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>}]

But now we can prove theorems about the quaternions. This generates a step proof that the 4th power of the generator we have called is the identity:

FindEquationalProof
&#FindEquationalProof[p[i, p[i, p[i, i]]] == e, %]

In addition to doing mathematical proofs, we can now use in Version to do general proofs with arbitrary predicates (or, more specifically, general first-order logic). Here’s a famous example of a syllogism, based on the predicates and . gives a proof of the assertion that Socrates is mortal:

FindEquationalProof
&#FindEquationalProof[ mortal[socrates], {ForAll[x, Implies[man[x], mortal[x]]], man[socrates]}]

I think it’s pretty neat that this is possible, but it must be admitted that the actual proof generated (which is 53 steps long in this case) is a bit hard to read, not least because it involves conversion to equational logic.

Still, can successfully automate lots of proofs. Here it&#;s solving a logic puzzle given by Lewis Carroll, that establishes (here with a step proof) that babies cannot manage crocodiles:

FindEquationalProof
&#FindEquationalProof[ Not[Exists[x, And[baby[x], manageCrocodile[x]]]], {ForAll[x, Implies[baby[x], Not[logical[x]]]], ForAll[x, Implies[manageCrocodile[x], Not[despised[x]]]], ForAll[x, Implies[Not[logical[x]], despised[x]]]}]

Geo-Everything

The Wolfram Language knows about many things. One of them is geography. And in Version we’ve substantially updated and expanded our sources of geographic data (as well as upgrading our server-based algorithms). So, for example, the level of detail available in typical maps has increased substantially:

GeographicData map

For many years now we’ve had outstanding geodetic computation in the Wolfram Language. And we also have excellent computational geometry for doing all sorts of computations on regions in Euclidean space. But of course the Earth is not flat, and one of the achievements of Version is to bring our region-computation capabilities to the geo domain, handling non-flat regions.

It&#;s an interesting exercise in geometry. We have things like the polygon of the United States defined in geo coordinates&#;as a lat-long region on the Earth. But to use our computational geometry capabilities we need to make it something purely Euclidean. But we can do that by using our geodesy capabilities to embed it in full 3D space.

So now we can just compute the centroid of the region that is the US:

RegionCentroid
&#RegionCentroid[Polygon[Entity["Country", "UnitedStates"]]]

That third element in the geo position is a depth (in meters), and reflects the curvature of the US polygon. And, actually, we can see this directly too:

DiscretizeRegion
&#DiscretizeRegion[Entity["Country", "UnitedStates"]["Polygon"]]

This is a 3D object, so we can rotate it to see the curvature more clearly:

alt

We can also work the other way around: taking geo regions and projecting them onto a flat map, then computing with them. One knows that Greenland looks very different sizes with different map projections. Here’s its “map area” in the Mercator projection (in units of degrees-squared):

Area
&#Area[GeoGridPosition[Entity["Country", "Greenland"]["Polygon"], "Mercator"]]

But here it is (also in degrees-squared) in an area-preserving projection:

Area
&#Area[GeoGridPosition[Entity["Country", "Greenland"]["Polygon"], "CylindricalEqualArea"]]

And as part of the effort to make “geo everything”, Version also includes and .

Advance of the Knowledgebase

Every second of every day there is new data flowing into the Wolfram Knowledgebase that powers Wolfram >], All], ExternalIdentifier["WikidataID", "P", < "Label" -> "coordinate location", "Description" -> "geocoordinates of the subject. For Earth, please note that \ only WGS84 coordinating system is supported at the moment" "Label" -> "hoax", "Description" -> "deliberately fabricated falsehood made to masquerade as the \ truth" >], "Association"]

What Is That Molecule? Advances in Chemical Computation

You have an image of a molecular structure diagram, say from a paper. But how can you get the molecule it represents in a computable form? Well, with Version all you need do is use :

MoleculeRecognize
&#MoleculeRecognize[CloudGet["mlbjerseyschina.us"]]

It’s the analog of , but for molecules. And what it produces is a Wolfram Language symbolic representation of the molecule. So, for example, you can then generate a 3D structure:

mol = MoleculeRecognize
&#mol = MoleculeRecognize[CloudGet["mlbjerseyschina.us"]];
MoleculePlot3D
&#MoleculePlot3D[mol]

Or you can compute the distribution of torsion angles of the structure:

Histogram
&#Histogram[MoleculeValue[mol, "TorsionAngle"], ]

You can also connect to the world of external identifiers:

MoleculeValue
&#MoleculeValue[mol, "PubChemCompoundID"]

But what’s really useful about is that it can be used programmatically. Take all the images of chemicals from a paper, “molecule OCR” them&#;then do things like check whether the molecules are equivalent, or make a word cloud of their 3D structures:

WordCloud
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Paclets and Paclet Development

Changes beginning in version &#;
The &#;Asset&#; extension type was formerly called &#;Resource&#;, and the function that gave a resource path was PacletManager`PacletResource[pacletObj, &#;assetname&#;]. That function will continue to work in current and future versions of the PacletManager. Here is what the previous mlbjerseyschina.us file would look like using the older, but still compatible, &#;Resource&#; extension:
PacletObject[<

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  • Numerous other feature enhancements, bug fixes and performance improvements
  • Represent a biological sequence such as DNA with BioSequence and perform operations on it with functions BioSequenceTranscribe, BioSequenceComplement, etc.
  • New PDE modeling framework for solving a wide variety of partial differential equations both symbolically and numerically
  • Analyze spatial data and work with point processes with new spatial statistics functions such as SpatialPointData, RandomPointConfiguration, Wolfram Mathematica 12.1.0 MAC Archives, MeanPointDensity, PointCountDistribution and more
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    >]]]]

    As another example, which gets a little more elaborate in terms of semantic querying, let’s ask for the opposites of things studied by philosophy, giving the result as an association:

    Wolfram Mathematica 12.1.0 MAC Archives rowspan="1">
    WikidataData
    &#WikidataData[ EntityClass[All, ExternalIdentifier["WikidataID", "P", < "Label" -> "opposite of", "Description" -> "item that is the opposite of this item"

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