Fix the grammar of a few sentences

docs/source/random_numbers.rst

@@ -10,48 +10,48 @@ a seed and a Random Number Generator (RNG) are used.
 Seed
 ----
 
-The ``random_seed`` is used to control the behaviour of the RNG. The seed
-can be ``None`` or an integer number.
-If you are **not** interested in the reproducibility of your results, you can
-set ``random_seed`` to ``None``. However, this is not recommended.
-In fact, this means that if you run multiple times a function or method where
-a RNG is used, e.g. for producing noise timelines, then
-the outputs will be **different** and you will not be able to re-obtain these
-results again.
-If instead you are interested in the reproducibility of your results, you can
-set ``random_seed`` to an integer number.
-With this choice, if you run multiple times a function or method where
-a RNG is used, then the outputs will be **the same** and
-you will re-obtain these results by re-running your script, as the random
-numbers produced by the generator will be exactly the same.
-
-How should you set the ``random_seed``? This parameter **must** be passed to
-the constructor of a :class:`.Simulation` class.
-If a :class:`.Simulation` instance is created without passing the seed, an
-error will be raised and your script will fail. This feature has been
-introduced on purpose in order to avoid automatic settings of the seed and
-unclear behaviour of the generation of random numbers.
-If you are running a parallel script using MPI, you do not have to
-worry about setting a different seed for different MPI ranks: by passing
-the same seed, the random numbers generated by the different ranks will be
-different. This is ensured by the random number generator. We want this
-behaviour as we do not want repetitions of, e.g., the same noise TOD if
-we split their computation on different MPI ranks. For example, in this
-way, if you split the TOD matrix of an :class:`.Observation` class by the
-time, you will not encounter the same noise after the samples generated
-by a certain rank; if you split the TOD matrix of an :class:`.Observation`
-class by the detectors, each one will have a different noise timestream.
-
-Regarding the reproducibility of the results in a parallel code, there is
-an **important thing** to bear in mind: if you set the seed to an integer
-number but run your script with a different number of MPI ranks, the
-outputs will be **different**! Think about a noise time stream of 4 samples.
-If you use 2 MPI ranks, then the first 2 samples will be generated by
-one RNG (the one of rank 0), while the last 2 samples will be generated
-by a different RNG (the one of rank 1). If you then run the same script
-with the same seed but with 4 MPI ranks, each of the samples will be
-generated by a different RNG and only the first sample will be the same
-for the two runs, as it is always the first one generated by rank 0's RNG.
+The ``random_seed`` is used to control the behavior of the RNG. The
+seed can be ``None`` or an integer number. If you are **not**
+interested in the reproducibility of your results, you can set
+``random_seed`` to ``None``. However, this is not recommended, as if
+you run many times a function or method that uses an RNG, e.g., for
+producing noise timelines, then the outputs will be **different**, and
+you will not be able to re-obtain these results again. If, instead,
+you are interested in the reproducibility of your results, you can set
+``random_seed`` to an integer number. With this choice, if you run a
+function or method multiple times using an RNG, then the outputs will
+be **the same**, and you will obtain these results again by re-running
+your script, as the random numbers produced by the generator will be
+the same.
+
+How should you set the ``random_seed``? This parameter **must** be
+passed to the constructor of a :class:`.Simulation` class. If you
+create a :class:`.Simulation` instance without passing the seed, the
+constructor will raise an error, and your script will fail. We
+implemented this feature to avoid automatic settings of the seed and
+unclear behavior of the generation of random numbers. If you run a
+parallel script using MPI, you do not have to worry about setting a
+different seed for different MPI ranks. The random number generator
+ensures that the random numbers generated by the ranks will produce
+orthogonal sequences. We want this behavior as we do not want
+repetitions of, e.g., the same noise TOD if we split their computation
+on different MPI ranks. For example, in this way, if you split the TOD
+matrix of an :class:`.Observation` class by the time, you will not
+encounter the same noise after the samples generated by a certain
+rank; if you split the TOD matrix of an :class:`.Observation` class by
+the detectors, each one will have a different noise timestream.
+
+Regarding the reproducibility of the results in a parallel code, there
+is an **important thing** to bear in mind. If you set the seed to an
+integer number but run your script with a different number of MPI
+ranks, the outputs will be **different**! Think about a noise time
+stream of 4 samples. If you use 2 MPI ranks, then the first two
+samples will be generated by one RNG (rank 0), while the last two
+samples will be generated by a different RNG (rank 1). If you then run
+the same script with the same seed but with 4 MPI ranks, each of the
+samples will be generated by a different RNG, and only the first
+sample will be the same for the two runs, as it is always the first
+one generated by rank 0’s RNG.
 
 The setting of the ``random_seed`` is as simple as this::
 
@@ -64,12 +64,12 @@ The setting of the ``random_seed`` is as simple as this::
       random_seed=12345, # Here the seed for the random number generator is set
   )
 
-During the execution of the :class:`.Simulation` constructor, the
-:meth:`.Simulation.init_random` method is run. There, the RNG is
-built and seeded with ``random_seed``. You can then use this RNG
-by calling ``sim.random``. There is also the possibility to
-change the random seed after creatinga :class:`.Simulation` instance.
-This is achieved by calling :meth:`.Simulation.init_random`::
+The :class:`.Simulation` constructor runs the
+:meth:`.Simulation.init_random` method, which builds the RNG and seeds
+it with ``random_seed``. You can then use this RNG by calling
+``sim.random``. There is also the possibility to change the random
+seed after creating a :class:`.Simulation` instance. You can achieve
+this by calling :meth:`.Simulation.init_random`::
 
   sim = lbs.Simulation(random_seed=12345)
   [...]