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The Bits and Bytes of Computer Networking Terms and definitions
Terms and definitions from The Bits and Bytes of Computer Networking
A record: The most common resource record, used to point a certain domain name at a certain IPv4 IP address
ACK flag: One of the TCP control flags. ACK is short for acknowledge. A value of one in this field means that the acknowledgment number field should be examined
Acknowledgement number: The number of the next expected segment in a TCP sequence
Ad-Hoc network: A network configuration without supporting network infrastructure. Every device involved with the ad-hoc network communicates with every other device within range, and all nodes help pass along messages
Address class system: A system which defines how the global IP address space is split up
Address Resolution Protocol (ARP): A protocol used to discover the hardware address of a node with a certain IP address
Anycast: A technique that's used to route traffic to different destinations depending on factors like location, congestion, or link health
Application layer: The layer that allows network applications to communicate in a way they understand
Application layer payload: The entire contents of whatever data applications want to send to each other
ARP table: A list of IP addresses and the MAC addresses associated with them
ASN: Autonomous System Number is a number assigned to an individual autonomous system
Asymmetric Digital Subscriber Line (ADSL): A device that establishes data connections across phone lines and different speeds for uploading and downloading data
Automatic allocation: A range of IP addresses is set aside for assignment purposes
Baud rate: A measurement of how many bits could be passed across a phone line in a second
Bit: The smallest representation of data that a computer can understand
Bluetooth: The most common short range wireless network
Border Gateway Protocol (BGP): A protocol by which routers share data with each other
Broadband: Any connectivity technology that isn't dial-up Internet
Broadcast: A type of Ethernet transmission, sent to every single device on a LAN
Broadcast address: A special destination used by an Ethernet broadcast composed by all Fs
Cable categories: Groups of cables that are made with the same material. Most network cables used today can be split into two categories, copper and fiber.
Cable modem termination system: Connects lots of different cable connections to an ISP's core network
Cable modem: A device that sits at the edge of a consumer's network and connects it to the cable modem termination syste
Cables: Insulated wires that connect different devices to each other allowing data to be transmitted over them
Caching and recursive name servers: They are generally provided by an ISP or your local network, and their purpose is to store domain name lookups for a certain amount of time.
Carrier-Sense Multiple Access with Collision Detection (CSMA/CD): CSMA/CD is used to determine when the communications channels are clear and when the device is free to transmit data.
Channels: Individual, smaller sections of the overall frequency band used by a wireless network
Client: A device that receives data from a server
CLOSE: A connection state that indicates that the connection has been fully terminated, and that no further communication is possible
CLOSE_WAIT: A connection state that indicates that the connection has been closed at the TCP layer, but that the application that opened the socket hasn't released its hold on the socket yet
Cloud computing: The concept and technological approach of accessing data, using applications, storing files, etc. from anywhere in the world as long as you have an internet connection
CNAME: A resource record used to map one domain to another
Collision domain: A network segment where only one device can communicate at a time
Computer networking: The full scope of how computers communicate with each other
Connection-oriented protocol: A data-transmission protocol that establishes a connection at the transport layer, and uses this to ensure that all data has been properly transmitted
Connectionless protocol: A data-transmission protocol that allows data to be exchanged without an established connection at the transport layer. The most common of these is known as UDP, or User Datagram Protocol
Copper cable categories : These categories have different physical characteristics like the number of twists in the pair of copper wires. These are defined as names like category (or cat) 5, 5e, or 6, and how quickly data can be sent across them and how resistant they are to outside interference are all related to the way the twisted pairs inside are arranged
Crosstalk: Crosstalk is when an electrical pulse on one wire is accidentally detected on another wire
Cyclical Redundancy Check (CRC): A mathematical transformation that uses polynomial division to create a number that represents a larger set of data. It is an important concept for data integrity and is used all over computing, not just network transmissions
Datalink layer: The layer in which the first protocols are introduced. This layer is responsible for defining a common way of interpreting signals, so network devices can communicate
Data offset field: The number of the next expected segment in a TCP packet/datagram
Data packet: An all-encompassing term that represents any single set of binary data being sent across a network link
Data payload section: Has all of the data of the protocols further up the stack of a frame
Demarcate: To set the boundaries of something
Demarcation point: Where one network or system ends and another one begins
Demultiplexing: Taking traffic that's all aimed at the same node and delivering it to the proper receiving service
Destination MAC address: The hardware address of the intended recipient that immediately follows the start frame delimiter
Destination network: The column in a routing table that contains a row for each network that the router knows about
Destination port: The port of the service the TCP packet is intended for
DHCP: A technology that assigns an IP address automatically to a new device. It is an application layer protocol that automates the configuration process of hosts on a network
DHCP discovery: The process by which a client configured to use DHCP attempts to get network configuration information
Dial-up: Uses POTS for data transfer, and gets its name because the connection is established by actually dialing a phone number
DNS zones: A portion of space in the Domain Name System (DNS) that is controlled by an authoritative name server
Domain: Used to demarcate where control moves from a top-level domain name server to an authoritative name server
Domain name: A website name; the part of the URL following www.
Domain Name System (DNS): A global and highly distributed network service that resolves strings of letters, such as a website name, into an IP address
Dotted decimal notation: A format of using dots to separate numbers in a string, such as in an IP address
DSL: Digital subscriber line was able to send much more data across the wire than traditional dial-up technologies by operating at a frequency range that didn't interfere with normal phone calls
DSLAM: Digital Subscriber Line Access Multiplexers are devices that connect multiple DSL connections to a high-speed digital communications channel
Duplex communication: A form of communication where information can flow in both directions across a cable
Duration field: Specifies how long the total frame is
Dynamic allocation: A range of IP addresses is set aside for client devices and one of these IPs is issued to these devices when they request one
Dynamic IP address: An IP address assigned automatically to a new device through a technology known as Dynamic Host Configuration Protocol
Error detection: The ability for a protocol or program to determine that something went wrong
Error recovery: The ability for a protocol or program to attempt to fix an error
ESTABLISHED: Status indicating that the TCP connection is in working order, and both sides are free to send each other data
Ethernet: The protocol most widely used to send data across individual links
Ethernet frame: A highly structured collection of information presented in a specific order
EtherType field: It follows the Source MAC Address in a dataframe. It's 16 bits long and used to describe the protocol of the contents of the frame
Exterior gateway: Protocols that are used for the exchange of information between independent autonomous systems
Fiber cable: Fiber optic cables contain individual optical fibers which are tiny tubes made of glass about the width of a human hair. Unlike copper, which uses electrical voltages, fiber cables use pulses of light to represent the ones and zeros of the underlying data
FIN: One of the TCP control flags. FIN is short for finish. When this flag is set to one, it means the transmitting computer doesn't have any more data to send and the connection can be closed
FIN_WAIT: A TCP socket state indicating that a FIN has been sent, but the corresponding ACK from the other end hasn't been received yet
Firewall: It is a device that blocks or allows traffic based on established rules
Five layer model: A model used to explain how network devices communicate. This model has five layers that stack on top of each other: Physical, Data Link, Network, Transport, and Application
Fixed allocation: Requires a manually specified list of MAC address and the corresponding IPs
Flag field: It is used to indicate if a datagram is allowed to be fragmented, or to indicate that the datagram has already been fragmented
Flat file: A collection of records/information that follow a consistent format with rules around stored values. On a host computer, one use is to have a list of network address and host name pairs (a hosts file)
Flow label field: 20-bit field that's used in conjunction with the traffic class field for routers to make decisions about the quality of service level for a specific datagram
Fragmentation: The process of taking a single IP datagram and splitting it up into several smaller datagrams
Fragmentation offset field: It contains values used by the receiving end to take all the parts of a fragmented packet and put them back together in the correct order
Frame check sequence: It is a 4-byte or 32-bit number that represents a checksum value for the entire frame
Frame control field: 16 bits long, it contains a number of sub-fields that are used to describe how the frame itself should be processed
Frequency band: A certain section of the radio spectrum that's been agreed upon to be used for certain communications
FTP: An older method used for transferring files from one computer to another, but you still see it in use today
FTTB: Fiber to the building, fiber to the business or even fiber to the basement, since this is generally where cables to buildings physically enter. FTTB is a setup where fiber technologies are used for data delivery to an individual building
FTTH: Fiber to the home. This is used in instances where fiber is actually run to each individual residents in a neighborhood or apartment building
FTTN: Fiber to the neighborhood. This means that fiber technologies are used to deliver data to a single physical cabinet that serves a certain amount of the population
FTTP: Fiber to the premises. FTTH and FTTB may both also be referred to as FTTP
FTTX: Stands for fiber to the X, where the X can be one of many things
Full duplex: The capacity of devices on either side of a networking link to communicate with each other at the exact same time
Fully qualified domain name: When you combine all the parts of a domain together
Half-duplex: It means that, while communication is possible in each direction, only one device can be communicating at a time
Handshake: A way for two devices to ensure that they're speaking the same protocol and will be able to understand each other
HDSL: High Bit-rate Digital Subscriber Lines. These are DSL technologies that provision speeds above 1.544 megabits per second
Header checksum field: A checksum of the contents of the entire IP datagram header
Header length field: A four bit field that declares how long the entire header is. It is almost always 20 bytes in length when dealing with IPv4
Hexadecimal: A way to represent numbers using a numerical base of 16
Hop limit field: An 8-bit field that's identical in purpose to the TTL field in an IPv4 header
Host file: It is a flat file that contains, on each line, a network address followed by the host name it can be referred to as
Hub: It is a physical layer device that broadcasts data to everything computer connected to it
Hybrid cloud: Used to describe situations where companies might run things like their most sensitive proprietary technologies on a private cloud or on premise while entrusting their less sensitive servers to a public cloud
Hypervisor: A piece of software that runs and manages virtual machines while also offering guests a virtual operating platform that's indistinguishable from actual hardware
IANA: The Internet Assigned Numbers Authority, is a non-profit organization that helps manage things like IP address allocation
ICMP: Internet control message protocol is used by router or remote hosts to communicate error messages when network problems prevent delivery of IP packets
ICMP payload: Piece of the packet which lets the recipient of the message knows which of their transmissions caused the error being reported
Identification field: It is a 16-bit number that's used to group messages together
Infrastructure as a Service (IaaS): A subset of cloud computing where a network and servers are provided for customers to run their services
Instantiation: The actual implementation of something defined elsewhere
Interface: For a router, the port where a router connects to a network. A router gives and receives data through its interfaces. These are also used as part of the routing table
Interior gateway: Interior gateway protocols are used by routers to share information within a single autonomous system
Internet Protocol (IP): The most common protocol used in the network layer
Internet Service Provider (ISP): A company that provides a consumer an internet connection
Internetwork: A collection of networks connected together through routers – the most famous of these being the Internet
IP datagram: a highly structured series of fields that are strictly defined
IP masquerading: The NAT obscures the sender's IP address from the receiver
IP options field: An optional field and is used to set special characteristics for datagrams primarily used for testing purposes
IPv6 tunnel: IPv6 tunnel servers on either end of a connection take incoming IPv6 traffic and encapsulate it within traditional IPv4 datagrams
IPv6 tunnel brokers: Companies that provide IPv6 tunneling endpoints for you, so you don't have to introduce additional equipment to your network
Line coding: Modulation used for computer networks
Link-local unicast address: Allow for local network segment communications and are configured based upon a host's MAC address
Listen: It means that a TCP socket is ready and listening for incoming connections
Local Area Network (LAN): A single network in which multiple devices are connected
Loopback address: An IP address that always points to itself. This type of address is used to test internal pathing through the TCP/IP protocols
MAC(Media Access Control) address: A globally unique identifier attached to an individual network interface. It's a 48-bit number normally represented by six groupings of two hexadecimal numbers
MAC filtering: Access points are configured to only allow for connections from a specific set of MAC addresses belonging to devices you trust
Mesh networks: Like ad-hoc networks, lots of devices communicate with each other device, forming a mesh if you were to draw lines for all the links between all the nodes
Metered connection: An internet connection where all data transfer usage is tracked. Cell phone plans that have a limit on data usage per month or that charge based on usage are examples of metered connections
Modulation: A way of varying the voltage of a constant electrical charge moving across a standard copper network cable
Multicast: A way of addressing groups of hosts all at once
Multicast frame: If the least significant bit in the first octet of a destination address is set to one, it means you're dealing with a multicast frame. A multicast frame is similarly set to all devices on the local network signal, and it will be accepted or discarded by each device depending on criteria aside from their own hardware MAC address
Multiplexing: It means that nodes on the network have the ability to direct traffic toward many different receiving services
MX record: It stands for mail exchange and this resource record is used in order to deliver email to the correct server
Name resolution: This process of using DNS to turn a domain name into an IP address
Network Address Translation (NAT): A mitigation tool that lets organizations use one public IP address and many private IP addresses within the network
Network layer: It's the layer that allows different networks to communicate with each other through devices known as routers. It is responsible for getting data delivered across a collection of networks
Network port: The physical connector to be able to connect a device to the network. This may be attached directly to a device on a computer network, or could also be located on a wall or on a patch panel
Network switch: It is a level 2 or data link device that can connect to many devices so they can communicate. It can inspect the contents of the Ethernet protocol data being sent around the network, determine which system the data is intended for and then only send that data to that one system
Next header field: Defines what kind of header is immediately after this current one
Next hop: The IP address of the next router that should receive data intended for the destination networking question or this could just state the network is directly connected and that there aren't any additional hops needed. Defined as part of the routing table
Node: Any device connected to a network. On most networks, each node will typically act as a server or a client
Non-metered connection: A connection where your data usage is not tracked or limited, instead you are charged a flat fee for unlimited and unrestricted usage. A Wi-Fi connection is an example of a non-metered connection
Non-routable address space: They are ranges of IPs set aside for use by anyone that cannot be routed to
NS record: It indicates other name servers that may also be responsible for a particular zone
NTP servers: Used to keep all computers on a network synchronized in time
Octet: Any number that can be represented by 8 bits
Optical Network Terminator: Converts data from protocols the fiber network can understand to those that are more traditional twisted pair copper networks can understand
Options field: It is sometimes used for more complicated flow control protocols
Organizationally Unique Identifier (OUI): The first three octets of a MAC address
OSI model: A model used to define how network devices communicate. This model has seven layers that stack on top of each other: Physical, Data Link, Network, Transport, Session, Presentation, and Application
Padding field: A series of zeros used to ensure the header is the correct total size
Pairing: When a wireless peripheral connects to a mobile device, and the two devices exchange information, sometimes including a PIN or password, so that they can remember each other
Patch panel: A device containing many physical network ports
Payload: The actual data being transported, which is everything that isn't a header
Payload length field: 16-bit field that defines how long the data payload section of the datagram is
Physical layer: It represents the physical devices that interconnect computers
Platform as a service: A subset of cloud computing where a platform is provided for customers to run their services
Point-To-Point VPN: Establishes a VPN tunnel between two sites but VPN tunneling logic is handled by network devices at either side, so that users don't all have to establish their own connections
Pointer resource record: It resolves an IP to a name
Port: It is a 16-bit number that's used to direct traffic to specific services running on a networked computer
Port forwarding: A technique where specific destination ports can be configured to always be delivered to specific nodes
Port preservation: A technique where the source port chosen by a client, is the same port used by the router
Preamble: The first part of an Ethernet frame, it is 8 bytes or 64 bits long and can itself be split into two sections
Presentation layer: It is responsible for making sure that the unencapsulated application layer data is actually able to be understood by the application in question
Private cloud: When a company owns the services and the rest of the cloud infrastructure, whether on-site or in a remote data center
Protocol: A defined set of standards that computers must follow in order to communicate properly is called a protocol
Protocol field: A protocol field is an 8-bit field that contains data about what transport layer protocol is being used
Proxy service: A server that acts on behalf of a client in order to access another service
PSH flag: One of the TCP control flags. PSH is short for push. This flag means that the transmitting device wants the receiving device to push currently- buffered data to the application on the receiving end as soon as possible
Public cloud: The cloud services provided by a third party
Public DNS servers: Name servers specifically set up so that anyone can use them for free
Quad A (AAAA) record: It is very similar to an A record except that it returns in IPv6 address instead of an IPv4 address
Receiving address: The MAC address of the access point that should receive the frame
Recursive name servers: Servers that perform full DNS resolution requests
Registrar: An organization responsible for assigning individual domain names to other organizations or individuals
Reverse lookup zone files: They let DNS resolvers ask for an IP, and get the FQDN associated with it returned
Reverse proxy: A service that might appear to be a single server to external clients, but actually represents many servers living behind it
Round robin: It is a concept that involves iterating over a list of items one by one in an orderly fashion
Router: A device that knows how to forward data between independent networks
Routing protocols: Special protocols the routers use to speak to each other in order to share what information they might have
RST flag: One of the TCP control flags. RST is short for reset. This flag means that one of the sides in a TCP connection hasn't been able to properly recover from a series of missing or malformed segments
Sequence control field: A field that is 16 bits long and mainly contains a sequence number used to keep track of ordering the frames
Sequence number: A 32-bit number that's used to keep track of where in a sequence of TCP segments this one is expected to be
Server: A device that provides data to another device that is requesting that data, also known as a client
Server or Service: A program running on a computer waiting to be asked for data
Service type field: A eight bit field that can be used to specify details about quality of service or QoS technologies
Session layer: The network layer responsible for facilitating the communication between actual applications and the transport layer
Short-range wireless network: It is what mobile devices uses to connect to their peripherals
Simplex communication: A form of data communication that only goes in one direction across a cable
Socket: The instantiation of an endpoint in a potential TCP connection
Software as a Service (SaaS): A way of licensing the use of software to others while keeping that software centrally hosted and managed
Source MAC address: The hardware address of the device that sent the ethernet frame or data packet. In the data packet it follows the destination MAC address
Source port: A high numbered port chosen from a special section of ports known as ephemeral ports
SRV record: A service record used to define the location of various specific services
Start Frame Delimiter (SFD): The last byte in the preamble, that signals to a receiving device that the preamble is over and that the actual frame contents will now follow
Start of authority: A declaration of the zone and the name of the name server that is authoritative for it
Static IP address: An IP address that must be manually configured on a node
Subnet mask: 32-bit numbers that are normally written as four octets of decimal numbers
Subnetting: The process of taking a large network and splitting it up into many individual smaller sub networks or subnets
Symmetric Digital Subscriber Line (SDSL): A device that establishes data connections across phone lines and has upload and download speeds that are the same
SYN flag: One of the TCP flags. SYN stands for synchronize. This flag is used when first establishing a TCP connection and make sure the receiving end knows to examine the sequence number field
SYN_RECEIVED: A TCP socket state that means that a socket previously in a listener state, has received a synchronization request and sent a SYN_ACK back
SYN_SENT: A TCP socket state that means that a synchronization request has been sent, but the connection hasn't been established yet
T-Carrier technologies: Technologies Invented to transmit multiple phone calls over a single link. Eventually, they also became common transmission systems to transfer data much faster than any dial-up connection could handle
TCP checksum: A mechanism that makes sure that no data is lost or corrupted during a transfer
TCP segment: A payload section of an IP datagram made up of a TCP header and a data section
TCP window: The range of sequence numbers that might be sent before an acknowledgement is required
Time-To-Live field (TTL): An 8-bit field that indicates how many router hops a datagram can traverse before it's thrown away
Top Level Domain (TLD): The top level of the DNS or the last part of a domain name. For example, the "com" in www.weather.com
Total hops: The total number of devices data passes through to get from its source to its destination. Routers try to choose the shortest path, so fewest hops possible. The routing table is used to keep track of this
Total length field: A 16-bit field that indicates the total length of the IP datagram it's attached to
Traffic class field: An 8-bit field that defines the type of traffic contained within the IP datagram and allows for different classes of traffic to receive different priorities
Transmission Control Protocol (TCP): The data transfer protocol most commonly used in the fourth layer. This protocol requires an established connection between the client and server
Transmitter address: The MAC address of whatever has just transmitted the frame
Transport layer: The network layer that sorts out which client and server programs are supposed to get the data
TTL: The lifetime limit of data given in seconds. This number can be configured by the owner of a domain name for how long a name server is allowed to cache in entry before it should discard it and perform a full resolution again
Twisted pair cable: The most common type of cabling used for connecting computing devices. It features pairs of copper wires that are twisted together
Two-factor authentication: A technique where more than just a username and password are required to authenticate. Usually, a short-lived numerical token is generated by the user through a specialized piece of hardware or software
TXT record: It stands for text and was originally intended to be used only for associating some descriptive text with a domain name for human consumption
Types of DNS servers: There are five primary types of DNS servers; caching name servers, recursive name servers, root name servers, TLD name servers, and authoritative name servers
Unicast transmission: A unicast transmission is always meant for just one receiving address
Urgent pointer field: A field used in conjunction with one of the TCP control flags to point out particular segments that might be more important than others
URG flag: One of the TCP control flags. URG is short for urgent. A value of one here indicates that the segment is considered urgent and that the urgent pointer field has more data about this
User Datagram Protocol (UDP): A transfer protocol that does not rely on connections. This protocol does not support the concept of an acknowledgement. With UDP, you just set a destination port and send the data packet
Version field: First field in an IP header that specifies the version of IP
Virtual LAN (VLAN): It is a technique that lets you have multiple logical LANs operating on the same physical equipment
Virtual Private Network (VPN): A technology that allows for the extension of a private or local network, to a host that might not work on that same local network
Virtualization: A single physical machine called a host runs many individual virtual instances called guests
VLAN header: A piece of data that indicates what the frame itself is. In a data packet it is followed by the EtherType
Wide area network: Acts like a single network but spans across multiple physical locations. WAN technologies usually require that you contract a link across the Internet with your ISP
Wi-Fi Protected Access (WPA): A security program that uses a 128-bit key to protect wireless computer networks, which makes it more difficult to crack than WEP
Wired Equivalence Privacy (WEP): An encryption technology that provides a very low level of privacy. WEP should really only be seen as being as safe as sending unencrypted data over a wired connection
Wireless access point: A device that bridges the wireless and wired portions of a network
Wireless LANS (WLANS): One or more access points act as a bridge between a wireless and a wired network
Wireless networking: Networks you connect to through radios and antennas
Zone Files: Simple configuration files that declare all resource records for a particular zone
Terms and definitions from The Bits and Bytes of Computer Networking A A record: The most common resource record, used to point a certain domain


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text:
ANAW, ZEDEKIAH B. ABRAHAM:
(Redirected from ZEDEKIAH BEN ABRAHAM.)
By: Moses Beer
Author of ritualistic works; younger brother of Benjamin b. Abraham Anaw; lived at Rome in the thirteenth century; received his Talmudic training not only in Rome, but also in Germany, where he was the pupil of R. Jacob of Würzburg, and possibly also of Abigdor Cohen of Vienna. He owes his great reputation not to any original research, but to a compilation on the ritual to which he gave the title "Shibbole ha-Leḳeṭ" (Ears of Gleaning). It is divided into three hundred and seventy-two paragraphs, included in the following twelve sections, treating of the laws, regulations, and ceremonies relating to prayers, Sabbath, benedictions, new moon, feast of dedication, Purim, Passover, semi-holy days, fasts, New-year, Day of Atonement, and Tabernacles. Appended to the work are several treatises and responsa on miscellaneous religious and legal matters, such as circumcision, mourning rites, fringes, slaughtering, inheritance, and interest. As the title indicates, and as the author never fails to point out, the work was culled from many older authorities, such as "Halakot Gedolot," "Pardes," Alfasi, Isaac b. Abba Mari, Zerahiah ha-Levi, Isaiah di Trani, etc. To these extracts from other authorities the work owes its vogue. The preface, written in a pure and vigorous Hebrew, is introduced by a short acrostic.
But Zedekiah did not restrict himself to the mere work of a compiler. He systematized his material skilfully, gave it a concise as well as popular form, and judiciously discriminated between conflicting opinions and decisions, giving preference to those that seemed to him true. For this procedure he apologized modestly in his preface with an anecdote, in substantially the following terms: A philosopher, when asked how he dared to oppose the great men of the past, answered, "We fully acknowledge the greatness of our old authorities and the insignificance of ourselves. But we are in the position of pygmies that ride on the shoulders of giants. Pygmies though we are, we see farther than the giants when we use their knowledge and experience."
Additions to the "Shibbole ha-Leḳeṭ" were made by Zedekiah himself, in a work the title of which is no longer known: these additions also contain a large number of responsa. It is usually cited, however, as "Issur wa-Hetter" (Things Forbidden and Allowed), and has not yet been printed.
A complete edition of the "Shibbole ha-Leḳeṭ" was published by Solomon Buber in 1886 at Wilna. The editor wrote a comprehensive introduction to it, containing an analysis of the work. Abridgments of it were published much earlier: Venice (Daniel Bomberg), 1545; Dubnov, 1793; Salonica, 1795. Furthermore, it was plagiarized and published in a condensed form under the title "Tanya," or "Tanya Rabbati," which went through four editions: Mantua, 1514; Cremona, 1565; Zolkiev, 1800; Szydlikov, 1836. A third abridgment, entitled "Ma'aseh ha-Geonim" (The Work of Old Authorities), circulated in manuscript and is extant in the Bodleian Library, Oxford.
Anaw was in correspondence with Abigdor Cohen, Meir of Rothenburg, and Abraham ben Joseph ofPesaro. Very often he mentions his senior contemporary, Isaiah di Trani (the Elder), to whose Bible commentary Anaw in 1297 wrote glosses.
Buber's Introduction to Shibbole ha-Leḳeṭ, Wilna, 1886;
Schorr, in Zion, i. 93 et seq.;
Vogelstein and Rieger, Gesch. d. Juden in Rom, i. 382 et seq.;
Berliner, Gesch. d. Juden in Rom, ii. 55;
Steinschneider, Cat. Bodl. No. 7449;
Michael, Or ha-Ḥayyim, No. 1169;
Güdemann, Gesch. des Erziehungswesens d. Juden in Italien, pp. 192, 193.
M. B.

redpajama_set_name: RedPajamaCommonCrawl


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text:
Guardiola ´not surprised at all´ by Man Utd challenge: ´It´s normal´
Pep Guardiola always expected Manchester United would challenge Manchester City again as the sides head into Saturday's derby separated by only four points.
That gap was eight points after their previous meeting in early October, which City sensationally won 6-3 at the Etihad Stadium.
But United are the Premier League's form team, winning 15 of their subsequent 18 games in all competitions and losing just once.
Erik ten Hag has improved the Old Trafford outfit to the point they could really pile the pressure on second-placed City with a victory this weekend.
Guardiola was asked about United's progress ahead of that match and replied: "It's normal. A new manager arrived, many things must be settled, then the improvement of the team is clear.
"It's not just in terms of results, it's everything."
United have improved "a lot", he accepted, but a Red Devils revival was always on the cards at some stage.
"What happened in the previous seasons is a Man United problem, so I don't know, but now it's normal that United is there," Guardiola said. "Still there, top, fighting.
"The position is there because it's the position Man United has to be.
6-3 #ManCity pic.twitter.com/zouHNXtFZ9
— Manchester City (@ManCity) October 2, 2022
"For me, did they surprise me? No. Must be there? Yes. Alongside the other teams? Yes. That's why this competition is so difficult, but I'm not surprised at all."
Guardiola, entering his 500th top-flight match as a coach, said earlier in the week he had "a few ideas, thoughts, ridiculous ones" for taking on United, but he was in less entertaining mood following an EFL Cup exit to Southampton.
"A joke," he said of those comments. "It was a joke."
With the defeat to Saints fresh in his mind, denying him a fifth success in that competition, Guardiola was similarly unimpressed by discussion of the title race as City head into the weekend five points behind leaders Arsenal.
"It's not a knockout game, so there are many games still to play," he answered on the topic.
"Yes, it's important, it's always important against United, but it's a football game like many that are going to happen in the future.


text:
\section{Introduction}
Quasi-arithmetic means were introduced as a generalization of Power Means in 1920s/30s in a series of nearly simultaneous papers \cite{Kno28,Def31,Kol30,Nag30}.
For an interval $I$ and a continuous and strictly monotone function $f \colon I \to \R$ (from now $I$ stands for an interval and $\CM$ is a family of continuous, strictly monotone functions on $I$) we define \emph{quasi-arithmetic mean} $\QA{f} \colon \bigcup_{n=1}^\infty I^n \to I$  by
\Eq{*}{
\QA{f}(v):=f^{-1}\left( \frac{f(v_1)+f(v_2)+\cdots+f(v_n)}{n} \right), \quad (n \in \N,\: v \in I^n)\:.
}
The function $f$ is called a \emph{generator} of quasi-arithmetic mean.

It it well known that for
$I=\R_+$, $\pi_p(x):=x^p$ for $p\ne 0$ and $\pi_0(x):=\ln x$, then mean $\QA{\pi_p}$ coincides with the $p$-th power mean (this fact had been already noticed by Knopp \cite{Kno28}).

There were a number of results related to quasi-arithmetic means. For example one can define  the preorder on $\CM$ as follows:
\Eq{*}{
f \prec g \iff \QA{f}(v) \le \QA{g}(v) \text{ for all } v \in \bigcup_{n \in \N}I^n.
}
It is well know (see for example \cite{HarLitPol34} ) that $\QA{f}=\QA{g}$ if and only if their generators are affine transformation of each other, i.e. there exist $\alpha,\,\beta \in \R$ such that $g=\alpha f + \beta$. Therefore it is natural to define relation $\sim$ on $\CM$ by
\Eq{*}{
f \sim g \iff \QA{f} = \QA{g}\:.
}

Furthermore it is easy to check that $\prec$ induces a partial order on $\CM /_\sim$\:. This order has a lot of interesting properties (see for example results by Cargo-Shisha \cite{CarShi64,CarShi69} and by the author \cite{Pas1710}).

As we are going to elaborate some lattice properties of quasi-arithmetic means, we need to introduce supremum and infimum of any subset in this family.
First of all, when we have one element only, we can naturally define the set of all functions generating a bigger quasi-arithmetic mean $U_f:=\{ s \in \CM \colon f \prec s\}$.
Then, for every subfamily $\calF \subset \CM$ we define a function $\UQA{\calF} \colon \bigcup_{n=1}^\infty I^n \to I$ by
\Eq{*}{
\UQA{\calF}(v) :=
\begin{cases}
\inf \big\{\QA{s}(v) \colon s \in \bigcap\limits_{f \in \calF} U_f\big\} & \text{ if }\bigcap\limits_{f \in \calF} U_f \ne \emptyset,\\[5mm]
\max(v) & \text{ otherwise.}
\end{cases}
}
Notice that $\UQA{\calF}$ is a mean, while $U_f$ is a family of functions. Similarly one can define $L_f:=\{ s \in \CM \colon f \succ s\}$ and, for $\calF \subset \CM$, a mean $\LQA{\calF} \colon \bigcup_{n=1}^\infty I^n \to I$ by
\Eq{*}{
\LQA{\calF}(v) :=
\begin{cases}
\sup \big\{\QA{s}(v) \colon s \in \bigcap\limits_{f \in \calF} L_f\big\} & \text{ if }\bigcap\limits_{f \in \calF} L_f \ne \emptyset,\\[5mm]
\min(v) & \text{ otherwise.}
\end{cases}
}

Obviously for every $\calF$ both $\LQA{\calF}$ and $\UQA{\calF}$ are monotone and symmetric. Furthermore whenever they are quasi-arithmetic means, then generatiors of $\LQA \calF$ and $\UQA \calF$ are infimum and supremum of $\calF$, respectively (with respect to the partial ordering $\prec$). Therefore it is very natural to ask about possible sufficient conditions to $\LQA\calF$ and $\UQA{\calF}$ to be a quasi-arithmetic mean and, knowing this, about its generator.

In the present paper we will focus on a family of means generated by $\mathcal{C}^2(I)$ functions having nowhere vanishing first derivative (from now on we will denote a family of all such functions by $\Sm$\,). Our main results states that whenever we have two such functions form $\Sm$ then they have both supremum and infimum. These supremum and infimum are quasi-arithmetic means generated by functions from $\Sm$. Furthermore we present a formulae to calculate them in term of ordinary diffential equations (equalities on Arrow-Pratt indexes).

\subsection{Properties of quasi-arithmetic means}
In this section we will present some selection of known results concerning quasi-arithmetic means. It fact there exists a rich literature in this area (see for example \cite[chap. 4]{Bul03} and references therein) and, as a natural consequence, we will present just these results which will be used later.

First of all we list some known comparability conditions for quasi-arithmetic means

\begin{prop}
\label{prop:comp}
 Let $f,\,g \in \CM$. Then $f\prec g$ if and only if one of the following (equivalent) conditions are valid
 \begin{enumerate}[label={(\roman*)},series=theoremconditions]
\item \label{CC:gf^-1} $g$ is increasing and $g \circ f^{-1}$ is convex or $g$ is decreasing and $g \circ f^{-1}$ is concave,
\item \label{CC:fg^-1} $f$ is increasing and $f \circ g^{-1}$ is concave or $f$ is decreasing and $f \circ g^{-1}$ is convex.
\end{enumerate}
Moreover if $f$ and $g$ are both differentiable and $f \cdot g' \ne 0$ we obtain equvalent statements
 \begin{enumerate}[resume*=theoremconditions]
  \item\label{CC:g'/f'} $f$ and $g$ are of the same monotonicity and $g'/f'$ is nondecreasing or $f$ and $g$ are of the opposite monotonicity and $g'/f'$ is nonincreasing;
  \item\label{CC:f'/g'} $f$ and $g$ are of the same monotonicity and $f'/g'$ is nonincreasing or $f$ and $g$ are of the opposite monotonicity and $f'/g'$ is nondecreasing.
  \end{enumerate}
Additionally if $f$ and $g$ are twice differentiable we have
 \begin{enumerate}[resume*=theoremconditions]
 \item \label{CC:Mikusinski} $\tfrac{f''(x)}{f'(x)} \le \tfrac{g''(x)}{g'(x)}$ for all $x \in I$.
 \end{enumerate}
\end{prop}

Let us stress two notions which are eqivalent to all conditions above: $g \in U_f$ and $f \in L_g$. Formally, these two are not conditions of comparability, and therefore they are not listed among others.

Notice that condition \ref{CC:Mikusinski} is defined for functions in $\Sm$ only. Now let us recall some recent results concerning smoothness properties implied by comparability enclosed in \cite{Pas1710}.

\begin{lem}[\cite{Pas1710}, Theorem 7]
 \label{lem:hnonvanish}
 Let $f,\,g \in \CM$ such that $f\prec g$.
 Then
 \begin{itemize}
  \item one-sided derivatives $f'_-$ and $g'_-$  (resp. $f'_+$ and $g'_+$) exist at the same points;
  \item $f'_-$ and $g'_-$ (resp. $f'_+$ and $g'_+$) vanish at the same points.
 \end{itemize}
\end{lem}

As a simple but useful conclusion, which was not worded in \cite{Pas1710}, we get the following corollary
\begin{cor}
 \label{cor:hnonvanish}
 Let $f,\,g \in \CM$ be two differentiable, strictly monotone functions such that $f \prec g$. Then
 \Eq{*}{
 \big\{x \in I \mid f'(x)=0\big\}= \big\{x \in I \mid g'(x)=0\big\}\:.
 }
\end{cor}

Furthermore in view of \cite[Theorem 12]{Pas1710} and Corollary~\ref{cor:hnonvanish} above we can establish the following lemma.

\begin{lem}
 \label{lem:diffnonz}
  Let $f,\,g,\,h \in \CM$ such that $f \prec g \prec h$.
  If $f$ and $h$ are both differentiable at some point $x_0 \in I$, then so is $g$.

  Moreover all equalities $f'(x_0)=0$, $g'(x_0)=0$, and $h'(x_0)=0$ are pairwise equivalent.
\end{lem}

At the end of this section let us recall two results concerning convergence of means. First of them was obtained in 1990s by P\'ales \cite{Pal91}.
\begin{lem}[\cite{Pal91}, Corollary~1]
 \label{lem:Pales}
Let $f$ and $f_n$, $n \in \N$, be continuous, strictly monotone functions defined on $I$. Then $\lim\limits_{n \to \infty} \QA{f_n}=\QA{f}$ pointwise if and only if
\Eq{*}{
\lim_{n \to \infty} \frac{f_n(x)-f_n(z)}{f_n(y)-f_n(z)}=\frac{f(x)-f(z)}{f(y)-f(z)} \qquad \text{for all }x,\,y,\,z \in I \text{ with } y \ne z.
}
\end{lem}

Second one was obtained by the author in 2013 \cite{Pas13} and strenghtened in both 2015 \cite{Pas15b} and 2018 \cite{Pas18a}, for the sake of brevity we will use the 2013's version.
\begin{lem}[\cite{Pas13}, Corollary 3]
\label{lem:Pas13} Let $f$ and $f_n$ for $n \in \N$ be a functions from $\Sm$.
If $f_n''/f_n' \to f''/f'$ in $L^1(I)$ then $\QA{f_n} \to \QA{f}$ uniformly.
\end{lem}

The rest of the present note is organized as follows -- in the next section we formulate our main result and present some of its applications. Its long proof  (jointly with all relevant technicalities) is shifted to the last section (section~\ref{sec:proofmain}).

\section{\label{sec:mainres} Main result}
It is known (see \ref{CC:Mikusinski}) that (under certain smoothness assumptions) comparability of quasi-arithmetic means are closely related with the so-called Arrow-Pratt index, i.e. the operator $f \mapsto f''/f'$. Moreover it is definitelly the easiest expression among all known conditions, as it reduces comparability of quasi-arithmetic means to comparability of single-variable functions associated with them. Moreover it can be proved (directly from \ref{CC:Mikusinski}) that
\Eq{*}{
f \sim g\iff \frac{f''}{f'}=\frac{g''}{g'}\:.
}
Thus quasi-arithmetic means are generalized rather by their Arrow-Pratt indexes than generators.

We will establish, in terms of Arrow-Pratt index, the value of $\UQA{}$ for two means (i.e $\UQA{\{f,g\}}$, which will be alternatively denoted as $\UQA{f,g}$).
\begin{lem}
\label{lem:main}
 For every $f,\,g \in \Sm$ there exists $h \in \Sm$ such that $\UQA{f,g}=\QA{h}$.
 Moreover
 \Eq{def:h}{
 \frac{h''}{h'}=\max\Big(\frac{f''}{f'},\,\frac{g''}{g'} \Big).
 }
\end{lem}

Let us stress that there are two important statements which are bind in this lemma. First of all it states that the value of $\UQA{f,g}$ is a quasi-arithmetic mean. Second, this quasi-arithmetic mean is generalized by a $\mathcal{C}^2$ function wiht nowhere vanishing derivatives. Having this proved, equality \eq{def:h} is an immediate corollary of \ref{CC:Mikusinski}.

Let us now turn to a slight generalization of Lemma~\ref{lem:main} which is in fact a main result in this paper. As Lemma~\ref{lem:main} deals with two means only, there appear a natural question -- what happens when a family $\calF$ is bigger. This generalization is presented in our main theorem.

\begin{thm}
 Let $\calF \subset \Sm$.
 \begin{enumerate}
 \item If the function $G \colon I \ni x \mapsto \sup_{f \in \calF} \frac{f''(x)}{f'(x)}$ is continuous then $\UQA \calF=\QA{g}$, where $g \in \Sm$ and $g''/g'=G$.
  \item If the function $H \colon I \ni x \mapsto \inf_{f \in \calF} \frac{f''(x)}{f'(x)}$ is continuous then $\LQA \calF=\QA{h}$, where $h \in \Sm$ and $h''/h'=H$.
  \end{enumerate}
 \end{thm}

 \begin{proof}
By \cite[Corollary 2.1]{CarShi69} we know that the family of quasi-arithmetic means defined on $I$ is a separable space. Therefore we may assume that $\calF$ is countable, i.e. $\calF=(f_n)_{n \in \N}$.
Here and below we do not claim all $f_n$-s to be different, in particular this proof covers a case where $\calF$ is a finite family.

Define $M_n:=\UQA{f_1,\dots,f_n}$ and a sequence of functions $(g_n)_{n \in \N}$ from $\Sm$ such that
\Eq{*}{
\frac{g_n''}{g_n'}=\max_{i \in \{1,\dots,n\}} \frac{f_i''}{f_i'} \quad (n \in \N).
}
We will prove by induction that $M_n=\QA{g_n}$ for all $n \in \N$. Indeed, for $n=1$ this equality is trivial.

Having this we know that $M_{n+1} \ge M_n$ and $M_{n+1} \ge \QA{f_{n+1}}$. It implies $M_{n+1} \ge \UQA{g_n,f_{n+1}}$. Which, in view of Lemma~\ref{lem:main} implies $M_{n+1} \ge \QA{g_{n+1}}$. On the other hand
\Eq{*}{
\QA{g_{n+1}} \ge \QA{f_k}\quad \text{ for all } k \in \{1,\dots,n+1\},
}
what implies $M_{n+1}=\QA{g_{n+1}}$. In particular $\UQA{\calF} \ge M_{n}$ for all $n \in \N$. Thus $\UQA{\calF} \ge \lim_{n \to \infty} M_n=\lim_{n \to \infty} \QA{g_n}$.

On the other hand, as $\QA{g} \ge \QA{f_n}$ for all $n \in \N$, we have $\UQA{\calF} \le \QA{g}$. Therefore
\Eq{*}{
\lim_{n \to \infty} \QA{g_n} \le \UQA{\calF} \le \QA{g}.
}
Finally in view of Lemma~\ref{lem:Pas13} we get $\lim_{n \to \infty} \QA{g_n}=\QA{g}$ and, consequently, $\UQA{\calF}=\QA{g}$.

To establish a second part we need to use a reflection of quasi-arithmetic means which were considered for example in \cite{PalPas18b}. For a given continuous, monotone function $f \colon I \to \R$ let $\hat f \colon (-I) \to \R$ be defined as $\hat f(x)=f(-x)$. Then, for every vector $v \in \bigcup_{n=1}^\infty I^n$ we have $\QA{f}(v)=-\QA{\hat f}(-v)$. As a trivial consequence, for every $f,\,g \colon I \to \R$, inequality $f \prec g$ is equivalent to $\hat f \succ \hat g$. In particular if we define
$\widehat \calF:=\{\hat f \mid f \in \calF\}$, then
\Eq{E:hat1}{
-\LQA{\calF}(-v)=\UQA{\widehat \calF}(v)=\QA{h_0}(v),
}
where by the first part $h_0 \colon I \to \R$ is a solution of a differential equation
\Eq{*}{
\frac{h_0''(x)}{h_0'(x)}=\sup_{f \in \widehat \calF} \frac{f''(x)}{f'(x)}=\sup_{f \in \calF} \frac{\hat f''(x)}{\hat f'(x)},\quad x \in (-I).
}
But $\hat u''(x)/\hat u'(x)=-u''(-x)/u'(-x)$ for all $u \in \Sm$. Additionally, in view of \eq{E:hat1}, $\LQA{\calF}=\QA{\hat h_0}$. Finally, for all $x \in I$,
\Eq{*}{
\frac{\hat h_0''(x)}{\hat h_0'(x)}
=-\frac{h_0''(-x)}{h_0'(-x)}
=-\sup_{f \in \calF} \frac{\hat f''(-x)}{\hat f'(-x)}
=\inf_{f \in \calF} \Big(-\frac{\hat f''(-x)}{\hat f'(-x)}\Big)
=\inf_{f \in \calF} \Big(\frac{f''(x)}{f'(x)}\Big),
}
what concludes the proof.
\end{proof}

Notice that Lemma~\ref{lem:main} implies that the set $\Sm/_\sim$ embedded with a partial ordering $\prec$ admit some type of lattice property. This statement has an alternative (significantly simpler) proof using \ref{CC:Mikusinski}. However, contrary to \ref{CC:Mikusinski}, Lemma~\ref{lem:main} considers the smallest upper bound of \emph{all} continuous and monotone functions, not only these from $\Sm$.

\section{Examples}
First, let us applied a result in Lemma~\ref{lem:main} in a simple example

\begin{exa}
 Let $I=(-\tfrac{\pi}2,\tfrac\pi{2})$, $f,\,g \colon I \to \R$ be given by $f(x)=\sin(x)$, $g(x)=\tan x$.

 Then $f''(x)/f'(x)=-\tan x$, and $g''(x)/g'(x)=2\tan x$. By Lemma~\ref{lem:main} we get, as $\tan$ is even, that $\UQA{f,g}=\QA{h}$, where
 \Eq{*}{
 h''(x)/h'(x)=
 \begin{cases}
 f''(x)/f'(x) & \text{ for }x \in (-\tfrac{\pi}2,0] \\
 g''(x)/g'(x) & \text{ for }x \in (0,\tfrac\pi{2}).
 \end{cases}
 }
Therefore $h$ is $\mathcal{C}^2$ function, which is an affine transformation of $\sin$ and $\tan$ on negative and positive elements, respectively. One can easy prove that
\Eq{*}{
 h(x)=
 \begin{cases}
 \sin(x) & \text{ for }x \in (-\tfrac{\pi}2,0] \\
 \tan(x) & \text{ for }x \in (0,\tfrac\pi{2}).
 \end{cases}
 }
Similarly $\LQA{f,g}=\QA{k}$, where
\Eq{*}{
 k(x)=
 \begin{cases}
 \tan(x) & \text{ for }x \in (-\tfrac{\pi}2,0] \\
 \sin(x) & \text{ for }x \in (0,\tfrac\pi{2}).
 \end{cases}
 }
 Observe that both $h$ and $k$ are $\mathcal{C}^2$ functions (otherwise we should make an affine tranformation in a merging point).
\end{exa}

Second, we show that the assumption that first derivative is nonvanishing is important.

\begin{exa}
 Let $f,\,g \colon (-1,1) \to \R$ be given by $f(x)=x$ and $g(x)=x^3$. Then $\UQA{f,g}=\max$ and $\LQA{f,g}=\min$.

 Indeed, in view of Lemma~\ref{lem:hnonvanish}, for every $h \colon (-1,1) \to \R$ such that $f \prec h$ we get $h'_+(0)$ exists and $h'_+(0) \ne 0$.
If we apply this lemma again we obtain that $\QA{g}$ and $\QA{h}$ are incomparable. Therefore $U_f \cap U_g=\emptyset$ and, consequently, $\UQA{f,g}=\max$. Prove of the second eqaulity is analogous.
\end{exa}

\section{\label{sec:proofmain} Proof of Lemma~\ref{lem:main}}
All this section will be devoted to prove the central lemma in the present paper. It is divided into three parts.
Relatively short proof of this statement will be presented in section~\ref{ssec:thm:main}. First, we introduce the notion of bilateral derivative. Having this, we present vary auxiliary result which will be useful in a main part. Sketch of the proof, as well as its division into lemmas is ilustrated on Figure~\ref{fig:1}.

\begin{figure}
 \definecolor{uuuuuu}{rgb}{0,0,0}
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\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm,scale=0.8]
\begin{axis}[
x=1cm,y=1cm,
axis lines=none,
ymajorgrids=false,
xmajorgrids=false,
xmin=-9.1,
xmax=5,
ymin=-8,
ymax=3.5,
]
\clip(-12.25,-11.7) rectangle (12.25,7.14);
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\fill[line width=2pt,color=zzttqq,fill=ttzzqq,fill opacity=0.25] (4.97,2.5) -- (-8.19,-2.56) -- (4.99,-5.38) -- cycle;
\draw [line width=0.5pt,color=zzttqq] (5,1)-- (-8.21,-4.08);
\draw [line width=0.5pt,color=zzttqq] (-8.21,-4.08)-- (4.99,-7.26);
\draw [line width=0.5pt,color=zzttqq] (4.97,2.5)-- (-8.19,-2.56);
\draw [line width=0.5pt,color=zzttqq] (-8.19,-2.56)-- (4.99,-5.38);
\draw [->,line width=2pt] (4.53,-3.1) -- (3.51,-3.1);
\draw [->,line width=2pt] (3.51,-3.1) -- (2.17,-3.1);
\draw [->,line width=2pt] (-1,-3.1) -- (-5.663242884807682,-3.1);
\begin{scriptsize}
\draw [fill=ududff] (-8.21,-4.08) circle (2.5pt);
\draw[color=ududff] (-8.05,-3.65) node {$g$};
\draw [fill=ududff] (-8.19,-2.56) circle (2.5pt);
\draw[color=ududff] (-8.03,-2.13) node {$f$};
\draw [fill=uuuuuu] (-5.663242884807682,-3.1006263326891) circle (2pt);
\draw[color=uuuuuu] (-5.51,-2.71) node {$h$};
\draw [fill=ududff] (4.53,-3.1) circle (2.5pt);
\draw[color=ududff] (4.69,-2.67) node {$s$};
\draw [fill=ududff] (3.51,-3.1) circle (2.5pt);
\draw[color=ududff] (3.75,-2.67) node {$s_2$};
\draw [fill=ududff] (2.17,-3.1) circle (02.5pt);
\draw[color=ududff] (2.41,-2.67) node {$s_3$};
\draw [fill=ududff] (-1,-3.1) circle (2.5pt);
\draw[color=ududff] (-0.85,-2.67) node {$k$};
\draw[color=black] (0.8,-3.1) node {\huge $\cdots$};
\draw[color=black] (1.765,-4) node {\large{Lemma~\ref{lem:seq}}};
\draw[color=black] (-2.8,-2.7) node {\large{Lemma~\ref{lem:C2C1}}};
\draw[color=ududff] (0,0) node {\Large{$U_f$}};
\draw[color=ududff] (0,-5.3) node {\Large{$U_g$}};
\draw[color=ududff] (3,-0.7) node {\Large{$U_f \cap U_g$}};
\draw [decorate,decoration={brace,amplitude=15pt},xshift=0pt,yshift=0pt]
(4.53,-3.1) -- (-1,-3.1) node [black,midway,yshift=0.6cm]
{};
\end{scriptsize}
\end{axis}
\end{tikzpicture}
\caption{\label{fig:1}Proof of Lemma~\ref{lem:main} and auxiliary results. Elements are ordered from left to right by $\prec$.}
\end{figure}

\subsection{Lower bilateral derivative and its basic properties}
In order to deal with a functions which are not necessarily differentiable we need to introduce certain generalization of derivative. For a function $f \colon I \to \R$ let us introduce the \emph{lower (bilateral) derivative}  \cite[p. 52]{Bru78}; \cite[Appendix I]{HabLalRou77} as
\Eq{*}{
\LDer f(x_0):= \liminf_{x \to x_0}\frac{f(x)-f(x_0)}{x-x_0} \qquad (x_0 \in I).
}

Notice that every function $f$ has a (possibly infinite) lower derivative.
Observe that for a lower derivative some
some version of quotient rule remains is valid (see \eq{E:Diniquo} below); proof of this equality is similar to the standard one and therefore omitted.
Analogously chain rule holds whenever at least one (out of two) function is differentiable.
Finally, the following lemma holds.

\begin{lem}[\cite{HabLalRou77}, pp. 345--347] \label{lem:BilMon} Let $f \colon I \to \R$ be a continuous function. Then $f$ is nondecreasing if and only if $\LDer f (x)\ge 0$ for all $x \in I$.
\end{lem}

In the same way one can define the notation of upper derivative, which is however not used in this paper.

\subsection{Auxiliary results}
In this section we will prove two technical results concerning comparability of means. Both of them are conncected with smoothness properties of quasi-arithmetic means.

Let us emphasize that Lemma~\ref{lem:seq} is simply implied by Lemma~\ref{lem:main}, however it will be used in its proof. In particular our Lemma~\ref{lem:main} can be seamed as its strenghtening.
Contrary to this Lemma~\ref{lem:convex} and Lemma~\ref{lem:C2C1} are obviously a separated statements, which could be appliciable in a number of different settings.

\begin{lem}
 \label{lem:convex}
 Let $f,\,g \colon I \to \R$ be two continuous, strictly monotone functions, $a,b,c \in I$ with $a<b<c$. If $f|_{(a,b)} \prec g|_{(a,b)}$, $f|_{(b,c)} \prec g|_{(b,c)}$
 and both $f'(b)$ and $g'(b)$ exists and are nonzero then $f|_{(a,c)} \prec g|_{(a,c)}$.
\end{lem}
\begin{proof}
 Assume that both $f$ and $g$ are increasing. Applying \ref{CC:gf^-1} we obtain $g \circ f^{-1}$ is convex on $(f(a),f(b))$ and $(f(b),f(c))$. As $g \circ f^{-1}$ is differentiable at $f(b)$, it can be easily proved that $g \circ f^{-1}$ is convex at $(f(a),f(c))$ (as its one-sided derivatives are monotone). As the property \ref{CC:gf^-1} can be inversed, it implies $f|_{(a,c)} \prec g|_{(a,c)}$.
\end{proof}

\begin{lem}\label{lem:seq}
 Let $f,\,g \in \CM$ be two differentiable functions such that $f' \cdot g' \ne 0$, and $s \in U_f \cap U_g$. Then there exists a differentiable function $k \in U_f \cap U_g$ such that $k'$ is nonvanishing and $k \prec s$.

\end{lem}
Idea of the prove is to correct a function $s$ iteratively, removing the nondiferentiability point one-by-one, and finally pass to the limit.
\begin{proof}
Take $s \in U_f \cap U_g$ arbitrarily. Assume without loss of generality that $f,\,g,\,s$ are all increasing.

By Lemma~\ref{lem:hnonvanish} we know that $s$ is one-sided differentiable at every point and, moreover, its one-sided derivatives are nowhere vanishing. In particular, as $s$ is convex with respect to $f$, we get that there exists a countable set $Z=(z_n)_{n \in \N}$, such that $s$ has a derivative at every point of $I \setminus Z$. Relative convexity of $s$ also implies that
\Eq{*}{
s'_-(z_n) \le s'_+(z_n) \quad \text{ for all } z_n \in Z.
}
Set $s_1:=s$ and define, for all $n \ge 1$,
\Eq{*}{
s_{n+1}(x):=\begin{cases}
            \dfrac{(s_n)'_+(z_n)}{(s_n)'_-(z_n)}(s_{n}(x)-s_{n}(z_n))+s_{n}(z_n) & \text{ for }x <z_n,\\[4mm]
            s_n(x) & \text{ for }x \ge z_n.
           \end{cases}
}
Then the mapping $n \mapsto s_n(x)$ is pointwise decreasing for all $x \in I$ and $s_{n_0}$ is differentiable at $I \setminus (z_n)_{n \ge n_0}$, for all $n_0 \in \N$.

Suppose that $s_n \succ f$. By \ref{CC:gf^-1}, as $s_n$ is increasing, $s_{n} \circ f^{-1}$ is convex on $f(I)$. Thus $s_{n+1} \circ f^{-1}$ is convex on $f\big(I\cap (-\infty,x_{n}) \big)$ and (separately) on
$f\big(I\cap (x_{n},\infty) \big)$. Furthermore
\Eq{*}{
(s_{n+1} \circ f^{-1})'_+(f(z_n))&=(s_n)'_+(z_n)\cdot (f^{-1})'(z_n),\\
(s_{n+1} \circ f^{-1})'_-(f(z_n))&=\frac{(s_n)'_+(z_n)}{(s_n)'_-(z_n)} \cdot (s_n)'_-(z_n)(f^{-1})'(z_n)\\
&=(s_n)'_+(z_n)(f^{-1})'(z_n).
}
Thus $s_{n+1} \circ f^{-1}$ is differentiable at $f(z_n)$ and, in view of Lemma~\ref{lem:convex}, convex. It implies $f \prec s_{n+1}$, similarly $g \prec s_{n+1}$. Thus $s_{n+1} \in U_f \cap U_g$.

Additionally we have $s_{n+1} \le s_{n}$ for all $n \in \N$.
Let $k \colon I \to \R$ be a pointwise limit of the sequence $(s_n)_{n \in \N}$, i.e. $k:= \lim_{n \to \infty} s_n$.

As $s_n \circ f^{-1}$ is convex for every $n \in \N$, its pointwise limit $k \circ f^{-1}$ is convex too. In particular $k \circ f^{-1}$ is continuous function defined of $f(I)$. Therefore $k$ being a composition of two continuous functions is continuous too.

Similarly $f \circ s_n^{-1}$ is concave for every $n \in \N$ and pointwise monotone (as a function of $n$). In particular it has a continuous limit $u$.
Then $k^{-1}=f^{-1} \circ u$, consequently $k$ is invertible. Therefore, as it $k$ also continuous, we obtain that it is strictly monotone.

It implies that $k$ generates a quasi-arithmetic mean. Moreover, in view of Lemma~\ref{lem:Pales}, we have $\lim_{n \to \infty} \QA{s_n}=\QA{k}$.

Furthermore, by Lemma~\ref{lem:convex}, for all $n \in \N$, the function
\Eq{*}{
s_{n+1} \circ s_n^{-1}(y)=
\begin{cases}
            y & \text{ for }y \ge s_n(x_n),\\
            \dfrac{(s_n)'_+(x_n)}{(s_n)'_-(x_n)}(y-s_{n}(x_n))+s_{n}(x_n) & \text{ for }y <s_n(x_n)\\
           \end{cases}
}
is concave. In view of \ref{CC:fg^-1}, it imples  $s_{n+1} \prec s_n$ for all $n \in \N$.
 In particular
 \Eq{*}{
 k=\lim_{n \to \infty} s_n \prec s_{n_0} \text{ for all }n_0 \in \N.
 }
 Moreover as $s_n \in U_f \cap U_g$ for all $n \in \N$, then $k$ being a limit of $s_n$ is an element of $U_f \cap U_g$ too. Furthermore $k \prec s_1=s$.

Notice that in view of Lemma~\ref{lem:diffnonz} for all $n_0 \in \N$, inequality $f \prec k \prec s_{n_0}$ implies that $k$ is differentiable at $I \setminus \{z_n\}_{n \ge n_0}$. As $n_0$ was taken arbitrarily we get that  $k$ is differentiable. Moreover, by Corollary~\ref{cor:hnonvanish}, as $f'$ is nowhere vanishing, so is $k'$.
\end{proof}

Next lemma provides a generalization of Mikusi\'nski comparability condition \ref{CC:Mikusinski} under a mixed smoothness assumption (one function is $\mathcal{C}^1$, second one is $\mathcal{C}^2$). Notice that, as assumptions of generating function are different, one can expect some dual results using the symmetry of quasi-arithmetic means described for example in \cite{PalPas18b}. Indeed, this is a case in this lemma. However, just to keep compactness of its wording it is omitted.

\begin{lem}
\label{lem:C2C1}
 Assume that $f,g \in \CM$. If $f$ is a $\mathcal{C}^2$ function and $k$ is an increasing $\mathcal{C}^1$ function, moreover $f'\cdot k' \ne 0$. Then $\QA{f} \le \QA{k}$ if and only if
\Eq{*}{
\dfrac{f''(x)}{f'(x)} \le \dfrac{\LDer(k')(x)}{k'(x)}\qquad \text{ for all }x \in I.
}\end{lem}

\begin{proof}
 Assume without loss of generality that $f$ is increasing.

Therefore, as $\QA{f} \le \QA{k}$ we get, by \ref{CC:g'/f'}, $k'/f'$ is nondecreasing. If we repeat the prove of quotient rule to lower derivative (recall that $f'$ is differentiable by the assumption) and apply Lemma~\ref{lem:BilMon} we obtain
\Eq{E:Diniquo}{
0 \le \LDer \Big(\frac{k'}{f'}\Big) (x)=\frac{\LDer(k')(x)f'(x)-f''(x)k'(x)}{f'(x)^2},
}
which simplifies to $
\LDer(k')(x)/k'(x) \ge {f''(x)}/{f'(x)}$.

As all implications were in fact equivalence we obtain that it is both necessary and sufficient condition.
\end{proof}

\subsection{\label{ssec:thm:main} Proof of Lemma~\ref{lem:main}}
First observe that there exists a function $h \in \mathcal{C}^2(I)$ sattisfying \eq{def:h}, as the differential equation $h''/h'=v$ has a solution for every continuous function $v$. Furthermore $h$ is defined up to an affine transformation (cf. \cite{Pas13} for details), in particular all solutions of \eq{def:h} generate the same quasi-arithmetic mean.

It is easy to verify that $h \in U_f \cap U_g$. We will show that in fact $h$ is the smallest element in $U_f \cap U_g$.

Take $s \in U_f \cap U_g$ arbitrarily.
Applying Lemma~\ref{lem:seq}, there exists $k \in U_f \cap U_g$ such that $k \prec s$, $k$ is differentiable, and $k' \ne 0$. Assume that $f$, $g$, and $k$ are increasing.

If we apply Lemma~\ref{lem:C2C1} for the pairs $(f,\,k)$ and $(g,\,k)$ and use the  max function we obtain
\Eq{*}{
\frac{\LDer(k')(x)}{k'(x)} &\ge \max \left( \frac{f''(x)}{f'(x)},\frac{g''(x)}{g'(x)}\right)=\frac{h''(x)}{h'(x)}\qquad (x \in I)\:.
}
Therefore, applying the oposite implication in Lemma~\ref{lem:C2C1}, we get $h \prec k$. As we also know that $k \prec s$, we get
\Eq{*}{
h \prec s \quad \text{ for all }\quad s \in U_f \cap U_g,
}
which implies that $h$ is the smallest element of $U_f \cap U_g$ (more precisely $[h]_\sim$ is the smallest element in the quotient set $(U_f \cap U_g) /_\sim$ embed with the induced partial ordering). This property is trivially equivalent to the statement of this lemma.

\subsection*{Acknowledgement} I am grateful to professor Roman Badora who suggest me to investigate lattice properties for means.

\def\cprime{$'$} \def\R{\mathbb R} \def\Z{\mathbb Z} \def\Q{\mathbb Q}
  \def\C{\mathbb C}

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text:
Merchants Register Road Victory Against Velocity
By Sam Laskaris
The Merchants Lacrosse Club hit the road for the first time this season on Saturday. And the Oshawa-based squad managed to defeat the host Kitchener Waterloo Velocity 11-9.The Ontario Series Lacrosse (OSL) match was staged at the Waterloo Memorial RecreationComplex.
Despite picking up the win, Merchants' head coach Jason Crosbie was not a happy camper following the match. That's because his squad, which led 10-4 after the second period, was outscored 5-1 in the final 20 minutes of action. Crosbie was upset with the complacency his charges displayed down the stretch. "We just didn't finish right," he said. "We just didn't play the right way to finish the game."
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The Merchants have four games remaining during the month of May. And all four of these contests will be at home, at Children's Arena in Oshawa. For starters, the Merchants will host the Velocity this coming Friday. The opening faceoff is set for 8 p.m. The Merchants will then square off against the defending OSL champions, Oakville Titans, on May 24 in a 7 p.m. start before battling the Six Nations Rivermen the following evening in a tilt which starts at 8 p.m.
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text:
Guibray est un quartier de la commune française de Falaise dans le département du Calvados.

Géographie

Histoire

Le site de Guibray est occupé dès l'époque mérovingienne par un sanctuaire remplacé par la suite par l'église Notre-Dame de Guibray.

Lors de la création des communes sous la Révolution, la paroisse de Guibray, devenue commune sous le nom révolutionnaire de Guibray-la-Montagne, est réunie à la commune de Falaise.

La foire de Guibray se tenait tous les ans à l'Assomption. Elle commençait le  et durait 15 jours. Une partie des anciennes loges existent encore dans le centre de Guibray. Elles font l'objet d'une inscription au titre des monuments historiques depuis le .

Références

Articles connexes

 Église Notre-Dame de Guibray
 La Foire de Guibray
 Versainville

Ancienne commune dans le Calvados
Commune ayant porté un nom révolutionnaire dans le Calvados
Quartier de ville en France
Falaise (Calvados)

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text:
Q: ImageReader makes Camera lag I'm facing a problem where I am using a ImageReader in addition to have a SurfaceView which is used to show the output of the camera. I have added the SurfaceView itself and the surface for the ImageReader like this (and added the ImageReader listeners in order to receive events on new images):
preview.addTarget(_surfaces.get(1)); //ImageReader surface
preview.addTarget(_surfaces.get(0)); //SurfaceView of the layout

However, if I remove the ImageReader the application is working fine (no lagging from the camera). But as soon as the ImageReader surface is added the camera i lagging a lot. Anyone have any ideas in order to makes the camera run more smooth? The ImageReader is created like this:
reader = ImageReader.newInstance(640, 480, ImageFormat.JPEG, 1);

I guess the lag could be due to that the image has to be rendered two times(?), both on the SurfaceView of the layout and the ImageReader itself

A: Turns out that the problem is the ImageFormat of the ImageReader. Changing the ImageFormat.JPEG to for example ImageFormat.YUV_420_888 fixed the lagging problem.

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UN peacekeepers have observed regular contacts between Nusra forces in the area and the Israeli troops stationed on the other side of the ceasefire line (Israel has illegally occupied part of the Golan since 1967). They also observed cargo of an unknown nature passing between the two sides from the Israelis.


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text:
Andy Serkis Could've Easily Brought An Evil Jar Jar Binks To Life In 'Star Wars: The Force Awakens'
Andrew RobertsEntertainment Writer
Count your blessings, folks. We could've possibly had Jar Jar Binks in The Force Awakens if we had cracked wise or shunned a new Star Wars movie. It's not like Disney cares, with their army of cash machines and animatronics. When a man in a Donald Duck costume shows up at your front door with a bat in hand, you know you've done the wrong thing.
But with Andy Serkis, they have a talent that could've played two characters at once in the new film. And they could've taken their punishment to a new level of pain by bringing back Jar Jar Binks. An evil Jar Jar Binks. I know what you're probably saying to yourself right now: Isn't Jar Jar Binks naturally evil? Sure. He's a being created out of pure spite by a man who never learned how to properly trim his beard. But Serkis whipped out a Jar Jar Binks impression on Conan and it's scary how good it is.
At first, I thought it was Gollum playing Jar Jar Binks as an evil Sith lord, but I soon found out it wasn't. And my mind wandered into dark territory, a place where Jar Jar starred in The Force Awakens and Chewie was constantly doing "foot stuff" with all the members of the cast. All that hair is going to need some cleaning.
(Via Team Coco)
Topics: #Jar Jar Binks, #Star Wars, #Star Wars: The Force AwakensTags: ANDY SERKIS, Jar Jar Binks, Star Wars, star wars: the force awakens
All The Best New R&B From This Week That You Need To Hear
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A newsletter from North American YMCA Development Organization Volume 29 | Number 4 | October 2015
IT'S THE MOST WONDERFUL TIME OF THE YEAR (FOR GIVING)!
As the end of the calendar year approaches, many charities are preparing to launch their year-end giving campaigns, which, in many cases, account for the lion's share of their annual fundraising budget. In fact, studies continue to demonstrate that more than one-third of total annual giving takes place in the last three months of the year, with the highest percentage – 17.5 percent – coming in December (Blackbaud Charitable Giving Report). While the traditional annual campaign is still "king" at most local Ys, it is imperative – from both a fundraising and awareness standpoint – that local Ys begin to develop and execute effective plans to capitalize on year-end giving opportunities. Thankfully, the opportunities for year-end giving engagement are as diverse and plentiful as our local Ys.
The key here is to start somewhere. Do what makes sense for your Y. But do it with concrete goals and a comprehensive plan. What follows are my top ten tips, suggestions and ideas from local Ys and philanthropy experts, to help you in planning for a successful year-end giving season:
Use National Philanthropy Day – November 15 – as an opportunity to kick off your year-end giving season while providing valuable recognition for your key donors and volunteers. Established by President Ronald Reagan in 1986, National Philanthropy Day (NPD) is a celebration of philanthropy – giving, volunteerism and charitable engagement – that highlights the accomplishments, large and small, that philanthropy – and all those involved in the philanthropic process – makes to our society and our world (Association of Fundraising Professionals). NPD presents a great opportunity for donor and volunteer stewardship, and can serve as a primer for making an ask at year-end. For more information, visit www.npdlove.com
Tips 2-5 apply to general year-end giving, as well as #GivingTuesday, a global movement designed to inspire people to give back by supporting their favorite charities. Established in 2012 by the 92nd Street Y and the United Nations Foundation, #GivingTuesday has grown tremendously in a three-year period, engaging 2,500 non-profits and raising $13.5 million in 2012, to engaging more than 15,000 charities and raising $45.7 million in 2014 (visit www.givingtuesday.org for more information).
As #GivingTuesday lives primarily in the digital space, the movement has inspired a huge increase in online giving during this single day of giving, which brings us to our next tip:
Get your websites in order! Online giving on #GivingTuesday has grown 93 percent in a two-year period and mobile giving doubled last year (Campbell & Company) but whether you're soliciting donations on #GivingTuesday or not, make sure that your website and online giving platform are up to date, simple to use, and provide a clear direction for prospective donors to find you and make a gift. Give philanthropy some prime real estate on your website (and on your branch sites and program pages), with a clear call to action and a visible "Donate Now" button.
Determine your goals and objectives. Don't just participate in #GivingTuesday because everyone else is doing it. Think strategically and determine what you want to accomplish. Your goals may not be all about dollars and cents. #GivingTuesday presents a great opportunity to engage staff, volunteers, donors and other ambassadors to share you Y's story.
Define the audiences you wish to reach. Who do you plan to engage in your year-end giving campaign? Current donors? Prospective donors? Program participants? Members who aren't currently giving? Whether it's one or all of the above, your plan should include strategies to reach each group. This can be done online with e-newsletters, or by using "snail mail," but be sure to tailor your message accordingly.
Create compelling content, in advance, to connect with current and prospective supporters on an emotional level, and inspire them to give. Jen Harman, Director of Donor Marketing & Communications with the YMCA of Greater Charlotte shared her Y's social media content calendar on a recent Y-USA webinar. The Charlotte Y's year-end giving campaign has grown over the past four years, with expanded focus on a number of key target groups, including members, donors and lapsed donors. "It's important to consider key stories and messaging ahead of time," shares Harman. "Our calendar starts with National Philanthropy Day and continues through January 31, so it's much easier to plan it out ahead of time so that we can focus on communicating with and stewarding our donors at year-end."
The next four tips are adapted from Gail Perry (www.gailperry.com), who recently published her "Top Five Tips for a Profitable Year-End Campaign."
Say "thank you" first. In the next 40 days, prior to National Philanthropy, rally your volunteers and staff to help you show gratitude to your donors. Make it fun by gathering groups for "thank-a-thons" aimed at not only thanking donors, but having meaningful conversations about why they give.
Prime your donors before you ask. In addition to saying thank you, let your donors know that you're getting ready for your year-end campaign. Via phone, e-mail or mailed letters and postcards, generate excitement by sharing stories of the people you're impacting and make the emotional connection.
Create a multi-channel campaign. Connect with your donors multiple times with compelling stories and appeals that carry a specific theme and design, perhaps using the new Y-USA Nationwide Campaign materials, located on the Brand Resource Center. It may take multiple hits to get their attention!
Include a specific ask. When you engage your donors, let them know what you want to accomplish, what you need to raise to make it a reality, and how their support will help. Ask for a specific amount whenever appropriate, and be sure to connect it to the people you serve.
Our tenth and final tip is to START SOMEWHERE! Virtually all charities are asking at year-end, and it is imperative that YMCAs become a part of the conversation. Don't leave money on the table. Your members and donors are likely making gifts to other charities, so it's important to remind them that the Y is an option for their charitable giving.
There is no shortage of resources to support you with year-end giving, and we've included some below. We wish you a healthy, happy and profitable year-end giving season!
For #GivingTuesday: For general year-end giving:
www.givingtuesday.org www.gailperry.com
www.blackbaud.com/givingtuesday www.networkforgood.org
www.salsalabs.com/giving-tuesday-salsa www.nonprofithub.org
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