This document provides a summary of three key terms from mathematics:
1. Algorithm - A sequence of instructions that tell how to accomplish a task, which must be specified exactly and have a finite number of steps.
2. Al-Khwarizmi - A Muslim mathematician from the 9th century who introduced modern numerals to Europe and whose works provided the source for the term "algebra".
3. Alternate interior angles - When a transversal cuts two lines, it forms two pairs of angles where the angles in each pair are equal according to Euclidian geometry.
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
This document discusses different types of straight lines and their equations in a plane. It covers lines parallel to coordinate axes, lines through a given point with a given slope, lines with a given slope and y-intercept, and lines passing through two given points. It also discusses finding the equation of the line perpendicular to a given line and passing through a given point, as well as finding the distance from a point to a line. Examples of finding various line equations are provided.
The document discusses different forms of equations for straight lines, including:
- Point-slope form, which defines a line through a known point with a given slope
- Slope-intercept form, which defines a line with a given slope and y-intercept
- Normal form, which defines a line based on its perpendicular distance from the origin and the angle it forms with the x-axis
- General form, which is the standard Ax + By + C = 0 equation for a line
It also covers how to find the distance from a point to a line, and how parallel and perpendicular lines can be identified based on the coefficients in their equations.
The document discusses asymptotes of functions, specifically linear asymptotes which are straight lines that a curve comes closer to without touching. Horizontal asymptotes are found by taking limits of a function as x approaches infinity. Vertical asymptotes occur at x-values where the function is undefined due to a zero in the denominator. Shortcuts are provided to determine horizontal asymptotes based on comparing the highest powers of the numerator and denominator. Examples demonstrate finding both horizontal and vertical asymptotes through limits or factorizing.
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
This document provides an introduction and overview of calculus topics for functions of several real variables, including:
1) Limits and continuity of functions in R3 space.
2) Derivatives of functions of several variables, including partial derivatives and gradients.
3) Total differential, divergence, and rotor.
4) Tangent planes and normal lines to surfaces.
The document defines scalar and vector functions, and discusses concepts like limits, continuity, derivatives, and geometrical interpretations in multi-dimensional spaces. It also outlines further calculus topics to be covered like integrals of functions with parameters, line and surface integrals.
This document provides a summary of three key terms from mathematics:
1. Algorithm - A sequence of instructions that tell how to accomplish a task, which must be specified exactly and have a finite number of steps.
2. Al-Khwarizmi - A Muslim mathematician from the 9th century who introduced modern numerals to Europe and whose works provided the source for the term "algebra".
3. Alternate interior angles - When a transversal cuts two lines, it forms two pairs of angles where the angles in each pair are equal according to Euclidian geometry.
This document provides instruction on applying derivatives to solve various types of application problems. It begins by outlining objectives of analyzing and solving application problems involving derivatives as instantaneous rates of change or tangent line slopes. Examples of application problems covered include writing equations of tangent and normal lines, curve tracing, optimization problems, and related rates problems involving time rates. The document then provides definitions and examples of using derivatives to find slopes of curves and tangent lines. It also covers concepts like concavity, points of inflection, maxima/minima, and solving optimization problems using derivatives. Finally, it gives examples of solving related rates problems involving time-dependent variables.
This document discusses concepts related to calculus including limits, continuity, and derivatives of functions. Specifically, it covers:
- Definitions and theorems related to limits, continuity, and derivatives of algebraic functions.
- Evaluating limits, determining continuity of functions, and taking derivatives of algebraic functions using basic theorems of differentiation.
- The objective is for students to be able to evaluate limits, determine continuity, and find derivatives of continuous algebraic functions in explicit or implicit form after discussing these calculus concepts.
This document discusses different types of straight lines and their equations in a plane. It covers lines parallel to coordinate axes, lines through a given point with a given slope, lines with a given slope and y-intercept, and lines passing through two given points. It also discusses finding the equation of the line perpendicular to a given line and passing through a given point, as well as finding the distance from a point to a line. Examples of finding various line equations are provided.
The document discusses different forms of equations for straight lines, including:
- Point-slope form, which defines a line through a known point with a given slope
- Slope-intercept form, which defines a line with a given slope and y-intercept
- Normal form, which defines a line based on its perpendicular distance from the origin and the angle it forms with the x-axis
- General form, which is the standard Ax + By + C = 0 equation for a line
It also covers how to find the distance from a point to a line, and how parallel and perpendicular lines can be identified based on the coefficients in their equations.
The document discusses asymptotes of functions, specifically linear asymptotes which are straight lines that a curve comes closer to without touching. Horizontal asymptotes are found by taking limits of a function as x approaches infinity. Vertical asymptotes occur at x-values where the function is undefined due to a zero in the denominator. Shortcuts are provided to determine horizontal asymptotes based on comparing the highest powers of the numerator and denominator. Examples demonstrate finding both horizontal and vertical asymptotes through limits or factorizing.
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
This document provides an introduction and overview of calculus topics for functions of several real variables, including:
1) Limits and continuity of functions in R3 space.
2) Derivatives of functions of several variables, including partial derivatives and gradients.
3) Total differential, divergence, and rotor.
4) Tangent planes and normal lines to surfaces.
The document defines scalar and vector functions, and discusses concepts like limits, continuity, derivatives, and geometrical interpretations in multi-dimensional spaces. It also outlines further calculus topics to be covered like integrals of functions with parameters, line and surface integrals.
This document provides an overview of Combinatory Categorial Grammar (CCG). CCG is a grammatical framework that assigns syntactic categories containing semantic information to words and phrases. These categories can then be combined through function application and composition rules to build syntactic structures paired with semantic interpretations. The document discusses CCG's categories, lexicon, combination rules including composition and type raising, and principles controlling the rules' application. It also briefly mentions CCG parsers and applications like semantic parsing for question answering.
The document discusses properties of similar figures and how to determine if two figures are similar. It provides examples of similar figures and how to use scale factors and proportional sides to determine missing side lengths. Some key points made include:
- Two figures are similar if corresponding angles have the same measure and ratios of corresponding sides are equal.
- The scale factor is the ratio of corresponding sides and can be used to determine unknown side lengths of similar figures.
- Examples show determining if figures are similar and calculating missing side lengths using scale factors and proportional sides.
This document discusses arithmetic sequences and series. It defines an arithmetic sequence as a sequence where each term is equal to the previous term plus a constant difference. Formulas are provided for calculating the nth term of a sequence and for finding the index of a given term. Formulas are also provided for calculating the nth partial sum of an arithmetic series. Examples are worked through demonstrating how to use the formulas to find terms, indices, and partial sums of arithmetic sequences and series.
5.5 parallel and perpendicular lines (equations) day 1bweldon
The document discusses writing equations of parallel and perpendicular lines, including identifying whether lines are parallel or perpendicular based on their slopes. It provides examples of writing equations of parallel and perpendicular lines given a point and the slope or line of another line. Guidance problems are also included to practice these skills.
We introduce plaque inverse limits of branched covering self-maps of simply-connected Riemann surfaces and study their local topology at various irregular points.
Graph theory has many applications including social networks, data organization, and communication networks. The document discusses Dijkstra's algorithm for finding the shortest path between nodes in a graph and its application to finding shortest routes between cities. It also discusses using graph representations for fingerprint classification, where fingerprints are modeled as graphs with nodes for fingerprint regions and edges between adjacent regions. Fingerprints are classified based on the structure of these graphs and compared to model graphs for matching.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
1) The document outlines the course outcomes for Calculus I with Analytic Geometry. It discusses fundamental concepts like analytic geometry, functions, limits, continuity, derivatives, and their applications.
2) The course aims to teach students to analyze and solve problems involving lines, circles, conics, transcendental functions, derivatives, tangents, normals, maxima/minima, and related rates.
3) The assessment tasks and grading criteria are also presented, including quizzes, classwork, and a final examination. Minimum averages for satisfactory performance are provided.
This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.
The document defines key concepts in graph theory including:
- Types of graphs such as simple graphs, connected graphs, and regular graphs.
- Graph terminology like vertices, edges, walks, paths, and subgraphs.
- Special graphs like Hamiltonian and Euler graphs.
- Graph coloring problems including vertex coloring and edge coloring.
- Examples are given to illustrate graph concepts and properties.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
This document is a lecture on model theory given by Erik A. Andrejko. It covers topics such as completeness, soundness, elementary submodels, the Tarski-Vaught test, the downward and upward Lowenheim-Skolem-Tarski theorems, definability, categoricity, and complete theories. The document presents definitions, facts, and theorems regarding these topics in model theory.
The document discusses various concepts relating to straight lines in mathematics including:
1) Calculating the gradient of a straight line between two points.
2) Horizontal and vertical lines having gradients of 0 or being undefined.
3) The relationship between gradient and angle of a line.
4) Finding the midpoint, collinearity of points, and gradients of perpendicular lines.
This document provides an overview of basic graph theory concepts. It defines graphs, subgraphs, and some special types of graphs. A graph is defined as a pair of sets (V,E) where V is the set of vertices and E is the set of edges. A subgraph is a graph whose vertex and edge sets are subsets of another graph. Complete graphs, empty graphs, bipartite graphs, and complement graphs are some special types of graphs discussed. The document also introduces concepts like isomorphic graphs, graph embeddings, planar vs. non-planar graphs, multigraphs, digraphs, and operations on graphs like edge/vertex removal.
This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
This document provides an introduction to graph theory concepts. It defines graphs as mathematical objects consisting of nodes and edges. Both directed and undirected graphs are discussed. Key graph properties like paths, cycles, degrees, and connectivity are defined. Classic graph problems introduced include Eulerian circuits, Hamiltonian circuits, spanning trees, and graph coloring. Graph theory is a fundamental area of mathematics with applications in artificial intelligence.
This document is a report on Complex Variable & Numerical Method prepared by 5 students and guided by Miss. Chaitali Shah. It contains the following topics: complex numbers, complex variable, basic definitions, limits, continuity, differentiability, analytic functions, Cauchy-Riemann equations, harmonic functions, Milne-Thomson method, and applications of complex functions/variables in engineering. Examples are provided to illustrate several concepts.
The document discusses relations and their application to databases in the relational data model. It defines binary and n-ary relations, and explains how databases can be represented as n-ary relations with records as n-tuples consisting of fields. Primary keys are introduced as fields that uniquely identify each record. Common relational operations like projection and join are explained, with examples provided to illustrate how they transform relations.
This document provides an overview of Combinatory Categorial Grammar (CCG). CCG is a grammatical framework that assigns syntactic categories containing semantic information to words and phrases. These categories can then be combined through function application and composition rules to build syntactic structures paired with semantic interpretations. The document discusses CCG's categories, lexicon, combination rules including composition and type raising, and principles controlling the rules' application. It also briefly mentions CCG parsers and applications like semantic parsing for question answering.
The document discusses properties of similar figures and how to determine if two figures are similar. It provides examples of similar figures and how to use scale factors and proportional sides to determine missing side lengths. Some key points made include:
- Two figures are similar if corresponding angles have the same measure and ratios of corresponding sides are equal.
- The scale factor is the ratio of corresponding sides and can be used to determine unknown side lengths of similar figures.
- Examples show determining if figures are similar and calculating missing side lengths using scale factors and proportional sides.
This document discusses arithmetic sequences and series. It defines an arithmetic sequence as a sequence where each term is equal to the previous term plus a constant difference. Formulas are provided for calculating the nth term of a sequence and for finding the index of a given term. Formulas are also provided for calculating the nth partial sum of an arithmetic series. Examples are worked through demonstrating how to use the formulas to find terms, indices, and partial sums of arithmetic sequences and series.
5.5 parallel and perpendicular lines (equations) day 1bweldon
The document discusses writing equations of parallel and perpendicular lines, including identifying whether lines are parallel or perpendicular based on their slopes. It provides examples of writing equations of parallel and perpendicular lines given a point and the slope or line of another line. Guidance problems are also included to practice these skills.
We introduce plaque inverse limits of branched covering self-maps of simply-connected Riemann surfaces and study their local topology at various irregular points.
Graph theory has many applications including social networks, data organization, and communication networks. The document discusses Dijkstra's algorithm for finding the shortest path between nodes in a graph and its application to finding shortest routes between cities. It also discusses using graph representations for fingerprint classification, where fingerprints are modeled as graphs with nodes for fingerprint regions and edges between adjacent regions. Fingerprints are classified based on the structure of these graphs and compared to model graphs for matching.
The document discusses methods for graphing quadratic equations. It explains that the graphs of quadratic equations are called parabolas, which are symmetric around a center line with a highest or lowest point called the vertex. It provides an example of graphing the equation y = x^2 - 4x - 12 by first finding the vertex, then making a table of symmetric x and y values around the vertex and plotting the points. The document also describes how to find the x-intercepts and y-intercepts of a parabola and provides an alternate two-step method for graphing a parabola.
1) The document outlines the course outcomes for Calculus I with Analytic Geometry. It discusses fundamental concepts like analytic geometry, functions, limits, continuity, derivatives, and their applications.
2) The course aims to teach students to analyze and solve problems involving lines, circles, conics, transcendental functions, derivatives, tangents, normals, maxima/minima, and related rates.
3) The assessment tasks and grading criteria are also presented, including quizzes, classwork, and a final examination. Minimum averages for satisfactory performance are provided.
This document defines key concepts related to straight lines, including their various forms of equations, slope, intercepts, and how to calculate them. It discusses the general, standard, point-slope, and intercept forms of linear equations. It also explains how to find the slope between two points using the slope formula, and how to determine the x-intercept and y-intercept from a line's equation. Examples are provided to illustrate these concepts.
The document defines key concepts in graph theory including:
- Types of graphs such as simple graphs, connected graphs, and regular graphs.
- Graph terminology like vertices, edges, walks, paths, and subgraphs.
- Special graphs like Hamiltonian and Euler graphs.
- Graph coloring problems including vertex coloring and edge coloring.
- Examples are given to illustrate graph concepts and properties.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
This document is a lecture on model theory given by Erik A. Andrejko. It covers topics such as completeness, soundness, elementary submodels, the Tarski-Vaught test, the downward and upward Lowenheim-Skolem-Tarski theorems, definability, categoricity, and complete theories. The document presents definitions, facts, and theorems regarding these topics in model theory.
The document discusses various concepts relating to straight lines in mathematics including:
1) Calculating the gradient of a straight line between two points.
2) Horizontal and vertical lines having gradients of 0 or being undefined.
3) The relationship between gradient and angle of a line.
4) Finding the midpoint, collinearity of points, and gradients of perpendicular lines.
This document provides an overview of basic graph theory concepts. It defines graphs, subgraphs, and some special types of graphs. A graph is defined as a pair of sets (V,E) where V is the set of vertices and E is the set of edges. A subgraph is a graph whose vertex and edge sets are subsets of another graph. Complete graphs, empty graphs, bipartite graphs, and complement graphs are some special types of graphs discussed. The document also introduces concepts like isomorphic graphs, graph embeddings, planar vs. non-planar graphs, multigraphs, digraphs, and operations on graphs like edge/vertex removal.
This document provides an overview of linear functions and equations. It defines linear equations as having the standard form Ax + By = C, with examples and how to identify linear vs. nonlinear equations. Linear functions are defined as having the form f(x) = mx + b. The document discusses slope, x-intercepts, y-intercepts, and how to graph linear equations from these components. It also covers representing linear functions in slope-intercept form as y = mx + b, and point-slope form as y - y1 = m(x - x1).
This document provides an introduction to graph theory concepts. It defines graphs as mathematical objects consisting of nodes and edges. Both directed and undirected graphs are discussed. Key graph properties like paths, cycles, degrees, and connectivity are defined. Classic graph problems introduced include Eulerian circuits, Hamiltonian circuits, spanning trees, and graph coloring. Graph theory is a fundamental area of mathematics with applications in artificial intelligence.
This document is a report on Complex Variable & Numerical Method prepared by 5 students and guided by Miss. Chaitali Shah. It contains the following topics: complex numbers, complex variable, basic definitions, limits, continuity, differentiability, analytic functions, Cauchy-Riemann equations, harmonic functions, Milne-Thomson method, and applications of complex functions/variables in engineering. Examples are provided to illustrate several concepts.
The document discusses relations and their application to databases in the relational data model. It defines binary and n-ary relations, and explains how databases can be represented as n-ary relations with records as n-tuples consisting of fields. Primary keys are introduced as fields that uniquely identify each record. Common relational operations like projection and join are explained, with examples provided to illustrate how they transform relations.
Compiler components and their generators traditional parsing algorithmsPeshrowKareem1
This document provides an overview of predictive parsing and LL parsing tables. It begins with a recap of formal languages and grammars. It then discusses predictive parsing, including recursive descent parsing and lookahead. The document explains how to fill an LL parsing table by determining nullable, FIRST, and FOLLOW sets. It provides an example grammar and shows how to calculate these sets and fill the table. Finally, it briefly discusses grammar classes like LL(k) and encoding operator precedence.
The document discusses lexical analysis and regular languages. It begins with an overview of lexical analysis and its components, including regular languages defined via regular grammars, regular expressions, and finite state automata. It then covers the equivalence between these formalisms for describing regular languages and how to construct a nondeterministic finite automaton from a regular expression.
This document contains a lesson on three-dimensional figures including vocabulary terms and examples. It defines 14 vocabulary terms related to 3D shapes such as polyhedron, face, edge, vertex, prism, pyramid, cylinder, cone, sphere, and more. It also provides two examples, one calculating the surface area and volume of a cone and another finding the surface area needed to make a cardboard box. The document aims to teach students how to identify, name, and calculate properties of three-dimensional figures.
The document discusses finding the right abstractions for reasoning problems. It describes Andreas Blass' insight about a category called PV that models problems and reductions between them. PV objects are binary relations representing problems, with morphisms describing reductions. The talk discusses using this framework and Dialectica categories to model Kolmogorov's theory of problems from 1932 and Veloso's theory. It provides examples of modeling geometry and tangent plane problems as Kolmogorov problems and reductions between them.
My talk about computational geometry in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
This document provides an introduction and outline for a course on Formal Language Theory. The course will cover topics like set theory, relations, mathematical induction, graphs and trees, strings and languages. It will then introduce formal grammars including regular grammars, context-free grammars and pushdown automata. The course is divided into 5 chapters: Basics, Introduction to Grammars, Regular Languages, Context-Free Languages, and Pushdown Automata. The Basics chapter provides an overview of formal vs natural languages and reviews concepts like sets, relations, functions, and mathematical induction.
This document discusses the relationships between orbits of linear maps and regular languages. It shows that the chamber hitting problem (CHP) and permutation filter realizability problem are Turing equivalent. It also shows that the injective filter and surjective filter realizability problems are decidable by reducing them to problems about orbits. However, the regular realizability problem for the track product of the periodic and permutation filters is undecidable, as it can reduce the undecidable zero in the upper right corner problem.
Representation of Integer Positive Number as A Sum of Natural SummandsIJERA Editor
In this paper the problem of representation of integer positive number as a sum of natural terms is considered. The new approach to calculation of number of representations is offered. Results of calculations for numbers from 1 to 500 are given. Dependence of partial contributions to total sum of number of representations is investigated. Application of results is discussed.
This document defines and explains basic geometric figures including points, lines, line segments, rays, planes, and angles. It provides descriptions of each figure and conventions for naming them properly. Examples are given to illustrate the definitions and naming. The document serves as an introduction to the fundamental concepts and terminology of geometry.
This document discusses the relationships between orbits of linear maps and regular languages. It shows that the chamber hitting problem (CHP) and permutation filter-realizability problem are Turing equivalent. It also shows that the injective filter-realizability problem and surjective filter-realizability problem are decidable, while the track product of the periodic and permutation filter-realizability problem is undecidable. The zero in the upper right corner problem, which is undecidable, can be reduced to the latter regular realizability problem.
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dug84356_ch09a.qxd 9/14/10 2:11 PM Page 557
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9.1
9.2
9.3
9.4
9.5
9.6
Radicals
Rational Exponents
Adding, Subtracting, and
Multiplying Radicals
Quotients, Powers,
and Rationalizing
Denominators
Solving Equations with
Radicals and Exponents
Complex Numbers
9
Radicals and Rational
Exponents
Just how cold is it in Fargo, North Dakota, in winter? According to local meteorol
ogists, the mercury hit a low of –33°F on January 18, 1994. But air temperature
alone is not always a reliable indicator of how cold you feel. On the same date,
the average wind velocity was 13.8 miles per hour. This dramatically affected how
cold people felt when they stepped outside. High winds along with cold temper
atures make exposed skin feel colder because the wind significantly speeds up
the loss of body heat. Meteorologists use the terms “wind chill factor,”“wind chill
index,” and “wind chill temperature” to take into account both air temperature
and wind velocity.
Through experimentation in Ant
arctica, Paul A. Siple developed a
formula in the 1940s that measures the
wind chill from the velocity of the wind
and the air temperature. His complex
formula involving the square root of
the velocity of the wind is still used
today to calculate wind chill temper
atures. Siple’s formula is unlike most
scientific formulas in that it is not
based on theory. Siple experimented
with various formulas involving wind
velocity and temperature until he
found a formula that seemed to predict
how cold the air felt.
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ra
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re
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20
15
10
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5
10
15
Wind velocity (mph)
5 10 15 20 25 30
Siple s formula is stated
and used in Exercises 111
and 112 of Section 9.1.
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 558
→
→
→
558 Chapter 9 Radicals and Rational Exponents 9-2
9.1 Radicals
In Section 4.1, you learned the basic facts about powers. In this section, you will
study roots and see how powers and roots are related.
In This Section
U1V Roots
U2V Roots and Variables
U3V Product Rule for Radicals
U4V Quotient Rule for Radicals U1V Roots
U5V Domain of a Radical We use the idea of roots to reverse powers. Because 32 = 9 and (-3)2 = 9, both 3 andExpression or Function
-3 are square roots of 9. Because 24 = 16 and (-2)4 = 16, both 2 and -2 are fourth
roots of 16. Because 23 = 8 and (-2)3 = -8, there is only one real cube root of 8 and
only one real cube root of -8. The cube root of 8 is 2 and the cube root of -8 is -2.
nth Roots
If a = bn for a positive integer n, then b is an nth root of a. If a = b2, then b is a
square root of a. If a = b3, then b is the cube root of a.
If n is a positive even integer and a is positive, then there are two real nth roots of
a. We call these roots even roots. The positive even root of a positive number is called
the prin ...
Graphs are useful tools for modeling problems and consist of vertices and edges. Breadth-first search (BFS) is an algorithm that visits the vertices of a graph starting from a source vertex and proceeding to visit neighboring vertices first, before moving to neighbors that are further away. BFS uses a queue to efficiently traverse the graph and discover all possible paths from the source to other vertices, identifying the shortest paths in an unweighted graph. The time complexity of BFS on an adjacency list representation is O(n+m) where n is the number of vertices and m is the number of edges.
Introduction to Neo4j - a hands-on crash courseNeo4j
This document provides an introduction to graphs and Neo4j. It covers what graphs are, why they are useful, the components of property graph databases, and how to query graphs using Cypher. Examples are provided to illustrate how to match nodes, extend matches, create nodes and relationships, and hands-on practice is encouraged on the Neo4j sandbox. Resources for continuing one's graph journey are also listed.
The document discusses the concept of parity in problem solving. It provides examples of problems involving parity, such as determining whether gears or dominoes can be arranged in certain ways. It also presents problems involving sums of odd and even numbers, and discusses how considering parity can help determine whether certain arrangements or solutions are possible. The document aims to illustrate how thinking about the parity of variables can help in solving mathematical problems.
The document provides an overview of Darmon points, which are conjectural points on elliptic curves over number fields. It begins by setting up the basic context, including defining an elliptic curve E over a number field F, and a quadratic extension K/F. It then discusses the Birch and Swinnerton-Dyer conjecture relating the order of L(E/K,s) at s=1 to the rank of E(K). The document presents Heegner points as a classical example when K/F is a CM extension, and notes Darmon's goal was to drop the CM hypothesis. It states that while there are no points on the modular curve X0(N) attached to a real quadratic K
Rough sets and fuzzy rough sets in Decision MakingDrATAMILARASIMCA
Rough sets, Fuzzy rough sets, lower approximation, upper approximation, positive region and reduct, Equivalence relation, dependency coefficient, Information system for road accident system
The document discusses learning timed automata using Cypher queries. It provides background on automaton learning, including representing systems as state machines and learning their behavior through observation and experimentation. The goal is to develop a hypothesis model that matches the internal operations of the system under learning. The document outlines the basic automaton learning process and describes an algorithm that repeats splitting inconsistent states, merging similar states, and coloring states to finalize the learned automaton. It also discusses some interesting queries that could be used as part of the learning process in Cypher, such as selecting longest paths and handling inconsistencies that can be transitive when merging states.
This document summarizes Hannes Voigt's presentation on graph abstraction. It discusses matching patterns to bind variables, and using those variables to construct new graph elements through production patterns. It provides examples of simple graph construction and aggregation through grouping. The goal is to introduce an intuitive way to abstract and construct new subgraphs through pattern matching and variable bindings.
This document discusses Cypher for Gremlin, which provides the ability to run Cypher queries against Gremlin databases. It covers the Gremlin and Cypher query languages, examples of translating Cypher to Gremlin, and the Java APIs for client-side and server-side translation of Cypher to Gremlin.
Academic research on graph processing: connecting recent findings to industri...openCypher
The document discusses graph processing and querying in the context of industrial technologies and academic research. It provides an overview of different types of graph queries, including OLTP, analytics, OLAP, local queries and global queries. It also describes benchmarks for evaluating graph databases, including the LDBC Social Network Benchmark and Graphalytics Benchmark. The document discusses techniques for efficiently evaluating graph queries, including using local search to match patterns in a graph.
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
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Building Retrieval-Augmented Generation (RAG) systems with open-source and custom AI models is a complex task. This talk explores the challenges in productionizing RAG systems, including retrieval performance, response synthesis, and evaluation. We’ll discuss how to leverage open-source models like text embeddings, language models, and custom fine-tuned models to enhance RAG performance. Additionally, we’ll cover how BentoML can help orchestrate and scale these AI components efficiently, ensuring seamless deployment and management of RAG systems in the cloud.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
How to Get CNIC Information System with Paksim Ga.pptxdanishmna97
Pakdata Cf is a groundbreaking system designed to streamline and facilitate access to CNIC information. This innovative platform leverages advanced technology to provide users with efficient and secure access to their CNIC details.
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During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
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See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
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- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
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“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
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GraphSummit Singapore | The Future of Agility: Supercharging Digital Transfor...Neo4j
Leonard Jayamohan, Partner & Generative AI Lead, Deloitte
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1. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Formal Specification of Cypher
Nadime Francis
University of Edinburgh
Wednesday, May, 10th
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2. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Property Graphs
Person, Postdoc
name : ‘Nadime’
institute : ‘UoE’
Person, Professor
name : ‘Leonid’
institute : ‘UoE’
knows
since : 2010
colleague
since : 2015
A property graph is a tuple G = (N, R, s, t, ι, λ, τ), where:
N ⊆ N: finite set of nodes
R ⊆ R: finite set of relationships
s : R → N: maps each relationship to its source
t : R → N: maps each relationship to its target
ι : (N ∪ R) × K → V: maps each x and k to x.k.
λ : N → 2L: associates a set of label to each node
τ : R → T : associates a type to each relationship
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3. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Records and Tables
A record is a tuple with named fields : (a1 : v1, . . . , an : vn).
A table is a bag of uniform records.
Example:
(a : 1, b : 3), (a : ‘oCIM 2’, b : ‘London’),
(a : ‘oCIM’, b : ‘Walldorf’), (a : 1, b : 3)
a b
1 3
‘oCIM 2’ ‘London’
‘oCIM’ ‘Walldorf’
1 3
=
a b
‘oCIM’ ‘Walldorf’
1 3
‘oCIM 2’ ‘London’
1 3
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4. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
An Example
MATCH (n : Person) − [: knows]−> (m : Person)
WHERE n.institute = m.institute
RETURN n.name, m.name, n.institute AS institute
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5. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
An Example
MATCH (n : Person) − [: knows]−> (m : Person)
WHERE n.institute = m.institute
RETURN n.name, m.name, n.institute AS institute
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6. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
An Example
MATCH (n : Person) − [: knows]−> (m : Person)
WHERE n.institute = m.institute
RETURN n.name, m.name, n.institute AS institute
n m
{name : ‘Nadime’, institute : ‘UoE’} {name : ‘Leonid’, institute : ‘UoE’}
{name : ‘Paolo’, institute : ‘UoE’} {name : ‘Nadime’, institute : ‘UoE’}
{name : ‘Nadime’, institute : ‘UoE’} {name : ‘Stefan’, institute : ‘Neo’}
{name : ‘Alastair’, institute : ‘Neo’} {name : ‘Stefan’, institute : ‘Neo’}
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7. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
An Example
MATCH (n : Person) − [: knows]−> (m : Person)
WHERE n.institute = m.institute
RETURN n.name, m.name, n.institute AS institute
n m
{name : ‘Nadime’, institute : ‘UoE’} {name : ‘Leonid’, institute : ‘UoE’}
{name : ‘Paolo’, institute : ‘UoE’} {name : ‘Nadime’, institute : ‘UoE’}
{name : ‘Alastair’, institute : ‘Neo’} {name : ‘Stefan’, institute : ‘Neo’}
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8. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
An Example
MATCH (n : Person) − [: knows]−> (m : Person)
WHERE n.institute = m.institute
RETURN n.name, m.name, n.institute AS institute
n.name m.name institute
‘Nadime’ ‘Leonid’ UoE
‘Paolo’ ‘Nadime’ UoE
‘Alastair’ ‘Stefan’ Neo
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9. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
Q =
(α) MATCH (n : Person) − [: knows]−> (m : Person)
(β) WHERE n.institute = m.institute
(γ) RETURN n.name, m.name, n.institute AS institute
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10. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
Q =
(α) MATCH (n : Person) − [: knows]−> (m : Person)
(β) WHERE n.institute = m.institute
(γ) RETURN n.name, m.name, n.institute AS institute
Operations
[[op]]G : Tables → Tables
Semantics of a query by composition
Ex: [[Q]]G = [[α]]G ◦ [[β]]G ◦ [[γ]]G
Answers to Q on G: [[Q]]G ({})
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11. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Operations and Expressions
Q =
(α) MATCH (n : Person) − [: knows]−> (m : Person)
(β) WHERE n.institute = m.institute
(γ) RETURN n.name, m.name, n.institute AS institute
Operations
[[op]]G : Tables → Tables
Semantics of a query by composition
Ex: [[Q]]G = [[α]]G ◦ [[β]]G ◦ [[γ]]G
Answers to Q on G: [[Q]]G ({})
Expressions
[[exp]]G,u ∈ V where u is a record, giving binding to variables
Ex: [[β]]G (T) = u ∈ T | [[n.institute = m.institute]]G,u = true
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12. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Pattern Matching
Rigid pattern satisfaction
Rigid path pattern: no variable length edge patterns.
Ex: (n : Person) − [: knows ∗ 2]−> () − [: likes]−> (m : Movie)
Unique way for a path p to satisfy a rigid pattern π wrt G, u.
Notation: (p, G, u) |= π
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13. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Pattern Matching
Rigid pattern satisfaction
Rigid path pattern: no variable length edge patterns.
Ex: (n : Person) − [: knows ∗ 2]−> () − [: likes]−> (m : Movie)
Unique way for a path p to satisfy a rigid pattern π wrt G, u.
Notation: (p, G, u) |= π
Variable-length paths and free variables
rigid(π) = {π | π is rigid and π π }
Ex: () − [∗2]−> () − [∗4]−> () () − [∗1..3]−> () − [∗]−> ()
free(π, u): all names that occur in π and not in u
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14. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Pattern Matching
Rigid pattern satisfaction
Rigid path pattern: no variable length edge patterns.
Ex: (n : Person) − [: knows ∗ 2]−> () − [: likes]−> (m : Movie)
Unique way for a path p to satisfy a rigid pattern π wrt G, u.
Notation: (p, G, u) |= π
Variable-length paths and free variables
rigid(π) = {π | π is rigid and π π }
Ex: () − [∗2]−> () − [∗4]−> () () − [∗1..3]−> () − [∗]−> ()
free(π, u): all names that occur in π and not in u
[[MATCH π]]G (T) =
π ∈rigid(π)
u∈T, p∈paths
(u, u )
u is uniform with free(π , u)
and (p, G, (u, u )) |= π
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16. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Nulls in Patterns
MATCH (n : Person {name : null})
RETURN (n)
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17. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Nulls in Patterns
MATCH (n : Person {name : null})
RETURN (n)
1 Every node n with a name property?
2 Every node n such that n.name IS NULL = true?
3 Nothing?
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18. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Nulls in Patterns
MATCH (n : Person {name : null})
RETURN (n)
1 Every node n with a name property?
2 Every node n such that n.name IS NULL = true?
3 Nothing!
Because Q is actually equivalent to:
MATCH (n : Person)
WHERE n.name = null
RETURN (n)
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19. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Map Comparisons
When does {k1 : v1, . . . , kn : vn} = { 1 : w1, . . . , m : wm} return
true, false or null?
{name : null} = {}
{name : null} = {name : null}
{a : 1, b : 2} = {b : 2, a : 1}
{a : 1, a : 2} = {a : 2}
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20. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Map Comparisons
When does {k1 : v1, . . . , kn : vn} = { 1 : w1, . . . , m : wm} return
true, false or null?
false
{name : null} = {}
true
{name : null} = {name : null}
true
{a : 1, b : 2} = {b : 2, a : 1}
true
{a : 1, a : 2} = {a : 2}
Neither purely syntactic, nor ∀k, m1.k = m2.k.
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21. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Setting Properties using a Map
WITH {name : null} AS map
CREATE (n)
SET n = map
RETURN (n)
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22. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Setting Properties using a Map
WITH {name : null} AS map
CREATE (n)
SET n = map
RETURN (n)
Returns n as {}.
The property map of n is not equal to the map it was set to.
In particular, n {.∗} = map returns false.
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23. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
MATCH with no Free Variables
MATCH ()
RETURN ∗
MATCH ()
RETURN 1
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24. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
MATCH with no Free Variables
Fail
MATCH ()
RETURN ∗
RETURN ∗ is not allowed with
no variable in scope.
Pass
MATCH ()
RETURN 1
Returns as many copies of 1
as nodes in the database.
After MATCH (), the active table is a bag containing multiple copies
of the empty record.
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25. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Incomplete and Inconsistent Cases
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26. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Repeating UNWINDs
UNWIND [1, 2, 3] AS r
UNWIND r AS s
RETURN s
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
RETURN s
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27. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Repeating UNWINDs
Fail
UNWIND [1, 2, 3] AS r
UNWIND r AS s
RETURN s
Type mismatch, expected List
but was Integer.
Pass
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
RETURN s
Returns a column with 1, 2
and 3 as rows.
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28. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Repeating UNWINDs
Fail
UNWIND [1, 2, 3] AS r
UNWIND r AS s
RETURN s
Type mismatch, expected List
but was Integer.
Pass
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
RETURN s
Returns a column with 1, 2
and 3 as rows.
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
UNWIND s AS t
UNWIND t AS u
RETURN u
Actually works, and returns a column with 1, 2 and 3 as rows.
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29. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Violating Cyphermorphism
Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
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30. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Violating Cyphermorphism
Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
Error: cannot use the same relationship variable ‘r’ for multiple
patterns.
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31. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Violating Cyphermorphism
Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
Error: cannot use the same relationship variable ‘r’ for multiple
patterns.
Q1 =
MATCH (x) − [r∗] − (y) − [r∗] − (z)
RETURN x, y, z
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32. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Violating Cyphermorphism
Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
Error: cannot use the same relationship variable ‘r’ for multiple
patterns.
Q1 =
MATCH (x) − [r∗] − (y) − [r∗] − (z)
RETURN x, y, z
Works and enforces the paths from x to y and from y to z to use
the same sequence of relationships, violating Cyphermorphism.
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33. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Violating Cyphermorphism
Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
Error: cannot use the same relationship variable ‘r’ for multiple
patterns.
Q1 =
MATCH (x) − [r∗] − (y) − [r∗] − (z)
RETURN x, y, z
Works and enforces the paths from x to y and from y to z to use
the same sequence of relationships, violating Cyphermorphism.
It is not included in the query Q2 below:
Q1 =
MATCH (x) − [r∗] − (y) − [s∗] − (z)
RETURN x, y, z
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34. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Two Small Issues: Nulls as Indices and Keys, and Naming
RETURN 1 AS ‘0‘, 0
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35. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Two Small Issues: Nulls as Indices and Keys, and Naming
Fail
RETURN 1 AS ‘0‘, 0
Multiple result columns with the same name are not supported.
Need to specify how expression naming is handled.
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36. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Two Small Issues: Nulls as Indices and Keys, and Naming
Fail
RETURN 1 AS ‘0‘, 0
Multiple result columns with the same name are not supported.
Need to specify how expression naming is handled.
[1, 2, 3][null] [1, 2, 3][null..4]
{name : ‘Nadime’} [null]
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37. Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Two Small Issues: Nulls as Indices and Keys, and Naming
Fail
RETURN 1 AS ‘0‘, 0
Multiple result columns with the same name are not supported.
Need to specify how expression naming is handled.
Fail
[1, 2, 3][null] [1, 2, 3][null..4]
{name : ‘Nadime’} [null]
No error, never terminates.
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