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Roman Cruz
Roman Cruz

Blank Calculus Single Variable


Blank was born in Montreal, Quebec, and earned an undergraduate degree in mathematics from McGill University.[2] He received his Masters and Ph.D. in 1980 from Cornell University, with Anthony Knapp as advisor.[3] His 20th century work involved harmonic analysis. He also co-authored a pair of calculus textbooks with his Washington University colleague, Steven Krantz. Titled Calculus: Single Variable and Calculus: Multivariable, the textbooks and workbooks used to be used in calculus classes at Washington University.




blank calculus single variable


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Section 4 presents details of the SPARQL query language's syntax. It is a companion to the full grammar of the language and defines how grammatical constructs represent IRIs, blank nodes, literals, and variables. Section 4 also defines the meaning of several grammatical constructs that serve as syntactic sugar for more verbose expressions.


A 'binding' is a pair (variable,RDF term). In this result set, there are threevariables:x, y and z (shown as column headers). Each solution is shown as one row in the body of the table. Here, there is a single solution, in which variable x is bound to "Alice", variabley is bound to , and variable z is not bound to an RDF term. Variables are not required to be bound in a solution.


The example below shows a SPARQL query to find the title of a book from the given data graph. The query consists of two parts:the SELECT clause identifies the variables to appear in the query results, and the WHERE clause provides the basic graph pattern to match against the data graph. The basic graph pattern in this example consists of a single triple pattern with a single variable (?title) in the object position.


The query below uses the graph with IRI to find the profile document for Bob; it then matches another pattern against that graph. The pattern in the second GRAPH clause finds the blank node (variable w) for the person with the same mail box (given by variable mbox) as found in the first GRAPH clause (variable whom), because the blank node used to match for variable whom from Alice's FOAF file is not the same as the blank node in the profile document (they are in different graphs).


The CONSTRUCT query form returns a single RDF graph specified by a graph template. The result is an RDF graph formed by taking each query solution in the solution sequence, substituting for the variables in the graph template, and combining the triples into a single RDF graph by set union.


The use of variable x in the template, which in this example will be bound to blank nodes with labels _:a and _:b in the data, causes different blank node labels (_:v1 and _:v2) in the resulting RDF graph.


The DESCRIBE form returns a single result RDF graph containing RDF data about resources. This data is not prescribed by a SPARQL query, where the query client would need to know the structure of the RDF in the data source, but, instead, is determined by the SPARQL query processor. The query pattern is used to create a result set. The DESCRIBE form takes each of the resources identified in a solution, together with any resources directly named by IRI, and assembles a single RDF graph by taking a "description" which can come from any information available including the target RDF Dataset. The description is determined by the query service. The syntax DESCRIBE * is an abbreviation that describes all of the variables in a query.


The resources to be described can also be taken from the bindings to a query variable in a result set. This enables description of resources whether they are identified by IRI or by blank node in the dataset:


Basic graph patterns form the basis of SPARQL pattern matching. A basic graph pattern is matched against the active graph for that part of the query. Basic graph patterns can be instantiated by replacing both variables and blank nodes by terms, giving two notions of instance. Blank nodes are replaced using an RDF instance mapping, σ, from blank nodes to RDF terms; variables are replaced by a solution mapping from query variables to RDF terms.


This definition allows the solution mapping to bind a variable in a basic graph pattern, BGP, to a blank node in G. Since SPARQL treats blank node identifiers in a SPARQL Query Results XML Format document as scoped to the document, they cannot be understood as identifying nodes in the active graph of the dataset. If DS is the dataset of a query, pattern solutions are therefore understood to be not from the active graph of DS itself, but from an RDF graph, called the scoping graph, which is graph-equivalent to the active graph of DS but shares no blank nodes with DS or with BGP. The same scoping graph is used for all solutions to a single query. The scoping graph is purely a theoretical construct; in practice, the effect is obtained simply by the document scope conventions for blank node identifiers.


Basic graph patterns stand in the same relation to triple patterns that RDF graphs do to RDF triples, and much of the same terminology can be applied to them. In particular, two basic graph patterns are said to be equivalent if there is a bijection M between the terms of the triple patterns that maps blank nodes to blank nodes and maps variables, literals and IRIs to themselves, such that a triple ( s, p, o ) is in the first pattern if and only if the triple ( M(s), M(p) M(o) ) is in the second. This definition extends that for RDF graph equivalence to basic graph patterns by preserving variable names across equivalent patterns.


(e) None of these conditions refer explicitly to instance mappings on blank nodes in BGP. For some entailment regimes, the existential interpretation of blank nodes cannot be fully captured by the existence of a single instance mapping. These conditions allow such regimes to give blank nodes in query patterns a 'fully existential' reading.


Some answers mention the fact that the function could be thought of a complex function of a real variable. This is nowhere specifically mentioned in the problem and if we really want to be imaginative enough, we may well consider it to be a function of a complex variable. And this changes the whole nature of the problem. Functions of a complex variable are altogether a very different beast from the usual functions of a real variable which are seen in introductory calculus courses.


Erland Sommarskog pointed out an interesting fluke of SQL Server on the Microsoft Q&A forum. Using the old Sybase syntax with SELECT is the only way to assign multiple variables in a single statement in T-SQL since SQL Server does not yet support row constructors. (However, someone pointed out that the FETCH statement can also assign values to multiple variables in one statement.) The ANSI/ISO standard row constructor syntax looks like this:


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