spectrum set of these operators will be investigated.

Let T be a completely regular topological space and C(T ) be the space of bounded, continuous real-valued functions on T . C(T ) is endowed with the strict topology (the topology generated by seminorms determined by continuous functions vanishing at infinity). R. Giles ([13], p. 472, Theorem 4.6) proved in 1971 that the dual of C(T ) can be identified with the space of regular Borel measures on T . We prove this result for positive, additive setvalued maps with values in the space of convex weakly compact non-empty subsets of a Banach space and we deduce from this result the theorem of R. Giles ([13], theorem 4.6, p.473).

We analytically study shock wave in the Josephson transmission line (JTL) in the presence of ohmic dissipation. When ohmic resistors shunt the Josephson junctions (JJ) or are introduced in series with the ground capacitors the shock is broadened. When ohmic resistors are in series with the JJ, the shock remains sharp, same as it was in the absence of dissipation. In all the cases considered, ohmic resistors don't influence the shock propagation velocity. We study an alternative to the shock wave -an expansion fan -in the framework of the simple wave approximation for the dissipationless JTL and formulate the generalization of the approximation for the JTL with ohmic dissipation. The concept that in a nonlinear wave propagation system the various parts of the wave travel with different velocities, and that wave fronts (or tails) can sharpen into shock waves, is deeply imbedded in the classical theory of fluid dynamics 1 . The methods developed in that field can be profitably used to study signal propagation in nonlinear transmission lines 2-18 . In the early studies of shock waves in transmission lines, the origin of the nonlinearity was due to nonlinear capacitance in the circuit 19-22 . Interesting and potentially important examples of nonlinear transmission lines are circuits containing Josephson junctions (JJs) 23 -Josephson transmission lines (JTLs) 24-27 . The unique nonlinear properties of JTLs allow to construct soliton propagators, microwave oscillators, mixers, detectors, parametric amplifiers, analog amplifiers 25-27 . Transmission lines formed by in series connected JJs were studied beginning from 1990s, though much less than transmission lines formed by JJs connected in parallel 28 . However, the former began to attract quite a lot of attention recently 29-36 , especially in connection with possible JTL traveling wave parametric amplification 37-46 . Although numerical calculations have revealed the existence of shock waves in series connected JTL 47-49 , the analytical solutions have not been successfully derived so far. In the present paper we study the propagation of shock waves in the JTLs analytically, paying especial attention to the influence of ohmic resistance which inevitably exists in the system. The rest of the paper is constructed as follows. In Section II we study the JTL equations in the absence of any ohmic dissipation. Shocks in the JTL with ohmic resistors are studied in Section III. In Section IV we generalize the so called simple wave approximation 22,50 , which, for a dissipationless JTL, reduces the wave equation to a pair of decoupled equations for right and left propagating waves, to the case of JTL with ohmic dissipation. We then study the shock formation in the framework of this approximation. Application of the results obtained in the paper and opportunities for their generalization are very briefly discussed in Section V. We conclude in Section VI. In Appendix A we rederive the JTL equations in the framework of Lagrange and Hamilton approaches. In Appendix B we write down equation for the travelling wave in the discrete JTL. In Appendix C we consider weak shocks in the JTL with large shunting capacitance. In Appendix D we study the symmetry of the modified Burgers equation proposed in Section IV . In Appendix E we study discontinuities formation in the JTL with ohmic resistors in series with the JJ.

If g is a map from a space X into R m and q is an integer, let B q,d,m (g) be the set of all planes This results complements an authors' result from . A parametric version of the above theorem, as well as a partial answer of a question from [4] and are also provided.

Characterizations of paracompact finite C-spaces via continuous selections avoiding Z σ -sets are given. We apply these results to obtain some properties of finite C-spaces. Factorization theorems and a completion theorem for finite C-spaces are also proved.

Characterizations of paracompact finite C-spaces via continuous selections avoiding Z σ -sets are given. We apply these results to obtain some properties of finite C-spaces. Factorization theorems and a completion theorem for finite C-spaces are also proved.

Two stage stochastic programs with random right hand side are considered Optimal values and solution sets are regarded as mappings of the expected re course functions and their perturbations respectively Conditions are identi ed implying that these mappings are directionally di erentiable and semidi eren tiable on appropriate functional spaces Explicit formulas for the derivatives are derived Special attention is paid to the role of a Lipschitz condition for solution sets as well as of a quadratic growth condition of the objective function

In this work, we study the possibility of inserting an increasing continuous lattice-valued function between two comparable semicontinuous functions on a preordered topological space. Depending on the monotonicity conditions imposed on the semicontinuous functions, new characterizations of di! erent classes of preordered topological spaces are obtained. Among them are char- acterizations of normally preordered and extremally preorder-disconnected spaces. Conditions for

We generalized the concepts in probability of Wijsman rough lacunary statistical by introducing the interval numbers of Weierstrass of fractional order, where α is a proper fraction and γ = (γmnk) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving Wijsman rough lacunary sequence θ of interval numbers and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related six dimensional triple geometric difference sequence spaces of interval numbers. In this study, we consider a generalization for Weierstrass rough six dimensional triple geometric difference sequence of these metric spaces by taking a ψ function, satisfying the following conditions. Let ψm,n,k be a positive function for all m,n, k ∈ N such that (i) limm,n,k→∞ ψmnk = 0, (ii) ∆ψmnk = ψmnk − ψm,n+1,k − ψm,n,k+1 + ψm,n+1,k+1 − ψm+1,n,k + ψm+1,n+1,k + ψm+1,n,k+1 − ψm+1,n+1,k+1 ≥ 0. or ψmnk = 1. Therefore, accordin...

Iranian Journal of Science & Technology, Transaction A, Vol. 32, No. A4 Printed in the Islamic Republic of Iran, 2008 © Shiraz University ... ON SOME CLASSES OF GENERALIZED DIFFERENCE PARANORMED SEQUENCE SPACES ASSOCIATED WITH MULTIPLIER SEQUENCES*

In this paper, we provide a theoretical analysis of effects of applying different forecast diversification methods on the structure of the forecast error covariance matrices and decomposed forecast error components based on the bias-variance-Bayes error decomposition of James and Hastie. We express the "diversity" of different forecasts in relation to different error components and propose a measure in order to quantify it. We illustrate and discuss typical inhomogeneities frequently occurring in the forecast error covariance matrices and show that previously proposed pooling based only on error variances cannot fully exploit the complementary information present in a set of diverse forecasts to be combined. If covariance values could be reliably calculated, they could be taken into account during the pooling process. We study the difficult case in which covariance information cannot be measured properly and propose a novel simplified representation of the covariance matrix, which is only based on knowledge about the forecast generation process. Finally, we propose a new pooling approach that avoids inhomogeneities in the forecast error covariance matrix by considering the information contained in the simplified covariance representation and compare it with the error-variance-based pooling approach introduced by Aiolfi and Timmermann. Applying our approach more than once leads to the generation of multistep and multilevel forecast combination structures, which have generated significantly improved forecasts in our previous extensive experimental work; the summary of which is also provided.

This paper defines new covering properties in tri-topological spaces called tri-Lindelöf space and the properties of this topological property and its relationship with some other types of tri-topological spaces will be studied. The effect of some types of functions on tri-Lindelöf spaces will be studied. This paper also investigates the necessary conditions through which the tri-topological space is reduced into a single topological space. Many and varied illustrative examples will be discussed and many well-known facts and theorems are generalized concerning Lindelöf spaces.

In this research work, we introduce the concept of countably compact spaces in an ideal topological space and study further properties of continuous functions. We prove that continuous function mapping a countably compact ideal topological space to the ideal space is an α-closed subset of the Cartesian product . We showed that the countably α-compact ideal space with weight is the continuous image of a closed subspace of the cube and illustrate that the continuous function where is countably compact can be extended over its domain under some constraints. Moreover, pseudocompact is defined in an ideal topological space and we proved that countably pseudocompactness is not hereditary with respect to closed sets and we showed that countab pseudocompactness is not finitely multiplicative. We find that if the ideal topological space is Tychonoff, is countably pseudocompact, and is continuous perfect function modulo , is countably psuedocompact.

In this research work, we introduce the concept of countably α-compact spaces in an ideal topological space (X,τ,I) and study further properties of α-continuous functions. We prove that α-continuous function mapping a countably α-compact ideal topological space (X,τ,I) to the ideal space (Y,σ,J) is an α-closed subset of the Cartesian product (X×Y,τ×σ,I×J). We showed that the countably α-compact ideal space (X,τ,I) with weight ∣X∣≥ ℵ₀ is the α-continuous image of a closed subspace of the cube D^({ℵ₀}) and illustrate that the α-continuous function f:(X,τ,I)→(Y,σ,J) where Y is countably α-compact can be extended over its domain under some constraints. Moreover, α-pseudocompact is defined in an ideal topological space (X,τ,I) and we proved that countably α-pseudocompactness is not hereditary with respect to α-closed sets and we showed that countab α-pseudocompactness is not finitely multiplicative. We find that if the ideal topological space (X,τ,I) is Tychonoff, (Y,σ,J) is countably α-...

Q-learning is a popular reinforcement learning algorithm. This algorithm has however been studied and analysed mainly in the infinite horizon setting. There are several important applications which can be modeled in the framework of finite horizon Markov decision processes. We develop a version of Q-learning algorithm for finite horizon Markov decision processes (MDP) and provide a full proof of its stability and convergence. Our analysis of stability and convergence of finite horizon Q-learning is based entirely on the ordinary differential equations (O.D.E) method. We also demonstrate the performance of our algorithm on a setting of random MDP along with an application on smart Grids.

introduced the idea of controlled metric type spaces, which is a new extension of b-metric spaces with addition of a controlled function α(x, y) of the righthand side of the b-triangle inequality. Phu [17] introduced the idea of rough convergence of sequences in a normed linear space. In this paper we have brought the idea of rough convergence of sequences in a controlled metric type space. We have proved several results associated with rough limit sets and some relevant results associated with such convergence.

We define a family of Khovanov-Lipshitz-Sarkar stable homotopy types for the homotopical Khovanov homology of links in thickened surfaces indexed by moduli space systems. This family includes the Khovanov-Lipshitz-Sarkar stable homotopy type for the homotopical Khovanov homology of links in higher genus surfaces (see the content of the paper for the definition). The question whether different choices of moduli spaces lead to the same stable homotopy type is open.

We study time-frequency localization operators of the form A ϕ 1 ,ϕ 2 a , where a is the symbol of the operator and ϕ 1 , ϕ 2 are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order p, is that a belongs to the modulation space M p,∞ (R 2d ) and the window functions to the modulation space M 1 . Here we prove a partial converse: if for every pair of window functions ϕ 1 , ϕ 2 ∈ S(R 2d ) with a uniform norm estimate, then the corresponding symbol a must belong to the modulation space M p,∞ (R 2d ). In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ and p = 2, we recapture earlier results, which were obtained by different methods.

This paper presents some important classes of the continuous functions defined from the set of real numbers to the space of complex intervals. These function spaces have an algebraic structure named as a quasilinear space which is suggested by Aseev in 1986. In this work, we analysis the quasilinear structure on the classes of the continuous and complex interval-valued functions. Further, we show that these spaces are the normed Ω-spaces. Finally, we examine the dimension of these function spaces.

In this paper, we give definition of quasiring as a new concept. Also, we introduce the notions of quasimodule and normed quasimodule defined on a quasiring. We should immediately note that a quasimodule is a generalization of quasilinear spaces. Similarly, normed quasimodule is a generalization of normed quasilinear spaces defined by Aseev, . Moreover, we obtain some results about the relationships between these concepts. We think that investigations on quasimodules may provide some important contributions to improvement of some branches of quasilinear functional analysis such as the duality theory of quasilinear spaces. Also we recognize that the notion of quasimodule is more suitable backdrop as regards theory of quasilinear spaces in examination of quasilinear functional.

The aim of this paper is to deal with BiHom-alternative algebras which are a generalization of alternative and Hom-alternative algebras, their structure is defined with two commuting multiplicative linear maps. We study cohomology and one-parameter formal deformation theory of left BiHom-alternative algebras. Moreover, we study central and $T_\theta$-extensions of BiHom-alternative algebras and their relationship with cohomology. Finally, we investigate generalized derivations and give some relevant results.

In the paper two-weighted norm estimates with general weights for Hardy-type transforms, maximal functions, potentials and Calderón-Zygmund singular integrals in variable exponent Lebesgue spaces defined on quasimetric measure spaces (X, d, µ) are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for appropriate inequalities. Examples of weights governing the boundedness of maximal, potential and singular operators in weighted variable exponent Lebesgue spaces are given.

The objective of this research was to determine the possible relation between deficits in spatial representation capability and vestibular function following cortical lesions. We thus investigated vestibulo-ocular behaviour in a group of 14 patients with unilateral cortical damage involving the occipito-temporo-parietal junction. Patients were divided in three sub-groups: (1) Group R+: five patients with right sided cortical lesions associated with a left hemi-neglect, (2) Group R-: four patients with right sided cortical lesions with no hemi-neglect and (3) Group L: five patients with left-sided cortical lesions. The patient groups were compared to a group of eight healthy age-matched subjects. The vestibulo-ocular reflex (VOR) was tested in complete darkness by rotating the subject around the vertical axis by sinusoidal rotation at different frequencies, and by steps of acceleration or deceleration. The nystagmus slow phase velocity was measured and plotted as a function of the head velocity and the VOR parameters including gain, bias, time constant and phase were calculated. The cortical lesions induced a significant VOR asymmetry in terms of: a directional preponderance of the VOR gain to the contralesion side, only during sinusoidal rotation, and, in contrast, a VOR bias and a directional preponderance of the VOR time constant and of the nystagmus frequency to the side of the cortical lesion. These latter VOR deficits were the most significant in the R+ group, i.e. in right cortical lesions with hemi-neglect syndrome. These results demonstrate in man, the existence of a cortical influence on vestibular function related to the mechanisms of spatial representation.

In this paper, we determine the upper and lower bounds for the norm of lower triangular matrix operators on Cesàro weighted (p, v)-fractional difference sequence spaces of modulus functions. We consider the matrix operators acting between ℓp(w) and Cp(v, ω, ∆ (η,ℓ) , F) and identify their bounds and vice-versa. We also investigate the same characteristics for Nörlund and weighted mean matrix operators.

We consider the relativistic Vlasov-Maxwell and Vlasov-Nordström systems which describe large particle ensembles interacting by either electromagnetic fields or a relativistic scalar gravity model. For both systems we derive a radiation formula analogous to the Einstein quadrupole formula in general relativity.