Simple Markov Process Research Articles

Energy changes of cosmic rays in the solar system

The energy (in the fixed frame of reference) of a cosmic-ray particle with velocity w in the frame of the solar wind fluctuates between 1 2 M(w + v) 2 and 1 2 M(w − v) 2 as the particle random walks in the magnetic field, where v is the wind velocity. This energy change cannot be treated as a simple Markov process, together with the random walk and adiabatic deceleration, in a fixed frame of reference. To describe the particle motion by a Fokker-Planck differential equation, the energy variable must be the particle energy in a frame of reference moving with the wind. The appropriate transformations of both the solutions and boundary conditions are presented to order v / w . The corrections due to this effect are ∼10% for 1 MeV particles and correspondingly less for higher energy particles. The energy balance between the solar wind and cosmic rays is examined and is shown to depend in a fundamental manner on these energy transformations. Although the solar wind does net work on the cosmic-ray gas, no particle gains more than 2 Mvw kinetic energy and the few particles which penetrate deeply into the solar system suffer a net energy loss because of adiabatic deceleration. Энepгия (B cTaциoнapнoй cиcTeMe кoopдинaT B пpocTpaнcTBe) чacTицы кocMичecкич лyчeй, co cкopocTью w W B cиcTeMe coлнeчнoгo BeTpa, кoлeблeTcя Meждy 1 2 m(w + v) 2 и 1 2 m(w − v) 2 , B To BpeMя кaк чacTицa бpoдиT B MaпниTнoM пoлe, V яBляeTcя cкopocTью BeTpa. Taкyю пepeMeиy эиepгии нe cлeдyeT paccMaTpиBaTь кaк oбыкиoBeииый пpoцecc MapкoBa, BMecTe c бpoжeниeM и aдиaбaTныM зaMeдлeниeM cкopocTи, B cTaциoнapнoй иcTeMe кoopдинaT B пpocTpaнcTBe. чToбы oпиcaTь дBижeниe чacTицы пpи пoMoщи диффepeнциaльнoгo ypaBнeния qFoккepПлaикa, нeoбчoдиMo, чToбы пepeMeннaя энepгии яBянлacь чacTицeй энepгии B пpoдBигaющeйcя BMecTe c BeTpoM cиcTeMe кoopдинaT B пpocTpaнcTBe. BыcTaBляюTcя copaзMepныe пpeBpaщeния кaк peщeний, Taк и гpaничиычycлoBий, cooTBeTcTBeннo пopядкy v / w . ПpoиcTeкaющиe oT эToгo эффeкTa пoпpaBки cocTaBляюT ~10% для чacTиц B 1. Mev и cooTBeTcTBeнныM oбpaзoM Meньщe для чacTиц бoлee Bыcoкoй энepгии. ЙccлeдyeTcя paBнoBecиe зиepгии Meждy coлнeчиыM BeTpoM и кocMичecкиMи лyчaMи, кoTopoe, пoBидиMoMy, элeMeнTapныM oбpaзoM зaBиcиT oT эTич пpeoбpaзoBaний зиepгии. ЧoTя coлиeчный BeTep нe oпepиpyeT нa гaзe кocMичecкич лyчeй, ни oднa чacTицa нe ycилиBaeTcя нa бoлee чeM 2 mvw кинeTичecкoй энepгии и Te нemиoгиe чacTицы, кoTopыe глyбoкo нpoникaюT B coлнeчнyю cиcTey иoдBepгa. oTcн чиcToй иoTepe зиepгии из-зa aдиaбaTнoгo зaMeдлeния cкopocTи.

Brand Preferences and Simple Markov Processes

Brand preference information combined with a simple two-dimensional Markov process can be used to study characteristics of market dynamics. Advertising activity can be used to alter temporarily the brand preference structure of the consuming public. Such changes generate interesting transient conditions that can be studied mathematically. Discussions of time relations, period-to-period changes in market shares, gains and losses resulting from promotional activity and rapidity of convergence to new steady-state values are all considered. In a concluding section, sensitivity characteristics of the relations are commented upon.

A Note on the Test of Serial Correlation Coefficients

In this note the author points out that in the case of stationary Guassian Markov process, i.e., autoregressive stochastic process, we can test the serial correlation coefficients by a method based on normal regression theory. Particularly, in the case of simple Markov process, we can find the confidence limits for its autocorrelation coefficient. In this method, so far as random variables are concerned, not all the information in the original data is used, with a consequence reduction of degrees of freedom. However, the other part of information is introduced as parameters in the distribution functions of random variables and in the statistic useful for tests.