Tetration Analytics - Network Analytics & Machine Learning Enhancing Data Center Security and Operations

Total Page:16

File Type:pdf, Size:1020Kb

Tetration Analytics - Network Analytics & Machine Learning Enhancing Data Center Security and Operations Tetration Analytics - Network Analytics & Machine Learning Enhancing Data Center Security and Operations Michael Herbert Principal Engineer INSBU BRKACI-2040 Okay what does Tetration Mean? • Tetration (or hyper-4) is the next hyperoperation after exponentiation, and is defined as iterated exponentiation • It’s bigger than a Google [sic] (Googol) • And yes the developers are a bunch of mathematical geeks BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 3 What if you could actually look at every process and every data packet header that has ever traversed the network without sampling? BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 4 Cisco Tetration Analytics Pervasive Sensor Framework Provides correlation of data sources across entire application infrastructure Enables identification of point events and provides insight into overall systems behavior Monitors end-to-end lifecycle of application connectivity BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 5 Cisco Tetration Analytics Policy Discovery and Observation APPLICATION WORKSPACES Public Cloud Private Cloud Cisco Tetration Analytics™ Application Segmentation Policy © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Profile and Context Driven Application Segmentation 1. Real-time Asset Tagging 2. Policy Workflows 3. Policy Enforcement (Role Based and Hierarchical) Cisco Tetration Application Insights (ADM) No Need to Tie Policy + to IP Address and Cisco Tetration Sensors Tag and Label-Based Add-on Policy Port (For Example, Mail Filters) Cisco Tetration Customer Defined Platform Performs the Translation Compliance Monitoring Enforcement Public Cloud Bare Metal Virtual Cisco ACITM* Traditional Network* © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Tetration Analytics: Open Access NORTHBOUND NORTHBOUND NORTHBOUND APPLICATION CONSUMERS CONSUMERS Kafka Broker Programmatic Message Tetration Interface Publish Apps Cisco Tetration Analytics Platform REST API Push Notification Tetration Apps Tetration flow search Out-of-box events Access to data lake Sensor management User defined events Write your own application © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Tetration Analytics Platform Architecture - Sensors Tetration Analytics Architecture Overview Data Collection Analytics Engine Open Access Software Sensor and Web GUI Enforcement Cisco Embedded REST API Network Sensors Tetration (Telemetry Only) Analytics Event Notification Cluster Third Party Sources (Configuration Data) Tetration Apps Self Managed Cluster No Hadoop / Data Science Background Needed Easy Integration via Open interfaces One Touch Deployment No External Storage Needed Open Data Lake (via Tetration Apps) BRKACI-2060 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Traditional Monitoring Is Showing Its Age Not suited for Modern Network and Security Operations Where Data Is Created Where Data Is Useful SNMP SNMP Server Non Syslog Real Syslog Collector time Storage & Analysis CLI Strong burden on Scripts back-end Normalize different encodings, transports, data models, timestamps BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 11 Data Granularity Needs to Improve One Minute SNMP Polling Telemetry – 10 Second Push SNMP – 1 Minute Polling © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Data Granularity Needs to Improve 10 Second SW Process Push Telemetry – 10 Second Push © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Data Granularity Needs to Improve Sub Second HW/SW Push © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Data Granularity Needs to Improve Type of Problems Customers are Looking to Address Workload Placement Service Level Monitoring ADM Security and Policy Enforcement Microburst Detection Traffic Engineering Capacity Planning Troubleshooting & Remediation (Self Driving) On-Change <= 1 sec ~10s sec ~minutes-hours Resolution = Frequency of Data Collection © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Processing on the Source Device is Expensive e.g. Consider Flow Collection Efficiency 512K Sampled Flow Cache with Flow Flow Data streaming export Table • Collect and Keep all Flow Data in the • Maintain a small ‘cache’ and Local Hardware or Software Flow export the cache at a high data Table • Sampling Flows Reduces rate • Size of the Table depends on the Cost of the Telemetry but • Shift the cost of aggregation to Data Rates and Connectivity Density Reduces Accuracy backend resources • BW is Growing Faster than Memory • Aggregate ‘Flow Table’ can be (Cost of Flow Entry per Gbps is not much larger flat) © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public The Richer the Data Sources the Better More Data == Better Interpolation Lamp Sensor Plug Sensor Heater © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public The Richer the Data Sources the Better You don’t always know what you need in advance • On-Box Filtering Loses Data • Can’t Change Your Mind About What’s Important Later • Can’t Scale Out Embedded Processing • Compression (Lossless) is Good • Massive Amounts of Data Motivate the Shift in Collection • Bulk Collection is Efficient • Bulk Processing/Export Not So Much © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Streaming Telemetry is a game changer Monitoring becomes a big data problem Where Data Is Created Where Data Is Useful Removing limitations and complexity • Streaming paradigm Real time • Dense Sensor Framework • Increased Data Granularity Volume – Scale of Data Velocity – Analysis of Streaming Data • Update on every event Variety – Different Forms of Data • Multiple Data Sources Big Data and Machine Learning Problem © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Pervasive Sensors Software Sensors Network Sensors Third Party Sources Available Now Next Generation 9K switches 3rd party Data Sources Linux VM Asset Tagging Nexus 9200-X Load Balancers Windows Server VM Bare Metal IP Address Management (Linux and Windows Server) Nexus 9300- CMDB Universal* EX/FX (Basic Sensor for other OS) … *Note: No per-packet Telemetry, Not an enforcement point New! Enforcement Point (Software agents) Low CPU Overhead (SLA enforced) Highly Secure (Code Signed, Authenticated) Low Network Overhead (SLA enforced) Every Flow (No sampling), NO PAYLOAD BRKACI-2060 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Software Sensor Tetration Sensor Application • Runs in the Host OS, not the Hypervisor libpcap Network Stack • Access to accurate state of the application and all connectivity Driver • Not in the data path • Sits in User Space • Designed by Kernel Developers NIC • Secure • Code Signed • SLA Enforcement • CPU and BW throttling BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 21 Software Sensor Enforcement Process High Privilege Collection • When leveraging the enforcement capability an additional component is downloaded by the Cluster to the existing sensors Low Privilege Monitoring • Monitoring and Enforcement are distinct functions with distinct threads (the enforcement code does not exist in the server until explicitly pushed Cluster Link • Agent will implement privilege separation • SSL libraries would run in low privilege space High Privilege Enforcement • /proc parsing in high privilege space • Enforcement in high privilege space Low Privilege Cluster Link © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public PKI within the Cluster/Sensor • Tetration Cluster runs an internal PKI • Root CA is per cluster, inserted at Image creation • Not accessible outside the cluster • Cannot connect to an external PKI • Certificate based authentication is performed for the Control Channel • CN of the certificate is the IP address • Certificates are rotated every 60 days • Sensors are code signed • Signature Authority is Cisco’s code signing certificate • Code Signature is validated at process start BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 23 How Sensor Communicate with the Cluster the First Time? Register with web server via ssl Assign UUID Rails Register with web server via ssl Sensor Download config Config Server Send meta data to collectors Collector BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 24 Components & Communication Software Sensor Agent Communication Unix Socket Control Channel TCP-SSL 443 Tetration Cluster Software Sensor/Agent Sensor Data TCP-SSL 5640 • When used policies pushed from the cluster are pairwise signed with TS (Replay protected) between Cluster and sensor agent LINUX/Windows/… • If rules changed on the end host – Enforcer restates the rules and sends a Notification to Controller 25 BRKDCN-2040 © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public Tetration Sensor Overhead (e.g. 2263 sensors) • CPU utilization on Host Sensor based on current deployments averages < 1% • Flow collection has zero impact on switch hardware sensor CPU • Network Overhead is ~1% of observed traffic load Presentation ID © 2017 Cisco and/or its affiliates. All rights reserved. Cisco Public 26 Tetration Host Sensor Has Three Rate Limiting Modes Top Adjusted Disabled • Uses no more CPU % than • Takes the provided limit and • Use in hosts where the given limit
Recommended publications
  • Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures
    mathematics Article Introduction to Dependence Relations and Their Links to Algebraic Hyperstructures Irina Cristea 1,* , Juš Kocijan 1,2 and Michal Novák 3 1 Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000 Nova Gorica, Slovenia; [email protected] 2 Department of Systems and Control, Jožef Stefan Institut, Jamova Cesta 39, 1000 Ljubljana, Slovenia 3 Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 10, 61600 Brno, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +386-533-153-95 Received: 11 July 2019; Accepted: 19 September 2019; Published: 23 September 2019 Abstract: The aim of this paper is to study, from an algebraic point of view, the properties of interdependencies between sets of elements (i.e., pieces of secrets, atmospheric variables, etc.) that appear in various natural models, by using the algebraic hyperstructure theory. Starting from specific examples, we first define the relation of dependence and study its properties, and then, we construct various hyperoperations based on this relation. We prove that two of the associated hypergroupoids are Hv-groups, while the other two are, in some particular cases, only partial hypergroupoids. Besides, the extensivity and idempotence property are studied and related to the cyclicity. The second goal of our paper is to provide a new interpretation of the dependence relation by using elements of the theory of algebraic hyperstructures. Keywords: hyperoperation; hypergroupoid; dependence relation; influence; impact 1. Introduction In many real-life situations, there are contexts with numerous “variables”, which somehow depend on one another.
    [Show full text]
  • Grade 7/8 Math Circles the Scale of Numbers Introduction
    Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero.
    [Show full text]
  • A Child Thinking About Infinity
    A Child Thinking About Infinity David Tall Mathematics Education Research Centre University of Warwick COVENTRY CV4 7AL Young children’s thinking about infinity can be fascinating stories of extrapolation and imagination. To capture the development of an individual’s thinking requires being in the right place at the right time. When my youngest son Nic (then aged seven) spoke to me for the first time about infinity, I was fortunate to be able to tape-record the conversation for later reflection on what was happening. It proved to be a fascinating document in which he first treated infinity as a very large number and used his intuitions to think about various arithmetic operations on infinity. He also happened to know about “minus numbers” from earlier experiences with temperatures in centigrade. It was thus possible to ask him not only about arithmetic with infinity, but also about “minus infinity”. The responses were thought-provoking and amazing in their coherent relationships to his other knowledge. My research in studying infinite concepts in older students showed me that their ideas were influenced by their prior experiences. Almost always the notion of “limit” in some dynamic sense was met before the notion of one to one correspondences between infinite sets. Thus notions of “variable size” had become part of their intuition that clashed with the notion of infinite cardinals. For instance, Tall (1980) reported a student who considered that the limit of n2/n! was zero because the top is a “smaller infinity” than the bottom. It suddenly occurred to me that perhaps I could introduce Nic to the concept of cardinal infinity to see what this did to his intuitions.
    [Show full text]
  • Simple Statements, Large Numbers
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2007 Simple Statements, Large Numbers Shana Streeks University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Streeks, Shana, "Simple Statements, Large Numbers" (2007). MAT Exam Expository Papers. 41. https://digitalcommons.unl.edu/mathmidexppap/41 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Master of Arts in Teaching (MAT) Masters Exam Shana Streeks In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Gordon Woodward, Advisor July 2007 Simple Statements, Large Numbers Shana Streeks July 2007 Page 1 Streeks Simple Statements, Large Numbers Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, as defined by Wikipedia (2007). Large numbers typically refer to large positive integers, or more generally, large positive real numbers, but may also be used in other contexts. Very large numbers often occur in fields such as mathematics, cosmology, and cryptography. Sometimes people refer to numbers as being “astronomically large”. However, it is easy to mathematically define numbers that are much larger than those even in astronomy. We are familiar with the large magnitudes, such as million or billion.
    [Show full text]
  • Categorical Comprehensions and Recursion Arxiv:1501.06889V2
    Categorical Comprehensions and Recursion Joaquín Díaz Boilsa,∗ aFacultad de Ciencias Exactas y Naturales. Pontificia Universidad Católica del Ecuador. 170150. Quito. Ecuador. Abstract A new categorical setting is defined in order to characterize the subrecursive classes belonging to complexity hierarchies. This is achieved by means of coer- cion functors over a symmetric monoidal category endowed with certain recur- sion schemes that imitate the bounded recursion scheme. This gives a categorical counterpart of generalized safe composition and safe recursion. Keywords: Symmetric Monoidal Category, Safe Recursion, Ramified Recursion. 1. Introduction Various recursive function classes have been characterized in categorical terms. It has been achieved by considering a category with certain structure endowed with a recursion scheme. The class of Primitive Recursive Functions (PR in the sequel), for instance, has been chased simply by means of a carte- sian category and a Natural Numbers Object with parameters (nno in the sequel, see [11]). In [13] it can be found a generalization of that characterization to a monoidal setting, that is achieved by endowing a monoidal category with a spe- cial kind of nno (a left nno) where the tensor product is included. It is also known that other classes containing PR can be obtained by adding more struc- ture: considering for instance a topos ([8]), a cartesian closed category ([14]) or a category with finite limits ([12]).1 Less work has been made, however, on categorical characterizations of sub- arXiv:1501.06889v2 [math.CT] 28 Jan 2015 recursive function classes, that is, those contained in PR (see [4] and [5]). In PR there is at least a sequence of functions such that every function in it has a more complex growth than the preceding function in the sequence.
    [Show full text]
  • The Notion Of" Unimaginable Numbers" in Computational Number Theory
    Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers.
    [Show full text]
  • Hyperoperations and Nopt Structures
    Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles.
    [Show full text]
  • I Jornada De Investigación Sistemas.Pdf
    ISBN: 978-9942-9902-6-6 1 1 Copyright: Dirección de Investigación de la Pontificia Universidad Católica del Ecuador Sede Esmeraldas, prohibida la reproducción total o parcial de este libro por ningún medio impreso o electrónico sin el permiso previo y por escrito del dueño del copyright. PONTIFICIA UNIVERSIDAD CATÓLICA DEL ECUADOR SEDE ESMERALDAS Dirección: Espejo y subida a Santa Cruz Casilla: 08-01-0065 Teléfonos: +593 (06) 2721983 – 2721595 Email: [email protected] www.pucese.edu.ec ESMERALDAS - ECUADOR PUBLICACIÓN ELECTRÓNICA ISBN: 978-9942-9902-6-6 Los artículos incluidos en esta publicación fueron sometidos a procedimientos de admisión y revisión por pares, llevados a cabo por un comité científico nacional e internacional de alto nivel. 2 La simulación en ingeniería, transcendiendo fronteras Coordinador de la Carrera de Sistemas y Computación Víctor Xavier Quiñonez Ku Director de Investigación PUCE Esmeraldas Ignacio Carazo Ortega Comité Organizador Víctor Xavier Quiñonez Ku Jaime Paúl Sayago Heredia Luis Alberto Herrera Izquierdo Comité Editorial Pablo Antonio Pico Valencia, Universidad de Granada – España Evelin Lorena Flores García, Universidad de Almería – España Cesar Raúl García Jacas, Universidad Nacional Autónoma de México José Luis Sampietro Saquicela, Universidad de Barcelona – España Luis Dionicio Rosales Romero, Universidad de los Andes – Venezuela Juan Luis Casierra Cavada, Pontificia Universidad Católica del Ecuador Pedro Roberto Suarez Suri, Pontificia Universidad Católica del Ecuador Víctor Xavier Quiñonez
    [Show full text]
  • New Thinking About Math Infinity by Alister “Mike Smith” Wilson
    New thinking about math infinity by Alister “Mike Smith” Wilson (My understanding about some historical ideas in math infinity and my contributions to the subject) For basic hyperoperation awareness, try to work out 3^^3, 3^^^3, 3^^^^3 to get some intuition about the patterns I’ll be discussing below. Also, if you understand Graham’s number construction that can help as well. However, this paper is mostly philosophical. So far as I am aware I am the first to define Nopt structures. Maybe there are several reasons for this: (1) Recursive structures can be defined by computer programs, functional powers and related fast-growing hierarchies, recurrence relations and transfinite ordinal numbers. (2) There has up to now, been no call for a geometric representation of numbers related to the Ackermann numbers. The idea of Minimal Symbolic Notation and using MSN as a sequential abstract data type, each term derived from previous terms is a new idea. Summarising my work, I can outline some of the new ideas: (1) Mixed hyperoperation numbers form interesting pattern numbers. (2) I described a new method (butdj) for coloring Catalan number trees the butdj coloring method has standard tree-representation and an original block-diagram visualisation method. (3) I gave two, original, complicated formulae for the first couple of non-trivial terms of the well-known standard FGH (fast-growing hierarchy). (4) I gave a new method (CSD) for representing these kinds of complicated formulae and clarified some technical difficulties with the standard FGH with the help of CSD notation. (5) I discovered and described a “substitution paradox” that occurs in natural examples from the FGH, and an appropriate resolution to the paradox.
    [Show full text]
  • Ever Heard of a Prillionaire? by Carol Castellon Do You Watch the TV Show
    Ever Heard of a Prillionaire? by Carol Castellon Do you watch the TV show “Who Wants to Be a Millionaire?” hosted by Regis Philbin? Have you ever wished for a million dollars? "In today’s economy, even the millionaire doesn’t receive as much attention as the billionaire. Winners of a one-million dollar lottery find that it may not mean getting to retire, since the million is spread over 20 years (less than $3000 per month after taxes)."1 "If you count to a trillion dollars one by one at a dollar a second, you will need 31,710 years. Our government spends over three billion per day. At that rate, Washington is going through a trillion dollars in a less than one year. or about 31,708 years faster than you can count all that money!"1 I’ve heard people use names such as “zillion,” “gazillion,” “prillion,” for large numbers, and more recently I hear “Mega-Million.” It is fairly obvious that most people don’t know the correct names for large numbers. But where do we go from million? After a billion, of course, is trillion. Then comes quadrillion, quintrillion, sextillion, septillion, octillion, nonillion, and decillion. One of my favorite challenges is to have my math class continue to count by "illions" as far as they can. 6 million = 1x10 9 billion = 1x10 12 trillion = 1x10 15 quadrillion = 1x10 18 quintillion = 1x10 21 sextillion = 1x10 24 septillion = 1x10 27 octillion = 1x10 30 nonillion = 1x10 33 decillion = 1x10 36 undecillion = 1x10 39 duodecillion = 1x10 42 tredecillion = 1x10 45 quattuordecillion = 1x10 48 quindecillion = 1x10 51
    [Show full text]
  • The Strange Properties of the Infinite Power Tower Arxiv:1908.05559V1
    The strange properties of the infinite power tower An \investigative math" approach for young students Luca Moroni∗ (August 2019) Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. Paul Lockhart { \A Mathematician's Lament" Abstract In this article we investigate some "unexpected" properties of the \Infinite Power Tower 1" function (or \Tetration with infinite height"): . .. xx y = f(x) = xx where the \tower" of exponentiations has an infinite height. Apart from following an initial personal curiosity, the material collected here is also intended as a potential guide for teachers of high-school/undergraduate students interested in planning an activity of \investigative mathematics in the classroom", where the knowledge is gained through the active, creative and cooperative use of diversified mathematical tools (and some ingenuity). The activity should possibly be carried on with a laboratorial style, with no preclusions on the paths chosen and undertaken by the students and with little or no information imparted from the teacher's desk. The teacher should then act just as a guide and a facilitator. The infinite power tower proves to be particularly well suited to this kind of learning activity, as the student will have to face a challenging function defined through a rather uncommon infinite recursive process. They'll then have to find the right strategies to get around the trickiness of this function and achieve some concrete results, without the help of pre-defined procedures. The mathematical requisites to follow this path are: functions, properties of exponentials and logarithms, sequences, limits and derivatives.
    [Show full text]
  • Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures • Irina Cristea Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures
    Symmetry Classical Algebraic Fuzzy in and Hypercompositional Structures Symmetry in Classical • Irina Cristea and Fuzzy Algebraic Hypercompositional Structures Edited by Irina Cristea Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures Special Issue Editor Irina Cristea MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Special Issue Editor Irina Cristea Center for Information Technologies and Applied Mathematics, University of Nova Gorica Slovenia Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) (available at: https://www.mdpi.com/journal/symmetry/special issues/Fuzzy Algebraic Hypercompositional Structures). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03928-708-6 (Pbk) ISBN 978-3-03928-709-3 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor .....................................
    [Show full text]