New in 9.0: Mathematics & Algorithms
Mathematica 9 adds major extensions and integration across the system. Probability and statistics adds survival and reliability analysis as well as a unified treatment for all random processes. Graphs and networks adds network flows, social network analysis, and major performance enhancements. Control systems adds PID autotuning as well as full support for descriptor and delay systems. Numerical differential equations now includes hybrid systems and parametric systems, as well as advanced support for differential algebraic equations.
Descriptive Statistics – Dependency Measures »
SpearmanRho — Spearman's rank correlation for data and distributions
KendallTau — Kendall's rank correlation for data and distributions
BlomqvistBeta ▪ GoodmanKruskalGamma ▪ HoeffdingD ▪ PillaiTrace ▪ WilksW
Hypothesis Tests – Independence Tests »
IndependenceTest — test whether two vectors are independent
CorrelationTest — test whether two vectors are uncorrelated
PearsonCorrelationTest ▪ SpearmanRankTest ▪ KendallTauTest ▪ HoeffdingDTest ▪ GoodmanKruskalGammaTest ▪ BlomqvistBetaTest ▪ PillaiTraceTest
New Parametric Distributions
HypoexponentialDistribution — hypoexponential distribution
VarianceGammaDistribution — variance gamma distribution
HyperexponentialDistribution ▪ CoxianDistribution ▪ MeixnerDistribution ▪ TsallisQExponentialDistribution ▪ TsallisQGaussianDistribution ▪ LogMultinormalDistribution
Derived Distributions »
SplicedDistribution — splicing several distributions such as different body and tail
CompoundPoissonDistribution — Poisson distribution with given jump size distribution
OrderDistribution — generalized to different and multivariate distributions
New Data Models
EventData — censored (left, right, interval) and truncated data
WeightedData — weighted data
Mean — descriptive statistics works for censored and weighted data
EstimatedDistribution — parametric distribution estimation from censored data
EmpiricalDistribution — nonparametric distributions from censored data
Survival Analysis »
SurvivalModelFit — survival functions with confidence bands from censored data
CoxModelFit — Cox proportional hazards model from censored data
LogRankTest — hypothesis test for equal hazard rates
Reliability Modeling and Analysis »
ReliabilityDistribution — reliability block diagram model for a system
FailureDistribution — fault tree model for a system
StandbyDistribution — standby reliability distribution (cold, hot, ...)
BirnbaumImportance — measure to pinpoint subsystem to improve for reliability
StructuralImportance ▪ BarlowProschanImportance ▪ ImprovementImportance ▪ RiskAchievementImportance ▪ RiskReductionImportance ▪ CriticalityFailureImportance ▪ CriticalitySuccessImportance ▪ FussellVeselyImportance
Random Processes »
RandomFunction — simulate a random process
TemporalData — represent one or several time series data
EstimatedProcess — process estimated from time series data
SliceDistribution — distribution for a time slice of a random process
StationaryDistribution ▪ CovarianceFunction ▪ ...
Parametric Processes »
RandomWalkProcess ▪ PoissonProcess ▪ WienerProcess ▪ ...
Finite Markov Processes »
DiscreteMarkovProcess ▪ ContinuousMarkovProcess ▪ ...
Queueing Processes »
QueueingProcess ▪ QueueingNetworkProcess ▪ ...
Time Series Processes »
ARMAProcess ▪ ARIMAProcess ▪ ...
Stochastic Differential Equation Processes »
ItoProcess ▪ StratonovichProcess ▪ ...
Differential Equations with Events »
WhenEvent — discrete actions in differential equations when an event occurs
NDSolve — now supports differential equations with events
NDSolveValue ▪ DiscreteVariables
Differential Equations with Parameters
ParametricNDSolveValue — solve differential equations depending on parameters
ParametricNDSolve ▪ ParametricFunction
Differential Algebraic Equations
NDSolve — can now solve high-index DAEs
NDSolveValue ▪ ParametricNDSolve ▪ ParametricNDSolveValue
Graphs & Networks »
FindPostmanTour — find a tour that traverses each edge at least once
WeaklyConnectedComponents ▪ EdgeConnectivity ▪ VertexConnectivity ▪ FindGraphPartition ▪ PlanarGraphQ ▪ ...
Social Network Analysis »
FindGraphCommunities — find communities within a graph
GlobalClusteringCoefficient — compute the global clustering coefficient
FindClique ▪ FindKClique ▪ FindKClan ▪ FindKPlex ▪ FindKClub ▪ LambdaComponents ▪ LuccioSamiComponents ▪ GraphReciprocity ▪ MeanClusteringCoefficient ▪ LocalClusteringCoefficient ▪ DegreeCentrality ▪ BetweennessCentrality ▪ ...
Network Flows and Cuts »
FindMaximumFlow — find the maximum flow from a source vertex to a target vertex
FindMinimumCostFlow — find the minimum cost flows
FindMinimumCut — find the minimum cut for a graph
OptimumFlowData ▪ FindEdgeCut ▪ FindVertexCut ▪ ...
Random Graphs »
GraphPropertyDistribution — automatically compute properties of random graphs
SpatialGraphDistribution — higher-performance random graph generation
Control Systems
PIDTune — automatic design of PID controllers
Models with Time Delays »
SystemsModelDelay — represent delays in state-space and transfer-function models
TransferFunctionModel ▪ StateSpaceModel ▪ SystemsModelDelayApproximate ▪ SmithDelayCompensator ▪ ...
Models with Algebraic Constraints »
DescriptorStateSpace — whether to use descriptor state-space representation
StateSpaceModel ▪ KroneckerModelDecomposition ▪ ...
Fast Symbolic Linear and Polynomial Algebra
LinearSolve — fast methods for sparse symbolic and exact systems
Expand — fast methods for polynomials with numeric coefficients
NullSpace ▪ MatrixRank ▪ Det ▪ Subresultants ▪ PolynomialExtendedGCD
New Polynomial Functions
SubresultantPolynomials ▪ SubresultantPolynomialRemainders
New Matrix Functions
MatrixLog — matrix log function
MatrixFunction — matrix function for a general function
New Algebraic Numbers
CubeRoot — real cube root
Surd — real 
th root
Vector Calculus »
Coordinate Systems
CoordinateChartData — properties of coordinate systems
TransformedField — transform a field between coordinate systems
CoordinateTransform ▪ CoordinateTransformData
Symbolic Tensor Algebra »
Arrays — domain of arrays with given properties
TensorRank ▪ TensorDimensions ▪ TensorSymmetry
TensorProduct ▪ TensorContract ▪ TensorTranspose ▪ ...
Symmetries and Symmetrized Arrays
SymmetrizedArray — compact representation of arrays with symmetry
SymmetrizedIndependentComponents ▪ SymmetrizedDependentComponents