'Axiom' diaporamas de présentation

Mathematical Ideas that Shaped the World

Mathematical Ideas that Shaped the World. Non-Euclidean geometry. Plan for this class . Who was Euclid? What did he do? Find out how your teachers lied to you Can parallel lines ever meet? Why did the answer to this question change the philosophy of centuries?

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Lecture 17 Software Security

Lecture 17 Software Security. modified from slides of Lawrie Brown. Secure programs. Security implies some degree of trust that the program enforces expected confidentiality, integrity, and availability. How can we look at software component and assess its security?. Secure programs.

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Making Mountains Out of Molehills The Banach-Tarski Paradox

Making Mountains Out of Molehills The Banach-Tarski Paradox. By Bob Kronberger Jay Laporte Paul Miller Brian Sikora Aaron Sinz. Introduction. Definitions Schroder-Bernstein Theorem Axiom of Choice Conclusion. Banach-Tarski Theorem.

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CS 363 Comparative Programming Languages

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Chapter 3

Chapter 3. Boolean Algebra and Logic Design Boolean Algebra Logic Gates Digital Design Implementation Technology ASICs Gate Arrays. Basic Algebraic Properties. A set is a collection of objects with a common property If S is a set and x is a member of the set S , then x ? S

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XL as an extension of the L-system formalism Tutorial "The language XL and the platform GroIMP as tools in plant mo

XL as an extension of the L-system formalism Tutorial "The language XL and the platform GroIMP as tools in plant modelling", Cottbus, December 19-20, 2007 Winfried Kurth Ole Kniemeyer Reinhard Hemmerling Gerhard Buck-Sorlin Chair for Graphics Systems, BTU Cottbus

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Axiomatic Semantics

Axiomatic Semantics. Will consider axiomatic semantics (A.S.) of IMP: <stmt> ::= skip | <assign> | <if> | <while> | <stmt>; <stmt> | <input> | <output> Only integer vars; no procedures/fns; vars declared implicitly References: Kurtz (ch. 11); Pagan (ch. 4.3)

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The Combinatorics of Voting Paradoxes

The Combinatorics of Voting Paradoxes. Noga Alon, Tel Aviv U. and Microsoft, Israel. IPAM, UCLA, Oct. 2009. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A. The Condorcet Paradox (1785): The majority may prefer A to B, B to C and C to A.

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Probability Of An Event

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4.10 Inner Product Spaces

4.10 Inner Product Spaces. In this section, we extend those concepts of R n such as: dot product of two vectors, norm of a vector, angle between vectors, and distance between points, to general vector space.

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Rank Aggregation Methods II Experiments

Rank Aggregation Methods II Experiments. CS728 Lecture 12. Recall the Rank Aggregation Problem. m candidates (a.k.a. “alternatives”) M = {1,…,m}: set of candidates n voters (a.k.a. “agents” or “judges”) N = {1,…,n}: set of voters Each voter i, has an ranking  i on M

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Proving Program Correctness

Proving Program Correctness. The Axiomatic Approach. What is Correctness?. Correctness: partial correctness + termination Partial correctness: Program implements its specification. Proving Partial Correctness. Goal: prove that program is partially correct

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Reasoning about Multiple Related Abstractions with MultiStar

Chair of Software Engineering. Reasoning about Multiple Related Abstractions with MultiStar. Stephan van Staden , Cristiano Calcagno. Introduction. Broad topic: reasoning about the correctness of OO programs In particular: well-behaved invariants Data abstractions pervade OO programs:

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Axioms

Common Notions. Axioms. Students will be able to identify the axioms used to solve algebraic equations Students will be able to solve geometric problems Axioms: a statement assumed to be true without proof. Objectives. Things that are equal to the same thing are equal to each other.

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Proofs in Algebra Honors Sec 2.7 Proving Statements

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Specifications and Morphisms

Morphism Partial-Order ? Integer E ? ? integer le ? ? ? axioms ? ? theorems. Specifications and Morphisms. Spec Partial-Order sort E op _ le_ : E , E ? Boolean axiom reflexive x le x axiom transitive x le y ? y le z ? x le z

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Synthetic Geometry

Synthetic Geometry. Leaving Certificate Higher Level Mathematics. Planes and Points. A plane is a flat surface that goes on for infinity in all directions. A point is a precise location on the plane. Usually represented by a dot. We denote points with capital letters.i.e. A,B,C etc

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Axioms

Axioms. A rule or a statement which we accept without any proof. Draw 2 points. Draw lines between these 2 points. How many can you draw? Axiom 1 : there is exactly one line through any 2 points. Ruler axiom. Draw a line between 2 points A and B. Measure from A to B and measure from B to A

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Axiom 1: Quality curriculum design is to develop and “deepen” student understanding. Quality curriculum is like a tree. The roots need to be deep to support strong growth on the surface. .

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